The distinction between rule-based and statistically-based translation may well be artificial and obscure what is really the interesting distinction in machine translation modules. The latter may well lie in the fact that some methods capture (at least partially) the semantics of a text, and are for example able to enumerate lemmas in the text, change the person of verbs or the gender of nouns, etc. In contrast, other translation methods do not capture the semantics of the text and only perform the translation. At least this type of classification seems to be relevant to artificial intelligence.

# Monthly Archives: January 2022

# Interjections

What are interjections (**Hello! Good evening! Merry Christmas! Happy Birthday!**…) in the present framework? They are words preceded by a punctuation mark (period, comma, exclamation mark, question mark, etc.) and followed by a punctuation mark.

# Viewpoint of a pole

Let us define the concept of point of view related to a given pole of an A/Ā duality: we get then, for example (at the level of the extension/restriction duality) the standpoint by extension, as well as the viewpoint by restriction. Similarly, the qualitative viewpoint or perspective results from it, as well as the quantitative point of view, etc.. (at the level of the qualitative/quantitative duality). Thus, when considering a given object *o* (either a concrete or an abstract object such as a proposition or a reasoning), we may consider it in relation to various dualities, and at the level of the latter, relative to each of its two dual poles.

The underlying idea inherent to the viewpoint relative to a given duality, or to a given pole of a duality, is that each of the two poles of the same duality, all things being equal, deserve an equal legitimacy. In this sense, if we consider an object *o* in terms of a duality A/Ā, one should not favour one of the poles with respect to the other. To obtain an objective point of view with respect to a given duality A/Ā, one should place oneself in turn from the perspective of the pole A, and then from that of the pole Ā. For an approach that would only address the viewpoint of one of the two poles would prove to be partial and truncated. The fact of considering in turn the perspective of the two poles, in the study of an object *o* and of its associated reference class allows to avoid a subjective approach and to meet as much as possible the needs of objectivity.

Franceschi, P., “Une classe de concepts” (in english), *Semiotica*, vol. 139 (1-4), 2002, pages 211-226.

Ambiguous images ** ▣** Arbitrary focus

**Bistable perception**

**▣****Complementarity relationship**

**▣****Conflict resolution**

**▣****▣**Conflict resolution with matrices of concepts

**▣**Conflict types relating to matrices of concepts

**▣**Contrary relationship

**▣**Courage

**▣**Dialectical contextualism

**▣**Dialectical monism

**▣**Dialectical monism in Aztec philosophy

**▣**Dialectical monism in Heraclitus

**▣**Dichotomic analysis

**▣**Dichotomic analysis applied to paradox resolution

**▣**Dichotomous reasoning

**▣**Disqualification of one pole

**▣**Disqualification of the positive

**▣**Doctrine of the mean

**▣**Doomsday argument

**▣**Dualities

**▣**Dual poles

**▣**Extreme opposition

**▣**General cognitive distortions

**▣**Instance of one-sidedness bias

**▣**Liar paradox

**▣**Matrix of concepts

**▣**Maximization

**▣**Mental filter

**▣**Minimization

**▣**Bistable cognition

**▣**Omission of the neutral

**▣**One-sidedness bias

**▣**One-sided viewpoint

**▣**Opposition relationship

**▣**Principle of dialectical indifference

**▣**Requalification into the other pole

**▣**Reference class

**▣**Reference class problem

**▣**Reference class problem in philosophical paradoxes

**▣**Reference class problem in the Doomsday argument

**▣**Reference class problem in Hempel’s paradox

**▣**Reference class problem in the surprise examination paradox

**▣**Selective abstraction

**▣**Sorites paradox

**▣**Specific cognitive distortions

**▣**Surprise examination paradox

**▣**System of taxa

**▣**Two-sided viewpoint

**▣**Viewpoint of a duality

**▣**Viewpoint of a pole

(c) Paul Franceschi

# Principle of dialectical indifference

(PRINCIPLE OF DIALECTICAL INDIFFERENCE) When considering a given object o and the reference class E associated with it, from the angle of duality A/Ā, all things being equal, it should be given equal weight to the viewpoint of the A pole and the viewpoint of the Ā pole.

The principle of dialectical indifference can be enunciated as follows: if we consider an object o under the angle of a given A/Ā duality, there is no reason to favour the viewpoint from A with regard to the viewpoint from Ā, and unless otherwise resulting from the context, we must weigh equally the viewpoints A and Ā. A direct consequence of this principle is that if one considers the perspective of the A pole, one also needs to take into consideration the standpoint of the opposite pole Ā (and vice versa). The need to consider both points of view, the one resulting from the A pole and the other associated with the Ā pole, meets the need of analysing the object o and the reference class associated with it from an objective point of view. This goal is achieved, as far as possible, by taking into account the complementary points of view which are those of the poles A and Ā. Each of these viewpoints has indeed, with regard to a given duality A/Ā, an equal relevance. Under such circumstances, when only the A pole or (exclusively) the pole Ā is considered, it consists then of a one-sided perspective. Conversely, the viewpoint which results from the synthesis of the standpoints corresponding to both poles A and Ā is of a two-sided type. Basically, this approach proves to be dialectical in essence. In effect, the step consisting of successively analysing the complementary views relative to a given reference class, is intended to allow, in a subsequent step, a final synthesis, which results from the joint consideration of the viewpoints corresponding to both poles A and Ā. In the present construction, the process of confronting the different perspectives relevant to an A/Ā duality is intended to build cumulatively, a more objective and comprehensive standpoint than the one, necessarily partial, resulting from taking into account those data that stem from only one of the two poles.

Further reading: Elements of dialectical contextualism

(c) Paul Franceschi

Ambiguous images ** ▣** Arbitrary focus

**Bistable perception**

**▣****Complementarity relationship**

**▣****Conflict resolution**

**▣****▣**Conflict resolution with matrices of concepts

**▣**Conflict types relating to matrices of concepts

**▣**Contrary relationship

**▣**Courage

**▣**Dialectical contextualism

**▣**Dialectical monism

**▣**Dialectical monism in Aztec philosophy

**▣**Dialectical monism in Heraclitus

**▣**Dichotomic analysis

**▣**Dichotomic analysis applied to paradox resolution

**▣**Dichotomous reasoning

**▣**Disqualification of one pole

**▣**Disqualification of the positive

**▣**Doctrine of the mean

**▣**Doomsday argument

**▣**Dualities

**▣**Dual poles

**▣**Extreme opposition

**▣**General cognitive distortions

**▣**Instance of one-sidedness bias

**▣**Liar paradox

**▣**Matrix of concepts

**▣**Maximization

**▣**Mental filter

**▣**Minimization

**▣**Bistable cognition

**▣**Omission of the neutral

**▣**One-sidedness bias

**▣**One-sided viewpoint

**▣**Opposition relationship

**▣**Principle of dialectical indifference

**▣**Requalification into the other pole

**▣**Reference class

**▣**Reference class problem

**▣**Reference class problem in philosophical paradoxes

**▣**Reference class problem in the Doomsday argument

**▣**Reference class problem in Hempel’s paradox

**▣**Reference class problem in the surprise examination paradox

**▣**Selective abstraction

**▣**Sorites paradox

**▣**Specific cognitive distortions

**▣**Surprise examination paradox

**▣**System of taxa

**▣**Two-sided viewpoint

**▣**Viewpoint of a duality

**▣**Viewpoint of a pole

# Glossary

Ambiguous images ** ▣** Arbitrary focus

**Bistable perception**

**▣****Complementarity relationship**

**▣****Conflict resolution**

**▣****▣**Conflict resolution with matrices of concepts

**▣**Conflict types relating to matrices of concepts

**▣**Contrary relationship

**▣**Courage

**▣**Dialectical contextualism

**▣**Dialectical monism

**▣**Dialectical monism in Aztec philosophy

**▣**Dialectical monism in Heraclitus

**▣**Dichotomic analysis

**▣**Dichotomic analysis applied to paradox resolution

**▣**Dichotomous reasoning

**▣**Disqualification of one pole

**▣**Disqualification of the positive

**▣**Doctrine of the mean

**▣**Doomsday argument

**▣**Dualities

**▣**Dual poles

**▣**Extreme opposition

**▣**General cognitive distortions

**▣**Instance of one-sidedness bias

**▣**Liar paradox

**▣**Matrix of concepts

**▣**Maximization

**▣**Mental filter

**▣**Minimization

**▣**Bistable cognition

**▣**Omission of the neutral

**▣**One-sidedness bias

**▣**One-sided viewpoint

**▣**Opposition relationship

**▣**Principle of dialectical indifference

**▣**Requalification into the other pole

**▣**Reference class

**▣**Reference class problem

**▣**Reference class problem in philosophical paradoxes

**▣**Reference class problem in the Doomsday argument

**▣**Reference class problem in Hempel’s paradox

**▣**Reference class problem in the surprise examination paradox

**▣**Selective abstraction

**▣**Sorites paradox

**▣**Specific cognitive distortions

**▣**Surprise examination paradox

**▣**System of taxa

**▣**Two-sided viewpoint

**▣**Viewpoint of a duality

**▣**Viewpoint of a pole

# Differential Cognitive Treatment of Polythematic Delusions and Generalised Anxiety Disorder

English translation of a paper published in french under the title “Traitement cognitif différentiel des délires polythématiques et du trouble anxieux généralisé“, in the *Journal de Thérapie Comportementale et Cognitive*, 2011, vol. 21-4, pp. 121-125.

Schizophrenia is often associated with other physical and mental problems. Generalized anxiety disorder is notably one of the comorbid disorders which is often linked to schizophrenia. The association of polythematic delusions and of ideas resulting from generalized anxiety disorder complicates the exercise of the corresponding cognitive therapy, for the resulting ideas are most often inextricably intertwined. In what follows, we endeavour to propose a methodology for the differential treatment of polythematic delusions inherent to schizophrenia when combined with ideas originating from generalized anxiety disorder. We propose, with regard to the corresponding content of delusions, an analysis which allows under certain conditions, to separate the content associated with polythematic delusions and the one that relates to generalized anxiety disorder, in order to facilitate the exercise of the corresponding cognitive therapy.

This article is cited in:

- Marc Willard, De l’économie de jetons à la remédiation cognitive, Journal de Thérapie Comportementale et Cognitive, Vol 22-3, 2012, p. 79-80

**Differential Cognitive Treatment of Polythematic Delusions and Generalised Anxiety Disorder**

Generalized anxiety disorder is a comorbid disorder which is commonly associated with schizophrenia. Such co-morbidity is likely to render more complex and difficult the corresponding cognitive therapy. In what follows, we strive to provide a methodology to allow for a differential treatment of polythematic delusions inherent to schizophrenia when associated with generalized anxiety disorder. We describe then, based on the content of the corresponding delusions, an analysis which allows, under certain conditions, to separate the content relating to the polythematic delusions and the one that concerns generalized anxiety disorder, in order to facilitate the implementation of the corresponding cognitive therapies.

**Schizophrenia and co-morbidity**

One of the difficulties inherent in the treatment of schizophrenia is the frequent co-morbidity that relates to the disease. This comorbidity bears either on physical conditions, or other psychiatric disorders. A co-morbidity of schizophrenia with physical [Sim et al affections. 2006] or neuropsychiatric disorders such as Tourette’s syndrome [Kerbeshiana et al. 2009], has then been observed. It was also found co-morbidity with other psychiatric disorders, which is very common [Bermanzohn 2000]. A significant co-morbidity has been ascertained in particular with depression (25%) [Bressan et al. 2003, Kim et al. 2008] and obsessive-compulsive disorder (26.50%) [Berman et al. 1995 Guillem et al. 2009]. Similarly, comorbidity within the sphere of anxiety disorders turns out to be quite common [Cosoff & Hafner, 1998, Braga et al. 2004]. It was thus demonstrated high rates of comorbidity with agoraphobia (8.20%) [Goodwin et al. 2002], panic disorder (13.80%) [Pallanti et al. 2004], generalized anxiety disorder (GAD), social phobia (13.30%) [Tibbo et al. 2003] or a specific phobia (13.60%) [Goodwin et al. 2002]. However, the co-morbidity of schizophrenia with certain disorders, such as intermittent explosive disorder, seems to have been little studied.

The comorbidity of schizophrenia with other psychiatric disorders does not lack to pose several problems in the treatment of the disease. First, such comorbidity is a factor that makes it difficult to improve the health status of the patient [Sim et al. 2006]. Second, the comorbidity of schizophrenia with other psychiatric disorders raises some specific issues that are important for cognitive and behavioural therapy of schizophrenia. Thus, the frequent comorbidity of schizophrenia with one or more associated disorders, suggests that it might be useful to adapt the corresponding cognitive therapy. In this sense, it may be useful to distinguish cognitive therapy targeted to polythematic delusions, from the therapy related to the comorbid disorders encountered in the patient. In any event, the introduction of a differential treatment of schizophrenia and comorbid disorders has specific aspects that should be highlighted. Such differential treatment requires for the therapist that the different disorders are well defined and especially that the therapist can distinguish within the patient’s speech what is relevant to the specific disorder that he/she aims to treat. In this context, the association of two or more disorders does not fail to create some confusion, for the patient’s delusions are often inextricably linked to ideas that arise from the co-morbid disorder(s). Thus, in the words of the patient, it is worth distinguishing between what is actual symptoms of schizophrenia (essentially polythematic delusions) and what results from the associated comorbid disorders, such as: depression, TAG, body dysmorphic disorder, social phobia, special phobia. This complex situation has the effect of making more difficult the corresponding cognitive therapy.

The difficulties just mentioned apply especially when schizophrenia is associated in patients with GAD, for which a co-morbidity rate of 12% [Cosoff & Hafner 1998] and 16.70% [Tibbo et al. 2003] were found. We propose, in what follows, to interest ourselves in the comorbidity of schizophrenia and TAG, and to proceed to describe in detail a methodology for distinguishing among the patient’s delusions, the content resulting from polythematic delusions proper from what is inherent to the TAG.

**Instances of differentiation: polythematic delusions and anxiogenous ideas**

The association of the content of these delusional ideas met in schizophrenia and anxiogenous ideas inherent to TAG may take different forms. For ease of analysis, we will work to identify certain stereotyped forms, among the mixed ideas resulting from this association. We shall use the classically defined delusions (delusions of reference, of telepathy, of thought-projection, of influence, or of control) and anxiogenous ideas whose structure is that of the projection of a negative future event. Let us consider then the mixed ideas likely to be encountered in the context of the association of polythematic delusions associated with schizophrenia and GAD (with the following abbreviations: R for *reference*, T for *telepathy*, P for *thought-projection*, I for *influence*, C for *control*):

(R1) | “Next week, TV presenters will again talk about me”. |

(T1) | “I am sure that in five minutes, the neighbour will again comment on my thoughts”. |

(P1) | “I am sure that soon, people in the street will again start yelling because they are disturbed by my thoughts”. |

(I1) | “Tomorrow, I will again create an accident, because of the disruption that I create in others with my bad mood”. |

(C1) | “I am sure that when they will arrive, the neighbours will still make me break things by controlling me”. |

These are mental constructs that combine both delusions which are specific to schizophrenia and anxiety ideas resulting from TAG (the structure of the latter being that of the anticipation of the occurrence of a future event of a negative nature). It is worth at this stage determining the structure of these mixed ideas in order to highlight what constitutes polythematic delusions proper and what is inherent in the TAG. Thus, in (R1) is contained, first, the delusion that the media talk about the patient, which is an instance of the delusion of reference. Second, the expectation that a negative event will occur, i.e. the media will speak again about the patient, is also present in (R1). Such an anticipation on a future event of a negative nature, has a special structure, which consists in the projection into the future of the occurrence of a negative event concerning the patient, even in the absence of an objective basis. This is one of the manifestations of the role played in TAG by expectations on indeterminate situations related to future events [Butler & Mathews 1987]. Most often, the patient considers a future event as certain, even though the probability of the event in question’s occurrence is much lower. Such anxiogenous idea has the structure below [Franceschi 2008a] (the patient’s anxiogenous idea occurs at time T_{0}, with T_{0} <T_{1}):

(A) | at time T_{1}, the event E of a negative nature, will occur | anxiogenous idea |

Given the above, we can now decompose the mixed idea (R1) in two separate ideas i.e. on the one hand, the delusion of reference and on the other hand, the projection of a negative future event:

(R) | “Television and the media speak about me” | delusion of reference |

(A_{R}) | at time T_{1} (“Next week”) the event E (“The presenters de television will speak about me”) of a negative nature, will occur | mixed anxiogenous idea |

At this stage, it is now possible to apply a principle of cognitive therapy specific to TAG to the resulting anxiogenous mixed idea, by considering alternative hypotheses to the occurrence of the negative event, by notably considering the hypothesis that other events, of a positive nature, may occur. It also turns out that the same analysis can be applied to mixed propositions (T1), (P1), (I1) and (C1), in the following way:

(T) | “The neighbours know the least of my thoughts” | delusion of telepathy |

(A_{T}) | at time T_{1} (“by five minutes”) the event E (“the neighbour will comment on my thoughts”) of a negative nature, will occur | mixed anxiogenous idea |

(P) | “People react according to what I think and start screaming” | thought projection delusion |

(A_{P}) | at time T_{1} (“in a moment”) the event E (“people in the street will begin to cry because they are disturbed by my thoughts”) of a negative nature, will occur | mixed anxiogenous idea |

(I) | “People are disturbed by my thoughts” | delusion of influence |

(A_{I}) | at time T_{1} (“tomorrow”) the event E (“I’m going to cause an accident, because of the disruption that I create in others with my bad mood”) of a negative nature, will occur | mixed anxiogenous idea |

(C) | “I have feelings and emotions according to what people do” | delusion of control |

(A_{C}) | at time T_{1} (“when they will arrive”) the event E (“the neighbours will make me break things by controlling me”) of a negative nature, will occur | mixed anxiogenous idea |

However, it turns out that the application of a principle cognitive therapy inherent to GAD to the mixed anxiety ideas resulting from the above analysis, is likely to present a problem. Indeed, a questioning of the form “Is it certain that the television will still talk to you tomorrow?” or “Isn’t it possible that television will not speak about you tomorrow? may give the impression that the therapist adheres to the patient’s delusional ideas of reference, which might be likely to strengthen them. To avoid this problem, it may be useful to eliminate the delusional content in the mixed anxiety-provoking idea. For once such removal is done, the principle of cognitive therapy inherent to GAD can then be applied directly to the residual anxiety idea without the aforementioned drawback begin faced. The methodology we propose to transform the mixed anxiety idea into a pure anxiety idea is based on the process of formation of the patient’s delusions. The development of the delusional ideas (R) (T) (P), (I) and (C) is carried starting from the primary delusional arguments, based on the attribution by the patient of a causal relationship when faced with the occurrence of two quasi-simultaneous events [Hemsley 1992 Franceschi 2008b]. Such primary delusional arguments have the following structure (the symbol **∴** denotes the conclusion):

(R1) | in T_{1} I was drinking an aperitif | premiss_{1} |

(R2) | in T_{2} the presenter of the show said: “Stop drinking !” | premiss_{2} |

(R3) | ∴ in T_{2} the presenter of the show said: “Stop drinking !” because in T_{1} I was drinking an aperitif | conclusion |

(T1) | in T_{1} I thought of Jacques “What an idiot !” | premiss_{1} |

(T2) | in T_{2} I heard Jacques say: “Enough!” | premiss_{2} |

(T3) | ∴ in T_{2} I heard Jacques say, “Enough! “Because in T_{1} I thought of him,”What an idiot!” | conclusion |

(P1) | in T_{1} I thought of someone who passed on the street “is badly dressed!” | premiss_{1} |

(P2) | in T_{2} I heard someone who passed on the street shout | premiss_{2} |

(P3) | ∴ in T_{2} I heard someone who passed in the street screaming because in T_{1} I thought of of him “He’s badly dressed!” | conclusion |

(I1) | in T_{1} I had a very bad mood | premiss_{1} |

(I2) | in T_{2} I heard there was a car accident in the street | premiss_{2} |

(I3) | ∴ in T_{2} there was a car accident in the street because in T_{1} I was in a very bad mood | conclusion |

(C1) | in T_{1} the neighbour has moved | premiss_{1} |

(C2) | in T_{2} I broke a glass | premiss_{2} |

(C3) | ∴ in T_{2} I broke a glass because in T_{1} the neighbour has moved | conclusion |

Such a structure from primary delusional arguments reveals that in instances of primary arguments of *reference*, of *telepathy*, of *thought projection* and of *influence*, an internal event to the patient (thought, emotion, feeling, action) slightly precedes an external event, in the following manner:

(α1) | in T_{1} the internal event E_{1} has occurred | premiss_{1} |

(α2) | in T_{2} the external event E_{2} has occurred | premiss_{2} |

(α3) | ∴ in T_{2} the external event E_{2} has occurred because in T_{1} the internal event E_{1} has occurred | conclusion |

In contrast, at the level of the instances of primary arguments of *control*, it is the event which is external to the patient that precedes an internal event:

(β1) | in T_{1} the external event E_{1} has occurred | premiss_{1} |

(β2) | in T_{2} the internal event E_{2} has occurred | premiss_{2} |

(β3) | ∴ in T_{2} the internal event E_{2} has occurred because in T_{1} the exernal event E_{1} has occurred | conclusion |

In this context, the elimination of the delusional content from mixed delusions can then be performed. For this, one eliminates from the mixed anxiogenous idea the mere idea of *causality*, by only retaining the event which constitutes the object of the anxiogenous idea, in the following way:

(A_{I}) | at time T1 (“tomorrow”) the event E (“I’m going to create an accident, because of the disturbance that I create in others with my bad mood”) of a negative nature, will occur | mixed anxiogenous idea |

(B_{I}) | at time T1 (“tomorrow”) the event E (“there will be an accident in the street”) of a negative nature, will occur | pure anxiogenous idea |

The methodology used here is thus to eliminate the delusional content in the speech of the patient and replace it with factual content, to which we can then apply a classical form of cognitive therapy for GAD, based on the consideration of alternative hypotheses: “Isn’t it possible that no accident occurs on the street tomorrow? “(I); “Isn’t it possible that tomorrow you could not break your glass? “(C); “Can’t we consider that no passer-by shouts in the street just now? “(P). Such formulation has thus the advantage of enabling the direct implementation of the very principle of the cognitive therapy inherent to TAG without facing the above-mentioned drawback.

**Conclusion**

In cognitive therapy of schizophrenia raises the question of the appropriate treatment of comorbid disorders associated with it. Regarding particularly GAD, which is often associated with schizophrenia, several questions arise as well. The first question is thus whether it is appropriate that two different therapists take care one of the therapy for GAD, and the other of the therapy for delusions. A second question, in this context, is whether it is better to implement the GAD therapy before that targeted at delusions [17,18]. The answer to these questions is beyond the scope of this study, but it may be important in the strategy implemented for cognitive therapy of schizophrenia.

At this point, it turns out that the usefulness of the above analysis it that it allows for simplifying the cognitive therapy in the case where there is a comorbid schizophrenia and TAG, in that it separates in the content of the original complex discourse of the patient, what is delusions proper and what is inherent in the TAG. This permits the isolation of a simplified discourse, to which can then be applied independently either the principle inherent to cognitive therapy for TAG, or the one that relates to delusions. This results in a second interest, in that it can help, if necessary, to two different therapists to take care of each cognitive therapy for GAD and delusions. Finally, a third interest is that it allows to make use of specific strategies. One such strategy is for example to implement cognitive therapy for GAD *before* cognitive therapy delusions. Is it better in effect when there exists in the patient a co-morbidity between schizophrenia and TAG, to implement cognitive therapy for GAD *before*, *after* or *at the same* the therapy for delusions? The above discussion does not lead to prefer one or other strategic option, but they can still be reformulated in terms of testable hypotheses. The first testable hypothesis that emerges is that the implementation of cognitive therapy for GAD, irrespective of cognitive therapy for delusions, could have a positive effect on symptoms of schizophrenia themselves. The second testable hypothesis is that the resulting cognitive therapy for delusions themselves could be more effective if it was implemented *after* a cognitive therapy for GAD has been achieved and demonstrated effective.

**References**

[1] Berman, I., Kalinowski, A., Berman, S.M., Lengua, J., Green, A.I. Obsessive and compulsive symptoms in chronic schizophrenia. Comprehensive Psychiatry 1995; 36: 6-10.

[2] Bermanzohn P.C., Porto L., Arlow P.B., Pollack S., Stronger R., Siris S.G. Hierarchical diagnosis in chronic schizophrenia: a clinical study of co-occurring syndromes. Schizophrenia Bulletin 2000; 26: 517–525.

[3] Braga R., Petrides G., Figueira I. Anxiety Disorders in Schizophrenia, Comprehensive Psychiatry 2004; 45(6): 460-468.

[4] Bressan, R.A., Chaves, A.C., Pilowsky, L.S., Shirakawa, I., Mari, J.J. Depressive episodes in stable schizophrenia: critical evaluation of the DSM-IV and ICD-110 diagnostic criteria. Psychiatry Research 2003; 117: 47–56.

[5] Butler G et Mathews A. Anticipatory anxiety and risk perception. Cognitive Therapy and Research 1987; 11: 551-565.

[6] Cosoff S.J., Hafner R.J. The prevalence of co-morbid anxiety in schizophrenia, schizoaffective disorder and bipolar disorder. Australian and New Zealand Journal of Psychiatry 1998; 32: 67-72.

[7] Franceschi P. Théorie des distorsions cognitives : application à l’anxiété généralisée, Journal de Thérapie Comportementale et Cognitive 2008a; 18: 127-131. English translation.

[8] Franceschi P. Une défense logique du modèle de Maher pour les délires polythématiques. Philosophiques 2008b; 35(2): 451-475. English translation.

[9] Goodwin R., Lyons J., McNally R. Panic attacks in schizophrenia. Schizophrenia Research 2002; 58: 213-220.

[10] Guillem F., Satterthwaite J., Pampoulova T., Stip E. Relationship between psychotic and obsessive compulsive symptoms in schizophrenia. Schizophrenia Research 2009; 115: 358-62.

[11] Hemsley D. Disorders of perception and cognition in schizophrenia. Revue européenne de Psychologie Appliquée 1992; 42(2): 105-114.

[12] Kerbeshiana J., Pengb C.Z., Burd L. Tourette syndrome and comorbid early-onset schizophrenia. Journal of Psychosomatic Research 2009; 67: 515-523.

[13] Kim S.W., Kim S.J., Yoon B.H., Kim J.M., Shin I.S., Hwang M., Yoon J.S. Diagnostic validity of assessment scales for depression in patients with schizophrenia. Psychiatry Research 2006; 144: 57-63.

[14] Kingdon D. et Turkington D. Cognitive-behavioural Therapy of Schizophrenia, New York: Guilford, 1994.

[15] Kingdon, D. et Turkington, D. Cognitive Therapy of Schizophrenia, New York, London: Guilford, 2005.

[16] Pallanti S., Quercioli L., Hollander E. Social anxiety in outpatients with schizophrenia: a relevant cause of disability. Am J Psychiatry 2004; 161: 53-58.

[17] Sim K., Chan Y.H., Chua T.H., Mahendran R., Chong S.A., McGorry P. Physical comorbidity, insight, quality of life and global functioning in first episode schizophrenia: A 24-month, longitudinal outcome study. Schizophrenia Research 2006; 88: 82-89.

[18] Tibbo P., Swainson J., Chue P., LeMelledo JM. Prevalence and relationship to delusions and hallucinations of anxiety disorders in schizophrenia. Depress Anxiety 2003;17: 65-72.

# Polythematic Delusions and Logico-Theoretical vs. Experimentalist Turn of Mind

*A paper published in the Journal for Neurocognitive Research, Vol. 2013, 55, No. 1-2.*

This article aims to contribute to cognitive therapy of polythematic delusions by proposing a preliminary step to the implementation of traditional cognitive therapy, based on the construction of alternative hypotheses to delusions and testing of the latter. This additional step resides in the construction in the patient of the necessary skills to use the general experimentalist method of knowledge acquisition. Such an approach is based on the contrast between the logico-theoretical and the experimentalist turn of mind. Some elements such as to allow any such construction in the patient are then described and analyzed.

This article is cited in:

**Polythematic Delusions and Logico-Theoretical vs. Experimentalist Turn of Mind**

Classical cognitive therapy targeted at polythematic delusions associated with schizophrenia is based on the search for evidence related to delusional ideas and the construction of alternative hypotheses to the latter. This article aims to contribute to cognitive therapy for polythematic delusions by proposing a preliminary step to this classical cognitive therapy. Such a step aims to strengthen the patient’s ability to use the general approach of experimentalist type for knowledge acquisition—an approach which is based on the opposition between the theoretical-logical and the experimentalist turn of mind. Some elements such as to enable the reinforcement of such a capability in the patient are thus described and analyzed.

**Theoretical-logical vs. experimentalist turn of mind**

Cognitive therapy of schizophrenia aimed at polythematic delusions includes a component mainly oriented toward teaching the patient skills for search of evidence relating to the ideas associated with polythematic delusions, as well as the construction of alternative hypotheses (Kingdon & Turkington, 1994, 2002). It consists thus, on the one hand, of teaching the patient how to construct alternative hypotheses to delusions such as “television is talking about me”, “a satellite sends me thoughts”, “aliens are plotting against me”, etc., and secondly to encourage him/her to test both delusions and the associated alternative hypotheses, in order to validate or invalidate them. Such an approach applies then, in a *specific* way, to polythematic delusions which are inherent to the patient, and the corresponding alternative hypotheses. At this point, we can observe that this approach is associated with a *general* methodology of the same nature, which is not based on the very polythematic delusions inherent to the patient. Such an approach proves then grounded on the acquisition of the general ability to build up alternative hypotheses and to carry out tests on different hypotheses. Unlike traditional therapy that bears specifically on the content of the polythematic delusions, such an approach presents a general nature and proves likely to refer to any type of hypotheses. We can describe the general nature of such an approach as *experimentalist*.

We can observe here that the fact that classical cognitive therapy is based on strengthening in the patient the ability to develop alternative hypotheses to delusions and to perform tests on them, implicitly relies on the fact that such an ability is low or deficient in the patient, at least with respect to the delusions developed by the latter. We suggest then to make the wider assumption that the overall ability to develop alternative hypotheses and tests in order to validate or invalidate ideas, could be low or deficient in the patient, and that the problem encountered with delusions represents the visible part of a more general problem that is inherent to the patient. We also propose that the two above-mentioned elements (test implementation and construction of alternative hypotheses) are also part of an overall ability that also includes additional elements, and can be defined as the ability to implement the experimentalist method of knowledge acquisition.

For the purpose of the present study, it is necessary to further clarify the very notion of general methodology of experimentalist nature. To this end, it is worth contrasting first the experimentalist turn of mind with the logico-theoretical one. Such opposition allows for a better understanding of the experimentalist approach itself. This opposition corresponds essentially to a classical opposition in science, which contrasts two particular styles that each lead to acquisition of knowledge. Whereas the method of experimentalist inspiration proceeds by elaborating hypotheses and testing them, the logico-theoretical method proceeds by logical deduction or induction from a set of knowledge of which the one who exerts it strives to maintain consistency. Both methods, on the scientific level, each have their supporters and detractors. However, advances in knowledge are to be credited to both methods, which ultimately appear as complementary and may eventually be regarded as two ways of accessing scientific knowledge.

The opposition between experimentalist vs. logico-theoretical turn of mind is not limited, however, to the scientific field. Indeed, such opposition has a more general scope and also applies to any body of knowledge, including that resulting from the process of acquiring information and knowledge concerning everyday life. In this context, the logico-theoretical turn of mind notably proceeds by deduction, by trying to acquire knowledge in a logical way; it is aimed at explaining and interpreting facts and phenomena. Such an approach is underpinned by a concern for consistency of the whole corresponding set of knowledge, by also trying to identify and remedy any internal contradiction. The logico-theoretical approach may also proceed by inductive reasoning, thus making use of inductive generalization. In addition, when an internal contradiction is found, thus rendering the whole set of knowledge inconsistent, the one who proceeds in a logico-theoretical way strives to quickly restore this consistency, by possibly modifying some elements that are part of the overall knowledge at his/her disposal. Conversely, the one who proceeds with the help of the experimentalist approach is basically concerned with validating theories and hypotheses, through experimentation, testing, and search for evidence. He/she is then concerned with only retaining ultimately that knowledge that has been validated by experience and whose strength lies in the evidence which has been thus collected.

Thus characterized through their opposition, it is no less apparent that each of the experimentalist or logico-theoretical method of knowledge acquisition has its drawbacks when pushed to the extreme. When applied in excess, the logico-theoretical method thus leads to risky speculation and lack of evidence, to unrealistic and disconnected from the reality viewpoints. Conversely, experimentation pushed to the extreme leads to knowledge that lacks power of abstraction, explanatory and predictive power, and does not allow understanding of the data and the phenomena. In order to better emphasize the related notions and also better highlight the relationship with the different neighboring concepts, it is worth providing some additional insight over the opposition between experimentalist and logico-theoretical methodology. We shall propose then to make use of the matrices of concepts described in Franceschi (2002), which allow to emphasize the relationships between some given concepts. A matrix of concepts thus consists of six concepts, distributed along two dual poles: A and Ā. Each of these poles admits of respectively a concept of neutral A^{0}, Ā^{0}, positive (A^{+}, Ā^{+}) and negative (A^{–}, Ā^{–}) nature. In total, the matrix consists of the six following concepts: A^{+}, A^{0}, A^{–}, Ā^{+}, Ā^{0}, Ā^{–}.

Among the main *relationships* that can be defined between concepts of the same matrix, it is worth citing: (i) the *duality* relation, between two neutral concepts of different poles: A^{0} and Ā^{0}; (ii) the relation of *antinomy* (or of *contrary*) between two concepts that are not neutral (that is to say, positive or negative) of opposite polarity and of different poles: A^{+} and Ā^{–}, as well as A^{–} and Ā^{+}; (iii) the *complementary* relationship between two concepts of same polarity (positive or negative) and of different poles: A^{+} and Ā^{+} as well as A^{–} and Ā^{–}. For example, the matrix of concepts corresponding to the concepts of *courage*, *cowardice* and *temerity* is thus as follows:

At this step, we are in a position to construct the matrix of concepts that applies to the opposition between theoretical-logical and experimentalist turn of mind. Each of these concepts presents an intrinsic neutral nature, but is likely to give rise to a positive and a negative form. The corresponding matrix of concepts is thus the following:

We can see here that the theoretical-logical turn of mind presents a positive form which leads to fruitful theorization, and a negative form that engenders unrealistic ideas and hazardous speculations. Conversely, the experimentalist turn of mind has a positive form which entails the solidity of knowledge and the search for evidence, and a negative form which leads to extreme empiricism, associated with lack of understanding and absence of explanation.

**General application of the experimentalist methodology**

The foregoing analysis leads to the hypothesis that it might be advisable to rebuild preliminarily in the patient the *general* ability to implement the experimentalist method of knowledge acquisition, before applying it later in a *specific* way to polythematic delusions. Cognitive therapy targeted at polythematic delusions would accordingly involve two phases: the first aimed at restoring in the patient the functional ability to the experimentalist general approach; and the second intended to implement the latter, in a specific way, by applying it to polythematic delusions. Several arguments seem to lean in favor of such an option. Firstly, it seems preferable that the patient be first convinced of the merits of the experimentalist method of acquiring knowledge, even before applying it to polythematic delusions. In other words, it seems better than the patient has himself/herself acquired first the belief that the theoretical-experimental method is effective and useful before applying it to the specific topics corresponding to delusions. Second, it also seems preferable that the patient would acquire a prior good practice and be exercised first to the experimentalist method on external, neutral and impersonal facts, before applying it to his/her own delusions which present for him/her an eminently emotional, personal and sensitive nature. Presumably also the choice of topics external to the patient should be such as to enable him/her to mobilize his/her cognitive abilities optimally. Thirdly, it is reasonable to think that the fact of applying the experimentalist methodology exclusively to the content of polythematic delusions, without possessing at the same time such a general ability might have some disadvantages. One of these drawbacks may lie in the fact that the patient could quickly rebuild some other delusions from other themes than those usual to him/her. Finally, it is worth mentioning that the acquisition and the general practice of the experimentalist methodology on topics that are completely independent of the content itself of the polythematic delusions should be non-confrontational, and likely to preserve the therapeutic alliance.

At this point, it is possible to sketch an outline of what could consist such a preliminary part of cognitive therapy for polythematic delusions. This last part would include a component targeted at learning the construction of alternative hypotheses, and a component designed to the learning of testing different hypotheses. However, in order to form a coherent whole, it is reasonable to think that this part of the therapy should also include an explanation of a number of related concepts, among which we can mention: the distinction between fact and hypothesis; the notion of proof; the distinction between evidence and conviction; the notion of validation and invalidation of a hypothesis; the distinction between fact and interpretation of fact; the distinction between fact and perception of fact; the distinction between fact and fact narration; the construction of alternative hypotheses; the development of tests with regard to a hypothesis; the causal relationship between facts; the proof of the causal relationship; the distinction between facts whose cause is intentional or non-intentional; the notion of explanation of a given phenomenon. Several of these elements, especially those related to the construction of alternative hypotheses, are integral part of the training program for metacognition developed by Moritz et al. (2010, 2011).

It is also useful to point out here several areas where the above-mentioned principles could be put into practice. These areas are potentially very diverse, but it is however possible to describe more accurately some of them, which relate to electronics and computer science. In the field of *electronics* first, consideration could be given to take an interest in electronic circuits (or computer simulations thereof) and their operation. Thus, the patient’s attention could focus on the operational problems of such and such circuit and especially on the search for causes of observed failures (e.g. the fact that a led indicator does not light up). It will then be necessary to formulate different hypotheses regarding the cause of the malfunction, which may relate to different circuit components (transistors, power supply, the led indicator itself, etc..) which can then be tested in order to be validated or invalidated, and later give rise, depending on the test results to other hypotheses, etc..

The field of *computer science*, second, could offer various fields of applications, especially in programming. We may notably consider a computer program that is supposed to produce a given result but has a defect in its execution, due to a “bug”. It will be then a matter of accurately determining the cause of this bug i.e., of finding the specific instruction within the program, which is responsible. The patient would thus be required to make assumptions regarding the specific instruction in the program among *instr*_{1}, *instr*_{2}, *instr*_{3}, …, *instr*_{n}, which is at the origin of the bug and to test successively the latter. This will lead for example to eliminate the instruction *instr*_{1} of which it will be assumed that it is the origin of the bug and to test the program without it, etc.. If the latter test invalidates the hypothesis, such an approach will then lead to test another instruction *instr*_{2}, and so on.

**Specific application of the experimentalist methodology to polythematic delusions**

Classical cognitive therapy of schizophrenia (Kingdon & Turkington, 1994; Beck & Rector, 2000; Kingdon & Turkington, 2002) aims to gradually reduce the degree of belief in the patient’s delusions. For this purpose, the therapist suggests to the patient, in a spirit of dialogue of Socratic inspiration to build alternative hypotheses. He/she also teaches to the patient the process of testing the various competing hypotheses by seeking evidence, thus allowing to confirm or refute them.

It seems useful, at this point, to describe the different stages that occur differently depending on the level—primary, secondary or tertiary—of the corresponding delusions (Franceschi (2008). We shall consider in turn each of these levels. We propose to analyze here the delusions of reference, given that the analysis can be transposed to the delusions of influence, of telepathy, of thought projection or of control.

A *primary* delusional argument of reference, first, has the following structure (the symbol **∴** denotes the conclusion):

(R1) | in T_{1} I was drinking an aperitif | premiss_{1} |

(R2) | in T_{2} the presenter of the show said: “Don’t drink!” | premiss_{2} |

(R3) | ∴ in T_{2} the presenter of the show said: “Don’t drink!” because in T_{1} I was drinking an aperitif | conclusion |

The corresponding delusional idea is that according to which the presenter said in T_{2}: “Don’t drink!” *because* the patient has been drinking an aperitif in T_{1}. The structure of such a delusional idea is as follows: the event E_{1} (in T_{1} I was drinking an aperitif) is the cause of the event E_{2} (in T_{2} the presenter of the show said: “Don’t drink!”). In this case, the logical structure of the alternative hypothesis to the delusional conclusion (R3) is that the event E_{1} which is internal to the patient *is not the cause* of the external event E_{2}. The different alternative hypotheses identify then themselves with *alternative causes* to the event E_{2}. Thus, the delusional conclusion (R3) may be confronted with an alternative hypothesis such as: the presenter said in T_{2}: “Don’t drink!” because the script of this television program contained it. Another alternative hypothesis is that it is the assistant presenter who suggested to say it, etc..

One may think, however, that the fact of proposing to the patient alternative hypotheses to the delusional conclusion (R3) just mentioned, could prove insufficient. In effect, the patient’s delusional idea that the event E_{1} internal to the patient is the cause of the external event E_{2}, not only has the nature of a hypothesis, but also has *explanatory* power, in the sense that it constitutes an explanation of the fact that appears bewildering to the patient that the presenter has said: “Do not drink!” *immediately after* the patient has been drinking an aperitif. In comparison, the fact that the event E_{2} internal to the patient is not the cause of the external event E_{2}, constitutes an alternative hypothesis, but proves devoid of such explanatory power. For this reason, we believe that the mere statement, under this form, of the latter alternative hypothesis should not suffice to gain the support of the patient. For it is necessary to submit to the latter an alternative hypothesis to the conclusion (R3), which is also able to provide an explanation for the rapid succession of events E_{1} and E_{2}. In this context, an alternative hypothesis that also allows to provide an explanation for the rapid succession of two phenomena, is the one according to which the external event E_{2} succeeded immediately after internal event E_{1}, by the effect of a *coincidence*. Under these conditions, the patient faces two competing hypotheses that may explain the rapid and disturbing sequence of events E_{1} and E_{2}: the first hypothesis being that E_{1} is the cause of E_{2}; and the second being that the rapid succession of E_{1} and E_{2} is but a coincidence.

Secondly, the structure of *secondary* delusional arguments of reference is as follows:

(R1) | in T_{2} the presenter of the show spoke according to what I was doing | premiss_{1} |

(R2) | in T_{4} the presenter of the show spoke according to what I felt | premiss_{2} |

(R3) | in T_{6} the presenter of the show spoke according to what I was doing | premiss_{3} |

(R…) | … | … |

(R10) | ∴ the presenters of the shows speak according to what I do or what I feel | conclusion |

The corresponding delusional idea of reference is then the conclusion (R10) that the presenters of the shows speak according to what the patient makes or feels. The conclusion (R10) is of an inductive nature and constitutes a generalization from the several instances (R1), (R2), (R3), … Here, the logical structure of the alternative hypothesis to the conclusion (R10) is that the presenters of the shows *do not speak* according to what the patient makes. But in the same way as above, such a hypothesis proves devoid of explanatory power. In contrast, the alternative hypothesis, which has an additional explanatory power, is the fact that by the effect of coincidences, the rapid succession of two events that may give the impression of the existence of a relationship causality, occurs frequently.

Finally, the *ternary* delusional arguments of reference exhibit the following structure:

(R10) | ∴ the presenters of the shows speak according to what I do or what I feel | premiss |

(R11) | ∴ television speaks about me | conclusion |

The ternary delusional idea of reference is the one according to which television speaks of the patient. The logical structure of the alternative hypothesis is the one under which television *does not speak* of the patient. However, in the same way as above, such a hypothesis does not possess in itself an explanatory power. For the conclusion (R11) has, in the patient’s mind, an explanatory function to the succession of events that he/she experienced. It proves thus necessary, at this stage, to propose an alternative explanation, which resides in the fact that through the effect of coincidences, it frequently happens that the patient’s internal events are immediately followed by external events, which can give the impression that there is a causal relationship between the two successive events. However, it may be pointed out to the patient, there is a much larger number of pairs of successive events that are not consistent with a causal relationship. It is indeed a common attitude to pay attention only to the succession of two events that could be meaningful, even though it occurs every day many more successions of two unmeaningful events and to which one does not pay any attention. This appears as a special case of *misinterpretation of random data* (Bressan, 2002).

**Conclusion**

At this point, it is worth translating the previous elements in terms of testable hypotheses by the clinician. This leads thus to test the hypothesis that cognitive therapy applied to polythematic delusions may be more effective if it included two successive steps: the first advocated by the present study that aims to reconstruct the patient’s general ability to acquisition of knowledge through the practice of the experimentalist method; and the second, classically defined by cognitive therapy of schizophrenia that leads to apply specifically the skills thus acquired to the content of polythematic delusions.

Finally, it is possible to synthesize the ideas expressed in Franceschi (2011) regarding the co-morbidity of schizophrenia with the elements resulting from the present study. We are thus able to define the different stages of the resulting process for cognitive therapy of delusions inherent to schizophrenia. This would mean thus, in a first step, determining the co-morbid disorders (specific phobias, generalized anxiety disorder, social phobia, intermittent explosive disorder, etc.). associated in the patient with the delusional ideas and to apply first a specific cognitive therapy. In a second step, it would mean applying the learning phase of the above-mentioned method experimentalist of a general nature. Finally, in a final phase, it should be proceeded as indicated by classical cognitive therapy, by applying specifically the experimentalist methodology to the content of delusions. This can be translated as follows in terms of testable hypotheses: a cognitive therapy of schizophrenia that would proceed according to these three successive stages could be more effective than classical cognitive therapy.

**References**

Beck, A. (2002). Delusions: A Cognitive perspective. Journal of Cognitive Psychotherapy, 16–4, 455–468.

Bressan, P. (2002). The connection between random sequences, everyday coincidences, and belief in the paranormal. Applied Cognitive Psychology, 16, 17–34.

Franceschi, P. (2002). Une classe de concepts. Semiotica, 139–1/4, 211–226. English translation.

Franceschi, P. (2008). Une défense logique du modèle de Maher pour les délires polythématiques. Philosophiques, 35-2, 451–475. English translation.

Franceschi, P. (2011). Traitement cognitif différentiel des délires polythématiques et du trouble anxieux généralisé. Journal de Thérapie Comportementale et Cognitive, 21–4, 121–125. English translation.

Kingdon, D. & Turkington, D. (1994). Cognitive-behavioural Therapy of Schizophrenia. New York: Guilford.

Kingdon, D. & Turkington, D. (Eds.) (2002). The Case Study Guide to Cognitive Behaviour Therapy of Psychosis. Chichester: Wiley.

Moritz, S., Woodward, T. S., & Metacognition Study Group. (2010). Metacognitive training for schizophrenia (MCT). Manual (4th ed.), Hamburg: VanHam Campus.

Moritz, S., Kerstan, A., Veckenstedt, R., Randjbar, S., Vitzthum, F., Schmidt, C., Heise, M., Woodward, T.S. (2011). Further evidence for the efficacy of a metacognitive group training in schizophrenia. Behaviour Research and Therapy, 49, 151–157.

# On the Plausibility of Psychotic Hallucinations

*A paper published in the Journal for Neurocognitive Research, Vol. 53, No 1-2 (2011).*

In this paper, we describe several factors that can contribute, from the patient’s viewpoint, to the plausibility of psychotic hallucinations. We sketch then a Plausibility of Hallucinations Scale, consisting of a 50-item questionnaire, which aims at evaluating the degree of plausibility of hallucinations. We also emphasize the utility of pointing out to the patient the several factors that contribute to the plausibility of his/her hallucinations, in the context of cognitive therapy for schizophrenia.

This paper is cited in:

- Mark Grimshaw, Tom Garner,
*Sonic Virtuality: Sound as Emergent Perception*, New York: Oxford University Press, 2015 - Charlotte Aynsworth, Daniel Collerton, Robert Dudley,
*Measures of visual hallucinations: Review and recommendations*, Clinical Psychology Review, Volume 57, 2017, Pages 164-182 - I. de Chazeron, B. Pereirae, I. Chereau-Boudete, G. Broussee, D. Misdrahie, G. Fénelone, A.-M. Tronchee, R. Schwane, C. Lançone, A. Marquese, B. Debillye, F. Durife, P.M. Llorca,
*Validation of a Psycho-Sensory Hallucinations Scale (PSAS) in schizophrenia and Parkinson’s disease*, Schizophrenia Research, Volume 161, Issues 2-3, Pages 269–276, 2015

**On the plausibility of psychotic hallucinations**

Cognitive therapy of hallucinations is part of cognitive therapy for schizophrenia. Several accounts of cognitive therapy of hallucinations have been described in the literature (Chadwick et al.,1996; Rector & Beck, 2002; Kingdon & Turkington, 2005). On the one hand, Chadwick et al. (1996) stress the importance of the ABC model for cognitive therapy of hallucinations: the hallucinations are the activating events, which engender cognitions, which in turn yield emotional distress and anger. By working on beliefs about the voices, they primary aim at reducing the negative emotions which are the consequences of automatic thoughts following the occurrence of hallucinations. Chadwick et al. also have a special emphasis on the omnipotence and omniscience of the voices. On the other hand, Kingdon & Turkington (2005) propose the cognitive model of hallucinations as an alternative explanation for the voices: auditory hallucinations are the patient’s automatic thoughts that are perceived as originating from outside the patient’s mind. Kingdon & Turkington weigh the available evidence for both competing explanations and finally work on reattribution of auditory hallucinations. Rector & Beck (2002) take a similar stance, and stress that the final aim of the therapy “is to help patients recognize that the voices simply reflect either their own attitudes about themselves or those they imagine others to have about them”.

The purpose of the present paper is to contribute to cognitive therapy for schizophrenia by focusing on the plausibility of psychotic hallucinations. Our concern will be with providing an account of complex hallucinations encountered in schizophrenia that stresses multiple factors which reinforce, from the patient’s viewpoint, the intrinsic plausibility of the hallucinations. The purpose of this paper is then to expose how hallucinations can seem plausible and credible to the patient. In section 1, we describe several factors that contribute to the plausibility of hallucinations occurring in schizophrenia. We sketch then in section 2 a scale which is designed to measure accurately the plausibility of hallucinations. In section 3, we point out what could be the impact on cognitive therapy for schizophrenia of the present account. Finally, we point out the limitations of the present study and some directions for further research.

1**. Factors of plausibility of hallucinations**

We shall enumerate in what follows several factors that can contribute, from the patient’s viewpoint, to the plausibility of the hallucinations that he/she experiences. Hallucinations are one major symptom of schizophrenia. According to DSM-IV, a hallucination is defined as “A sensory perception that has the compelling sense of reality of a true perception but that occurs without external stimulation of the relevant sensory organ”. (DSM-IV, p. 767). By *plausibility*, we mean the fact that the patient’s abnormal perceptions are seemingly attributable to an external source (usually, other people). The plausibility that results from certain phenomenological features of auditory hallucinations has notably been hinted at by Stephane et al. (2003):

(…) hearing “multiple voices” is associated with attribution of the “voices” to others, which is plausible intuitively as well. This indicates that the patients’ experiences of hallucinations could be understood, intuitively, based on common sense experiences of the world.

In this paper, we shall expand this idea, by pointing out that multiple factors are susceptible of congruently strengthening the patient’s conviction that his/her abnormal perceptions come from the external world.

Hallucinations come in a variety of modalities. In order to shed light on the factors that can lead to the plausibility of hallucinations occurring in schizophrenia, it is worth drawing first some useful distinctions.

*1.1 Unimodal and multi-modal hallucinations*

Let us consider, to begin with, the distinction between unimodal and multi-modal hallucinations. Unimodal hallucinations can be classified into five types, corresponding to our five sensory pathways: auditory, visual, olfactory, tactile and gustatory. Multi-modal hallucinations are made up of unimodal hallucinations of different types which occur simultaneously (or quasi-simultaneously). There are accordingly 26 combinations of multi-modal hallucinations (plus 5 unimodal ones). The latter can be enumerated exhaustively as follows (we also describe an instance of some common multi-modal cases, since it can be useful for explanatory purposes):

(i) *1-modal*: *auditory*, *visual*, *olfactory*, *tactile*, *gustatory*

(ii) *2-modal*: *auditory-visual* (“I saw *x* sitting on my bed and I heard him saying ‘Bastard!’”), *auditory-olfactory* (“I heard *x* saying ‘I will smoke a cigar! ’ and at this very moment I smell a taste of tobacco”), *auditory-tactile* (“I heard *x* saying ‘You will be stung by a scorpion!’ and at this very moment I felt a sharp sting of pain on my left arm”), *auditory-gustatory*, *visual-olfactory* (“I saw *x* on my bed smoking a cigar and I also smell the taste of tobacco”), *visual-tactile* (“I saw a scorpion on my left arm and at this very moment I felt a sharp sting of pain there”), *visual-gustatory* (“I saw blood dripping from my finger and it had the taste of blood when I put it on my tongue”), *olfactory-tactile*, *olfactory-gustatory*, *tactile-gustatory*

(iii) *3-modal*: *auditory-visual-olfactory*, *auditory-visual-tactile* (“I heard *x* saying ‘You will be stung by a scorpion!’ and at this very moment I saw a scorpion on my left arm while feeling a sharp sting of pain there”), *auditory-visual-gustatory* (“I heard ‘I will harm you’ and at this very moment I saw blood dripping from my finger and it had the taste of blood when I put it on my tongue”), *auditory-olfactory-tactile*, *auditory-olfactory-gustatory*, *auditory-tactile-gustatory*, *visual-olfactory-tactile*, *visual-olfactory-gustatory*, *visual-tactile-gustatory*, *olfactory-tactile-gustatory*

(iv) *4-modal*: *auditory-visual-olfactory-tactile*, *auditory-visual-olfactory-gustatory*, *auditory-visual-tactile-gustatory* (“I heard ‘I will harm you’ and then I saw blood dripping from my finger, while I felt a sharp pain there. It had the taste of blood when I put it on my tongue”), *auditory-olfactory-tactile-gustatory*, *visual-olfactory-tactile-gustatory*

(v) *5-modal*: *auditory-visual-olfactory-tactile-gustatory*

At this step, it should be noted that multi-modal hallucinations retain their force from the plausibility that results from the simultaneous (or quasi-simultaneous) occurrence of two or more unimodal hallucinations of different types. For this reason, multi-modal hallucinations retain considerable power with regard to unimodal ones. The sense of reality that results from multi-modal hallucinations is due to the fact that several sensory pathways are congruently involved in the hallucinatory process. If we consider, for example, bimodal hallucinations of the auditory-visual type, it proves to be that the simultaneous occurrence of an additional visual hallucination strongly reinforces the sense of reality that results from the auditory hallucination. As the above examples illustrate, multi-modal hallucinations are seemingly highly more plausible and realistic than unimodal ones, and result in a much greater sense of realism. More generally, it illustrates how (*n* + 1)-modal hallucinations are seemingly much more realistic than *n*-modal ones, a supplementary sensory pathway being involved in the hallucinatory process.

*1.2 Factors of plausibility of auditory hallucinations*

It is worth mentioning, second, several factors that can contribute to the intrinsic plausibility of auditory hallucinations:

(i) *structured* versus *unstructured* auditory hallucinations: structured sounds notably consist of comments on the patient’s thoughts or activities, conversations of several persons, or commands ordering the patient to do things, etc., while on the other hand, unstructured sounds consist of ringing, buzzing, whistling, etc.

(ii) auditory hallucinations having an *external* versus an *internal* origin: auditory hallucinations seemingly coming out from outer space could reinforce the idea that the voices have an external origin, e.g. are attributable to other people.

(iii) the *locus* (Chadwick et al., 1996, p. 103) – i.e. the space location – of auditory hallucinations is also susceptible of reinforcing their intrinsic plausibility. We can consider, for example, a patient who hears the voice of the presenter of the show saying ‘Bastard!’. Now this sounds more realistic if the locus of the voice is the television device rather than the ashtray. Let us suppose now that the patient hears a voice saying ‘I can read your thoughts’. Now it sounds more likely to the patient if the voice comes out from the telephone than from the halogen lamp.

It is worth noting here that this criterion is susceptible to vary from culture to culture.^{1} In effect, depending on the individuals, a speaking tree or a speaking animal could be, in certain cases, consistent with the patient’s cultural background.

(iv) *bilateral *versus* unilateral* auditory hallucinations: auditory hallucinations coming indifferently from the patient’s right or from his/her left are more plausible than unilateral ones.^{2}

(v) time location *related* versus *unrelated* to the patient’s thoughts, emotions or actions (Stephane et al. 2003 make mention of the “relation to the moment”). In this regard, auditory hallucinations that are simultaneous with the patients internal phenomena gain more plausibility.

(vi) *phrases* versus *single words*: in this context, phrases, conversations, elaborate sentences gain are more plausible than single words.

(vii) *multiple voices* versus *single voice* (Stephane et al., 2003).

(viii) auditory hallucinations* fitting *versus *not fitting with the patient’s desires or fears*: this factor consists of whether the hallucinations experienced by a patient fit adequately or not with his/her individual fears or desires. For in the affirmative, it would greatly increase the plausibility of the corresponding hallucinations. Let us take an example. The patient is very anxious about the evolution of his/her illness. He/she hears a voice that says: ‘You will relapse next month’. Now the content of this auditory hallucination fits adequately with the patient’s own fears. The reason why auditory hallucinations fitting with desires or fears are more plausible, is that they are *coherent* with the patient’s belief system. By contrast, had the content of auditory hallucinations been unrelated or contradictory with the patient’s desires and fears, the corresponding information would have then resulted in a lack of coherence with the patient’s belief system (this is in line with the approach to hallucinations exposed in Rector & Beck, 2002, which is concerned with: “(…) how the specific voice content and beliefs about the voices reflect the person’s prehallucinatory fears, concerns, interests, preoccupations and fantasies”).

(ix) *interactive *versus *non-interactive voices*: whether the patient can interact or not with voices, i.e. discuss or engage in dialog with them.

*1.3 Factors of plausibility of visual hallucinations*

Several factors can contribute, third, to the intrinsic plausibility of visual hallucinations:

(i) *formed* versus *unformed* visual hallucinations: formed hallucinations are made up of figures, faces, morphing objects or scenes. By contrast, unformed hallucinations consist of dots, lines, geometrical figures, flashes, etc.

(ii) *ordinary* versus *bizarre* or extraordinary visual hallucinations: for obvious reasons, objects that look ordinary gain more likeliness than bizarre, unreal objects.

(iii) objects in *color* versus in *black and white*.

(iv) visual hallucinations *fitting *versus *not fitting with surroundings*: as noted by Teunisse et al. (1996), the relationship to surroundings could play an important role in the plausibility of complex hallucinations. Such or such unimodal hallucination could fit well (e.g. a person lying on a bed, a scorpion walking on the ground) or not (a figure on the ceiling) with surroundings. Now it should be apparent that fitting with surroundings visual hallucinations are consistent with the patient’s knowledge of the physical world. This renders, from the patient’s viewpoint, the hallucination very plausible. By analogy with the locus of auditory hallucinations, fitting with surroundingscan be assimilated to the *locus* – i.e. space location – of visual hallucinations.

(v) *bilateral *versus* unilateral* visual hallucinations.

(vi) time location of visual hallucinations *related* versus *unrelated* to the patient’s thoughts, emotions or actions (e.g. the patient thinks to a scorpion and at this very moment he/she sees a scorpion on the ground).

(vii) *animated* versus *static* images.

*1.4 Factors of plausibility of olfactory hallucinations*

Several factors can contribute, fourth, to the plausibility of olfactory hallucinations:

(i) *bilateral *versus* unilateral* olfactory hallucinations.

(ii) olfactory hallucinations* fitting *versus *not fitting with the patient’s desires or fears*: the patient fears of being killed and smells a poisonous gas in his/her room.

(iii) *transient* versus *permanent* olfactory hallucinations: some patients experience olfactory hallucinations that occur any time of day and also last for hours (Tousi & Frankel 2004).

*1.5 Factors of plausibility of tactile hallucinations*

Certain factors can contribute, fifth, to the plausibility of tactile hallucinations:

(i) *bilateral *versus* unilateral* tactile hallucinations.

(ii) tactile hallucinations* fitting *versus *not fitting with the patient’s desires or fears*: the patient fears of being murdered and feels an electric-shock sensation.

*1.6 Factors of plausibility of gustatory hallucinations*

Some factors can contribute, sixth, to the plausibility of gustatory hallucinations:

(i) *common *versus* strange* gustatory hallucinations: in some cases, the patient may find that his/her food tastes strange. This could decrease the plausibility of the corresponding hallucination, in contrast with gustatory hallucinations where the patient experiences normal and common tastes.

(ii) gustatory hallucinations* fitting *versus *not fitting with the patient’s desires or fears*: the patient fears of being poisoned and feels the taste of poison in his/her mouth.

2**. Plausibility of hallucinations scale**

From the above, it results that it could be useful to measure accurately the plausibility of the hallucinations occurring in schizophrenia. For this purpose, we shall now sketch a 50-item scale, which is targeted at evaluating the plausibility of hallucinations experienced by a patient. This binary scale consists of a questionnaire which allows for yes/no answers (each yes answer weighting 2 points):

It is worth noting that this 50-item scale can be regarded as non-specific to psychotic hallucinations. It is also suited to other disorders or illnesses involving hallucinations. Among these are other mental illnesses, but also Charles Bonnet syndrome (Teunisse et al., 1996; Menon et al., 2003), epilepsy (Sachdev, 1998; Schwartz & Marsh, 2000), etc. In particular, the Charles Bonnet syndrome occurs in the elderly and is usually associated with ocular pathology and severe visual impairment. The Charles Bonnet syndrome is characterized by the presence of complex and persistent visual hallucinations. The syndrome is usually associated with an absence of hallucinations in other sensory modalities. It is worth noting that the Charles Bonnet syndrome affects psychologically normal individuals with full or partial insight and the patients are accordingly non-delusional. On the other hand, auditory hallucinations are frequently associated with temporal lobe epilepsy, where hallucinations in other modalities can also occur.

It is worth mentioning that the Plausibility of Hallucinations Scale could also be used in association with other instruments for measuring insight, such as the Beck Cognitive Insight Scale (Beck et al., 2003) in order to gain more accurate knowledge of the patient’s state. For schizophrenia is usually associated with lack of insight into the internal origin of the hallucinations. By contrast, in other illnesses such as Charles Bonnet syndrome or in pseudo-hallucinations related to brain trauma or PTSD (Stephane et al., 2004), the patient usually retains insight into the internal origin of his/her hallucinations.

3**. Impact on Cognitive Therapy**

We suggest that the above emphasis on the plausibility of hallucinations could be usefully incorporated into the process of cognitive-behavior therapy of schizophrenia (Kingdon & Turkington, 1994, 2005; Chadwick et al., 1996; Rector & Beck, 2002). The general idea would be to point out to the patient who experiences highly plausible hallucinations those factors that confer to his/her hallucinations their intrinsic plausibility. Hopefully, this could insert itself well into the process of cognitive-behavior therapy, whose primary goal is to help the patient gaining more insight into the nature of his/her hallucinations and in particular to understand that they do not originate from an external source. In this context, stressing to the patient the plausibility of his/her hallucinations, could help him/her understand better how hallucinations can be self-deceiving.

It is worth focusing, to begin with, on multi-modal hallucinations. In this context, a first step would be to point out to the patient that multi-modal hallucinations are capable of seeming very plausible and realistic. It could then be argued and explained to the patient that multi-modal hallucinations are more plausible than unimodal ones. This could be illustrated through some examples. This latter strategy could make use of “what if statements” (Ellis & Dryden, 1997). Along these lines, it could be pointing out to the patient that if someone, instead of experiencing one single auditory hallucination, would experience simultaneously one supplementary visual hallucination, then the resulting multi-modal (of the auditory-visual type) hallucination would sound much more realistic. Along these lines, it could be pointed out to the patient that the particular case of multi-modal hallucination that he/she experiences is potentially very realistic and inherently capable of deceiving him/her.

Once the patient familiar with the concept of multi-modal hallucinations, another goal could be to learn the patient how to use by himself/herself the preceding taxonomy of multi-modal hallucinations and to apply it when he/she experiences these types of complex hallucinations. He/she would then be capable of identifying the corresponding case at hand. Hopefully, this could help the patient rationalize his/her abnormal perceptions and perhaps accept better the internal origin of his/her hallucinations as an alternative explanation.

The fact of classifying multi-modal hallucinations would be helpful to the patient, it seems, to help him/her rationalize and explain the phenomena he/she experiences. For we should bear in mind that the patient experiences abnormal phenomena, which are unfamiliar to psychologically normal people. In this context, helping the patient rationalize, classify and describe accurately the phenomena of his/her own internal world, proves then to be a valuable practical goal to attain. Accordingly, identifying, recognizing and labeling a given type of multi-modal hallucination could help lessen its associated omnipotence (Chadwick et al., 1996). This could be helpful to the patient, we suggest, who ordinarily faces an unexplained and upsetting phenomenon. More generally, the fact of identifying the various factors that render his/her complex hallucinations so plausible could help the patient gaining more insight into the internal origin of his/her hallucinations. At this step, it should be noted that the present account is notably in line – for what concerns the delusion that consists in attributing an external origin to the hallucinations – with the views advocated by Brendan Maher (1988, 1999), who sees delusions as a patient’s attempt to explain some perplexing and puzzling phenomena. According to Maher, delusions arise from normal (mainly rational but occasionally irrational) reasoning applied to abnormal phenomena. Among these abnormal phenomena which are very perplexing to the patient are the hallucinations.

4**. Limitations and directions for further research**

The main limitation of the present study is that the psychometric properties of the Plausibility of Hallucinations Scale have not been tested. However, given the number of items of the scale, one can expect a good sensibility. On the other hand, the reliability and validity of the scale remain to be tested.

Finally, the above developments suggest several questions, which could be usefully the subject of further study, based on the Plausibility of Hallucinations Scale. A first question that arises is the following: (i) Is the plausibility of hallucinations rate higher in schizophrenia than in other illnesses involving hallucinations, e.g. other mental illnesses, Charles Bonnet syndrome, temporal lobe epilepsy, etc.? A comparison of the plausibility of hallucinations rate occurring in schizophrenia and other illnesses involving hallucinations could be made accordingly. We suggest that such comparison could provide some useful information about the relationships of these illnesses (Sachdev, 1998). Although schizophrenia (paranoid subtype) should prima facie involve a higher rating, it seems that an accurate measure of the degree of plausibility of hallucinations could result in some interesting information. Along these lines, a comparison of the plausibility of hallucinations ratings occurring in different subtypes of schizophrenia could also be informative.

The above Plausibility of Hallucinations Scale is also designed to allow for comparisons between different chronological states in the same patient. This suggests a second type of question: (ii) Does the plausibility of hallucinations rating evolve during the course of schizophrenia? Along these lines, Nayani & David (1996) observed an increase in the complexity of auditory hallucinations over time, seemingly related to lesser distress and better coping. A similar question could be raised for other illnesses involving hallucinations. In this context, Menon et al. (2003) reported accordingly that “Elementary hallucinations may progressively evolve into complex visual hallucinations” in the Charles Bonnet syndrome.

Lastly, a third interesting question goes as follows: (iii) Is the plausibility rate of hallucinations occurring in schizophrenia correlated with the I.Q. of the patient, i.e. do patients with a high I.Q. more frequently experience complex hallucinations with a high level of plausibility? In other words, is the following hypothesis confirmed: The higher the I.Q., the higher the plausibility of hallucinations rating? Hopefully, the answer to these questions will provide some information that might well be useful to the understanding of the illness and to cognitive-behavior therapy of schizophrenia.

# Acknowledgments

We thank Peter Brugger, Paul Gilbert and Hélène Verdoux for very useful comments on earlier drafts.

# References

American Psychiatric Association (1994). Diagnostic and Statistical Manual of Mental Disorders (4th edition). American Psychiatric Association: Washington. |

Beck, A. T., Rector, N. A. (2003). A Cognitive Model of Hallucinations. Cognitive Therapy and Research, 27(1), 19-52. |

Beck, A. T., Baruch, E., Balter, J. M., Steer, R. A., Warman, D. M. (2004). A new instrument for measuring insight: the Beck Cognitive Insight Scale. Schizophrenia Research, 68(2-3), 319-329. |

Chadwick, P., Birchwood, M. & Trower, P. (1996). Cognitive Therapy for Delusions, Voices, and Paranoia. Chichester: Wiley. |

Ellis, A., Dryden, W. (1997). The practice of rational emotive behaviour therapy, London: Free Association Books. |

Kingdon, D. & Turkington, D. (1994). Cognitive-behavioural Therapy of Schizophrenia. Guilford: New York. |

Kingdon, D., & Turkington, D. (2005). Cognitive Therapy of Schizophrenia. New York, London: Guilford. |

Maher, B. A. (1988). Anomalous experiences and delusional thinking: the logic of explanations. In: T.F. Oltmanns & B.A. Maher (Eds.), Delusional Beliefs,pp. 15-33.Wiley: New York. |

Maher, B. A. (1999). Anomalous experience in everyday life: Its significance for psychopathology. The Monist, 82, 547-570. |

Menon, G. J., Rahman, I., Menon, S. J., Dutton, G. N. (2003). Complex visual hallucinations in the visually impaired: the Charles Bonnet Syndrome. Survey of Ophthalmology, 48, 58-72. |

Nayani, T. H., David, A. S. (1996). The auditory hallucination: a phenomenological survey. Psychological Medicine, 26(1), 177-189. |

Rector, N. A., Beck A. T. (2002). Cognitive Therapy for Schizophrenia: From Conceptualization to Intervention. Canadian Journal of Psychiatry, 47(1), 41-50. |

Sachdev, P. (1998). Schizophrenia-Like Psychosis and Epilepsy: The Status of the Association. American Journal of Psychiatry, 155(3), 325-336. |

Schwartz, J. M., Marsh, L. (2000). The Psychiatric Perspectives of Epilepsy. Psychosomatics, 41(1), 31-38. |

Siddle, R. (2002). Communications from my parents. In: D. Kingdon & D. Turkington (Eds.), The Case Study Guide to Cognitive Behaviour Therapy of Psychosis, pp. 109-121. Chichester: Wiley. |

Stephane, M., Thuras, P., Nasrallah, H., Georgopoulos, A. P. (2003). The internal structure of the phenomenology of auditory verbal hallucinations. Schizophrenia Research, 61, 185-193. Stephane, M., Hill, T., Matthew, E., & Folstein, M. (2004) New phenomenon of abnormal auditory perception associated with emotional and head trauma: Pathological confirmation by SPECT scan. Brain and Language, 89, 503-507. |

Teunisse, R. J., Cruysberg, J. R., Hoefnagels, W. H. (1996). Visual hallucinations in psychologically normal people: Charles Bonnet’s syndrome. Lancet, 347, 794-797. |

Toone, B. K. (2000). The psychoses of epilepsy. Journal of Neurology, Neurosurgery and Psychiatry, 69(1), 1-3. |

Tousi, B. & Frankel, M. (2004). Olfactory and visual hallucinations in Parkinson’s disease, Parkinsonism and Related Disorders, 10, 253-254. |

1 We thank Paul Gilbert for the suggestion of taking into account cultural beliefs with regard to this specific criterion.

2 We owe the suggestion to include the bilateral/unilateral distinction related to hallucinations in all modalities to Peter Brugger.

# A Two-Sided Ontological Solution to the Sleeping Beauty Problem

*Preprint published on the PhilSci archive.*

I describe in this paper an ontological solution to the Sleeping Beauty problem. I begin with describing the hyper-entanglement urn experiment. I restate first the Sleeping Beauty problem from a wider perspective than the usual opposition between halfers and thirders. I also argue that the Sleeping Beauty experiment is best modelled with the hyper-entanglement urn. I draw then the consequences of considering that some balls in the hyper-entanglement urn have ontologically different properties from normal ones. In this context, drawing a red ball (a Monday-waking) leads to two different situations that are assigned each a different probability, depending on whether one considers “balls-as-colour” or “balls-as-object”. This leads to a two-sided account of the Sleeping Beauty problem.

This account supersides my previous preprints on this topic. Please do no cite previous work.

**A Two-Sided Ontological Solution to the Sleeping Beauty Problem**

1**. The hyper-entanglement urn**

Let us consider the following experiment. In front of you is an urn. The experimenter asks you to study very carefully the properties of the balls that are in the urn. You go up then to the urn and begin to examine its content carefully. You notice first that the urn contains only red or green balls. By curiosity, you decide to take a sample of a red ball in the urn. Surprisingly, you notice that while you pick up this red ball, another ball, but a green one, also moves simultaneously. You decide then to replace the red ball in the urn and you notice that immediately, the latter green ball also springs back in the urn. Intrigued, you decide then to catch this green ball. You notice then that the red ball also goes out of the urn at the same time. Furthermore, while you replace the green ball in the urn, the red ball also springs back at the same time at its initial position in the urn. You decide then to withdraw another red ball from the urn. But while it goes out of the urn, nothing else occurs. Taken aback, you decide then to undertake a systematic and rigorous study of all the balls in the urn.

At the end of several hours of a meticulous examination, you are now capable of describing precisely the properties of the balls present in the urn. The latter contains in total 1000 red balls and 500 green balls. Among the red balls, 500 are completely normal balls. But 500 other red balls have completely astonishing properties. Indeed, each of them is linked to a different green ball. When you remove one of these red balls, the green ball which is associated with it also goes out at the same time from the urn, as if it was linked to the red ball by a magnetic force. Indeed, if you remove the red ball from the urn, the linked green ball also disappears instantly. And conversely, if you withdraw from the urn one of the green balls, the red ball which is linked to it is immediately removed from the urn. You even try to destroy one of the balls of a linked pair of balls, and you notice that in such case, the ball of the other colour which is indissociably linked to it is also destroyed instantaneously. Indeed, it seems to you that relative to these pairs of balls, the red ball and the green ball which is linked to it behave as one single object.

The functioning of this urn leaves you somewhat perplexed. In particular, your are intrigued by the properties of the pairs of correlated balls. After reflection, you tell yourself that the properties of the pairs of correlated balls are finally in some respects identical to those of two entangled quantum objects. Entanglement (Aspect & al. 1982) is indeed the phenomenon which links up two quantum objects (for example, two photons), so that the quantum state of one of the entangled objects is correlated or anti-correlated with the quantum state of the other, whatever the distance where the latter is situated. As a consequence, each quantum object can not be fully described as an object per se, and a pair of entangled quantum objects is better conceived of as associated with a single, entangled state. It also occurs to you that perhaps a pair of correlated balls could be considered, alternatively, as a ubiquitous object, i.e. as an object characterised by its faculty of occupying two different locations at the same time, with the colours of its two occurrences being anti-correlated. Setting this issue aside for the moment, you prefer to retain the similarity with the more familiar quantum objects. You decide to call “hyper-entanglement urn” this urn with its astonishing properties. After reflection, what proves to be specific to this urn, is that it includes at the same time some normal and some hyper-entangled balls. The normal red balls are no different from our familiar balls. But hyper-entangled balls do behave in a completely different way. What is amazing, you think, is that nothing seemingly differentiates the normal red balls from the red hyper-entangled ones. You tell yourself finally that it could be confusing.

Your reflection on the pairs of hyper-entangled balls and their properties also leads you to question the way the balls which compose the pairs of hyper-entangled balls are to be counted. Are they to be counted as normal balls? Or do specific rules govern the way these pairs of hyper-entangled balls are to be counted? You add a normal red ball in a hyper-entanglement urn. It is then necessary to increment the number of red balls present in the urn. On the other hand, the total number of green balls is unaffected. But what when you add in the hyper-entanglement urn the red ball of a pair of hyper-entangled balls? In that case, the linked green ball of the same pair of hyper-entangled balls is also added instantly in the urn. Hence, when you add a red ball of a pair of hyper-entangled balls in the urn, it also occurs that you add at the same time its associated green ball. So, in that case, you must not only increment the total number of red balls, but also the total number of green balls present in the urn. In the same way, if you withdraw a normal red ball from the urn, you simply decrement the total number of red balls of the urn, and the number of green balls in the urn is unaffected. But if you remove the red ball (resp. green) of a pair of hyper-entangled balls, you must decrement the total number of red balls (resp. green) present in the urn as well as the total number of green balls (resp. red).

At this very moment, the experimenter happens again and withdraws all balls from the urn. He announces that you are going to participate in the following experiment:

*The hyper-entanglement urn* A fair coin will be randomly tossed. If the coin lands Heads, the experimenter will put in the urn a normal red ball. On the other hand, if the coin lands Tails, he will put in the urn a pair of hyper-entangled balls, composed of a red ball and a green ball, both indissociably linked. The experimenter also adds that the room will be put in absolute darkness, and that you will therefore be completely unable to detect the colour of the balls, no more that you will be able to know, when you will have withdrawn a ball from the urn, whether it is a normal ball, or a ball which is part of a pair of hyper-entangled balls. The experimenter tosses then the coin. While you catch a ball from the urn, the experimenter asks you to assess the likelihood that the coin felt Heads.

2**. The Sleeping Beauty problem**

Consider now the well-known *Sleeping Beauty problem* (Elga 2000, Lewis 2001). Sleeping Beauty learns that she will be put into sleep on Sunday by some researchers. A fair coin will be tossed and if the coin lands Heads, Beauty will be awakened once on Monday. On the other hand, if the coin lands Tails, Beauty will be awakened twice: on Monday and Tuesday. After each waking, she will be put into sleep again and will forget that waking. Furthermore, once awakened, Beauty will have no idea of whether it is Monday or Tuesday. On awakening on Monday, what should then be Beauty’s credence that the coin landed Heads?

At this step, one obvious first answer (I) goes as follows: since the coin is fair, the initial probability that the coin lands Head is 1/2. But during the course of the experiment, Sleeping Beauty does not get any novel information. Hence, the probability of Heads still remains 1/2.

By contrast, an alternative reasoning (II) runs as follows. Suppose the experiment is repeated many times, say, to fix ideas, 1000 times. Then there will be approximately 500 Heads-wakings on Monday, 500 Tails-wakings on Monday and 500 Tails-wakings on Tuesday. Hence, this reasoning goes, the probability of Heads equals 500/1500 = 1/3.

The argument for 1/2 and the argument for 1/3 yield conflicting conclusions. The Sleeping Beauty problem is usually presented accordingly as a problem arising from contradicting conclusions resulting from the two above-mentioned competing lines of reasoning aiming at assigning the probability of Heads once Beauty is awakened. I shall argue, however, that this statement of the Sleeping Beauty problem is somewhat restrictive and that we need to envisage the issue from a wider perspective. For present purposes, the Sleeping Beauty problem is the issue of calculating properly (i) the probability of Heads (resp. Tails) once Beauty is awakened; (ii) the probability of the day being Monday (resp. Tuesday) on awakening; and (iii) the probability of Heads (resp. Tails) on waking on Monday. From the halfer perspective, the probability of the day being Monday on awakening equals 3/4, and the probability of the day being Tuesday on awakening is 1/4. By contrast, from the thirder’s perspective, the probability of the day being Monday on awakening equals 2/3 and the probability of the day being Tuesday on awakening is 1/3.

But the argument for 1/2 and for 1/3 also have their own account of conditional probabilities. To begin with, the probability of Heads on waking on Tuesday is not a subject of disagreement, for it equals 0 in both accounts. The same goes for the probability of Tails on waking on Tuesday, since it equals 1 from the halfer’s or from the thirder’s viewpoint. But agreement stops when one considers the probability of Heads on waking on Monday. For it equals 2/3 from a halfer’s perspective. However, from a thirder’s perspective, it amounts to 1/2. On the other hand, the probability of Tails on waking on Monday is 1/3 from a halfer standpoint, and 1/2 for a thirder.

3**. The urn analogy**

In what follows, I shall present an ontological solution to the Sleeping Beauty problem, which rests basically on the *hyper-entanglement urn* experiment. A specific feature of this account is that it incorporates insights from the halfer and thirder standpoints, a line of resolution initiated by Nick Bostrom (2007) that has recently inspired some new contributions (Groisman 2008, Delabre 2008)^{1}.

The argument for 1/3 and the argument for 1/2 rest basically on an urn analogy. This analogy is made explicit in the argument for 1/3 but is less transparent in the argument for 1/2. The argument for 1/3, to begin with, is based on an urn analogy which associates the situation related to the Sleeping Beauty experiment with an urn that contains, in the long run (assuming that the experiment is repeated, say, 1000 times), 500 red balls (Heads-wakings on Monday), 500 red balls (Tails-wakings on Monday) and 500 green balls (Tails-wakings on Tuesday), i.e. 1000 red balls and 500 green balls in total. In this context, the probability of Heads upon awakening is determined by the ratio of the number of Heads-wakings to the total number of wakings. Hence, P(Heads) = 500/1500 =1/3. The balls in the urn are normal ones and for present purposes, it is worth calling this sort of urn a “standard urn”.

On the other hand, the argument for 1/2 is also based on an urn analogy, albeit less transparently. The main halfer proponent grounds his reasoning on calculations (Lewis 2001), but for the sake of clarity, it is worth rendering the underlying associated analogy more apparent. For this purpose, let us recall how the calculation of the probability of drawing a red ball is handled by the argument for 1/2. If the coin lands Heads then the probability of drawing a red ball is 1, and if the coin lands Tails then this latter probability equals 1/2. We get then accordingly the probability of drawing a red ball (Monday-waking): P(R) = 1 x 1/2 + 1/2 x 1/2 = 3/4. By contrast, if the coin lands Tails, we calculate as follows the probability of drawing a green ball (Tuesday-waking): P(G) = 0 x 1/2 + 1/2 x 1/2 = 1/4. To sum up, according to the argument for 1/3: P(R) = 3/4 and P(G) = 1/4. For the sake of comparison, it is worth transposing this reasoning in terms of an urn analogy. Suppose then that the Sleeping Beauty experiment is iterated. It proves then that the argument for 1/2 is based on an analogy with a standard urn that contains 3/4 of red balls and 1/4 of green ones. These balls are also normal ones and the analogy underlying the argument for 1/2 is also with a “standard urn”. Now assuming as above that the experiment is repeated 1000 times, we get accordingly an urn that contains 500 red balls (Heads-wakings on Monday), 250 red balls (Tails-wakings on Monday) and 250 green balls (Tails-wakings on Tuesday), i.e. 750 red balls and 250 green balls in total. Such content of the urn results directly from Lewis’ calculation. However, as it stands, this analogy would arguably be a poor argument in favour of the halfer’s viewpoint. But at this step, we should pause and consider that Lewis’ argument for 1/2 did not rely on this urn analogy, though the latter is a consequence of Lewis’ calculation. We shall now turn to the issue of whether the standard urn is the correct analogy for the Sleeping Beauty experiment.

In effect, it turns out that the argument for 1/3 and the argument for 1/2 are based on an analogy with a standard urn. But at this stage, a question arises: is the analogy with the standard urn well-suited to the Sleeping Beauty experiment? In other terms, isn’t another urn model best suited? In the present context, this alternative can be formulated more accurately as follows: isn’t the situation inherent to the Sleeping Beauty experiment better put in analogy with the *hyper-entanglement urn*, rather than with the *standard urn*? I shall argue, however, that the analogy with the standard urn is mistaken, for it fails to incorporate an essential feature of the experiment, namely the fact that Monday-Tails wakings are indissociable from Tuesday-Tails wakings. For in the Tails case, Beauty cannot wake up on Monday without also waking up on Tuesday and reciprocally, she cannot wake up on Tuesday without also waking up on Monday.

When one reasons with the standard urn, one feels intuitively entitled to add red-Heads (Heads-wakings on Monday), red-Tails (Tails-wakings on Monday) and green-Tails (Tails-wakings on Tuesday) balls to compute frequencies. But red-Heads and red-Tails balls prove to be objects of an essentially different nature in the present context. In effect, red-Heads balls are in all respects similar to our familiar objects, and can be considered properly as single objects. By contrast, it turns out that red-Tails balls are quite indissociable from green-Tails balls. For we cannot draw a red-Tails ball without picking up the associated green-Tails ball. And conversely, we cannot draw a green-Tails ball without picking up the associated red-Tails ball. In this sense, red-Tails balls and the associated green-Tails balls do not behave as our familiar objects, but are much similar to entangled quantum objects. For Monday-Tails wakings are indissociable from Tuesday-Tails wakings. On Tails, Beauty cannot be awakened on Monday (resp. Tuesday) without being also awakened on Tuesday (resp. Monday). From this viewpoint, it is mistaken to consider red-Tails and green-Tails balls as separate objects. The correct intuition, I shall argue, is that the red-Tails and the associated green-Tails ball can be assimilated to a pair of hyper-entangled balls and constitute but one single object. In this context, red-Tails and green-Tails balls are best seen intuitively as constituents and mere parts of one single object. In other words, red-Heads balls and, on the other hand, red-Tails and green-Tails balls, cannot be considered as objects of the same type for probability purposes. And this situation justifies the fact that one is not entitled to add unrestrictedly red-Heads, red-Tails and green-Tails balls to compute probability frequencies. For in this case, one adds objects of intrinsically different types, i.e. one single object with the mere part of another single object.

Given what precedes, the correct analogy, I contend, is with a hyper-entanglement* urn* rather than with a normal urn. As will become clearer later, this new analogy incorporates the strengths of both above-mentioned analogies with the standard urn. And we shall now consider the Sleeping Beauty problem in light of this new perspective.

4**. Consequences of the analogy with the hyper-entanglement urn**

At this step, it is worth drawing the consequences of the analogy with the *hyper-entanglement urn*, that notably result from the ontological properties of the balls. Now the key point proves to be the following one. Recall that nothing seemingly distinguishes normal balls from hyper-entangled ones within the *hyper-entanglement urn*. And among the red balls, half are normal ones, but the other half is composed of red balls that are each hyper-entangled with a different green ball. If one considers the behaviour of the balls, it turns out that normal balls behave as usual. But hyper-entangled ones do behave differently, with regard to statistics. Suppose I add the red ball of a hyper-entangled pair into the *hyper-entanglement urn*. Then I also add instantly in the urn its associated green ball. Suppose, conversely, that I remove the red ball of a hyper-entangled pair from the urn. Then I also remove instantly its associated green ball.

At this step, we are led to the core issue of calculating properly the probability of drawing a red ball from the *hyper-entanglement urn*. Let us pause for a moment and forget temporarily the fact that, according to its classical formulation, the Sleeping Beauty problem arises from conflicting conclusions resulting from the argument for 1/3 and the argument for 1/2 on calculating the probability of Heads once Beauty is awakened. For as we did see it before, the problem also arises from the calculation of the probability of the day being Monday on awakening (drawing a red ball), since conflicting conclusions also result from the two competing lines of reasoning. In effect, Elga argues for 2/3 and Lewis for 3/4. Hence, the Sleeping Beauty problem could also have been formulated alternatively as follows: once awakened, what probability should Beauty assign to her waking on Monday? In the present context, this is tantamount to the probability of drawing a red ball from the *hyper-entanglement urn*.

What is then the response of the present account, based on the analogy with the *hyper-entanglement urn*, to the issue of calculating the probability of drawing a red ball? In the present context, “drawing a red ball” turns out to be somewhat ambiguous. For according to the ontological properties of the balls within the *hyper-entanglement urn*, one can consider red balls either from the viewpoint of colour-ness, or from the standpoint of object-ness^{2}. Hence, in the present context, “drawing a red ball” can be interpreted in two different ways: either (i) “drawing a red ball-as-colour”; or (ii) “drawing a red ball-as-object”. Now disambiguating the notion of drawing a red ball, we should distinguish accordingly between two different questions. First, (i) what is the probability of drawing a red ball-as-colour (Monday-waking-as-time-segment)? Let us denote by P(R↑) the latter probability. Second, (ii) what is the probability of drawing a red ball-as-object (Monday-waking-as-object)? Let us denote it by P(R→). This distinction makes sense in the present context, since it results from the properties of the hyper-entangled balls. In particular, this richer semantics results from the case where one draws a green ball of a hyper-entangled pair from the urn. For in the latter case, this green ball is not a red one, but it occurs that one also picks up a red ball, since the associated red ball is withdrawn simultaneously.

Suppose, on the one hand, that we focus on the colour of the balls, and that we consider the probability P(R↑) of drawing a red ball-as-colour. It occurs now that there are 2/3 of red balls-as-colour and 1/3 of green balls-as-colour in the urn. Accordingly, the probability P(R↑) of drawing a red ball-as-colour equals 2/3. On the other hand, the probability P(G↑) of drawing a green ball-as-colour equals 1/3.

Assume, on the other hand, that we focus on balls as objects, considering that one pair of hyper-entangled balls behaves as one single object. Now we are concerned with the probability P(R→) of drawing a red ball-as-object. On Heads, the probability of drawing a red ball-as-object is 1. On Tails, we can either draw the red or the green ball of a hyper-entangled pair. But it should be pointed out that if we draw on Tails the green ball of a hyper-entangled pair, we also pick up instantly the associated red ball. Hence, the probability of drawing a red ball on Tails is also 1. Thus, P(R→) = 1 x 1/2 + 1 x 1/2 = 1. Conversely, what is the probability P(G→) of drawing a green ball-as-object (a waking on Tuesday)? The probability of drawing a green ball-as-object is 0 in the Heads case, and 1 in the Tails case. For in the latter case, we either draw the green or the red ball of a hyper-entangled pair. But even if we draw the red ball of the hyper-entangled pair, we draw then instantly its associated green ball. Hence, P(G→) = 0 x 1/2 + 1 x 1/2 = 1/2. To sum up: P(R→) = 1 and P(G→) = 1/2. The probability of drawing a red ball-as-object (a waking on Monday) is then 1, and the probability of drawing a green ball-as-object (a waking on Tuesday) is 1/2. Now it turns out that P(R→) + P(G→) = 1 + 1/2 = 1.5. In the present account, this results from the fact that drawing a red ball-as-object and drawing a green ball-as-object from a hyper-entangled pair are not* exclusive events* for probability purposes. For we cannot draw the red-Tails (resp. green-Tails) ball without drawing the associated green-Tails (resp. red-Tails) ball.

To sum up now. It turns out that the probability P(R↑) of drawing a red ball-as-colour (Monday-waking-as-time-segment) equals 2/3. And the probability P(G↑) of drawing a green ball-as-colour (Tuesday-waking-as-time-segment) equals 1/3. On the other hand, the probability P(R→) of drawing a red ball-as-object (Monday-waking-as-object) equals 1; and the probability P(G→) of drawing a green ball-as-object (Tuesday-waking-as-object) equals 1/2.

At this step, we are led to the issue of calculating properly the number of balls present in the urn. Now we should distinguish, just as before, according to whether one considers balls-as-colour or balls-as-object. Suppose then that we focus on the colour of the balls. Then we have grounds to consider that there are in total 2/3 of red balls and 1/3 of green balls in the *hyper-entanglement urn*, i.e. 1000 red ones and 500 green ones. This conforms with the calculation that results from the thirder’s standpoint. Suppose, that we rather focus on balls as single objects. Things go then differently. For we can consider first that there are 1000 balls as objects in the urn, i.e. 500 (red) normal ones and 500 hyper-entangled ones. Now suppose that the 500 (red) normal balls are removed from the urn. Now there only remain hyper-entangled balls within the urn. Suppose then that we pick up one by one the remaining balls from the urn, by removing alternatively one red ball and one green ball from the urn. Now it turns out that we can draw 250 red ones and 250 green ones from the urn. For once we draw a red ball from the urn, its associated green ball is also withdrawn. And conversely, when we pick up a green ball from the urn, its associated red ball is also withdrawn. Hence, inasmuch as we consider balls as objects, there are in total 750 red ones and 250 green ones in the urn. At this step, it should be noticed that this corresponds accurately to the composition of the urn which is associated with Lewis’ halfer calculation. But this now makes sense, as far as the analogy with the *hyper-entanglement urn* is concerned. The above-mentioned analogy with the urn associated with Lewis’ halfer calculation was a poor argument inasmuch as the urn was a standard one, but things go differently when one considers now the analogy with the *hyper-entanglement urn*.

5**. A two-sided account**

From the above, it results that the line of reasoning which is associated with the balls-as-colour standpoint corresponds to the thirder’s reasoning. And conversely, the line of thought which is associated with the balls-as-object viewpoint echoes the halfer’s reasoning. Hence, the balls-as-colour/balls-as-object dichotomy parallels the thirder/halfer opposition. Grounded though they are on an unsuited analogy with the standard urn, the argument for 1/3 and the argument for 1/2 do have, however, their own strengths. In particular, the analogy with the urn in the argument for 1/3 does justice to the fact that the Sleeping Beauty experiment entails that 2/3 of Monday-wakings will occur in the long run. On the other hand, the analogy with the urn in the argument for 1/2 handles adequately the fact that one Heads-waking is put on a par with two Tails-wakings. In the present context however, these two analogies turn out to be *one-sided* and fail to handle adequately the probability notion of drawing a red ball (waking on Monday). But in the present context, the probability P(R↑) of drawing a red ball-as-colour corresponds to the thirder’s insight. And the probability P(R→) of drawing a red ball-as-object corresponds to the halfer’s line of thought. At this step, it turns out that the present account is *two-sided*, since it incorporates insights from the argument for 1/3 and from the argument for 1/2.

Finally, it turns out that the standard urn which is classically used to model the Sleeping Beauty problem does not allow for two possible interpretations of the probability of drawing a red ball. Rather, in the standard urn model, the two interpretations are exclusive of one another and this yields the classical contradiction between the argument for 1/3 and the argument for 1/2. But as we did see it, with the *hyper-entanglement urn* model, this contradiction dissolves, since two different interpretations of the probability of drawing a red ball (waking on Monday) are now allowed, yielding then two different calculations. In the latter model, these probabilities are no more exclusive of one another and the contradiction dissolves into complementarity.

Now the same ambiguity plagues the statement of the Sleeping Beauty problem, and its inherent notion of “waking”. For shall we consider “wakings-as-time-segment” or “wakings-as-object”? The initial statement of the Sleeping Beauty problem is ambiguous about that, thus allowing the two competing viewpoints to develop, with their respective associated calculations. But once we diagnose accurately the source of the ambiguity, namely the ontological status of the wakings, we allow for the two competing lines of reasoning to develop in parallel, thus dissolving the initial contradiction^{3}.

In addition, what precedes casts new light on the argument for 1/3 and the argument for 1/2. For given that the Sleeping Beauty experiment, is modelled with a standard urn, both accounts lack the ability to express the difference between the probability P(R↑) of drawing a red ball-as-colour (a Monday-waking-as-time-segment) and the probability P(R→) of drawing a red ball-as-object (a Monday-waking-as-object), for it does not make sense with the standard urn. Consequently, there is a failure to express this difference with the standard urn analogy, when considering drawing a red ball. But such distinction makes sense with the analogy with the *hyper-entanglement urn*. For in the resulting richer ontology, the distinction between P(R↑) and P(R→) yields two different results: P(R↑) = 2/3 and P(R→) = 1.

At this step, it is worth considering in more depth the balls-as-colour/balls-as-object opposition, that parallels the thirder/halfer contradiction. It should be pointed out that “drawing a red ball-as-colour” is associated with an indexical (“this ball is red”), somewhat internal standpoint, that corresponds to the thirder’s insight. Typically, the thirder’s viewpoint considers things from the inside, grounding the calculation on the indexicality of Beauty’s present waking. On the other hand, “drawing a red ball-as-object” can be associated with a non-indexical (“the ball is red”), external viewpoint. This corresponds to the halfer’s standpoint, which can be viewed as more general and external.

As we did see it, the calculation of the probability of drawing a red ball (waking on Monday) is the core issue in the Sleeping Beauty problem. But what is now the response of the present account on conditional probabilities and on the probability of Heads upon awakening? Let us begin with the conditional probability of Heads on a Monday-waking. Recall first how the calculation goes on the two concurrent lines of reasoning. To begin with, the probability P(Heads|G) of Heads on drawing a green ball is not a subject of disagreement for halfers and thirders, since it equals 0 on both accounts. The same goes for the probability P(Tails|G) of Tails on drawing a green ball, since it equals 1 from the halfer’s or the thirder’s viewpoint. But agreement stops when one considers the probability P(Heads|R) of Heads on drawing a red ball. For P(Heads|R) = 1/2 from the thirder’s perspective and P(Heads|R) = 2/3 from the halfer’s viewpoint. On the other hand, the probability P(Tails|R) of Tails on drawing a red ball is 1/2 for a thirder and 1/3 for a halfer.

Now the response of the present account to the calculation of the conditional probability of Heads on drawing a red ball (waking on Monday) parallels the answer made to the issue of determining the probability of drawing a red ball. In the present account, P(Heads|G) = 0 and P(Tails|G) = 1, as usual. But we need to disambiguate how we interpret “drawing a red ball” by distinguishing between P(Heads|R↑) and P(Heads|R→), to go any further. For P(Heads|R↑) is the probability of Heads on drawing a red ball-as-colour. And P(Heads|R→) is the probability of Heads on drawing a red ball-as-object. P(Heads|R↑) is calculated in the same way as in the thirder’s account. Now we get accordingly: P(Heads|R↑) = 1/2. On the other hand, P(Heads|R→) is computed in the same way as from the halfer’s perspective, and we get accordingly: P(Heads|R→) = [P(Heads) x P(R→|Heads)] / P(R→) = [1/2 x 1] / 1 = 1/2.

Now the same goes for the probability of Heads upon awakening. For there are two different responses in the present account, depending on whether one considers P(R↑) or P(R→). If one considers balls-as-colour, the probability of Heads upon awakening is calculated in the same way as in the argument for 1/3, and we get accordingly: P(Heads↑) = 1/3 and P(Tails↑) = 2/3. On the other hand, if one is concerned with balls-as-object, it ensues, in the same way as with the halfer’s account, that there is no shift in the prior probability of Heads. As Lewis puts it, Beauty’s awakening does not add any novel information. It follows accordingly that the probability P(Heads→) of Heads (resp. Tails) on awakening still remains 1/2.

Finally, the above results are summarised in the following table:

halfer | thirder | present account | |

P(Heads↑) | 1/3 | 1/3 | |

P(Tails↑) | 2/3 | 2/3 | |

P(Heads→) | 1/2 | 1/2 | |

P(Tails→) | 1/2 | 1/2 | |

P(drawing a red ball-as-colour) ≡ P(R↑) | 2/3 | 2/3 | |

P(drawing a green ball-as-colour) ≡ P(G↑) | 1/3 | 1/3 | |

P(drawing a red ball-as-object) ≡ P(R→) | 3/4 | 1 | |

P(drawing a green ball-as-object) ≡ P(G→) | 1/4 | 1/2 | |

P(Heads| drawing a red ball-as-colour) ≡ P(Heads|R↑) | 1/2 | 1/2 | |

P(Tails| drawing a red ball-as-colour) ≡ P(Tails|R↑) | 1/2 | 1/2 | |

P(Heads| drawing a red ball-as-object) ≡ P(Heads|R→) | 2/3 | 1/2 | |

P(Tails| drawing a red ball-as-object) ≡ P(Tails|R→) | 1/3 | 1/2 |

At this step, it is worth recalling the diagnosis of the Sleeping Beauty problem put forth by Berry Groisman (2008). Groisman attributes the two conflicting responses to the probability of Heads to an ambiguity in the protocol of the Sleeping Beauty experiment. He argues that the argument for 1/2 is an adequate response to the probability of Heads on awakening, under the setup of coin tossing. On the other hand, he considers that the argument for 1/3 is an accurate answer to the latter probability, under the setup of picking up a ball from the urn. Groisman also considers that putting a ball in the box and picking up a ball out from the box are two different events, that lead therefore to two different probabilities. Roughly speaking, Groisman’s “coin tossing/picking up a ball” distinction parallels the present balls-as-colour/balls-as-object dichotomy. However, in the present account, putting a ball in the urn is no different from picking up a ball from the urn. For if we put in the urn a red ball of a hyper-entangled pair, we also immediately put in the urn its associated green ball. Rather, from the present standpoint, drawing (resp. putting in the urn) a red ball-as-colour from the urn is probabilistically different from picking up a red ball-as-object. The present account and Groisman’s analysis share the same overall direction, although the details of our motivations are significantly different.

Finally, the lesson of the Sleeping Beauty Problem proves to be the following: our current and familiar objects or concepts such as balls, wakings, etc. should not be considered as the sole relevant classes of objects for probability purposes. We should bear in mind that according to an unformalised axiom of probability theory, a given situation is classically modelled with the help of urns, dices, balls, etc. But the rules that allow for these simplifications lack an explicit formulation. However in certain situations, in order to reason properly, it is also necessary to take into account somewhat unfamiliar objects whose constituents are pairs of indissociable balls or of mutually inseparable wakings, etc. This lesson was anticipated by Nelson Goodman, who pointed out in *Ways of Worldmaking* that some objects which are prima facie completely different from our familiar objects also deserve consideration: “we do not welcome molecules or concreta as elements of our everyday world, or combine tomatoes and triangles and typewriters and tyrants and tornadoes into a single kind”.^{4} As we did see it, in some cases, we cannot add unrestrictedly an object of the Heads-world with an object of the Tails-world. For despite the appearances, objects of the Heads-world may have ontologically different properties from objects of the Tails-world. And the status of our probabilistic paradigm object, namely a ball, proves to be world-relative, since it can be a whole in the Heads-world and a part in the Tails-world. Once this goodmanian step accomplished, we should be less vulnerable to certain subtle cognitive traps in probabilistic reasoning.

**Acknowledgements**

I thank Jean-Paul Delahaye and Claude Panaccio for useful discussion on earlier drafts. Special thanks are due to Laurent Delabre for stimulating correspondence and insightful comments.

**References**

Arntzenius, F. (2002). Reflections on Sleeping Beauty. *Analysis*, 62-1, 53-62

Aspect, A., Dalibard, J. & Roger, G. (1982). Physical Review Letters. 49, 1804-1807

Black, M. (1952). The Identity of Indiscernibles. *Mind* 61, 153-164

Bostrom, N. (2002). *Anthropic Bias: Observation Selection Effects in Science and Philosophy*. (New York: Routledge)

Bostrom, N. (2007). Sleeping Beauty and Self-Location: A Hybrid Model. *Synthese*, 157, 59-78

Bradley, D. (2003). Sleeping Beauty: a note on Dorr’s argument for 1/3. *Analysis*, 63, 266-268

Delabre, L. (2008). La Belle au bois dormant : débat autour d’un paradoxe. Manuscript

Elga, A. (2000). Self-locating Belief and the Sleeping Beauty Problem. *Analysis*, 60, 143-147

Goodman, N. (1978). *Ways of Worldmaking.* (Indianapolis: Hackett Publishing Company)

Groisman, B. (2008). The End of Sleeping Beauty’s Nightmare. *British Journal for the Philosophy of Science*, 59, 409-416

Leslie, J. (2001). *Infinite Minds* (Oxford & New York: Oxford University Press)

Lewis, D. (2001). Sleeping Beauty: Reply to Elga. *Analysis*, 61, 171-176

Monton, B. (2002). Sleeping Beauty and the Forgetful Bayesian. *Analysis*, 62, 47-53

White, R. (2006). The generalized Sleeping Beauty problem: A challenge for thirders. *Analysis*, 66, 114-119

1 Bostrom opens the path to a third way out to the Sleeping Beauty problem: “At any rate, one might hope that having a third contender for how Beauty should reason will help stimulate new ideas in the study of self-location”. In his account, Bostrom sides with the halfer on P(Heads) and with the thirder on conditional probabilities, but his treatment has some counter-intuitive consequences on conditional probabilities.

2 This issue relates to the identity of indiscernibles and is notably hinted at by Max Black (1952, p. 156) who describes a universe composed of two identical spheres: “Isn’t it logically possible that the universe should have contained nothing but two exactly similar spheres? We might suppose that each was made of chemically pure iron, had a diameter of one mile, that they had the same temperature, colour, and so on, and that nothing else existed. Then every quality and relational characteristic of the one would also be a property of the other.” In the present context, it should be pointed out that the colours of the hyper-entangled balls are anti-correlated. John Leslie (2001, p. 153) also raises a similar issue with his paradox of the balls: “Here is a yet greater paradox for Identity of Indiscernibles to swallow. Try to picture a cosmos consisting just of three qualitatively identical spheres in a straight line, the two outer ones precisely equidistant from the one at the centre. Aren’t there plain differences here? The central sphere must be nearer to the outer spheres than these are to each other. Identity of Indiscernibles shudders at the symmetry of the situation, however. It holds that the so-called two outer spheres must really be only a single sphere. And this single sphere, which now has all the same qualities as its sole surviving partner, must really be identical to it. There is actually just one sphere!”.

3 It is worth noting that the present treatment of the Sleeping Beauty problem, is capable of handling several variations of the original problem that have recently flourished in the literature. For the above solution to the Sleeping Beauty problem applies straightforwardly, I shall argue, to these variations of the original experiment. Let us consider, to begin with, a variation were on Heads, Sleeping Beauty is not awakened on Monday but instead on Tuesday. This is modelled with a *hyper-entanglement urn* that receives one normal green ball (instead of a red one in the original experiment) in the Heads case.

Let us suppose, second, that Sleeping Beauty is awakened two times on Monday in the Tails case (instead of being awakened on both Monday and Tuesday). This is then modelled with a hyper-entanglement* urn* that receives one pair of hyper-entangled balls which are composed of two red balls in the Tails case (instead of a pair of hyper-entangled balls composed of a red and a green ball in the original experiment).

Let us imagine, third, that Beauty is awakened two times – on Monday and Tuesday – in the Heads case, and three times – on Monday, Tuesday and Wednesday – in the Tails case. This is then modelled with a hyper-entanglement* urn* that receives one pair of hyper-entangled balls composed of one red ball and one green ball in the Heads case; and in the Tails case, the *hyper-entanglement urn* is filled with one triplet of hyper-entangled balls, composed of one red, one green and one blue ball.

4 Goodman (1978, p. 21).

# Elements of Dialectical Contextualism

*Posprint in English (with additional illustrations) of an article appeared in French in the collective book (pages 581-608) written on the occasion of the 60th birthday of Pascal Engel.*

**Abstract** In what follows, I strive to present the elements of a philosophical doctrine, which can be defined as *dialectical contextualism*. I proceed first to define the elements of this doctrine: dualities and polar contraries, the principle of dialectical indifference and the one-sidedness bias. I emphasize then the special importance of this doctrine in one specific field of meta-philosophy: the methodology for solving philosophical paradoxes. Finally, I describe several applications of this methodology on the following paradoxes: Hempel’s paradox, the surprise examination paradox and the Doomsday Argument.

In what follows, I will endeavour to present the elements of a specific philosophical doctrine, which can be defined as *dialectical contextualism*. I will try first to clarify the elements that characterise this doctrine, especially the dualities and dual poles, the principle of dialectical indifference and the one-sidedness bias. I will proceed then to describe its interest at a meta-philosophical level, especially as a methodology to assist in the resolution of philosophical paradoxes. Finally, I will describe an application of this methodology to the analysis of the following philosophical paradoxes: Hempel’s paradox , the surprise examination paradox and the Doomday Argument.

The dialectical contextualism described here is based on a number of constitutive elements which have a specific nature. Among these are: the dualities and dual poles, the principle of dialectical indifference and the one-sidedness bias. It is worth analysing in turn each of these elements.

1**. Dualities and dual poles**

To begin with, we shall focus on defining the concept of *dual poles* (*polar opposites*)^{1}. Although intuitive, this concept needs to be clarified. Examples of dual poles are *static*/*dynamic*, *internal*/*external*, *qualitative*/*quantitative*, etc.. We can define the dual poles as concepts (which we shall denote by A and Ā), which come in pairs, and are such that each of them is defined as the opposite of the other. For example, *internal* can be defined as the opposite of *external* and symmetrically, *external* can be defined as the contrary of *internal*. In a sense, there is no primitive notion here and neither A nor Ā of the dual poles can be regarded as the primitive notion. Consider first a given duality, that we can denote by A/Ā, where A and Ā are dual concepts. This duality is shown in the figure below:

At this point, we can also provide a list (which proves to be necessarily partial) of dualities:

Internal/External, Quantitative/Qualitative, Visible/Invisible, Absolute/Relative Abstract/Concrete, Static/Dynamic, Diachronic/Synchronic, Single/Multiple, Extension/Restriction, Aesthetic/Practical, Precise/Vague, Finite/Infinite, Single/compound, Individual/Collective, Analytical/Synthetic, Implicit/Explicit, Voluntary/Involuntary

In order to characterize more accurately the dual poles, it is worth distinguishing them from other concepts. We shall stress then several properties of the dual poles, which allow to differentiate them from other related concepts. The dual poles are neutral concepts, as well as simple qualities; in addition, they differ from vague notions. To begin with, two dual poles A and Ā constitute *neutral* concepts. They can thus be denoted by A^{0} and Ā^{0}. This leads to represent both concepts A^{0} and Ā^{0} as follows:

The dual poles are neutral concepts, i.e. concepts that present no ameliorative or pejorative nuance. In this sense, *external*, *internal*, *concrete*, *abstract*, etc.., are dual poles, unlike concepts such as *beautiful*, *ugly*, *brave*, which present either a ameliorative or pejorative shade, and are therefore non-neutral. The fact that the dual poles are neutral has its importance because it allows to distinguish them from concepts that have a positive or negative connotation. Thus, the pair of concepts *beautiful*/*ugly* is not a duality and therefore *beautiful* and *ugly* do not constitute dual poles in the sense of the present construction. Indeed, *beautiful* has a positive connotation and *ugly* has a pejorative connotation. In this context, we can denote them by *beautiful*^{+} and *ugly*^{–}.

It should be emphasised, second, that the two poles of a given dual duality correspond to *simple qualities*, as opposed to *composite qualities*. The distinction between single and composite qualities can be made in the following manner. Let A_{1} and A_{2} be simple qualities. In this case, A_{1} ∧ A_{2}, and A_{1} ∨ A_{2} are composite qualities. To take an example, *static*, *qualitative*, *external* are simple qualities, while *static and qualitative*, *static and external*, *qualitative and external* are composite qualities. A more general definition is as follows: let B_{1} and B_{2} be single or composite qualities, then B_{1} ∧ B_{2} and B_{1} ∨ B_{2} are composite qualities. Incidentally, this also highlights why the pairs of concepts *red*/*non-red*, *blue*/*non-blue* concepts can not be considered as dual poles. Indeed, *non-red* can thus be defined as follows as a composite quality: *violet* ∨ *indigo* ∨ *blue* ∨ *green* ∨ *yellow* ∨ *orange* ∨ *white* ∨ *black*. In this context, one can assimilate *non-blue* to the *negation-complement* of *blue*, such complement negation being defined with the help of composite qualities.

Given the above definition, we are also in a position to distinguish the dual poles from *vague* objects. We can first note that dual poles and vague objects have certain properties in common. Indeed, vague objects come in pairs in the same way as dual poles. Moreover, vague concepts are classically considered as having an extension and an anti-extension, which are mutually exclusive. Such a feature is also shared by the dual poles. For example, *qualitative* and *quantitative* can be assimilated respectively to an extension and an anti-extension, which also have the property of being mutually exclusive, and the same goes for *static* and *dynamic*, etc.. However, it is worth noting the differences between the two types of concepts. A first difference (i) lies in the fact that the union of the extension and the anti-extension of vague concepts is not exhaustive in the sense that they admit of borderline cases (and also borderline cases of borderline cases, etc., giving rise to a hierarchy of higher-order vagueness of order *n*), which is a penumbra zone. Conversely, the dual poles do not necessarily have such a characteristic. Indeed, the union of the dual poles can be either exhaustive or non-exhaustive. For example, the *abstract*/*concrete* duality is then intuitively exhaustive, since there does not seem to exist any objects that are neither abstract nor concrete. The same goes for the *vague*/*precise* duality: intuitively, there does no exist indeed objects that are neither vague nor precise, and that would belong to an intermediate category. Hence, there are dual poles whose extension and anti-extension turns out to be exhaustive, unlike vague concepts, such as the two poles of the *abstract*/*concrete* duality. It is worth mentioning, second, another difference (ii) between dual poles and vague objects. In effect, dual poles are simple qualities, while vague objects may consist of simple or compound qualities. There exist indeed some vague concepts which are termed multi-dimensional vague objects, such as the notion of *vehicle*, of *machine*, etc.. A final difference between the two categories of objects (iii) lies in the fact that some dual poles have an inherently precise nature. This is particularly the case of the *individual*/*collective* duality, which is susceptible to give rise to a very accurate definition.

2**. The principle of dialectical indifference**

From the notions of duality and of dual poles which have been just mentioned, we are in a position to define the notion of a *viewpoint* related to a given duality or dual pole. Thus, we have first the notion of viewpoint corresponding to a given A/Ā duality: it consists for example in the standpoint of the *extension*/*restriction* duality, or of the *qualitative*/*quantitative* duality or of the *diachronic*/*synchronic* duality, etc.. It also follows the concept of point of view related to a given pole of an A/Ā duality: we get then, for example (at the level of the *extension*/*restriction* duality) the standpoint by *extension*, as well as the viewpoint by *restriction*. Similarly, the *qualitative* viewpoint or perspective results from it, as well as the *quantitative* point of view, etc.. (at the level of the *qualitative*/*quantitative* duality). Thus, when considering a given object *o* (either a concrete or an abstract object such as a proposition or a reasoning), we may consider it in relation to various dualities, and at the level of the latter, relative to each of its two dual poles.

The underlying idea inherent to the viewpoints relative to a given duality, or to a given pole of a duality, is that each of the two poles of the same duality, *all things being equal*, deserve an equal legitimacy. In this sense, if we consider an object *o* in terms of a duality A/Ā, one should not favour one of the poles with respect to the other. To obtain an objective point of view with respect to a given duality A/Ā, one should place oneself in turn from the perspective of the pole A, and then from that of the pole Ā. For an approach that would only address the viewpoint of one of the two poles would prove to be partial and truncated. The fact of considering in turn the perspective of the two poles, in the study of an object *o* and of its associated reference class allows to avoid a subjective approach and to meet as much as possible the needs of objectivity.

As we can see it, the idea underlying the concept of point of view can be formalized in a *principle of dialectical indifference*, in the following way:

(PRINCIPLE OF DIALECTICAL INDIFFERENCE) When considering a given object *o* and the reference class E associated with it, from the angle of duality A/Ā, all things being equal, it should be given equal weight to the viewpoint of the A pole and the viewpoint of the Ā pole.

This principle is formulated in terms of a *principle of indifference*: if we consider an object *o* under the angle of an A/Ā duality, there is no reason to favour the viewpoint from A with regard to the viewpoint from Ā, and unless otherwise resulting from the context, we must weigh equally the viewpoints A and Ā. A direct consequence of this principle is that if one considers the perspective of the A pole, one also needs to take into consideration the standpoint of the opposite pole Ā (and vice versa). The need to consider both points of view, the one resulting from the A pole and the other associated with the Ā pole, meets the need of analysing the object *o* and the reference class associated with it from an objective point of view. This goal is achieved, as far as possible, by taking into account the complementary points of view which are those of the poles A and Ā. Each of these viewpoints has indeed, with regard to a given duality A/Ā, an equal relevance. Under such circumstances, when only the A pole or (exclusively) the pole Ā is considered, it consists then of a *one-sided* perspective. Conversely, the viewpoint which results from the synthesis of the standpoints corresponding to both poles A and Ā is of a *two-sided* type. Basically, this approach proves to be dialectical in essence. In effect, the step consisting of successively analysing the complementary views relative to a given reference class, is intended to allow, in a subsequent step, a final synthesis, which results from the joint consideration of the viewpoints corresponding to both poles A and Ā. In the present construction, the process of confronting the different perspectives relevant to an A/Ā duality is intended to build cumulatively, a more objective and comprehensive standpoint than the one, necessarily partial, resulting from taking into account those data that stem from only one of the two poles.

The definition of the dialectical principle of indifference proposed here refers to a *reference class* E, which is associated with the object *o*. The reference class^{2} is constituted by a number of phenomena or objects. Several examples can be given: the class of human beings who ever lived, the class of future events in the life of a person, the class of body parts of a given person, the class of ravens, etc.. We shall consider in what follows, a number of examples. Mention of such a reference class has its importance because its very definition is associated with the above-mentioned duality A/Ā. In effect, the reference class can be defined either from the viewpoint of A or from the viewpoint of Ā. Such a feature needs to be emphasized and will be useful in defining the bias which is associated with the very definition of the principle of dialectical indifference: the *one-sidedness bias*.

3**. Characterisation of the one-sidedness bias**

The previous formulation of the principle of dialectical indifference suggests straightforwardly an error of reasoning of a certain type. Informally, such a fallacy consists in focusing on a given standpoint when considering a given object, and of neglecting the opposite view. More formally, in the context described above, such a fallacy consists, when considering an object *o* and the reference class associated with it, in taking into account the viewpoint of the A pole (respectively Ā), while completely ignoring the viewpoint corresponding to its dual pole Ā (respectively A) to define the reference class. We shall term *one-sidedness bias* such type of fallacy. The conditions of this type of bias, in violation of the principle of dialectical indifference, needs however to be clarified. Indeed, in this context, we can consider that there are some cases where the two-sidedness with respect to a given duality A/Ā is not required. Such is the case when the elements of the context do not presuppose conditions of objectivity and exhaustiveness of views. Thus, a lawyer who would only emphasise the evidence in defence of his/her client, while completely ignoring the evidence against him/her does not commit the above-mentioned type of error of reasoning. In such a circumstance, in fact, the lawyer would not commit a faulty one-sidedness bias, since it is his/her inherent role. The same would go in a trial for the prosecutor, who conversely, would only focus on the evidence against the same person, by completely ignoring the exculpatory elements. In such a situation also the resulting one-sidedeness bias would not be inappropriate, because it follows well from the context that it consists well of the limited role assigned to the prosecutor. By contrast, a judge who would only take into account the evidence against the accused, or who would commit the opposite error, namely of only considering the exculpatory against the latter, would well commit an inappropriate one-sidedness bias because the mere role of the judge implies that he/she takes into account the two types of elements, and that his/her judgement is the result of the synthesis which is made.

In addition, as hinted at above, the mention of a reference class associated with the object *o* proves to be important. In effect, as we will have the opportunity to see it with the analysis of the following examples, the definition itself is associated with an A/Ā duality. And the reference class can be defined either from the viewpoint of A, or from the viewpoint of Ā. Such feature has the consequence that all objects are not likely to give rise to a one-sidedness bias. In particular, the objects that are not associated with a reference class that is itself likely to be envisaged in terms of an A/Ā duality, do not give rise to any such one-sidedness bias.

Before illustrating the present construction with the help of several practical examples, it is worth considering, at this stage, the one-sidedness bias which has been just defined, and which results from the very definition of the principle of dialectical indifference, in the light of several similar concepts. In a preliminary way, we can observe that a general description of this type of error of reasoning had already been made, in similar terms, by John Stuart Mill (On Liberty, II):

He who knows only his own side of the case, knows little of that. His reasons may be good, and no one may have been able to refute them. But if he is equally unable to refute the reasons on the opposite side; if he does not so much know what they are, he has no ground for preferring either opinion.

In the recent literature, some very similar concepts have also been described. It consists in particular of the *dialectic bias* notably described by Douglas Walton (1999). Walton (999, pp. 76-77) places then himself in the framework of the dialectical theory of bias, which opposes one-sided to two-sided arguments:

The dialectical theory of bias is based on the idea […] that an argument has two sides. […] A *one-sided argument* continually engages in pro-argumentation for the position supported and continually rejects the arguments of the opposed side in a dialogue. A *two-sided* (*balanced*) argument considers all arguments on both sides of a dialogue. A balanced argument weights each argument against the arguments that have been opposed to it.

Walton describes thus the *dialectical bias* as a one-sided perspective that occurs during the course of the argument. Walton emphasizes, though, that dialectic bias, which is universally common in human reasoning, does not necessarily constitute an error of reasoning. In line with the distinction between “good” and “bad” bias due to Antony Blair (1988), Walton considers that the dialectic bias is incorrect only under certain conditions, especially if it occurs in a context that is supposed to be balanced, that is to say where the two sides of the corresponding reasoning are supposed to be mentioned (p. 81):

Bad bias can be defined as “pure (one-sided) advocacy” in a situation where such unbalanced advocacy is normatively inappropriate in argumentation.

A very similar notion of one-sidedness bias is also described by Peter Suber (1998). Suber describes indeed a fallacy that he terms *one-sidedness fallacy*. He describes it as a fallacy which consists in presenting one aspect of the elements supporting a judgement or a viewpoint, by completely ignoring the other aspect of the relevant elements relating to the same judgement:

The fallacy consists in persuading readers, and perhaps ourselves, that we have said enough to tilt the scale of evidence and therefore enough to justify a judgment. If we have been one-sided, though, then we haven’t yet said enough to justify a judgment. The arguments on the other side may be stronger than our own. We won’t know until we examine them.

The error of reasoning consists then in taking only into account one viewpoint relating to the judgement in question, whereas the other viewpoint could as well prove to be decisive with regard to the conclusion to be drawn. Suber also undertakes to provide a characterization of the one-sidedness fallacy and notes in particular that the fallacy of one-sidedness constitutes a valid argument. For its conclusion is true if its premises are true. Moreover, Suber notes, it appears that the argument is not only valid but sound. For when the premises are true, the conclusion of the argument can be validly inferred. However, as hinted at by Suber, the argument is defective due to the fact that a number of premises are lacking. This is essential because if the missing premises are restored within the argument, the resulting conclusion can be radically different.

4**. An instance of the one-sidedness bias**

To illustrate the above concepts, it is worth at this stage providing an example of the one-sidedness bias. To this end, consider the following instance, which is a form of reasoning, mentioned by Philippe Boulanger (2000, p. 3)^{3}, who attributes it to the mathematician Stanislaw Ulam. The one-sidedness bias shows up in a deductive form. Ulam estimates that if a company were to achieve a level of workforce large enough, its performance would be paralysed by the many internal conflicts that would result. Ulam estimates that the number of conflicts between people would increase according to the square of the number *n* of employees, while the impact on the work that would result would only grow as a function of *n*. Thus, according to this argument, it is not desirable that the number of employees within a company becomes important. However, it turns out that Ulam’s reasoning is fallacious, as Boulanger points it out, for it focuses exclusively on the conflictual relations between employees. But the *n*^{2} relationships among the company employees can well be confrontational, but may include as well collaborative relationships that are quite beneficial for the company. And so there is no reason to favour conflictual relationships with respect to collaborative ones. And when among *n*^{2} relationships established between the company employees, some are genuine collaborative relationships, the effect is, instead, of improving business performance. Therefore, we can not legitimately conclude that it is not desirable that the workforce of a company reaches a large size.

For the sake of clarity, it is worth formalizing the above reasoning. It turns out thus that Ulam’s reasoning can be described as follows:

(D1_{Ā} ) if <a company has a large workforce>

(D2_{Ā} ) then <*n*^{2} conflictual relationships will result>

(D3_{Ā} ) then negative effects will result

(D4_{Ā} ) **∴** the fact that <a company has a large workforce> is bad

This type of reasoning has the structure of a one-sidedness bias, since it focuses only on conflicting relationships (the *dissociation* pole of the *association*/*dissociation* duality), by ignoring a parallel argument with the same structure that could legitimately be raised, focusing on collaborative relationships (the *association* pole), which is the other aspect relevant to this particular topic. This parallel argument goes as follows:

(D1_{A}) if <a company has a large workforce>

(D2_{A}) then <*n*^{2} collaborative relationships will result>

(D3_{A}) then positive effects will result

(D4_{A}) **∴** the fact that <a company has a large workforce> is good

This finally casts light on how the two formulations of the argument lead to conflicting conclusions, i.e. (D4_{Ā}) and (D4_{A}). At this point, it is worth noting the very structure of the conclusion of the above reasoning, which is as follows:

(D5_{Ā} ) the situation *s* is bad from the viewpoint of Ā (*dissociation*)

while the conclusion of the parallel reasoning is as follows:

(D5_{A}) the situation *s* is good from the viewpoint of A (*association*)

But if the reasoning had been complete, by taking into account the two points of view, a different conclusion would have ensued:

(D5_{Ā} ) the situation *s* is bad from the viewpoint of Ā (*dissociation*)

(D5_{A}) the situation *s* is good from the viewpoint of A (*association*)

(D6_{A/Ā}) the situation *s* is bad from the viewpoint of Ā (*dissociation*) and good from the viewpoint of A (*a**ssociation*)

(D7_{A/Ā}) the situation *s* is neutral from the viewpoint of the duality A/Ā (*association/dissociation*)

And such a conclusion turns out to be quite different from that resulting from (D5_{Ā} ) and (D5_{A}).

Finally, we are in a position to replace the one-sidedness bias which has just been described in the context of the present model: the object *o* is the above reasoning, the reference class is that of the relationships between the employees of a business, and the corresponding duality – allowing to define the reference class – is the *dissociation*/*association* duality.

5**. Dichotomic analysis and meta-philosophy**

The aforementioned principle of dialectic indifference and its corollary – one-sidedness bias – is likely to find applications in several domains^{4}. We shall focus, in what follows, on its applications at a meta-philosophical level, through the analysis of several contemporary philosophical paradoxes. Meta-philosophy is that branch of philosophy whose scope is the study of the nature of philosophy, its purpose and its inherent methods. In this context, a specific area within meta-philosophy is the method to use to attach oneself to resolve, or make progress towards the resolution of philosophical paradoxes or problems. It is within this specific area that falls the present construction, in that it offers *dichotomous analysis* as a tool that may be useful to assist in the resolution of paradoxes or philosophical problems.

The dichotomous analysis as a methodology that can be used to search for solutions to some paradoxes and philosophical problems, results directly from the statement of the principle of dialectical indifference itself. The general idea underlying the dichotomous approach to paradox analysis is that two versions, corresponding to one and the other pole of a given duality, can be untangled within a philosophical paradox. The corresponding approach then is to find a reference class which is associated with the given paradox and the corresponding duality A/Ā, as well as the two resulting variations of the paradox that apply to each pole of this duality. Nevertheless, every duality is not suitable for this, as for many dualities, the corresponding version of the paradox remains unchanged, regardless of the pole that is being considered. In the dichotomous method, one focuses on finding a reference class and a relevant associated duality, such that the viewpoint of each of its poles actually lead to two structurally different versions of the paradox , or the disappearance of paradox from the point of view of one of the poles. Thus, when considering the paradox in terms of two poles A and Ā, and if it has no effect on the paradox itself, the corresponding duality A/Ā reveals itself therefore, from this point of view, irrelevant.

The dichotomous analysis is not by far a tool that claims to solve all philosophical problems, but only constitutes a methodology that is susceptible of shedding light on some of them. In what follows, we shall try to illustrate through several works of the author, how dichotomous analysis can be applied to progress towards the resolution of three contemporary philosophical paradoxes: Hempel’s paradox, the surprise examination paradox and the Doomsday argument.

In a preliminary way, we can observe here that in the literature, there is also an example of dichotomous analysis of a paradox in David Chalmers (2002). Chalmers attempts then to show how the *two-envelope paradox* leads to two fundamentally distinct versions, one of which corresponds to a *finite* version of the paradox and the other to an *infinite* version. Such an analysis, although conceived of independently of the present construction can thus be characterized as a dichotomous analysis based on the *finite*/*infinite* duality.

*The dual poles in David Chalmers’ analysis of the two-envelope paradox*

6**. Application to the analysis of the philosophical paradoxes**

At this point, it is worth applying the foregoing to the analysis of concrete problems. We shall illustrate this through the analysis of several contemporary philosophical paradoxes: Hempel’s paradox, the surprise examination paradox and the Doomsday argument. We will endeavour to show how a problem of one-sidednessn bias associated with a problem of definition of a reference class can be found in the analysis of the aforementioned philosophical paradoxes. In addition, we will show how the very definition of the reference class associated with each paradox is susceptible of being qualified with the help of the dual poles A and Ā of a given duality A/Ā as they have just been defined.

*6**.1. Application **to the analysis of **Hempel**‘s paradox*

Hempel’s paradox is based on the fact that the two following assertions:

(H) All ravens are black

(H*) All non-black things are non-ravens

are logically equivalent. By its structure (H*) presents itself indeed as the contrapositive form of (H). It follows that the discovery of a black raven confirms (H) and also (H*), but also that the discovery of a non-black thing that is not a raven such as a red flame or even a grey umbrella, confirms (H*) and therefore (H). However, this latter conclusion appears paradoxical.

We shall endeavour now to detail the dichotomous analysis on which is based the solution proposed in Franceschi (1999). The corresponding approach is based on finding a reference class associated with the statement of the paradox, which may be defined with the help of an A/Ā duality. If we scrutinise the concepts and categories that underlie propositions (H) and (H*), we first note that there are four categories: ravens, black objects, non-black objects and non- ravens. To begin with, a *raven* is precisely defined within the taxonomy in which it inserts itself. A category such as that of the ravens can be considered well-defined, since it is based on a precise set of criteria defining the species *corvus corax* and allowing the identification of its instances. Similarly, the class of *black* objects can be accurately described, from a taxonomy of colours determined with respect to the wave lengths of light. Finally, we can see that the class of *non-black* objects can also be a definition that does not suffer from ambiguity, in particular from the specific taxonomy of colours which has been just mentioned.

However, what about the class of *non-ravens*? What does constitute then an instance of a non-raven? Intuitively, a blue blackbird, a red flamingo, a grey umbrella and even a natural number, are non-ravens. But should we consider a reference class that goes up to include abstract objects? Should we thus consider a notion of *non-raven* that includes abstract entities such as integers and complex numbers? Or should we limit ourselves to a reference class that only embraces the animals? Or should we consider a reference class that encompasses all living beings, or even all concrete things, also including this time the artefacts? Finally, it follows that the initial proposition (H*) is susceptible of giving rise to several variations, which are the following:

(H_{1}*) All that is non-black among the *corvids* is a non-raven

(H_{2}*) All that is non-black among the *birds* is a non-raven

(H_{3}*) All that is non-black among the *animals* is a non-raven

(H_{4}*) All that is non-black among the *living beings* is a non-raven

(H_{5}*) All that is non-black among the *concrete things* is a non-raven

(H_{6}*) All that is non-black among the *concrete and abstract objects* is a non-raven

Thus, it turns out that the statement of Hempel’s paradox and in particular of proposition (H*) is associated with a *reference class*, which allow to define the *non-ravens*. Such a reference class can be assimilated to corvids, birds, animals, living beings, concrete things, or to concrete and abstract things, etc.. However, in the statement of Hempel’s paradox, there is no objective criterion for making such a choice. At this point, it turns out that one can choose such a reference class *restrictively*, by assimilating it for example to corvids. But in an equally legitimate manner, we can choose a reference class more *extensively*, by identifying it for example to the set of concrete things, thus notably including umbrellas. Why then choose such or such reference class defined in a restrictive way rather than another one extensively defined? Indeed, we are lacking a criterion allowing to justify the choice of the reference class, whether we proceed by *restriction* or by *extension*. Therefore, it turns out that the latter can only be defined *arbitrarily*. But the choice of such a reference class proves crucial because depending on whether you choose such or such class reference, a given object such as a grey umbrella will confirm or not (H*) and therefore (H). Hence, if we choose the reference class by extension, thus including all concrete objects, a grey umbrella will confirm (H). On the other hand, if we choose such a reference class by restriction, by assimilating it only to corvids, a grey umbrella will not confirm (H). Such a difference proves to be essential. In effect, if we choose a definition by extension of the reference class, the paradoxical effect inherent to Hempel’s paradox ensues. By contrast, if we choose a reference class restrictively defined, the paradoxical effect disappears.

*The dual poles in the reference class of the non-ravens within Hempel’s paradox*

The foregoing permits to describe accurately the elements of the preceding analysis of Hempel’s paradox in terms of one-sidedness bias such as it has been defined above: to the paradox and in particular to proposition (H*) are associated the reference class of *non-ravens*, which itself is susceptible of being defined with regard to the *extension*/*restriction *duality. However, for a given object such as a grey umbrella, the definition of the reference class by extension leads to a paradoxical effect, whereas the choice of the latter by restriction does not lead to such an effect.

*6.2. Application to the analysis of the surprise examination paradox*

The classical version of the surprise examination paradox (Quine 1953, Sorensen 1988) goes as follows: a teacher tells his students that an examination will take place on the next week, but they will not know in advance the precise date on which the examination will occur. The examination will thus occur surprisingly. The students reason then as follows. The examination cannot take place on Saturday, they think, otherwise they would know in advance that the examination would take place on Saturday and therefore it could not occur surprisingly. Thus, Saturday is eliminated. In addition, the examination can not take place on Friday, otherwise the students would know in advance that the examination would take place on Friday and so it could not occur surprisingly. Thus, Friday is also ruled out. By a similar reasoning, the students eliminate successively Thursday, Wednesday, Tuesday and Monday. Finally, every day of the week is eliminated. However, this does not preclude the examination of finally occurring by surprise, say on Wednesday. Thus, the reasoning of the students proved to be fallacious. However, such reasoning seems intuitively valid. The paradox lies here in the fact the students’ reasoning is apparently valid, whereas it finally proves inconsistent with the facts, i.e. that the examination can truly occur by surprise, as initially announced by the professor.

In order to introduce the dichotomous analysis (Franceschi 2005) that can be applied to the surprise examination paradox, it is worth considering first two variations of the paradox that turn out to be structurally different. The first variation is associated with the solution to the paradox proposed by Quine (1953). Quine considers then the student’s final conclusion that the examination can not take place surprisingly on any day of the week. According to Quine, the student’s error lies in the fact of not having envisaged from the beginning that the examination could take place on the last day. Because the fact of considering precisely that the examination will not take place on the last day finally allows the examination to occur by surprise on the last day. If the student had also considered this possibility from the beginning, he would not have been committed to the false conclusion that the examination can not occur surprisingly.

The second variation of the paradox that proves interesting in this context is the one associated with the remark made by several authors (Hall 1999, p. 661, Williamson 2000), according to which the paradox emerges clearly when the number *n* of units is large. Such a number is usually associated with a number *n* of days, but we may as well use hours, minutes, seconds, etc.. An interesting feature of the paradox is indeed that it emerges intuitively more significantly when large values of *n* are involved. A striking illustration of this phenomenon is thus provided by the variation of the paradox that corresponds to the following situation, described by Timothy Williamson (2000, p 139).

Advance knowledge that there will be a test, fire drill, or the like of which one will not know the time in advance is an everyday fact of social life, but one denied by a surprising proportion of early work on the Surprise Examination. Who has not waited for the telephone to ring, knowing that it will do so within a week and that one will not know a second before it rings that it will ring a second later?

The variation described by Williamson corresponds to the announcement made to someone that he/she will receive a phone call during the week, but without being able to determine in advance at what exact second the latter event will occur. This variation highlights how surprise may occur, in a quite plausible way, when the value of *n* is high. The unit of time considered here by Williamson is the second, in relation with a time duration that corresponds to one week. The corresponding value of *n* here is very high and equal to 604800 (60 x 60 x 24 x 7) seconds. However, it is not necessary to take into account a value as large of *n*, and a value of *n *equal to 365, for example, should also be well-suited.

The fact that two versions of the paradox that seem a priori quite different coexist suggests that two structurally different versions of the paradox could be inextricably intertwined within the surprise examination paradox. In fact, if we analyse the version of the paradox that leads to Quine’s solution, we find that it has a peculiarity: it is likely to occur for a value of *n* equal to 1. The corresponding version of the professor’s announcement is then as follows: “An examination will take place tomorrow, but you will not know in advance that this will happen and therefore it will occur surprisingly.” Quine’s analysis applies directly to this version of the paradox for which *n* = 1. In this case, the student’s error resides, according to Quine, in the fact of having only considered the hypothesis: (i) “the examination will take place tomorrow and I predict that it will take place.” In fact, the student should also have considered three cases: (ii) “the examination will not take place tomorrow, and I predict that it will take place” (iii) “the examination will not take place tomorrow and I do not predict that it will take place” (iv) “the examination will take place tomorrow and I do not predict that it will take place.” And the fact of having envisaged hypothesis (i), but also hypothesis (iv) which is compatible with the professor’s announcement would have prevented the student to conclude that the examination would not finally take place. Therefore, as Quine stresses, it is the fact of having only taken into account the hypothesis (i) that can be identified as the cause of the fallacious reasoning.

As we can see it, the very structure of the version of the paradox on which Quine’s solution is based has the following features: first, the non-surprise may actually occur on the last day, and second, the examination may also occur surprisingly on the last day. The same goes for the version of the paradox where *n* = 1: the non-surprise and the surprise may occur on day *n*. This allows to represent such structure of the paradox with the following matrix S[*k*, *s*] (where *k* denotes the day on which the examination takes place and S[*k*, *s*] denotes whether the corresponding case of non-surprise (*s* = 0) or surprise (*s* = 1) is possible (in this case, S[*k*, *i*] = 1) or not (in this case, S[*k*, *i*] = 0)):

day | non-surprise | surprise |

1 | 1 | 1 |

2 | 1 | 1 |

3 | 1 | 1 |

4 | 1 | 1 |

5 | 1 | 1 |

6 | 1 | 1 |

7 | 1 | 1 |

*Matrix structure of the version of the paradox corresponding to Quine’s solution for n = 7 (one week)*

day | non-surprise | surprise |

1 | 1 | 1 |

*Matrix structure of the version of the paradox corresponding to Quine’s solution for n = 1 (one day)*

Given the structure of the corresponding matrix which includes values that are equal to 1 in both cases of non-surprise and of surprise, for a given day, we shall term *joint* such a matrix structure.

If we examine the above-mentioned variation of the paradox set by Williamson, it presents the particularity, in contrast to the previous variation, of emerging neatly when *n* is large. In this context, the professor’s announcement corresponding for example to a value of *n* equal to 365, is the following: “An examination will take place in the coming year but the date of the examination will be a surprise.” If such a variation is analysed in terms of the matrix of non-surprise and of surprise, it turns out that this version of the paradox has the following properties: the non-surprise cannot occur on the first day while the surprise is possible on this very first day; however, on the last day, the non-surprise is possible whereas the surprise is not possible.

day | non-surprise | surprise |

1 | 0 | 1 |

… | … | … |

365 | 1 | 0 |

*Matrix structure of the version of the paradox corresponding to Williamson’s variation for n = 365 (one year)*

The foregoing allows now to identify precisely what is at fault in the student’s reasoning, when applied to this particular version of the paradox. Under these circumstances, the student would then have reasoned as follows. The surprise cannot occur on the last day but it can occur on day 1, and the non-surprise can occur on the last day, but cannot occur on the first day. These are proper instances of non-surprise and of surprise, which prove to be disjoint. However, the notion of surprise is not captured exhaustively by the extension and the anti-extension of the surprise. But such a definition is consistent with the definition of a vague predicate, which is characterized by an extension and an anti-extension which are mutually exclusive and non-exhaustive. Thus, the notion of surprise associated with a disjoint structure is that of a *vague* notion. Thus, the student’s error of reasoning at the origin of the fallacy lies in not having taken into account the fact that the surprise is in the case of a disjoint structure, a vague concept and includes therefore the presence of a penumbra corresponding to *borderline* cases between non-surprise and surprise. Hence, the mere consideration of the fact that the surprise notion is here a vague notion would have prohibited the student to conclude that S[*k*, 1] = 0, for all values of *k*, that is to say that the examination can not occur surprisingly on any day of the period.

Finally, it turns out that the analysis leads to distinguish between two independent variations with regard to the surprise examination paradox. The matrix definition of the cases of non-surprise and of surprise leads to two variations of the paradox, according to the *joint*/*disjoint* duality. In the first case, the paradox is based on a *joint* definition of the cases of non-surprise and of surprise. In the second case, the paradox is grounded on a *disjoint* definition. Both of these variations lead to a structurally different variation of the paradox and to an independent solution. When the variation of the paradox is based on a *joint* definition, the solution put forth by Quine applies. However, when the variation of the paradox is based on a *disjoint* definition, the solution is based on the prior recognition of the vague nature of the concept of surprise associated with this variation of the paradox.

*The dual poles in the class of the matrices associated with the surprise examination paradox*

As we finally see it, the dichotomous analysis of the surprise examination paradox leads to consider the class of the matrices associated with the very definition of the paradox and to distinguish whether their structure is joint or disjoint. Therefore, it follows an independent solution for each of the resulting two structurally different versions of the paradox.

*6.3. Application to the analysis of the Doomsday Argument*

The *Doomsday argument*, attributed to Brandon Carter, was described by John Leslie (1993, 1996). It is worth recalling preliminarily its statement. Consider then proposition (A):

(A) The human species will disappear before the end of the XXIst century

We can estimate, to fix ideas, to 1 on 100 the probability that this extinction will occur: P(A) = 0.01. Let us consider also the following proposition:

(Ā) The human species will not disappear at the end of the XXIst century

Let also E be the event: I live during the 2010s. We can also estimate today to 60 billion the number of humans that ever have existed since the birth of humanity. Similarly, the current population can be estimated at 6 billion. One calculates then that one human out of ten, if event A occurs, will have known of the 2010s. We can then estimate accordingly the probability that humanity will be extinct before the end of the twenty-first century, if I have known of the 2010s: P(E, A) = 6×10^{9}/6×10^{10} = 0.1. By contrast, if humanity passes the course of the twenty-first century, it is likely that it will be subject to a much greater expansion, and that the number of human will be able to amount, for example to 6×10^{12}. In this case, the probability that humanity will not be not extinct at the end of the twenty-first century, if I have known of the 2010s, can be evaluated as follows: P(E, Ā) = 6×10^{9}/6×10^{12} = 0,001. At this point, we can assimilate to two distinct urns – one containing 60 billion balls and the other containing 6,000,000,000,000 – the total human populations that will result. This leads to calculate the posterior probability of the human species’ extinction before the end of the XXIst century, with the help of Bayes’ formula: P'(A) = [P(A) x P(E, A)] / [P(A) x P(E, A) + P(Ā) x P(E, Ā )] = (0.01 x 0.1) / (0.01 x 0.1 + 0.99 x 0.001) = 0.5025. Thus, taking into account the fact that I am currently living makes pass the probability of the human species’ extinction before 2150 from 1% to 50.25 %. Such a conclusion appears counter-intuitive and is in this sense, paradoxical.

It is worth now describing how a dichotomous analysis (Franceschi, 1999, 2009) can be applied to the Doomsday Argument. We will endeavour, first, to point out how the Doomsday Argument has an inherent *reference class*^{5} problem definition linked to a duality A/Ā. Consider then the following statement:

(A) The human race will disappear before the end of the XXIst century

Such a proposition presents a dramatic, apocalyptic and tragic connotation, linked to the imminent extinction of the human species. It consists here of a prediction the nature of which is catastrophic and quite alarming. However, if we scrutinise such a proposition, we are led to notice that it conceals an inaccuracy. If the time reference itself – the end of the twenty-first century – proves to be quite accurate, the term “human species” itself appears to be ambiguous. Indeed, it turns out that there are several ways to define it. The most accurate notion in order to define the“’human race” is our present scientific taxonomy, based on the concepts of genus, species, subspecies, etc.. Adapting the latter taxonomy to the assertion (A), it follows that the ambiguous concept of “human species” is likely to be defined in relation to the genus, the species, the subspecies, etc.. and in particular with regard to the *homo* genus, the *homo sapiens* species, the *homo sapiens sapiens* subspecies, etc.. Finally, it follows that assertion (A) is likely to take the following forms:

(A_{h}) The *homo* genus will disappear before the end of the XXIst century

(A_{hs}) The *homo sapiens* species will disappear before the end of the XXIst century

(A_{hss}) The *homo sapiens sapiens* subspecies will disappear before the end of the XXIst century

At this stage, reading these different propositions leads to a different impact, given the original proposition (A). For if (A_{h}) presents well in the same way as (A) a quite dramatic and tragic connotation, it is not the case for (A_{hss}). Indeed, such a proposition that predicts the extinction of our current subspecies *homo sapiens sapiens* before the end of the twenty-first century, could be accompanied by the replacement of our present human race with a new and more advanced subspecies than we could call *homo sapiens supersapiens*. In this case, the proposition (A_{hss}) would not contain any tragic connotation, but would be associated with a positive connotation, since the replacement of an ancient race with a more evolved species results from the natural process of evolution. Furthermore, by choosing a reference class even more limited as that of the humans having not known of the computer (*homo sapiens sapiens antecomputeris*), we get the following proposition:

(A_{hss}_{s}) The infra-subspecies *homo sapiens sapiens* *antecomputeris* will disappear before the end of the XXIst century

which is no longer associated at all with the dramatic connotation inherent to (A) and proves even quite normal and reassuring, being devoid of any paradoxical or counterintuitive nature. In this case, in effect, the disappearance of the infra-subspecies *homo sapiens sapiens antecomputeris* is associated with the survival of the much-evolved infra-subspecies *homo sapiens sapiens postcomputeris*. It turns out then that a restricted class of reference coinciding with an infra-subspecies goes extinct, but a larger class corresponding to a subspecies (*homo sapiens sapiens*) survives. In this case, we observe well the Bayesian shift described by Leslie, but the effect of this shift proves this time to be quite innocuous.

Thus, the choice of the reference class for proposition (A) proves to be essential for the paradoxical nature of the conclusion associated with the Doomsday Argument. If one chooses then an extended reference class for the very definition of humans, associated with e.g. the *homo* genus, one gets the dramatic and disturbing nature associated with proposition (A). By contrast, if one chooses such a reference class restrictively, by associating it for example with the infra-subspecies *homo sapiens sapiens antecomputeris*, a reassuring and normal nature is now associated with the proposition (A) underlying the Doomsday Argument.

Finally, we are in a position to replace the foregoing analysis in the present context. The very definition of the reference class of the “humans” associated with the proposition (A) inherent to the Doomsday Argument is susceptible of being made according to the two poles of the *extension*/*restriction* duality. An analysis based on a two-sided perspective leads to the conclusion that the choice by extension leads to a paradoxical effect, whereas the choice by restriction of the reference class makes this paradoxical effect disappear.

*The dual poles within the reference class of “humans” in the Doomsday Argument*

The dichotomous analysis, however, as regards the Doomsday argument, is not limited to this. Indeed, if one examines the argument carefully, it turns out that it contains another reference class which is associated with another duality. This can be demonstrated by analysing the argument raised by William Eckhardt (1993, 1997) against the Doomsday argument. According to Eckhardt, the human situation corresponding to DA is not analogous to the two-urn case described by Leslie, but rather to an alternative model, which can be termed the *consecutive token dispenser*. The consecutive token dispenser is a device that ejects consecutively numbered balls at regular intervals: “(…) suppose on each trial the consecutive token dispenser expels either 50 (early doom) or 100 (late doom) consecutively numbered tokens at the rate of one per minute.” Based on this model, Eckhardt (1997, p. 256) emphasizes that it is impossible to make a random selection, where there are many individuals who are not yet born within the corresponding reference class: “How is it possible in the selection of a random rank to give the appropriate weight to unborn members of the population?”. The strong idea of Eckhardt underlying this diachronic objection is that it is impossible to make a random selection when there are many members in the reference class who are not yet born. In such a situation, it would be quite wrong to conclude that a Bayesian shift in favour of the hypothesis (A) ensues. However, what can be inferred rationally in such a case is that the initial probability remains unchanged.

At this point, it turns out that two alternative models for modelling the analogy with the human situation corresponding to the Doomsday argument are competing: first, the synchronic model (where all the balls are present in the urn when the draw takes place) recommended by Leslie and second, Eckhardt’s diachronic model, where the balls can be added in the urn after the draw. The question that arises is the following: is the human situation corresponding to the Doomsday argument in analogy with (i) the synchronic urn model, or with (ii) the diachronic urn model? In order to answer, the following question arises: does there exist an objective criterion for choosing, preferably, between the two competing models? It appears not. Neither Leslie nor Eckhardt has an objective motivation allowing to justify the choice of their own favourite model, and to reject the alternative model. Under these circumstances, the choice of one or the other of the two models – whether synchronic or diachronic – proves to be arbitrary. Therefore, it turns out that the choice within the class of the models associated with the Doomsday argument is susceptible of being made according to the two poles of the *synchronic*/*diachronic* duality. Hence, an analysis based on a two-sided viewpoint leads to the conclusion that the choice of the synchronic model leads to a paradoxical effect, whereas the choice of the diachronic model makes this latter paradoxical effect disappear.

*The dual poles within the models’ class of the Doomsday Argument*

Finally, given the fact that the above problem related to the reference class of the *humans* and its associated choice within the *extension*/*restriction* duality only concerns the synchronic model, the structure of the dichotomous analysis at two levels concerning the Doomsday Argument can be represented as follows:

As we can see it, the foregoing developments implement the form of dialectical contextualism that has been described above by applying it to the analysis of three contemporary philosophical paradoxes. In Hempel’s paradox, the reference class of the non-ravens is associated with proposition (H*), which itself is susceptible of being defined with regard to the *extension*/*restriction* duality. However, for a given object *x* such as a grey umbrella, the definition of the reference class by extension leads to a paradoxical effect , whereas the choice of the latter reference class by restriction eliminates this specific effect. Secondly, the matrix structures associated with the surprise examination paradox are analysed from the angle of the *joint*/*disjoint* duality, thus highlighting two structurally distinct versions of the paradox , which themselves admit of two independent resolutions. Finally, at the level of the Doomsday argument, a double dichotomic analysis shows that the class of humans is related to the *extension*/*restriction *duality, and that the paradoxical effect that is evident when the reference class is defined by extension, dissolves when the latter is defined by restriction. It turns out, second, that the class of models can be defined according to the *synchronic*/*diachronic* duality; a paradoxical effect is associated with the synchronic view, whereas the same effect disappears if we place ourselves from the diachronic perspective.

**Acknowledgements**

This text is written starting from some entirely revised elements of my habilitation to direct research work report, presented in 2006. The changes introduced in the text, comprising in particular the correction of a conceptual error, follow notably from the comments and recommendations that Pascal Engel had made to me at that time.

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1Such notion is central to the concept of *matrices of concepts* introduced in Franceschi (2002), of which we can consider that it constitutes the core, or a simplified form. In this paper that bears more specifically on the elements of dialectical contextualism and their application for solving philosophical paradoxes, merely presenting the dual poles proves to be sufficient.

2The present construction also applies to objects that are associated with several classes of reference. We shall limit ourselves here, for the sake of simplicity, to one single reference class.

3Philippe Boulanger says (personal correspondence) that he heard Stanislaw Ulam develop this particular point in a conference at the University of Colorado.

4An application of the present model to the cognitive distortions introduced by Aaron Beck (1963, 1964) in the elements of cognitive therapy, is provided in Franceschi (2007). Cognitive distortions are conventionally defined as fallacious reasoning that play a key role in the emergence of a number of mental disorders. Cognitive therapy is based in particular on the identification of these cognitive distortions in the usual reasoning of the patient, and their replacement by alternative reasoning. Traditionally, cognitive distortions are described as one of the twelve following methods of irrational reasoning: 1. Emotional reasoning 2. Hyper-generalization 3. Arbitrary inference 4. Dichotomous reasoning. 5. Should statements (Ellis 1962) 6. Divination or mind reading 7. Selective abstraction 8. Disqualifying the positive 9. Maximization and minimization 10. Catastrophism 11. Personalisation 12. Labelling.

5The analysis of the Doomsday Argument from the perspective of the reference class problem is performed in detail by Leslie (1996). But Leslie’s analysis aims at showing that the choice of the reference class, by extension or restriction does not affect the conclusion of the argument itself.

# A Logical Defence of Maher’s Model of Polythematic Delusions

*English translation of a paper published in French in Philosophiques, autumn 2008, under the title “Une défense logique du modèle de Maher pour les délires polythématiques”.*

In this paper, we proceed to describe a model for the formation and maintenance of polythematic delusions encountered in schizophrenia, which is in adequation with Brendan Maher’s account of delusions. Polythematic delusions are considered here as the conclusions of arguments triggered by apophenia that include some very common errors of reasoning such as post hoc fallacy and confirmation bias. We describe first the structure of reasoning which leads to delusions of reference, of telepathy and of influence, by distinguishing between the primary, secondary, tertiary and quaternary types of delusional arguments. These four levels of arguments correspond to a stage the nature of which is respectively instantial, inductive, interpretative at a monothematic level and interpretative at a polythematic level. We also proceed to identify accurately the fallacious steps in the corresponding reasoning. We expose then the role of apophenia in the elaboration of delusional ideas. Lastly, we describe the role played by the hallucinations in the present model.

This article is cited in:

- How delusion is formed? Park J.S., Kang U.G., Medical Hypotheses, 24 Dec 2015, 87: pages 61-65

**A Logical Defence of Maher’s Model of Polythematic Delusions**

**Paul FRANCESCHI**

Classically, the term of “delusion” applies to two fundamentally distinct forms: monothematic and polythematic delusions (Davies & Coltheart 2000; Bortolotti 2005). Monothematic delusions present an unique topic and are usually associated with cerebral lesions. Among the latter, one can mention Capgras’ delusion (by virtue of which the patient thinks that one of his/her fellows has been replaced by an impostor), Fregoli’s delusion (when the patient is persuaded that he/she is followed by one or several persons whom he cannot identify because they are dressed up) or Cotard’s delusion (when the patient is persuaded that he/she is died). Conversely, polythematic delusions have numerous topics, which are most often interconnected and usually linked to psychotic disturbances. Among polythematic delusions, one can notably mention: delusion of reference, delusion of grandeur, delusion of influence, delusion of persecution, delusion of control, delusion of telepathy.

In what follows, I will set out to introduce a new, as far as I know, model of the mechanism which leads to the formation of polythematic delusions^{i} met in schizophrenia. This model, which takes place in the recent development of psycho-pathological philosophy (Faucher, 2006), offers to describe the mechanism which leads, on the one hand, to the formation of delusional ideas and on the other hand, to their maintenance. In this model, delusions are the result of the patient’s cognitive activity in response to a specific form of abnormal perception. Even though the corresponding reasoning appears essentially normal, it includes however the repetition of some typical errors of reasoning. This leads to consider polythematic delusions as the conclusions of fallacious arguments, worked out in response to a particular type of abnormal perception, among which one can distinguish between primary, secondary, tertiary and quaternary delusional arguments. These four levels of arguments correspond, as we shall see it, to some functional stages the nature of which is respectively instantial (based on several instances), inductive (generalising the conclusion of each of the instances), interpretative at a monothematic level and finally, interpretative at a polythematic level.

It is worth mentioning, moreover, that the notion of delusion has important philosophical underpinnings. In particular, the understanding of delusions proves to be essential for the notions of belief (Engel 2001, Bayne & Pacherie 2005), of justification of beliefs, of knowledge, of rationality (Campbell 2001, Bortolotti 2005) and also of self-deception.

**1. Cognitive models of delusions**

Before describing in detail the present model, it is worth presenting the main cognitive models for delusions encountered in the literature. Some authors described then a cognitive model for delusional ideas observed in schizophrenia. As Chadwick & al. (1996) underline it, as well as Fowler & al. (1995) who set out to review these types of models, there does not truly exist a unique cognitive model for delusions, for it rather consists of a family of models.

A first cognitive model was described by Chadwick & al. (1996), who set out to introduce an application to delusions of Albert Ellis’ ABC-analysis. The original model described by Ellis (1962) consists of a diagram which plays a preponderant role in the emergence of mental disorders. Ellis distinguishes thus between three types of events: A, B and C. The As (for Activating event) are external facts or events of the patient’s internal life, such as thoughts or bodily feelings. The Bs (for Belief) are thoughts relating to the same events, which themselves are possibly rational in nature or not. Moreover, the corresponding cognitive process can be more or less conscious. Finally, the Cs (for Consequence) are emotional states such as anger, sadness, joy, frustration, etc. which can be of a positive or negative nature. Thus, the Cs that follow irrational thoughts are most often of a negative nature. The succession of events of type A, B and C plays a predominant role in the emergence of mental disorders: activating events trigger irrational thoughts, which themselves engender negative emotions. The type of therapy recommended by Ellis consists precisely in rendering the patient aware of this mechanism and in controlling the corresponding process. Adapting Ellis’ original model to psychosis, Chadwick & al. point out thus how the As constitute events that are external or internal to the patient, the Bs are his/her thoughts and the Cs are constituted by the emotions triggered by the patient’s thoughts. This specific framework allows to provide a cognitive ABC-analysis of the main types of delusions. For example, the delusion of persecution finds its origin in the external fact that the patient hears the noise of a car’s horn; this triggers in the patient the thought according to which his/her enemies come to kill him/her; it follows then in the patient’s a negative emotion of fright and of anxiety.

A second cognitive model of delusions was put forth by Brendan Maher (1974; 1988; 1999). Maher did suggest that delusions are the result – in the context of schizophrenia of paranoid subtype – of a broadly normal interpretation of the abnormal phenomena undergone by the patient (hallucinations, loss of audition, impairment in the intensity of perception, etc.). According to Maher, it is not therefore by his/her reasoning that the patient differs from a normal individual, but by his/her own altered perceptions. Delusional ideas are thus conceived of as a response to abnormal and emotionally disturbing phenomena experienced by the patient. Such disturbing phenomena lead the patient to search an explanation, which is at the origin of the delusional construction. According to Maher, the patient’s intellectual process is a product of normal reasoning and does not differ significantly from the one which is shown in every individual, or even in every scientist, when confronted with an unexplained phenomenon: “It is the core of the present hypothesis that the explanations (i.e. the delusions) of the patient are derived by cognitive activity that is essentially indistinguishable from that employed by non-patients, by scientists and by people generally.” (Maher 1974, 103). By normal reasoning, Maher means here a basically logical reasoning, but which occasionally includes some errors of reasoning of a common type. It is worth mentioning here that Maher’s model has led to several objections. Above all, this model was criticised on the grounds that it did not allow to account for the fact that delusions can also develop in seemingly normal conditions (Chapman & Chapman 1988). Secondly, it was objected to Maher’s model (Davies & Coltheart 2000, p. 8) that it did not explain how delusional beliefs are adopted and maintained in spite of their implausible nature. It is worth mentioning lastly that an important philosophical problem emerges within Maher’s model. It is what Pascal Engel termed the “paradox of delusions” (Engel 2001), and that can be formulated as follows: if the delusional construction is underlain by essentially normal reasoning and therefore by rationality, how it is possible to explain that the delusions’ conclusions are so manifestly wrong and contrary to evidence. The present analysis takes place in such context, and aims thus at proposing a solution to this paradox of delusions.

While Maher’s conception mentions abnormal perception as a unique factor at the origin of delusional ideas, another approach, notably put forth by Martin Davies and Max Coltheart (2000, 2001), describe two factors in the genesis and the maintenance of delusions. The first factor resides, as well as for Maher, in abnormal perception. And the second factor resides, according to Davies and Coltheart, in the patient’s disability to reject a hypothesis given its incoherent and implausible nature with regard to the patient’s rest of knowledge. Davies and Coltheart criticises thus Maher’s model by notably pointing out that it does not allow to provide an account of the maintenance of delusions, even though their conclusion turns out to be completely implausible.

It is worth also mentioning a third type of cognitive model, which stresses that several cognitive biases (Garety & al. 2001) can be observed in the thinking mode of patients suffering from schizophrenia. Among these biases is the patient’s tendency to *jump-to-conclusions* (Garety & Hemsley 1994). Experimental studies (Fear & Healy 1997; Garety & al. 1991) showed thus that patients had a more marked tendency than others to conclude very fast, starting from a limited group of information at their disposal. A second type of cognitive bias is an *attribution* or *externalist bias* which consists for the patient to attribute an external motive to events of a negative type which he/she undergoes. The patient favours then arbitrarily an external reason with regard to an internal and personal motive, when he/she sets out to determine the origin of an event which relates to him/her. Such conclusion notably results from the work of Bentall & Kaney (1989), and Kaney & al. (1989), who noticed that patients suffering from delusions of persecution were more prone than normal individuals to attribute both an external cause to negative events which they undergo and an internal motive to positive events which concerned them. This type of cognitive bias is also related to an *attentional bias*, which was noticed (Bentall & al. 1989; Kaney & al. 1989) in patients suffering from schizophrenia of paranoid subtype, who demonstrate as well a more marked tendency than others to turn their attention to menacing elements, among a group of stimuli, especially if the latter are related with themselves. Lastly, Aaron Beck (2002) also underlined how the reasoning of patients suffering from schizophrenia revealed an *egocentric bias*. This type of cognitive bias leads the patient to relate most external events with the elements of his/her personal life. Noise, sounds, smells, and generally facts and external phenomena, are thus bearing for the patient a hidden or explicit meaning, and which concerns him/her directly.

Finally, it is worth mentioning a cognitive model which sets out to define accurately the nature of delusions, by especially emphasising the fact that the latter do not constitute beliefs in the usual sense or, possibly, constitute beliefs of a special type. Such model made the subject of recent developments (Sass,1994; Young, 1999; Engel, 2001; Stephens & Graham, 2006) tending to question the classical definition of delusions, based on abnormal belief.

**2. Apophenia**

I will now endeavour to describe the present model and to expose accurately the mechanism which leads to the formation and maintenance of delusional ideas. In Maher’s model, delusions constitute a cognitive development elaborated by the patient in response to abnormal perception. The present model inserts itself within such conception: polythematic delusions constitute then conclusions of fallacious arguments worked out by the patient, in response to a particular type of abnormal perception: apophenia. Finally, although the reasoning which leads to delusions proves to be fallacious, it can however be considered as normal, because it includes errors of reasoning which turn out to be very common.

Before describing the structure of reasoning which leads to polythematic delusions, it is worth considering preliminarily the apophenia. One finds then mention, especially in the observations made by some patients in remission relating their psychotic experience (Stanton & David 2000), of a very specific feeling, which can be characterised as a *feeling of interconnection* with the ambient world. Such feeling is not felt in normal conditions and truly presents a bizarre nature. Schneider had already (1930) pointed out how in an individual suffering from schizophrenia, a meaningful interconnection was created between facts that are internal to the patient and external events (“Meaningful connections are created between temporary coincident external impressions … or perceptions with thoughts that happen to be present, or events and recollections happening to occur in consciousness at the same time”). Isabel Clarke (2000) also mentions in the patient a very particular feeling of fusion and of universal interconnection (“an exhilarating feeling of unity and interconnectedness”). Brundage (1983) also evokes a phenomenon of the same nature which manifests itself by a particular feeling of connection with all events that occur in the surroundings (“there is a connection to everything that happens”) as well as the feeling that the slightest things have a meaning (“every single thing means something”). It appears then that the patient experiences a strong feeling of interconnection between on the one hand, his/her internal phenomena and on the other hand, the external phenomena.

The role of such feeling of interconnection was recently underlined notably by Peter Brugger (2001). Brugger terms then *apophenia* the tendency to see connections between objects or ideas which are a priori without any relationship (“the propensity to see connections between seemingly unrelated objects or ideas”) and attributes the creation of this term to K. Conrad (1958). In the present context, one can consider a slightly more restrictive definition of apophenia, for it suffices here to characterise apophenia as the feeling in the patient that there is a narrow relationship between his/her internal phenomena (thought, feelings, emotions, acts) and external phenomena.

One can notice here that Maher does not mention explicitly apophenia when he enumerates abnormal perceptions which are susceptible of being experienced by the patient. However, he mentions a phenomenon which appears as closely related to apophenia. Among the abnormal perceptions undergone by the patient, Maher mentions indeed (Maher 1999) that it can consist, for example, of the fact that the patient perceives as salient some normally insignificant sensory data, of unrecognised defects in the sensory system of the patient such as a loss of audition, of temporary impairments in the intensity of perception, of hallucinations, of difficulties of concentration of neurological origin, etc. Maher includes then among abnormal perceptions the fact that the patient perceives as salient some ordinarily insignificant sensory data, what can be considered as closely related to apophenia.

At this step, it is worth describing more accurately the cognitive mechanism which, in relation with apophenia, leads to the formation of polythematic delusions. This will allow to cast more light on the role and the nature of apophenia itself.

**3. Formation and maintenance of delusional ideas**

In the present context, the reasoning that leads to delusional ideas is considered as a specific case of fallacious argument, i.e. as reasoning the conclusion of which is not logically justified by its premises, which are worked out in response to a particular type of abnormal perception: apophenia. In general, this type of reasoning leads to an erroneous conclusion. But it can happen very exceptionally that the resulting conclusion turns out to be true (for example if a patient suffering from schizophrenia with delusion of persecution was mistakenly spied on notably because he/she had been confused with a high diplomat). Another feature of the type of fallacious reasoning which leads to polythematic delusions is that it includes errors of reasoning of a normal type, i.e. very common. Finally, it is worth pointing out that in spite of their patently false conclusion, the task which consists in diagnosing accurately the fallacious steps in the reasoning which leads to delusional ideas proves to be far from easy.

The fallacious reasoning which leads to polythematic delusions presents a particular structure, as we will see it, within which it is worth distinguishing several functional steps, which take place successively within the elaboration of delusions ideas: primary, secondary, tertiary and quaternary steps. The primary step, first, is of an instantial nature, in the sense that it is based on some instances. The secondary step presents an inductive nature, which proceeds by generalisation of the conclusions resulting from each of the preceding instances. And the tertiary step is of an interpretative nature at a monothematic level. Finally, the quaternary step has an interpretative function, but this time at a polythematic level. The distinction of these four *successive* steps is of interest in the understanding of the mechanism which leads to the formation of delusional ideas, for it allows to describe its intrinsic structure, at the level of both its formation and maintenance. On the other hand, as we shall see it later, cognitive therapy of psychosis can apply differently to each of these specific steps.

In what follows, we shall especially be concerned with delusional ideas of reference, of telepathy, of influence and of grandeur, which correspond to polythematic delusions commonly met in schizophrenia. It is also worth mentioning that the corresponding model can be easily extended to other types of delusional ideas, especially to thought-broadcasting delusions or delusions of persecution. At this stage, it worth drawing a distinction between the mechanism which leads to the formation of delusional ideas, and the one which concurs to their maintenance.

*3.1 Formation of delusional ideas*

Classically, one distinguishes in schizophrenia the following types of delusions: delusion of reference, delusion of influence, delusion of control, delusion of telepathy, delusion of grandeur, delusion of persecution. The present model will set out first to describe the mechanism which leads to the formation of these main types of delusions, by setting out to introduce a reconstruction of the specific cognitive process in a patient at the beginning of psychosis.

Let us begin with delusions of *reference*. Let us consider the following argument, which leads the patient to conclude that television speaks about him/her, and therefore to delusional ideas of reference (T_{1} and T_{2} denote here two successive temporal positions, with a very short time interval between T_{1} and T_{2}; the symbol denotes the conclusion; and R is taken for *reference*):

(R1) in T_{1} I was drinking an aperitif

(R2) in T_{2} the presenter of the show said: “Stop drinking!”

(R3) ∴ in T_{2} the presenter said: “Stop drinking!” because in T_{1} I was drinking an aperitif

(R4) in T_{3} I was upset and anxious

(R5) in T_{4} the presenter of the show said “Stop stressing”

(R6) ∴ in T_{4} the presenter of the show said “Stop stressing!” because in T_{3} I was upset and anxious

(R7) in T_{5} I was smoking a cigarette

(R8) in T_{6} I heard the presenter saying “That is not good !“

(R9) ∴ in T_{6} the presenter said “That is not good !” because in T_{5} I was smoking a cigarette

(R10) in T_{7} I felt fine and lucid and I was relaxed

(R11) in T_{8} the presenter of the show said: “We are in great form!

(R12) ∴ in T_{8} the presenter said “We are in great form!” because in T_{7} I felt fine and lucid and I was relaxed

(R…) …

(R13) ∴ the presenters of the shows speak according to what I do or what I feel

(R14) ∴ television speaks about me

One can distinguish within the structure of this reasoning several parts the function of which turns out to be specific. These different parts correspond respectively to *primary* delusional arguments (it consists of the steps (R1)-(R3), (R4)-(R6), (R7)-(R9) and (R10)-(R12)), to *secondary* delusional arguments (the steps (R3), (R6), (R9), (R12) and (R13)) and *tertiary* delusional arguments (the steps (R13) and (R14)). It is worth considering in turn each of these arguments. Let us begin with *primary* delusional arguments, that correspond to an instantial step, in the sense that it is made up of several different instances. *Primary* delusional arguments are constituted here by four different instances, i.e. the steps (R1)-(R3), (R4)-(R6), (R7)-(R9) and (R10)-(R12). These four primary delusional arguments lead the patient to conclude that at a given time, the TV presenters spoke according to his/her acts or to what he/she felt.

Let us consider now the following stage (R13), which constitutes the conclusion of a *secondary* delusional argument, and is of a different nature. Its premises are the conclusion (R3), (R6), (R9), (R12) of the four previous instances of primary delusional arguments of reference. The patient generalises from the latter to the conclusion that the TV presenters speak according to what he/she is doing or to what he/she is feeling. The overall structure of this type of secondary delusional argument is then as follows:

(R3) in T_{2} the presenter of the show spoke according to what I was doing

(R6) in T_{4} the presenter of the show spoke according to what I was feeling

(R9) in T_{6} the presenter of the show spoke according to what I was doing

(R12) in T_{8} the presenter of the show spoke according to what I felt

(R…) …

(R13) ∴ the presenters of the shows speak according to what I do or feel

On can then term *inductive* this type of secondary delusional argument because it has the form of an *enumerative induction*, where the patient constructs his/her conclusion by generalising, in an inductive way, from the conclusions of several instances of primary delusional arguments. Thus, secondary delusional arguments correspond to a step the nature of which proves to be *inductive*.

At this stage, it is also worth mentioning the third step, which leads to delusion of reference. It consists of the tertiary delusional argument of reference, constituted by steps (R13) and (R14), the premise (R13) of which being the conclusion of the secondary delusional argument of reference:

(R13) ∴ the presenters of the shows speak according to what I do or feel

(R14) ∴ television speaks about me

In such argument, the patient takes into account the conclusion of the inductive step that the presenters of the shows speak according to his/her acts or to his/her internal state, and interprets it by concluding that television speaks about him/her. It consists, as we did see it, of a step the function of which is merely *interpretative*, in the sense that it aims at making sense of the inductive conclusion which results from the secondary delusional argument. Tertiary delusional arguments are therefore the product of an interpretative step, which situates itself at a monothematic level (here, the specific topic is that of delusion of reference).

A structurally identical mechanism leads to delusional ideas of *telepathy*. Several instances of primary delusional arguments of influence lead first the patient to conclude that his/her own thoughts are at the origin of acts that are accomplished by other persons. By an inductive step, the patient is then led to the conclusion that people act according to his/her thoughts. Finally, in an interpretative step, the patient concludes that other people can read his/her thoughts (or that they can hear them). It consists there, in the patient’s mind, of an attempt at explaining the very disturbing conclusion which results from the inductive step according to which other persons act according to his/her thoughts.

The same mechanism also engenders the formation of delusional ideas of *influence*. In that case, several instances of primary delusional arguments of influence lead the patient to conclude that his/her own thoughts are at the origin of annoyances caused to other persons. An inductive step leads then the patient to the conclusion that people react negatively in function of his/her thoughts. Finally, an interpretative step leads the patient to conclude that he/she perturbs and disturbs other people.

Moreover, such mechanism leads to the formation of delusional ideas of *control*. They find their origin in the instances of primary delusional arguments of control. Such instances have the same structure as that of the instances of primary delusional arguments of reference, of telepathy or of influence, with however this difference that the temporal order of both types of events – internal and external, with regard to the patient – is now reversed. Within the primary delusional arguments of reference, of telepathy or of influence, an internal event with regard to the patient *precedes* an external event, whereas it is the opposite with regard to a primary delusional argument of control: the external event precedes then the internal event. Thus, several instances of primary delusional arguments of control lead the patient to conclude inductively that some external events have an effect on his/her thoughts, his/her emotions or his/her acts. The interpretative step leads then the patient to think that he/she is controlled by external beings or objects such as robots or a satellite.

Finally, it is worth specifying the role played by *quaternary* delusional arguments. The premises of the latter arguments are conclusions of tertiary delusional arguments. Quaternary delusional arguments are more general arguments, which present, as well as tertiary delusional arguments, an interpretative nature. But unlike tertiary delusional arguments which turn out to be interpretative at a monothematic level, quaternary delusional arguments are interpretative at a polythematic level. They indeed take into account jointly the conclusions of tertiary delusional arguments of reference, of telepathy, of influence, etc. by striving to make sense, *globally*,of them and to interpret them. The reasoning below constitutes then a quaternary delusional argument leading to ideas of *grandeur*:

(R15) television and the media speak about me

(T16) people can read my thoughts

(I17) I influence other people’s behaviour

(18) ∴ I am someone exceptional

(19) ∴ I am an extraterrestrial

At a quaternary level, the patient takes then into account the different conclusions resulting from tertiary delusional arguments, the function of which is interpretative at the level of a given delusional topic and attempts this time to interpret the set constituted by the latter. The resulting conclusion constitutes veritably, for the patient, an overall theory the function of which is to make sense and to explain all the abnormal phenomena which he/she experiences.

*3.2 Maintenance of delusional ideas*

It is worth considering now the mechanism which leads to the maintenance of delusional ideas. Let us place ourselves at the level of secondary delusional arguments which, at the level of the formation of delusional ideas, are of an inductive nature. Consider then especially the form that take secondary delusional arguments of reference, at the stage of the maintenance. At this step, the conclusion (R13) which results from secondary delusional arguments, in virtue of which the presenters of the shows speak according to what the patient makes or feels, was already established at the stage of the formation of delusional ideas. And the corresponding reasoning takes then into account a new instance of primary delusional argument (R20) of reference, in the following way:

(R20) in T_{100} the presenter of the show spoke according to what I was doing

(R21) ∴ this confirms that television speaks according to what I do

On can notice here that the inductive generalisation (R13) was already established at the stage of the formation of the secondary delusional argument, and that the new instance of primary delusional argument constitutes then, in the patient’s mind, a case of *confirmation* of the latter generalisation. As we can see, the role of the new instance of primary delusional argument is to confirm and therefore to reinforce, at the stage of the maintenance, a generalisation which was already established at the previous stage of the formation of delusional ideas.

**4. Analysis of delusional arguments**

At this stage, it is worth analysing in detail the structure of the type of reasoning which has been just described, in order to identify accurately the fallacious steps and to determine the role played by apophenia. Let us consider in turn primary, secondary, tertiary and quaternary delusional arguments. Let us scrutinise first the following instance of primary delusional argument of telepathy, which manifests itself at the level of the *formation* of delusions:

(T1) in T_{1} I thought of Michael “What an idiot!”

(T2) in T_{2} I heard Michael shout

(T3) ∴ in T_{2} I heard Michael shout because in T_{1} I thought of him “What an idiot!”

It appears here that the two premises (T1) and (T2) constitute genuine facts and therefore turn out to be true. Conversely, the conclusion (T3) that concludes to the existence of a *relation of causality* between the two consecutive facts F_{1} (in T_{1} I thought of Michael “What an idiot!”) and F_{2} (in T_{2 }I heard Michael shout) is it justified? It appears not. Indeed, both premises are only establishing a relation of *anteriority* between the two consecutive facts F_{1} and F_{2}. And the conclusion (T3) which deducts from it a *causality* relationship turns out therefore to be too strong. The corresponding reasoning presents then a fallacious nature. The corresponding error of reasoning, which concludes to a relation of causality while there is only a simple relation of anteriority, is traditionally called *post hoc fallacy*, according to the Latin sentence “Post hoc, ergo propter hoc” (thereafter, hence because of it). This is a very common type of fallacious reasoning, which is notably at the root of many superstitions (Martin 1998; Bressan 2002). David Hemsley (1992) notably mentions such type of reasoning in clinical observation: “A patient of the present author, recalling his psychotic experiences noted that the co-occurrence of two events often led immediately to an assumption of a causal relationship between them”. Finally, one can notice that in the context of *cognitive distortions*, the type of error of reasoning corresponding to post hoc fallacy can be considered a specific case of *arbitrary inference*.

Let us also proceed to analyse the type of reasoning which underlies secondary delusional arguments, and that presents at the stage of the *formation* of delusional ideas, as we did see it, the following inductive structure:

(T22) in T_{2} Michael spoke according to my thoughts

(T23) in T_{4} the neighbour spoke according to my thoughts

(T24) in T_{6} the radio presenter spoke according to my thoughts

(…)

(T25) ∴ people act according to my thoughts

Such type of reasoning appears prima facie completely correct. It consists here of a reasoning based on an inductive generalisation, in which the patient simply builds a more general conclusion from several instances. Such reasoning is completely correct, for its conclusion can be considered as true, inasmuch as its premises are true. However, a scrutiny reveals that the patient only takes into account here a limited number of instances, i.e. those instances that are based on the concordance of both premises, at the stage of primary delusional arguments. The patient then directs his/her attention exclusively to those instances that include two premises of which the internal event (premise 1) and the external event (premise 2) turn out to be concordant and render thus plausible a relation of causality. The corresponding turn of mind can be described as a *concordance bias*. In effect, the patient does not take into account at this stage those instances which could possibly be composed of two discordant premises. The latter are likely to come under two different forms. An instance of the first form is as follows:

(T1) in T_{1} I thought of Michael “What an idiot!”

(26) in T_{2} Michel was quiet

And an instance of the second form is :

(27) in T_{1} I didn’t think of Michael

(T2) in T_{2} I heard Michael shout

In these two types of cases, one can notice a discordance between the two premises, which goes directly contrary to the idea of causality between the two events. As we see it finally, the flaw in the patient’s reasoning resides essentially in the fact of only taking into account those instances where the concordance between an internal event and an external event renders plausible a causality relationship. But if the patient had taken into account at the same time the concordant and the discordant instances, he/she would have been led to conclude that the concordant instances represent only a small part of the set constituted by the class of relevant instances, and are only therefore the result of a random process. In such context, as we see it, the concordant instances in fact constitute but mere coincidences.

If one places oneself now at the stage of the *maintenance* of delusional ideas, one can observe the presence of a mechanism of the same nature. At the stage of the emergence of delusional ideas, secondary delusional arguments present, as we did see it, an inductive nature. On the other hand, at the stage of the maintenance of delusional ideas, the latter come under the form of arguments which lead to the confirmation of an inductive generalisation. Consider then the following instance, where the conclusion (T25) according to which people act according to the patient’s thoughts results from a secondary delusional argument and was already established at the stage of the formation of delusional ideas:

(T28) in T_{100} my sister spoke according to my thoughts

(T29) ∴ this confirms that people act according to my thoughts

This type of argument appears completely valid, for the conclusion results directly from its premises. However, the latter argument is also at fault by default, for it does not take into account some premises, which turn out to be as much as relevant as the instance (T28). As we can see, the error of reasoning consists then in taking only into account those instances which confirm the generalisation (T25), while ignoring those instances which disconfirm the latter. Hence, this type of argument reveals a *confirmation bias*, i.e. a tendency to favour those instances which confirm a generalisation, whereas it would be necessary to take into consideration at the same time those which confirm and those which disconfirm it. One can notice however that such type of cognitive bias presents a very common nature (Nickerson 1998, Jonas et al. 2001).

It is worth considering, third, tertiary delusional arguments. Consider then the following tertiary delusional argument of telepathy (a similar analysis also applies to tertiary delusional arguments of reference and of influence):

(T30) ∴ people act in function of my thoughts

(T31) ∴ people can read my thoughts (people can hear my thoughts)

One can notice here that if premise (T30) is true, then the conclusion (T31) constitutes a credible explanation. This type of argument presents then an interpretative nature and the conclusion (T31) according to which people can read the patient’s thoughts appears finally plausible, inasmuch as it is considered as true that people act according to his/her thoughts. As we can see, such argument is motivated by the patient’s concern of explaining and of interpreting the disturbing generalisation which results from the repetition of the many concordant above-mentioned instances.

Finally, the following quaternary delusional argument aims, in the same way, at making sense of the conclusions which result from the conjunction of conclusions of different tertiary delusional argument:

(R15) television and the media speak about me

(T16) people can read my thoughts

(I17) I influence people’s behaviour

(18) ∴ I am someone exceptional

(19) ∴ I am an extraterrestrial

As we can see it, the conclusion (18) results here directly from the three premises (R15), (T16) and (I17) and the corresponding reasoning which leads the patient to conclude that he/she is someone exceptional can also be considered as valid. On the other hand, the conclusion (19) appears here too strong with regard to premise (18).

Given what precedes, it appears that a number of steps in the reasoning which leads to delusional ideas in schizophrenia are characterised by a reasoning which appears mainly normal. By normal reasoning, one intends here a broadly logical and rational reasoning, but also including some errors of logic of a very common type. Such viewpoint corresponds to the one put forth by Maher (1988; 1999) who considers, as we did see it, that the delusional construction in schizophrenia is nothing else than normal reasoning worked out by the patient to try to explain the abnormal phenomena which he/she experiences.

However, one can notice that in the above-mentioned structure of reasoning, one part of the reasoning cannot a priori be truly considered as normal. It consists here of the different instances of primary delusional arguments. The latter are based, as we did see it, on errors of reasoning corresponding to post hoc fallacies. This type of error of reasoning arguably turns out to be extremely common. However, the instances of primary delusional arguments mentioned above present an unusual nature, in the sense that they put in relationship the patient’s thoughts (or his/her emotions, feelings or actions) with external phenomena. Prima facie, such type of reasoning cannot be considered as normal. For why is the patient led to put his/her thoughts in relationship with external phenomena? One can formulate the question more generally as follows: why does the patient put in relationship the phenomena of his/her internal and personal life (his/her thoughts, emotions, feelings, etc.) with mere external phenomena? This distinguishes itself indeed significantly from the behaviour of a normal person, for whom it exists a very clear-cut intuitive separation between on one hand, his/her own internal world, and on the other hand, the external phenomena.

The answer to the previous question can be found here in the role of apophenia. Due to apophenia, the feeling indeed imposes itself to the patient that his/her internal world is closely linked up with the external world. So, his/her thoughts, emotions, feelings and acts appear to him/her to be closely linked up with the external phenomena that he/she perceives, such as ambient noise and dialogues, the words of the presenters of television or of radio, the dialogues of the characters of comic strips, the movements of the wings of a butterfly or of a bird, the natural phenomena such as the wind or the rain, etc. In the context which results from apophenia, the repeated instances that constitute primary delusional arguments can then take place naturally. For since the patient lives with a permanent feeling of interconnection between events that relate to him/her specifically and those which occur in the world which surrounds him, he/she is then led to observe many concordances between events related to him/her and external facts. In such context, primary delusional arguments can then take place naturally.

In the present context, the role of apophenia can be considered as fundamental. And this leads to suggest that considering its specificity and considering the leading role that it plays in the development of primary delusional arguments and therefore of all the characteristic delusional ideas of schizophrenia, apophenia could be counted among the criteria of the illness^{ii}.

As we can see it, the process which gives rise to delusional arguments from the phenomenological experience constituted by apophenia proves finally to be in line with Maher’s account. And one finds here a clear explanation of delusions as the patient’s response to the abnormal phenomena which he/she experiences, among which one can then mention apophenia, as well as hallucinations.

Given what precedes, polythematic delusions can be defined as conclusions of arguments triggered by apophenia and that include some very common errors of reasoning such as post hoc fallacy and confirmation bias. Hence, apophenia *and* a normal reasoning including the type of aforementioned errors of reasoning turn out to be necessary and sufficient conditions for the development of polythematic delusions. This double condition notably explains why we are not all delusional. For if errors in reasoning based on post hoc fallacy and confirmation bias turn out indeed to be very common, they only trigger primary delusional arguments when they are associated with the abnormal perception which consists in apophenia. It is worth pointing out, moreover, that such model leaves also room for more stronger conditions. For if apophenia constitutes one of the two sufficient conditions for the development of polythematic delusions, the latter can also take place in conditions where abnormal perception is constituted not only by apophenia, but also by other abnormal perceptions such as hallucinations. And also, whereas the second condition which is sufficient for delusions identifies itself with normal reasoning including post hoc fallacy and confirmation bias, it proves that the development of delusions can also be made by means of a reasoning which deviates more or less from normal reasoning. But the essential characteristic of the present model resides in the fact that apophenia and normal reasoning including the aforementioned very common errors, constitute necessary and sufficient conditions for the development of polythematic delusions.

**5. The role of the hallucinations**

At this stage, it is worth highlighting the role played by hallucinations, the other major symptom of schizophrenia, in the process which has just been described. I will set out to describe in more detail here the role played by auditory hallucinations – given that the corresponding analysis can be easily extended to hallucinations relating to other sensory modalities, i.e. visual, tactile, olfactory and gustatory.

Auditory hallucinations are susceptible, first, of playing a role at the level of primary delusional arguments. In this type of case, the primary delusional argument presents the same structure as the one described above, with the only difference that an auditory hallucination – in place of a real external event – constitutes then the second premise of the primary delusional argument. The following instance constitutes then an example of primary delusional argument of reference, but it is there an auditory hallucination, by which the patient hears the voice of the presenter of the show saying “Clumsy!” while he/she watches TV, that constitutes the support of the second premise of the argument:

(32) in T_{1} I dropped my pen

(33) in T_{2} I heard the voice of the presenter of the show saying “Clumsy!”

(34) ∴ in T_{2} the presenter of the show said “Clumsy!” because in T_{1} I dropped my pen

In the same way, the following instance constitutes a case of primary delusional argument of telepathy. In that case, it is an auditory hallucination, by which the patient hears the voice of his neighbour saying “Calm down!”, that serves as a basis for the second premise of the argument:

(35) in T_{1} I was very upset

(36) in T_{2} I heard the voice of my neighbour saying “Calm down!”

(37) ∴ in T_{2} my neighbour said “Calm down!” because in T_{1} I was very upset

It is worth mentioning, second, the role that can be played by auditory hallucinations at the level of secondary delusional arguments. In such case, the corresponding generalisations develop from instances of primary delusional arguments which also include auditory hallucinations. In the example below, the patient generalises from the conclusions of three instances of primary delusional arguments of reference. But while the two latter instances (39) and (40) are based on real external phenomena, the first instance (38) is founded on hallucinated content, by which the patient heard the TV presenter saying “Clumsy!”:

(38) in T_{2} the TV presenter said “Clumsy!” because in T_{1} I dropped my pen

(39) in T_{4} the presenteress said “Calm down!” because in T_{3} I was upset

(40) in T_{6} the presenter of the show said “Thank you” because in T_{5} I thought “I love this presenter”

(…) …

(41) ∴ the TV presenters speak according to what I do or feel

As we see it, auditory hallucinations contribute in this way to increase the number of primary delusional arguments, by creating thus additional instances which add up themselves to the different types of standard instances previously defined. This gives then more weight to the inductive generalisations made by the patient from multiple instances of primary delusional arguments. Besides, it has also the effect of reinforcing the coherence of the patient’s delusional system and of rendering it then more resistant to contrary argumentation.

It is worth mentioning, lastly, another type of role which can be played by auditory hallucinations. Such is notably the case when the content of the hallucinations proves to be consistent with the conclusions that result from secondary, tertiary or quaternary arguments. Auditory hallucinations have then the effect of reinforcing the latter conclusions. The instance below constitutes a case where an auditory hallucination comes to reinforce the conclusion of a *tertiary* delusional argument of telepathy. In that case, the hallucinated content resides in the fact that the patient hears the voice of his friend Joseph saying “I know the slightest of your thoughts”:

(42) ∴ in T_{50} I thought that people know of my thoughts

(43) in T_{100} I heard Joseph saying: “I know the slightest of your thoughts”

(44) ∴ this confirms that people know of my thoughts

In a similar way, the following instance has the effect of reinforcing the conclusion which results from a *quaternary* delusional argument, where the hallucinated content consist of a voice heard by the patient that says: “You come from the planet Mars”:

(19) ∴ in T_{50} I thought I was an extraterrestrial

(45) in T_{100} I heard a voice saying : “You come from the planet Mars”

In a general way, we see here how hallucinations constitute an element which has the effect of reinforcing considerably the conclusions resulting from delusional arguments. The hallucinations have then the effect of reinforcing the strength and the consistency of the beliefs’ system of the patient, thus contributing to its maintenance, and rendering then his/her ideas more resistant to contrary argumentation.

**6. Comparison with other cognitive models of delusions**

The present model, as we can see it, mainly emphasises a cognitive approach of delusions encountered in schizophrenia. This model introduces a fundamental cognitive element, but also leaves room to a neurophysiological element (at the origin of apophenia), the role of which proves to be essential. One can notice finally that the model which has just been described turns out to be compatible with some other accounts of delusional ideas met in schizophrenia.

The present analysis, to begin with, is susceptible of inserting itself as part of the adaptation of Albert Ellis’ ABC-analysis described by Chadwick et al. (1996). In this context, the internal and external events with regard to the patient, that are the premises of primary delusional arguments, constitute the As. The primary, secondary, tertiary and quaternary delusional arguments, can also be assimilated to the Bs. Lastly, the negative emotions (anger, anxiety, frustration, etc.) felt by the patient, that result there from the conclusions of delusional tertiary and quaternary arguments, constitute the Cs. As we can see it, the present analysis leads, in comparison with the standard ABC-analysis, to distinguish several steps at the level of the Bs. This distinction is important, since it allows to distinguish several steps whose function is different, within the reasoning that leads to delusional ideas. Thus, the B_{1}s (primary delusional arguments) are instances that lead to the attribution of a causality relationship between internal and external (to the patient) phenomena; the B_{2}s (secondary delusional arguments) result from a generalisation of inductive nature; the B_{3}s (tertiary delusional arguments) correspond to an interpretative step at a monothematic level; finally, the B_{4}s (quaternary delusional arguments) are characteristic of a step of interpretation at a polythematic level, the conclusion of which truly constitutes a global explicative theory of the abnormal phenomena undergone by the patient. On the other hand, we are led there to distinguish between those parts of the patient’s reasoning which are globally valid (the B_{2}s, B_{3}s and B_{4}s) and the part which is invalid (the B_{1}s, based on post hoc fallacy). Such nuanced point of view should be likely to preserve – what constitutes one of the key points of cognitive and behaviour therapy – the therapeutic alliance, i.e., the relation of collaboration between the patient and the therapist aiming at shared objectives in the struggle against the illness. As we can see it, the present analysis leads to especially emphasise post hoc fallacy, which constitutes the weakness in the patient’s reasoning, but the repeated instances of which, triggered by apophenia, truly constitute the building block of the delusional construction.

The present model also has number of affinity with the approaches which are at the root of cognitive therapy of schizophrenia (Kingdon & Turkington 1994; Kingdon & Turkington 2005; Chadwick & al. 1996; Beck & Rector 2000). In this type of approach, the therapist sets out to reduce progressively the patient’s degree of belief in his/her delusional polythematic ideas. To this end, the therapist suggests the patient, in a spirit of dialogue of Socratic inspiration, to elaborate alternative hypotheses; he also teaches the patient the approach which consists in searching elements likely to confirm or to disconfirm his/her own hypotheses, as well as to build out alternative hypotheses. The contribution of the present analysis with regard to cognitive therapy of schizophrenia is likely to manifest itself in several ways. It proves to be useful to specify then, for the clinician, what could be such contribution, and to also provide a specific framework in which the present model will possibly be tested. The distinction of different steps in the development of delusions allows first to distinguish between different hypotheses corresponding to the conclusions of primary, secondary, ternary and quaternary arguments. The degree of belief associated with each of these levels of hypotheses is also susceptible of being evaluated separately, by notably allowing to determinate then at which level resides the strongest degree of conviction. In the same way, each of the conclusions of the primary, secondary, tertiary or quaternary arguments, will possibly be tested (confirmed/infirmed) and give room for the elaboration of alternative hypotheses (David Kingdon, personal communication). For example, at the level of primary delusional arguments, it will be possible to consider the belief according to which the presenter said in T_{2}: “You should not drink!” *because* the patient was drinking an aperitif in T_{1}; this hypothesis will possibly give rise to a search for evidence, and then confronted with an alternative hypothesis such as: the presenter said in T_{2}: “You should not drink!” because it was scheduled in the script of the television program. At the level of tertiary delusional arguments, the hypothesis according to which television speaks about the patient will also possibly be the object of a search for evidence, etc.

The interest for the clinician of the present approach resides, second, in the fact that it provides the patient with an alternative global explanation of the abnormal phenomena that he/she undergoes. The delusional construction of the patient constitutes, as we did see it, a theory which allows him/her to explain all the abnormal phenomena that he/she experiences. On can assume in this respect, that the fact for the patient to get a satisfactory theory explaining all abnormal phenomena which he/she experiences, is also likely to play an important role in the *maintenance* of his/her delusional system. In this context, the present analysis allows to propose to the patient an alternative explicative theory, grounded on apophenia and the different steps of reasoning which result from it. Such theory distinguishes itself from the explicative theory with which the patient is usually confronted (according to the common elliptical point of view, the latter is “mad”) and proves to be less stigmatising, since the reasoning which leads to delusions is notably considered here as normal. For this reason, one can assume that the patient could be more willing to adhere to the present alternative theory, as a global explanation of the abnormal phenomena which he/she experiences.

As we did see it, the present model conforms mainly with the one developed by Brendan Maher (1974; 1988; 1999), based on the fact that delusions result from a broadly normal interpretation of the abnormal phenomena undergone by the patient. The present analysis also specifies with regard to Maher’s model that apophenia (eventually associated with hallucinations) constitutes an abnormal perception which is enough for giving rise to delusional polythematic ideas met in schizophrenia. It has then been objected, as we did see it, to Maher’s model, that it did not allow to account for the fact that delusions can also take place in seemingly normal conditions, especially in a patient not suffering from hallucinations. But the present model points out that such conditions are not normal, since apophenia is present in such a patient. Since apophenia leads to abnormal perceptions, the essential factor described by Maher at the origin of delusional ideas, is therefore present as well. On the second hand, the present model also provides some elements of response with regard to the second objection, formulated against Maher’s model, by Davies and Coltheart (2000), according to which it does not allow to describe how delusional beliefs are adopted and maintained in spite of their implausible nature. The present model, however, sets out first to describe step-by-step the type of reasoning which leads to the adoption of polythematic delusions. By its structure, this type of reasoning appears mainly normal. It proceeds by enumeration of some instances, and then by generalisation and lastly, by interpretation. The present model also provides, as far as I can see, an answer to the criticism raised by Davies and Coltheart with regard to Maher’s model, who blame the latter for not describing how delusional beliefs are maintained in the patient’s belief system, in spite of their implausible nature. In the present model, as we did see it, it is the fact that new instances are generated every day which explains that beliefs are maintained. For when delusional beliefs are established in the patient’s belief system at the end of the stage of their formation, they are then maintained because apophenia continues to trigger every day^{iii} new instances of primary delusional argument. The latter come, in the patient’s mind, to *confirm* the conclusions of delusional arguments at a secondary, tertiary and quaternary level, already established at the stage of the formation of delusional ideas. From this point of view, there is no essential difference in the present model in the way that the formation and the maintenance delusional ideas take place. For as we did see it, the building block of the delusional construction is constituted there by the instances of primary delusional arguments, triggered by apophenia. And these instances which concur to the formation of delusional ideas, also ensure their maintenance every day, by confirming the conclusions of secondary, tertiary and quaternary arguments, which are already established at the stage of the formation of delusional ideas^{iv}.

Finally, the model which has just been described provides, as far as I can see, in comparison with Maher’s model, an element which proves to be necessary in the context of an explicative model of polythematic delusions. This element consists of an answer to the question of *why* the content of delusional ideas in schizophrenia identifies itself most often with delusions of reference, of telepathy, of thought insertion, of influence and of control. As it was exposed above, the answer provided by the present model is that a mechanism of the same nature, grounded on post hoc fallacy, leads to the development of these different delusional topics. In primary delusional arguments of reference, of telepathy or of influence, an event which is internal to the patient precedes an external event. And in the case of a primary delusional argument of control, this structure is simply reversed: it is an external event which precedes an event of the patient’s internal life.

We see it finally, the preceding analysis allows to justify and to reinforce Maher’s initial model. In this context, one can notice that one of the consequences of the present model is that the sole apophenia, associated with normal reasoning, turns out to be sufficient to give rise to the emergence and the maintenance of a delusional system^{v}.

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i Monothematic delusions are not included into the scope of the present study.

ii One can notice that a neurophysiological explanation on the origin of apophenia is provided by Manfred Spitzer (1995, p. 100). He describes then how the latter is linked to the level of activity of dopamine and of norepinephrine, which have an influence on the value of the signal/noise ratio that is at the root of the activation of neural circuits: “if the signal to noise ratio is too high, (…) small environmental signals (i.e. perceptions to which we would normally pay little or no attention at all) may become amplified to a degree that is much higher than usual. This could result in experiences of “significant events” when merely ordinary events were in fact happening”. Spitzer shows then how apophenia can be the consequence of an imbalance at the dopamine level. By placing normally insignificant events (among which the patient’s thoughts) in the foreground, the modification of the signal/noise ratio allows then the particular feeling of interconnection that constitutes apophenia to occur. Under these conditions, one can notably conceive of how the patient’s thoughts can appear to him/her as prominent so as to be put on the same plan, then put in relationship with external facts such as the words pronounced by a TV presenter.

iii As an anonymous referee for *Philosophiques* suggests, it would be necessary to quantify precisely the frequency of these instances. This could be made in a separate study.

iv These elements of response with regard to the way the maintenance of polythematic delusions takes place need to be supplemented, especially as regards the way the conclusions of quaternary delusional arguments are put in coherence with the rest of the patient’s beliefs, takes place. Such analysis, which requires subsequent work, is however beyond the scope of the present study.

v I am very grateful to David Kingdon, Albert Ellis, Eugen Fischer, Robert Chapman and two anonymous referees for *Philosophiques* for very helpful comments on ancestor versions and earlier drafts.

# On a Class of Concepts

*Posprint in English (with additional illustrations) of a paper published in French in Semiotica, vol. 139 (1-4), 2002, 211-226, under the title “Une Classe de Concepts”.*

This article describes the construction, of philosophical essence, of the class of the matrices of concepts, whose structure and properties present an interest in several fields. The paper emphasises the applications in the field of paradigmatic analysis of the resulting taxonomy and proposes it as an alternative to the semiotic square put forth by Greimas.

This paper is cited in:

- Andris Teikmanis, The ‘use’ of’semiotic modelling as a research tool of art history, Tartu Summer School of Semiotics, August 2011
- Andris Teikmanis, Toward Models of Socialism Realism
- Eric Thivant, Jocelyne Ientile-Yalenios & Alain Roger, Discours managériaux et analyse sémiotique. Comparaison de guides d’entretien dans des contextes culturels différents, Revue Management et Avenir, n° 55, 2012
- Paul Franceschi, Polythematic Delusions and Logico-Theoretical vs. Experimentalist Turn of Mind, Journal for neurocognitive Research, 2013, vol. 55, n° 1-2
- Paul Franceschi, Éléments d’un contextualisme dialectique, dans Liber Amicorum Pascal Engel, J. Dutant, G. Fassio & A. Meylan (éd.), Université de Genève, 2014, p. 581-608
- Isis Truck, Nesrin Halouani, & Souhail Jebali (2016) Linguistic negation and 2-tuple fuzzy linguistic representation model : a new proposal, pages 81–86, in Uncertainty Modelling in Knowledge Engineering and Decision Making, The 12th International FLINS Conference on Computational Intelligence in Decision and Control, Eds. Xianyi Zeng, Jie Lu, Etienne E Kerre, Luis Martinez, Ludovic Koehl, 2016, Singapore: World Scientific Publishing.

**On a Class of Concepts**

Classically, in the discussion relating to *polar opposites*^{1}, one primarily directs his interest to the common and lexicalized concepts, i.e. for which there exists a corresponding word in the vocabulary inherent to a given language. This way of proceeding tends to generate several disadvantages. One of them resides in the fact (i) that such concepts are likely to vary from one language to another, from one culture to another. Another (ii) of the resulting problems is that certain lexicalized concepts reveal a nuance which is either meliorative or pejorative, with degrees in this type of nuances which prove difficult to appreciate. Finally, another problem (iii) lies in the fact that certain concepts, according to semiotic analysis^{2} are regarded as *marked* with regard to others concepts which are *unmarked*, the status of unmarked concept conferring a kind of precedence, of pre-eminence to the concepts in question.

In my view, all the above-mentioned disadvantages arise from the fact that one traditionally works primarily, from the lexicalized concepts. The methodology implemented in the present study is at the opposite of this way of proceeding. Indeed, one will begin here to construct concepts in an abstract way, without consideration of whether these concepts are lexicalized or not. This construction being performed, one will then be able to verify that some of the concepts thus constructed correspond indeed to lexicalized concepts, whereas some others cannot be put in correspondence with any existing word in the common language. This latter methodology allows, I think, to avoid the above-mentioned disadvantages.

It will finally appear that the construction described below will make it possible to propose a taxonomy of concepts which constitutes an alternative to the one based on the *semiotic square* which has been proposed by Greimas.

# 1. Dualities

Let us consider the class of dualities, which is made up of concepts corresponding to the intuition that these latter:

(i) are different one from the other

(ii) are minimal or irreducible, i.e. can no more reduce themselves to some other more simple semantic elements

(iii) present themselves under the form of pairs of dual concepts or contraries

(iv) are predicates

Each of the concepts composing a given duality will be termed a *pole*. I shall present here a list, which does not pretend to be exhaustive, and could if necessary, be supplemented. Consider then the following enumeration of *dualities*^{3}:

Analytic/Synthetic, Animate/Inanimate, Exceptional/Normal, Antecedent/Consequent, Existent/Inexistent, Absolute/Relative, Abstract/Concrete, Accessory/Principal, Active/Passive, Aleatory/Certain, Discrete/Continuous, Deterministic/Indeterministic, Positive/Negative, True/False, Total/Partial, Neutral/Polarized, Static/Dynamic, Unique/Multiple, Container/Containing, Innate/Acquired (Nature/Nurture), Beautiful/Ugly, Good/Ill, Temporal/Atemporal, Extended/Restricted, Precise/Vague, Finite/Infinite, Simple/Composed, Attracted/Repulsed, Equal/Different, Identical/Contrary, Superior/Inferior, Internal/External, Individual/Collective, Quantitative/Qualitative, Implicit/Explicit^{4}, …

At this step, it should be observed that certain poles present a nuance which is either meliorative (*beautiful*, *good*, *true*), or pejorative (*ugly*, *ill*, *false*), or simply neutral (*temporal*, *implicit*).

Let us denote by A/Ā a given duality. If words of the common language are used to denote the duality, capital letters will be then used to distinguish the concepts used here from the common concepts. For example: the Abstract/Concrete, True/False dualities.

It should be noted lastly that several questions^{5} immediately arise with regard to dualities. Do dualities exist (i) in a finite or infinite number? In the same way, does there exist (ii) a logical construction which makes it possible to provide an enumeration of the dualities?

# 2. Canonical poles

Starting from the class of the *dualities*, we are now in a position to construct the class of the *canonical poles*. At the origin, the lexicalized concepts corresponding to each pole of a duality reveal a nuance^{6} which is respectively either meliorative, neutral, or pejorative. The class of the canonical poles corresponds to the intuition that, for each pole of a given duality A/Ā, one can construct 3 concepts: a positive, a neutral and a negative concept. In sum, for a given duality A/Ā, one thus constructs 6 concepts, thus constituting the class of the *canonical poles*. Intuitively, *positive canonical poles* respond to the following definition: positive, meliorative form of ; *neutral canonical poles* correspond to the neutral, i.e. neither meliorative nor pejorative form of ; and *negative canonical poles *correspond to the negative, pejorative form of . It should be noted that these 6 concepts are exclusively constructed with the help of logical concepts. The only notion which escapes at this step to a logical definition is that of duality or base.

For a given duality A/Ā, we have thus the following canonical poles: {A^{+}, A^{0}, A^{–}, Ā^{+}, Ā^{0}, Ā^{–}}, that we can also denote respectively by (A/Ā, 1, 1), (A/Ā, 1, 0) , (A/Ā, 1, -1) , (A/Ā, -1, 1) , (A/Ā, -1, 0) , (A/Ā, -1, -1).

A capital letter for the first letter of a *canonical pole* will be used, in order to distinguish it from the corresponding lexicalized concept. If one wishes to refer accurately to a canonical pole whereas the usual language lacks such a concept or well appears ambiguous, one can choose a lexicalized concept, to which the exponent corresponding to the chosen neutral or polarized state will be added. To highlight the fact that one refers explicitly to a canonical pole – positive, neutral or negative – the notations A^{+}, A^{0} et A^{–} will be used. We have thus for example the concepts Unite^{+}, Unite^{0}, Unite^{–} etc. Where Unite^{+} = Solid, Undivided, Coherent and Unite^{–} = Monolithic^{–}. In the same way, Rational^{0} designates the neutral concept corresponding to the term *rational* of the common language, which reveals a slightly meliorative nuance. In the same way, Irrationnal^{0} designates the corresponding neutral state, whereas the common word *irrational* reveals a pejorative nuance. One will proceed in the same way, when the corresponding lexicalized word proves ambiguous. One distinctive feature of the present construction is that one begins by constructing the concepts logically, and puts them afterwards in adequacy with the concepts of the usual language, insofar as these latter do exist.

The constituents of a *canonical pole* are:

– a *duality* (or *base*) A/Ā

– a *contrary component* *c* {-1, 1}

– a *canonical polarity* *p* {-1, 0, 1}

A *canonical pole* presents the form: (A/Ā, *c*, *p*).

Furthermore, it is worth distinguishing, at the level of each duality A/Ā, the following derived classes:

– the *positive canonical poles*: A^{+}, Ā^{+}

– the *neutral canonical poles*: A^{0}, Ā^{0}

– the *negative canonical poles*: A^{–}, Ā^{–}

– the *canonical matrix* consisting of the 6 canonical poles: {A^{+}, A^{0}, A^{–}, Ā^{+}, Ā^{0}, Ā^{–}}. The 6 concepts constituting the canonical matrix can also be denoted under the form of a 3 x 2 matrix.

Let also be a canonical pole, one will denote by ~ its *complement*, semantically corresponding to *non*-. We have thus the following complements: ~A^{+}, ~A^{0}, ~A^{–}, ~Ā^{+}, ~Ā^{0}, ~Ā^{–}. The notion of a complement entails the definition of a universe of reference U. Our concern will be thus with the complement of a given canonical pole in regard to the corresponding matrix^{7}. It follows then that: ~A^{+} = {A^{0}, A^{–}, Ā^{+}, Ā^{0}, Ā^{–}}. And a definition of comparable nature for the complements of the other concepts of the matrix ensues.

It should be noted lastly that the following questions arise with regard to canonical poles. The construction of the matrix of the canonical poles of the Positive/Negative duality: {Positive^{+}, Positive^{0}, Positive^{–}, Negative^{+}, Negative^{0}, Negative^{–}} ensues. But do such concepts as Positive^{0}, Negative^{0} and especially Positive^{–}, Negative^{+} exist (i) without contradiction?

In the same way, at the level of the Neutral/Polarized duality, the construction of the matrix {Neutral^{+}, Neutral^{ 0}, Neutral^{–}, Polarized^{+}, Polarized^{0}, Polarized^{–}} ensues. But do Neutral^{+}, Neutral^{–} exist (ii) without contradiction? In the same way, does Polarized^{0} exist without contradiction?

This leads to pose the question in a general way: does any neutral canonical pole admit (iii) without contradiction a corresponding positive and negative concept? Is there a general rule for all dualities or well does one have as many specific cases for each duality?

# 3. Relations between the canonical poles

Among the combinations of relations existing between the 6 canonical poles (A^{+}, A^{0}, A^{–}, Ā^{+}, Ā^{0}, Ā^{–}) of a same duality A/Ā, it is worth emphasizing the following relations (in addition to the *identity* relation, denoted by I).

Two canonical poles _{1}(A/Ā, *c*_{1}, *p*_{1}) and _{2}(A/Ā, *c*_{2}, *p*_{2}) of a same duality are dual or *antinomical* or opposites if their contrary components are opposite and their polarities are opposite^{8}.

Two canonical poles _{1}(A/Ā, *c*_{1}, *p*_{1}) and _{2}(A/Ā, *c*_{2}, *p*_{2}) of a same duality are *complementary* if their contrary components are opposite and their polarities are equal^{9}.

Two canonical poles _{1} (A/Ā, *c*_{1}, *p*_{1}) et _{2}(A/Ā, *c*_{2}, *p*_{2}) of a same duality are *corollary* if their contrary components are equal and their polarities are opposite^{10}.

Two canonical poles _{1} (A/Ā, *c*_{1}, *p*_{1}) and _{2}(A/Ā, *c*_{2}, *p*_{2}) of a same duality are *connex* if their contrary components are equal and the absolute value of the difference in their polarities is equal to 1 ^{11}.

Two canonical poles _{1} (A/Ā, *c*_{1}, *p*_{1}) and _{2}(A/Ā, *c*_{2}, *p*_{2}) of a same duality are *anti-connex* if their contrary components are opposite and the absolute value of the difference in their polarities is equal to 1.^{12}^{, }^{13}

The following questions then arise, with regard to the relations between the canonical poles. Does there exist (i) one (or several) canonical pole which is its own opposite? A priori, it is not possible without contradiction for a positive pole or a negative pole. But the question remains for a neutral pole.

In the same way, does there exist (ii) one (or several) canonical pole which is its own complementary? The following two questions then ensue: does there exist a positive canonical pole which is its own complementary? And also: does there exist a negative canonical pole which is its own complementary?

The questions (i) and (ii) can be formulated in a more general way. Let R be a relation such that R {I, c, , j, g, }. Does there exist (iii) one (or several) canonical pole a verifying a = Ra?

# 4. Degrees of duality

One constructs the class of the *degrees of duality*, from the intuition that there is a *continuous* succession of concepts from A^{+} to Ā^{–}, from A^{0} to Ā^{0} and from A^{–} to Ā^{+}. The continuous component of a *degree of duality* corresponds to a *degree* in the corresponding dual pair. The approach by degree is underlied by the intuition that there is a continuous and regular succession of degrees, from a *canonical pole* A^{p} to its contrary Ā^{-p}.^{14} One is thus led to distinguish 3 classes of *degrees of duality*: (i) from A^{+} to Ā^{–} (ii) from A^{0} to Ā^{0} (iii) from A^{–} to Ā^{+}.

A *degree of duality* presents the following components:

– a *dual pair* A^{p}/Ā^{-p} (corresponding to one of the 3 cases: A^{+}/Ā^{–}, A^{0}/Ā^{0} or A^{–}/Ā^{+})

– a *degree* *d* Î [-1; 1] in this *duality*

A degree of duality has thus the form: (A^{+}/Ā^{–}, *d*), (A^{0}/Ā^{0}, *d*) or (A^{–}/Ā^{+}, *d*).

On the other hand, let us call *neutral point* a concept pertaining to the class of the *degrees of duality*, whose degree is equal to 0. Let us denote by ^{0} such a concept, which is thus of the form (A^{p}/Ā^{-p}, 0) with *d*[^{0}] = 0. Semantically, a neutral point ^{0} corresponds to a concept which responds to the following definition: *neither* A^{p} *nor* Ā^{-p}. For example, (True/False, 0) corresponds to the definition: *neither True nor False*. In the same way (Vague/Precise, 0) corresponds the following definition: *neither Vague nor Precise*. Lastly, when considering the Neutral/Polarized and Positive/Negative dualities, one has then: Neutral^{0} = (Negative^{0}/Positive^{0}, 0) = (Neutral^{0}/Polarized^{0}, 1).

It is worth noting that this construction does not imply that the neutral point thus constructed is the unique concept which corresponds to the definition *neither* A^{p} *nor* Ā^{-p}. It will appear on the contrary that several concepts and even hierarchies of concepts can correspond to this latter definition.

The following property of the neutral points then ensue, for a given duality A/Ā: (A^{+}/Ā^{–}, 0) = (A^{0}/Ā^{0}, 0) = (A^{–}/Ā^{+}, 0).

At this point, it is worth also taking into account the following derived classes:

– a discrete and truncated class, built from the degrees of duality, including only those concepts whose degree of duality is such that *d* {-1, -0.5, 0, 0.5, 1}.

– the class of the degrees of complementarity, the degrees of corollarity, etc. The class of the *degrees of duality* corresponds to the relation of *antinomy*. But it is worth considering, in a general way, as many classes as there exists relations between the canonical poles of a same duality. This leads to as many classes of comparable nature for the other relations, corresponding respectively to degrees of *complementarity*, *corollarity*, *connexity* and *anti-connexity*.

It is worth noting finally the following questions, with regard to degrees of duality and neutral points. Does there exist (i) one (or several) canonical pole which is its own neutral point? A priori, it is only possible for a neutral pole.

Does any duality A/Ā admit (ii) a neutral point or trichotomic zero? One can call this question the *problem of the general trichotomy*. Is it a general rule^{15} or well does there exists some exceptions? It seems a priori that the Abstract/Concrete duality does not admit a neutral point. It appears to be the same for the Finite/Infinite or the Precise/Vague duality. Intuitively, these latter dualities do not admit an intermediate state.

Does the concept corresponding to the neutral point (Neutral^{0}/Polarized^{0}, 0) and responding to the definition: *neither neutral nor polarized* exist (iii) without contradiction in the present construction?

# 5. Relations between the canonical poles of a different duality: includers

It is worth also considering the relation of *includer* for the canonical poles. Consider the following pairs of dual canonical poles: A^{+} and Ā^{+}, A^{0} and Ā^{0}, A^{–} and Ā^{–}. We have then the following definitions: a *positive* *includer* ^{+} is a concept such that it is itself a positive canonical pole and corresponds to the definition ^{+} = A^{+} Ā^{+}. A *neutral includer* ^{0}is a neutral canonical pole such that ^{0} = A^{0} Ā^{0}. And a *negative includer* ^{–} is a negative canonical pole such that ^{–} = A^{–} Ā^{–}. Given these definitions, it is clear that one assimilates here the includer to the minimum includer. Examples: Determinate^{0} is an includer for True^{0}/False^{0}. And Determinate^{0} is also a pole for the Determinate^{0}/Indeterminate^{0} duality. In the same way, Polarized^{0}is an includer for Positive^{0}/Negative^{0}.

More generally, one has the relation of *n*-includer (*n* > 1) when considering the hierarchy of (*n* + 1) matrices. One has also evidently, the reciprocal relation of *includer *and of *n*-includer.

## Let us also consider the following derived classes:

## – *matricial includers*: they consist of concepts including the set of the canonical poles of a same duality. They respond to the definition: ^{0} = A^{+} A^{0} A^{–} Ā^{+} Ā^{0} Ā^{–}.

– *mixed* includers: they consist of concepts responding to the definition _{1} = A^{+} Ā^{–} or well _{2} = A^{–} Ā^{+}

It is worth also considering the *types of relations* existing between the canonical poles of a different duality. Let A and E be two matrices whose canonical poles are respectively {A^{+}, A^{0}, A^{–}, Ā^{+}, Ā^{0}, Ā^{–}} and {E^{+}, E^{0}, E^{–}, Ē^{+}, Ē^{0}, Ē^{–}} and such that E is an includer for A/Ā i.e. such that E^{+} = A^{+} Ā^{+}, E^{0} = A^{0} Ā^{0} and E^{–} = A^{–} Ā^{–}. One extends then the just-defined relations between the canonical poles of a same matrix, to the relations of comparable nature between two matrices presenting the properties of A and E. We has then the relations of 2-*antinomy*, 2-*complementarity*, 2-*corollarity*, 2-connexity, 2-*anti-connexity*^{16}. Thus, for example, A^{0} is 2-contrary (or trichotomic contrary) to Ē^{0}, 2-connex (or trichotomic connex) to E^{+} and E^{–} and 2-anti-connex (or trichotomic anti-connex) to Ē^{+} and Ē^{–}. In the same way, A^{+} and Ā^{+} are 2-contrary to Ē^{–}, 2-complementary to Ē^{+}, 2-corollary to E^{–}, 2-connex to E^{0} and 2-anti-connex to Ē^{0}, etc.

Let us consider also the following *property* of neutral points and includers. Let A and E be two matrices, such that one of the neutral poles of E is an includer for the neutral dual pair of a: E^{0} = A^{0} Ā^{0}. We has then the following property: the canonical pole Ē^{0} for the matrix E is a neutral point for the duality A^{0}/Ā^{0}. Thus, the neutral point for the duality A^{0}/Ā^{0} is the dual of the includer E^{0} of A^{0} and Ā^{0}. Example: Determinate^{0} = True^{0} False^{0}. Here, the neutral point for the True/False duality corresponds to the definition: *neither True nor False*. And we have then (True^{0}/False^{0}, 0) = (Determinate^{0}/Indeterminate^{0}, -1).

This last property can be generalized to a hierarchy of matrices A_{1}, A_{2}, A_{3}, …, A_{n}, such that one of the poles _{2} of A_{2} of polarity *p* is an includer for a dual pair of A_{1}, and that one of the poles _{3} of A_{3} is an includer for a dual pair of A_{2}, …, and that one of the poles _{n} of A_{n} is an includer for a dual pair of A_{n-1}. It follows then an infinite construction of concepts.

One also notes the emergence of a hierarchy, beyond the sole neutral point of a given duality. It consists of the hierarchy of the neutral points of order *n*, constructed in the following way from the dual canonical poles A_{0} and Ā_{0}:

– A_{0}, Ā_{0}

– A_{1} = neither A_{0} nor Ā_{0}

– A_{21} = neither A_{0} nor A_{1}

– A_{22} = neither Ā_{0} nor A_{1}

– A_{31} = neither A_{0} nor A_{21}

– A_{32} = neither A_{0} nor A_{22}

– A_{33} = neither A_{0} nor A_{21}

– A_{34} = neither Ā_{0} nor A_{22}

– …

One can also consider the emergence of this hierarchy under the following form^{17}:

– A_{0}, Ā_{0}

– A_{1} = neither A_{0} nor Ā_{0}

– A_{2} = neither A_{0} nor Ā_{0} nor A_{1}

– A_{3} = neither A_{0} nor Ā_{0} nor A_{1} nor A_{2}

– A_{4} = neither A_{0} nor Ā_{0} nor A_{1} nor A_{2} nor A_{3}

– A_{5} = neither A_{0} nor Ā_{0} nor A_{1} nor A_{2} nor A_{3} nor A_{4}

– …

Classically, one constructs this infinite hierarchy for True/False by considering I_{1} (Indeterminate), I_{2}, etc. It should be noticed that in this last construction, no mention is made of the includer (Determinate) of True/False. Neither does one make mention of the hierarchy of includers.

The notion of a *complement* of a canonical pole corresponds semantically to *non*-. One has the concept of a 2-complement of a canonical pole , defined with regard to a universe of reference U that consists of the 2-matrix of . One has then for example: ~A^{+} = {A^{0}, A^{–}, Ā^{+}, Ā^{0}, Ā^{–}, Ē^{+}, Ē^{0}, Ē^{–}}. And also, ~A^{+} = {Ā^{+}, E^{0}, E^{–}, Ē^{+}, Ē^{0}, Ē^{–}}, etc. More generally, one has then the notion of a *n*-complement (*n* > 0) of a canonical pole with regard to the corresponding *n*-matrix.

The following questions finally arise, concerning includers. For certain concepts, does there exist (i) one maximum includer or well does one have an infinite construction for each duality? Concerning the True/False duality in particular, the analysis of the semantic paradoxes has led to the use of a logic based on an infinite number of truth-values^{18}.

Does any duality admit (ii) one neutral includer? Certain dualities indeed seem not to admit of an includer: such is in particular the case for the Abstract/Concrete or Finite/Infinite duality. It seems that Abstract constitutes a maximum element. Admittedly, one can well construct formally a concept corresponding to the definition *neither Abstract nor Concrete*, but such a concept appears very difficult to justify semantically.

Does there exist (iii) a canonical pole which is its own minimum includer?

Does there exist (iv) a canonical pole which is its own non-minimum includer? One can formulate this problem equivalently as follows. At a given level, does one not encounter a canonical pole which already appeared somewhere in the structure? It would then consist of a structure comprising a loop. And in particular, does one not encounter one of the poles of the first duality?

# 6. Canonical principles

Let be a *canonical pole*. Intuitively, the class of the *canonical principles* corresponds to the concepts which respond to the following definition: *principle corresponding to what is* . Examples: Precise Precision; Relative Relativity; Temporal Temporality. The canonical principles can be seen as 0-ary predicates, whereas the canonical poles are *n*-ary predicates (*n* > 0). The lexicalized concepts corresponding to canonical principles are often terms for which the suffix –*ity* (or –*itude*) has been added to the radical corresponding to a canonical pole. For example: Relativity^{0}, Beauty^{+}, Activity^{0}, Passivity^{0}, Neutrality^{0}, Simplicity^{0}, Temporality^{0}, etc. A list (necessarily non-exhaustive) of the canonical principles is the following:

Analysis^{0}/Synthesis^{0}, [Animate^{0}]/[Inanimate^{0}], [Exceptional^{0}]/Normality^{0}, [Antecedent^{0}]/[Consequent^{0}], Existence^{0}/Inexistence^{0}, Absolute^{0}/Relativity^{0}, Abstraction^{0}/[Concrete], [Accessory^{0}]/[Principal^{0}], Activity^{0}/Passivity^{0}, [Random^{0}]/Certainty^{0}, [Discrete^{0}]/[Continuous^{0}], Determinism^{0}/Indeterminism^{0}, [Positive^{0}]/[Negative^{0}], Truth^{0}/Falsity^{0}, Attraction^{0}/Repulsion^{0}, Neutrality^{0}/Polarization^{0}, [Static^{0}]/Dynamic^{0}, Unicity^{0}/Multiplicity^{0}, Contenance^{0}/[Containing^{0}], Innate^{0}/Acquired^{0}, Beauty^{+}/Ugliness^{–}, Good^{+}/Evil^{–}, Identity^{0}/Contrary^{0}, Superiority^{0}/Inferiority^{0}, Extension^{0}/Restriction^{0}, Precision^{0}/Vagueness^{0}, Finitude^{0}/Infinitude^{0}, Simplicity^{0}/Complexity^{0}, [Internal^{0}]/[External^{0}], Equality^{0}/Difference^{0}, Whole^{0}/Part^{0}, Temporality^{0}/Atemporality^{0}, Individuality^{0}/Collectivity^{0}, Quantity^{0}/Quality^{0}, [Implicit^{0}]/[Explicit^{0}], …

It should be noticed that a certain number of canonical principles are not lexicalized. The notations A^{+}, A^{0}, A^{–} will be used to denote without ambiguity a canonical principle which is respectively positive, neutral or negative. One could also use the following notation: being a canonical pole, then -ity (or -itude) is a canonical principle. The following notation could then be used: Abstract^{0}–*ity*, Absolute^{0}–*ity*, Acessory^{0}–*ity*, etc. or as above [Abstract^{0}], [Absolute^{0}], etc.

The constituents of the canonical principles are the same ones as for the class of the canonical poles.

It is worth distinguishing finally the following derived classes:

– *positive canonical principles*

– *neutral canonical principles*

– *negative canonical principles*

– *polarized canonical principles*

with some obvious definitions^{19}.

# 7. Meta-principles

Let a^{0} be a neutral canonical principle^{20}. The class of the *meta-principles* corresponds to a disposition of the mind directed towards what is a^{0}, to an interest with regard to what is a^{0}. Intuitively, a meta-principle corresponds to a point of view, a perspective, an orientation of the human mind. Thus, the attraction for Abstraction^{0}, the interest for Acquired^{0}, the propensity to place oneself from the viewpoint of Unity^{0}, etc. constitute *meta-principles*. It should be noted that this construction makes it possible in particular to construct some concepts which are not lexicalized. This has the advantage of a better exhaustiveness and leads to a better and richer semantics.

Let a^{0} be a neutral canonical principle. Let us also denote by ^{}^{p} a meta-principle (*p* {-1, 0, 1}). One denotes thus a *positive* meta-principle by ^{}^{+}, a *neutral* meta-principle by ^{}^{0} and a *negative* meta-principle by ^{}^{–}. We have then the enumeration of the meta-principles, for a given duality: {A^{}^{+}, A^{}^{0}, A^{}^{–}, Ā^{}^{+}, Ā^{}^{0}, Ā^{}^{–}}. Moreover, one will be able to denote by a-*ism* a meta-principle. Example: Unite Unite-ism. We have then Internalism, Externalism, Relativism, Absolutism, etc. which correspond in particular to dispositions of the mind. A capital letter will preferably be used here to distinguish the meta-principles from the lexicalized concepts, and in particular to differentiate them from the corresponding philosophical doctrines, which often have very different meanings. It will be however possible to make use of the classical terms when they exist to designate the corresponding meta-principle. Thus All-ism corresponds to Holism.

One can term *Ultra*-a-*ism* or *Hyper*-a*-ism* the concept corresponding to ^{}^{–}. This latter form corresponds to an exclusive, excessive, exaggerated use of the viewpoint corresponding to a given principle. One has thus for example: Externalism^{–} = Hyper-externalism.

The constituents of the meta-principles are:

– a *polarity* *p* Î {-1, 0, 1}

– a *neutral canonical principle* composed of:

– a *duality* (or *base*) A/Ā

– a *contrary component* *c* {-1, 1}

– a *neutral polarity* *q* = 0

The *positive*, *neutral*, *negative* *canonical meta-principles* are respectively of the form ((A/Ā, *c*, 0), 1), ((A/Ā, *c*, 0), 0), ((A/Ā, *c*, 0), -1).

Between the canonical meta-principles of a same duality, one has the same relations as for the canonical poles.

One has lastly the derived classes consisting in:

– the *positive meta-principles* (*p* > 0)

– the *neutral meta-principles* (*p* = 0)

– the *negative meta-principles* (*p* < 0)

– the *polarized meta-principles* which include the *positive* and *negative meta-principles*

– the *matrix* of the canonical meta-principles, consisting of 6 meta-principles applicable to a given duality{A^{}^{+}, A^{}^{0}, A^{}^{–}, Ā^{}^{+}, Ā^{}^{0}, Ā^{}^{–}}.

– the *degrees of canonical meta-principles*. Intuitively, such concepts are more or less positive or negative. The polarity is regarded here as a *degree of polarity*. These concepts are such that *p* Î [-1; 1].

– the class of the *behavioral principles*. Intuitively, the class of the *behavioral principles* constitutes an extension of that of the meta-principles. While the meta-principle constitutes a disposition of the human mind, the concepts concerned here are those which aim to describe, in a more general way, the tendencies of the human behavior^{21}. Among the lexicalized concepts corresponding to the *behavioral principles*, one can mention: *courage*, *prudence*, *pessimism*, *rationality*, *avarice*, *fidelity*, *tendency to analysis*, *instability*, *objectivity*, *pragmatism*, etc. A first analysis reveals (i) that a certain number of them reveal a meliorative nuance: *courage*, *objectivity*, *pragmatism*; that (ii) others, by contrast, present a pejorative, unfavorable connotation: *cowardice*, *avarice*, *instability*; and finally (iii) that certain concepts present themselves under a form which is neither meliorative nor pejorative: *tendency to analysis*^{22}. One has here the same classes as for the meta-principles, and in particular the *degrees* of behavioral principles. Example: *coward* is more negative than *apprehensive*; in the same way, *bravery* is more positive than *courage*.

# Conclusion

The concepts constructed with the help of the present theory need to be distinguished in several regards from those resulting from the application of the *semiotic square* described by Greimas (1977, p. 25). This last theory envisages in effect four concepts: S1, S2, ~S1, ~S2. On the one hand, it appears that the semiotic square is based on two lexicalized concepts S1 and S2 that constitute a dual pair. It does not distinguish, when considering the dual concepts, whether these latter are positive, neutral or negative. By contrast, the present theory considers six concepts, lexicalized or not.

On the other hand, the present analysis differs from the semiotic square by a different definition of the complement-negation. Indeed, the semiotic square comprises two concepts corresponding to the complement-negation: non-S1 and non-S2. By contrast, in the present context, the negation is defined with regard to a universe of reference U, which can be defined with regard to the corresponding matrix, or well to the 2-matrix…, to the *n* -matrix. For each canonical pole, there is thus a hierarchy of concepts corresponding to non-S1 and non-S2.

One sees it, the present taxonomy of concepts differs in several respects from the one conceived of by Greimas. Implemented from the dualities and the logical concepts, the present theory has the advantage of applying itself to lexicalized concepts or not, and also of being freed [affranchie] from the definitions of concepts inherent to a given culture. In this context, the classification which has been just described constitutes an alternative to the one based on the semiotic square which has been proposed by Greimas.

**References**

FINE, Kit (1975). Vagueness, Truth and Logic. Synthese 30: 265-300 |

GREIMAS, A. J. (1977). Elements of a Narrative Grammar, Diacritics 7: 23-40 |

JAKOBSON, Roman (1983). Dialogues, Cambridge MA: MIT Press |

PEACOCKE, C. A. B. (1981). Are Vague Predicates Incoherent?. Synthese 46: 121-141 |

RESCHER, Nicholas (1969). Many-Valued Logic, New York: McGraw Hill |

1 Or *polar contraries*.

2 Cf. Jakobson (1983).

3 In the same way, it would have been possible to define a more restricted class, including only half of the semantic poles, by retaining only one of the two dual predicates, and by constructing the others with the *contrary* relation. However, the choice of either of the dual poles would have been arbitrary, and I have preferred to avoid it. The following construction would have then resulted. Let Contrary be the semantic pole and a whatever semantic pole, not necessarily distinct from Contrary; the concept resulting from the composition of Contrary and a is a *semantic pole*. It should also be noted that this type of construction would have led to:

Contrary° Contrary **=** Identical.

Contrary° Identical **=** Contrary.

Contrary^{n} = Identical (for *n* even).

Contrary^{n} = Contrary (for *n* odd).

In this context, it is worth noting that Contrary constitutes a specific case. In effect, if one seeks to build a class of the *canonical poles* which is minimal, it is worth noting that one can dispense oneself from Identical, whereas one cannot dispense oneself from Contrary. There is here an asymmetry. In effect, one can construct Identical with the help of Contrary, by using the property of *involution*: Contrary° Contrary **=** Identical. For other dualities, one can indifferently choose either of the concerned semantic poles.

4 It is worth noting that one could have drawn here a distinction between *unary* and *binary* poles, by considering that they consist of predicates. But a priori, such a distinction does not prove very useful for the resulting construction.

5 In what follows, the questions relating to the various classes are only mentioned. It goes without saying that they require an in-depth treatment which goes far beyond the present study.

6 With variable degrees in the nuance.

7 When it is defined with regard to a dual pair, the complement of the pole of a given duality identifies itself with the corresponding dual pole.

8 Formally *c*_{1} = –*c*_{2}, *p*_{1} = – *p*_{2} ® _{1}(A/Ā, *c*_{1}, *p*_{1}) = _{2}(A/Ā, *c*_{2}, *p*_{2}).

9 Formally *c*_{1} = – *c*_{2}, *p*_{1} = *p*_{2} ® _{1}(A/Ā, *c*_{1}, *p*_{1}) = f_{2}(A/Ā, *c*_{2}, *p*_{2}).

10 Formally *c*_{1} = *c*_{2}, *p*_{1} = – *p*_{2} ® _{1}(A/Ā, *c*_{1}, *p*_{1}) = c_{2}(A/Ā, *c*_{2}, *p*_{2}).

11 Formally *c*_{1} = *c*_{2}, |*p*_{1} – *p*_{2}| = 1 ® _{1}(A/Ā, *c*_{1}, *p*_{1}) = g_{2}(A/Ā, *c*_{2}, *p*_{2}).

12 Formally *c*_{1} = – *c*_{2}, |*p*_{1} – *p*_{2}| = 1 ® _{1}(A/Ā, *c*_{1}, *p*_{1}) = b_{2}(A/Ā, *c*_{2}, *p*_{2}).

13 We have then the following properties, with regard to the above-mentioned relations. The relation of identity constitutes a relation of equivalence. Antinomy, complementarity and corollarity are symmetrical, anti-reflexive, non-associative, involutive.

The operation of composition on the relations {*identity*, *corollarity*, *antinomy*, *complementarity*} defines an *abelian group* of order 4. With *G* = {I, c, , j}:

°IcjIIcjccIjjIcjjcI

where for all A Î *G*, A^{-1} = A, and A ° I = A, I being the neutral element. It should be noted that the group properties make it possible in particular to give straightforwardly a valuation to any propositions of the form: *the contrary concept of the complementary of* a_{1} *is identical to the corollary of the complementary of* a_{2}.

14 This construction of concepts can be regarded as an application of the *degree theory*. Cf. in particular Fine (1975), Peacocke (1981). The present theory however is not characterized by the preferential choice of the degree theory, but considers simply this latter theory as one of the methods of construction of concepts.

15 Some common trichotomies are: {*past*, *present*, *future*}, {*right*, *center*, *left*}, {*high*, *center*, *low*}, {*positive*, *neutral*, *negative*}.

16 There is a straightforward generalization to *n* matrices (*n* > 1) of this construction with the relations of *n*-antinomy, *n*-complementarity, *n*-corollarity, *n*-connexity, *n*-anti-connexity.

17 One can assimilate the two just-described hierarchies to only one single hierarchy. It suffices to proceed to the following assimilation:

– A_{2} = A_{21} or A_{22}

– A_{3} = A_{31} or A_{32} or A_{33} or A_{34}

– A_{4} = A_{41} or A_{42} or A_{43} or A_{44} or A_{45} or A_{46} or A_{47} or A_{48}

– …

18 *Infinite-valued logics*. Cf. Rescher (1969).

19 Furthermore, it should be noted that some other concepts can be thus constructed. Let also be a canonical pole. We have then the classes of concepts responding to the following definition: *to render* (Example: Unite Unify; Different Differentiate); *action of rendering* (Unite Unification; Different Differentiation); *that it is possible to render* (Unite Unitable; Different Differentiable), etc. These concepts are not however of interest in the present context.

20 It should be observed that we could have taken alternatively as a basis for the definition of the meta-principles a canonical principle, without distinguishing whether this latter is positive, neutral or negative. But it seems that such a definition would have engendered more complexity, without giving in return a genuine semantic interest.

21 This particular class would require however a much finer analysis than the one which is summarily presented here. I am only concerned here with showing that a many concepts pertaining to this category can be the subject of a classification whose structure is that of the meta-principles.

22 One can consider the following – necessarily partial – enumeration corresponding to the *behavioral principles*, in the order (A^{+}), (A^{0}), (A^{–}), (Ā^{+}), (Ā^{0}), (Ā^{–}):

firmness, propensity to repress, severity, leniency, propensity to forgive, laxism

defense, refusal, violence, pacifism, acceptance, weakness

pride, self-esteem, hyper-self-esteem, modesty, withdrawal of the ego, undervaluation of self

expansion, search of quantity, excess, perfectionism, search of quality, hyper-selectivity

delicacy, sensitivity, sentimentality, coolness, impassibility, coldness

objectivity, to be neutral being, impersonality, to be partisan, parti pris

uprightness, to act in a direct way, brusqueness, tact, to act in an indirect way, to flee the difficulties

combativeness, disposition to attack, aggressiveness, protection, disposition to defense, tendency to retreat

receptivity, belief, credulity, incredulity, doubt, excessive skepticism

expansion, oriented towards oneself, selfishness, altruism, oriented towards others, to render dependent

sense of economy, propensity to saving, avarice, generosity, propensity to expenditure, prodigality

mobility, tendency to displacement, instability, stability, tendency to stay at the same place, sedentariness

logical, rationality, hyper-materialism, imagination, irrationality, inconsistency

sense of humour, propensity to play, lightness, serious, propensity to the serious activity, hyper-serious

capacity of abstraction, disposition to the abstract, dogmatism, pragmatism, disposition to the concrete, prosaicness

audacity, tendency to risk, temerity, prudence, tendency to avoid the risks, cowardice

discretion, to keep for oneself, inhibition, opening, to make public, indiscretion

optimism, to apprehend the advantages, happy optimism, mistrust, to see the disadvantages, pessimism

sense of the collective, to act like the others, conformism, originality, to demarcate oneself from others, eccentricity

resolution, tendency to keep an opinion, pertinacity, flexibility of spirit, tendency to change opinion, fickleness

idealism, tendency to apprehend the objectives, quixotism, realism, tendency to apprehend the means, prosaicness

taste of freedom, to be freed, indiscipline, obedience, to subject oneself to a rule, servility

reflexion, interiorization, inhibition, sociability, exteriorisation, off-handednes

spontaneousness, tendency to react immediately, precipitation, calm, tendency to differ one’s reaction, slowness

eclecticism, multidisciplinarity, dispersion, expertise, mono-disciplinarity, bulk-heading

revival, propensity to change, rupture, safeguarding of the assets, propensity to maintenance, conservatism

motivation, passion, fanaticism, moderation, reason, tepidity

width of sights, tendency to synthesis, overflight, precision, tendency to analysis, to lose oneself in the details

availability, propensity to leisure, idleness, activity, propensity to work, overactivity

firmness, tendency not to yield, intransigence, diplomacy, tendency to make concessions, weakness

causticity, tendency to criticism, denigration, valorization, tendency to underline qualities, angelism

authority, propensity to command, authoritarianism, docility, propensity to obey, servility

love, tendency to be attracted, exaggerate affection, tendency to know to take one’s distances, repulsion, hatred

conquest, greed, bulimia, sobriety, to have the minimum, denudement

# A Brief Introduction to N-universes

*Preprint. * I present in this paper the basic elements of the n-universes, from an essentially pragmatic standpoint, i.e. by describing accurately the step-by-step process which leads to the modelling of a thought experiment.

**A Brief Introduction to N-universes**

Paul Franceschi

The n-universes are a methodological tool whose scope proves to be general and which find to apply in the thought experiments underlying the philosophical problems. The n-universes have been introduced in Franceschi (2001) in the context of the study of Goodman’s paradox and described in more detail in Franceschi (2002), in the context of their application to the paradoxes related to the Doomsday argument. I propose here to present the basic elements of n-universes, from a fundamentally practical standpoint, i.e. by describing the step-by-step process which leads to the modelisation of a thought experiment.

**Presentation of n-universes**

The n-universes are simplified models of the physical universe corresponding to a real situation put forth in a thought experiment. Making application of Occam’s razor, the n-universes thus make it possible to model a real situation with the help of the simplest model of universe, while nevertheless preserving the intrinsic structure of the corresponding real situation. Let us describe then the fundamental elements of the n-universes, from a basically operational standpoint. When one proceeds thus to model a concrete situation with a n-universe, one can determine its structure by means of the successive answers to the following questions:

1. *does the n-universe have constants or variables?*

The first task consists in determining what are the *criteria* of the n-universe corresponding to a given situation. These latter criteria include both constants and variables. Among the most common criteria, one can thus mention the temporal (Time) and spatial (Loc) criteria, but also the criteria of colour (Col), shape, temperature, polarisation, etc. Usually, the n-universe corresponding to a given thought experiment comprises at least the criteria of time and space. This can be illustrated through the following examples:

– the ΩObj_{0}Time_{0}Loc_{0}: a n-universe comprising a unique object, a temporal constant and a spatial constant

– the ΩObjColTime_{0}Loc: a n-universe comprising multiple objects, a colour variable, a temporal constant and a space variable

2. *does the n-universe comprises a unique or multiple objects?*

It is also worth drawing a distinction according to whether the given n-universe comprises either one single (ΩObj_{0}) or multiple objects (ΩObj). To cite a few examples:

– the ΩObj_{0}TimeLoc_{0}: a n-universe comprising a unique object, a temporal variable and a spatial constant

– the ΩObjTimeLoc: a n-universe comprising multiple objects, a temporal variable and a space variable.

3. *is a given variable-criterion demultiplied or not?*

This distinction only relates to the variable-criterions of a given n-universe, and does not apply to constant-criterions. A given variable-criterion (time, space, colour, etc.) of a n-universe can be demultiplied (*) or not (). If the variable-criterion is demultiplied, an object in this type of n-universe can exemplify several taxa of the criterion . In our physical universe, objects have a property of temporal persistence: they exemplify thus several successive temporal positions. The corresponding model is a n-universe where the objects are demultiplied with regard to the temporal criterion (ΩObjTime*). To take an example:

– the ΩObjTime*ObjLoc_{0}: a n-universe with multiple objects comprising a temporal variable and a spatial constant; the objects are also demultiplied with regard to the temporal criterion, so that a given object can thus exemplify several different temporal positions.

4. *are the multiple objects in a one-one or in a many-one relation with regard to a given criterion?*

This last distinction does not apply to the n-universes comprising one single object and only relates to those comprising *multiple* objects. Among these latter n-universes, one can then distinguish two cases. Firstly, when the same taxon of a given criterion is exemplified by several objects, these latter are in *many*-one relation with the criterion (ΩObj*). By contrast, when every taxon of a given criterion is only exemplified by one single object, the objects are in relation *one*-one with this last criterion (ΩObj). To give a few examples:

– the ΩObj*Col_{0}Obj*Time_{0}ObjLoc: this type of n-universe with multiple objects comprises a colour constant (Col_{0}), a temporal constant (Time_{0}) and a spatial variable (Loc); moreover, the objects are in a *many*-one relation with the colour constant, so that all objets share the same colour; in addition, the objects are in a *many*-one relation with the time constant, so that several objects can exist at a the unique temporal position; on the other hand, the objects are in relation *one*-one with the space criterion, so that only one object can exist at a given space location (*Fig. 1*)

– the ΩObjColObj*Time_{0}ObjLoc: this type of n-universe with multiple objects comprises a colour variable (Col), a temporal constant (Time_{0}) and a spatial variable (Loc); moreover, the objects are in a *one*-one relation with the colour variable, so that all objects can have a different colour; in addition, the objects are in a *many*-one relation with the time constant, so that several objects can exist at a the unique temporal position; on the other hand, the objects are in relation *one*-one with the space criterion, so that only one object can exist at a given space location (*Fig. 2*)

– the ΩObj*ColObj*Time_{0}ObjLoc: this type of n-universe with multiple objects comprises a colour variable (Col), a temporal constant (Time_{0}) and a spatial variable (Loc); moreover, the objects are in a *many*-one relation with the colour variable, so that several objets can have the same colour; in addition, the objects are in a *many*-one relation with the time constant, so that several objects can exist at a the unique temporal position; lastly, the objects are in relation *one*-one with the space criterion, so that only one object can exist at a given space location (*Fig. 3*)

– the ΩObj*Col_{0}Obj*Time_{0}Obj*Loc: this type of n-universe with multiple objects comprises a colour constant (Col_{0}), a temporal constant (Time_{0}) and a spatial variable (Loc); moreover, the objects are in a *many*-one relation with the colour constant, so that several objets can share the same colour; in addition, the objects are in a *many*-one relation with the time constant, so that several objects can exist at a the unique temporal position; on the other hand, the objects are in relation *many*-one with the space criterion, so that only several objects can occupy the same space location (*Fig. 4*)

**First steps with n-universes**

At this step, we are in a position to illustrate what precedes through a concrete example. Consider then the following experiment, described by John Leslie (1996, p. 191):

You develop amnesia in a windowless room. Where should you think yourself more likely to be: in Little Puddle with a tiny situation, or in London? Suppose you remember that Little Puddle’s population is fifty while London’s is ten million, and suppose you have nothing but those figures to guide you. (…) Then you should prefer to think yourself in London. For what if you instead saw no reason for favouring the belief that you were in the larger of the two places? Forced to bet on the one or on the other, suppose you betted you were in Little Puddle. If everybody in the two places developed amnesia and betted as you had done, there would be ten million losers and only fifty winners. So, it would seem, betting on London is far more rational. The right estimate of your chances of being there rather than in Little Puddle, on the evidence on your possession, could well be reckoned as ten million to fifty.

Let us proceed now to model the situation corresponding to the *Little Puddle/London experiment* in terms of n-universes. What are thus the criteria of the corresponding n-universe? It appears first that the corresponding situation characterises itself with the presence of multiple individuals: there are indeed 50 inhabitants in Little Puddle and 10 million in London. Consequently, the corresponding n-universe comprises multiple objects (ΩObj). It also appears that the Little Puddle/London experiment takes place at one single temporal position. Thus, the corresponding n-universe has a constant-time (ΩTime_{0}). Moreover, several inhabitants exist simultaneously at the unique temporal position Time_{0}. Hence, the objects are in relation *many*-one with the temporal constant (ΩObj*Time_{0}). Moreover, two space locations are explicitly distinguished: Little Puddle (Loc_{1}) and London (Loc_{2}). The corresponding situation can thus be modelled in a n-universe comprising a space variable (ΩLoc) which includes two different locations: Loc_{1} and Loc_{2}. Furthermore, it proves that each inhabitant is either in Little Puddle or in London, so as a given inhabitant cannot occupy several space locations at the same time. Thus the space criterion is not demultiplied. Lastly, one can observe that several people can be at the same time at a given space location: there are thus 50 inhabitants in Little Puddle and 10 million in London. Consequently, the objects are in relation *many*-one with the space variable (ΩObj*Loc). Taking into account what precedes, it follows that the situation corresponding to the Little Puddle/London experiment can be modelled in a ΩObj*Time_{0}Obj*Loc, a n-universe with multiple objects, comprising a temporal constant and a space variable, where the objects are in relation *many*-one with the time-constant and the space variable.

**Conclusion**

The n-universes, as we have just seen through the above illustration, make it possible to model the situations described in thought experiments, by simplifying their intrinsic elements in virtue of Occam’s razor. The n-universes then often allow to remove the inherent ambiguity and complexity which renders more difficult the reasoning applied to thought experiments.

**Works cited**

Franceschi, Paul. 2001. A Solution to Goodman’s paradox, e-print Philsci: 602, English translation of Une solution pour le paradoxe de Goodman, Dialogue 40: 99-123. |

Franceschi, Paul. 2002. Une application des n-univers à l’argument de l’Apocalypse et au paradoxe de Goodman. doctoral dissertation, Corté: University of Corsica |

Leslie, John. 1996. The End of the World: the science and ethics of human extinction, London: Routledge. |

# A Third Route to the Doomsday Argument

*A paper published (2009) in English in the Journal of Philosophical Research, vol. 34, pages 263-278 (with significant changes with regard to the preprint).*

In this paper, I present a solution to the Doomsday argument based on a third type of solution, by contrast with, on the one hand, the Carter-Leslie view and on the other hand, the Eckhardt et al. analysis. I begin by strengthening both competing models by highlighting some variations of their ancestors models, which renders them less vulnerable to several objections. I describe then a third line of solution, which incorporates insights from both Leslie and Eckhardt’s models and fits more adequately with the human situation corresponding to the Doomsday argument. I argue then that the resulting two-sided analogy casts new light on the reference class problem. This leads finally to a novel formulation of the argument that could well be more consensual than the original one.

This paper is cited in:

- Alasdair Richmond,
*The Doomsday Argument*, Philosophical Books Vol. 47 No. 2 April 2006, pp. 129–142 - Robert Northcott,
*A Dilemma for the Doomsday Argument*, Ratio, Volume29-3, September 2016, pages 268-282 - William Poundstone,
*How to Predict Everything: The Formula Transforming What We Know About Life and the Universe*, 2019, Oneworld

**A Third Route to the Doomsday Argument**

In what follows, I will endeavor to present a solution to the problem arising from the Doomsday argument (DA). The solution thus described constitutes a third way out, compared to, on the one hand, the approach of the promoters of DA (Leslie 1993 and 1996) and on the other hand, the solution recommended by its detractors (Eckhardt 1993 and 1997, Sowers 2002).^{i}

**I. ****The Doomsday Argument and the Carter-Leslie model**

For the sake of the present discussion, it is worth beginning with a brief presentation of DA. This argument can be described as reasoning which leads to a Bayesian shift, starting from an analogy between what was has been called the *two-urn case*^{ii} and the corresponding human situation.

Let us consider first the *two-urn case experiment* (adapted from Bostrom 1997):

*The* *two-urn case experiment* An opaque urn^{iii} is in front of you. You know that it contains either 10 or 1000 numbered balls. A fair coin has been tossed at time T_{0} and if the coin landed tails, then 10 balls were placed in the urn; on the other hand, if the coin landed heads, 1000 balls were placed in the urn. The balls are numbered 1,2,3,…. You formulate then the assumptions H_{few} (the urn contains only 10 balls) and H_{many} (the urn contains 1000 balls) with the initial probabilities P (H_{few}) = P (_{Hmany}) = 1/2.

Informed of all the preceding, you randomly draw a ball at time T_{1} from the urn. You get then the ball #5. You endeavor to estimate the number of balls that were contained at T_{0} in the urn. You conclude then to an upward Bayesian shift in favor of the H_{few} hypothesis.

The* two-urn case experiment* is an uncontroversial application of Bayes’ theorem. It is based on the two following concurrent assumptions:

(H1_{few}) | the urn contains 10 balls |

(H2_{many}) | the urn contains 1000 balls |

and the corresponding initial probabilities: P (H1) = P (H2) = 1/2. By taking into account the fact that E denotes the evidence according to which the randomly drawn ball carries the #5 and that P (E|H1) = 1/10 and P (E|H2) = 1/1000, an upward Bayesian shift follows, by a straightforward application of Bayes’ theorem. Consequently, the posterior probabilities are such that P'(H1) = 0.99 and P'(H2) = 0.01.

Let us consider, on the second hand, the *human situation corresponding to DA*. While being interested in the total number of humans that humankind will finally count, it is worth considering the two following concurrent hypotheses:

(H3_{few}) | the total number of humans having ever lived will amount to 10^{11} (Apocalypse near) |

(H4_{many}) | the total number of humans having ever lived will amount to 10^{14} (Apocalypse far) |

It appears now that every human being has his own birth rank, and that yours, for example, is about 60×10^{9}. Let us also assume, for the sake of simplicity, that the initial probabilities are such as P(H3) = P(H4) = 1/2. Now, according to Carter and Leslie, the *human situation corresponding to DA* is analogous to the *two urn case*.^{iv} If we denote by E the fact that our birth rank is 60×10^{9}, an application of Bayes’ theorem, by taking into account the fact that P(E|H3) = 1/10^{11} and that P(E|H4) = 1/10^{14}, leads to an important Bayesian shift in favor of the hypothesis of a near Apocalypse, i.e., P'(H3) = 0.999. The importance of the Bayesian shift which results from this reasoning, associated with a very worrying situation related to the future of humankind, from the only recognition of our birth rank, appears counter-intuitive. This intrinsic problem requires that we set out to find it a solution.

In such context, it appears that a solution to DA has to present the following characteristics. On the one hand, it must point out in which ways the human situation corresponding to DA is similar to the two-urn case or possibly, to an alternative model, the characteristics of which are to be specified. On the second hand, such solution to DA must point out in which ways one or several models on analogy with the human situation corresponding to DA are associated with a frightening situation for the future of humankind.

In what follows, I will endeavor to present a solution to DA. In order to develop it, it will be necessary first to build up the space of solutions for DA. Such a construction is a non-trivial task that requires the consideration of not only several objections that have been raised against DA, but also the reference class problem. Within this space of solutions, the solutions advocated by the supporters as well as critics of DA, will naturally be placed. I will finally show that within the space of solutions thus established, there is room for a third way out, which is in essence a different solution from that offered by the proponents and opponents of DA.

**II. ****Failure of an alternative model based on the incremental objection of Eckhardt et al.**

DA is based on the matching of a probabilistic model – the two-urn case – with the human situation corresponding to DA. In order to build the space of solutions for DA, it is necessary to focus on the models that constitute an alternative to the two-urn case, which can also be put in correspondence with the human situation corresponding to DA. Several alternative models have been described by the opponents to DA. However, for reasons that will become clearer later, not all these models can be accepted as valid alternative models to the two-urn case, and take a place within the space of solutions for DA. It is therefore necessary to distinguish among these models proposed by the detractors of DA, between those which are not genuine alternative models, and those which can legitimately be included within the space of solutions for DA.

A certain number of objections to DA were formulated first by William Eckhardt (1993, 1997). For the sake of the present discussion, it is worth distinguishing between two objections, among those which were raised by Eckhardt, and that I will call respectively: the *incremental* objection and the *diachronic* objection. With each one of these two objections is associated an experiment intended to constitute an alternative model to the two-urn case.

Let us begin with the *incremental* objection mentioned in Eckhardt (1993, 1997) and the alternative model associated with it. Recently, George Sowers (2002) and Elliott Sober (2003) have echoed this objection. According to this objection, the analogy with the urn that is at the root of DA, is ungrounded. Indeed, in the two-urn case experiment, the number of the balls is randomly chosen. However, these authors emphasize, in the case of the human situation corresponding to DA, our birth rank is not chosen at random, but is indeed indexed on the corresponding time position. Therefore, Eckhardt stresses, the analogy with the two-urn case is unfounded and the whole reasoning is invalidated. Sober (2003) develops a similar argument,^{v} by stressing that no mechanism designed to randomly assign a time position to human beings, can be highlighted. Finally, such an objection was recently revived by Sowers. The latter focused on the fact that the birth rank of every human being is not random because it is indexed to the corresponding time position.

According to the viewpoint developed by Eckhardt et al., the human situation corresponding to DA is not analogous to the *two-urn case* experiment, but rather to an alternative model, which may be called the *consecutive token dispenser*. The consecutive token dispenser is a device, originally described by Eckhardt^{vi}, that ejects consecutively numbered balls at regular intervals: “(…) suppose on each trial the consecutive token dispenser expels either 50 (early doom) or 100 (late doom) consecutively numbered tokens at the rate of one per minute”. A similar device – call it the *numbered balls dispenser* – is also mentioned by Sowers, where the balls are ejected from the urn and numbered in the order of their ejection, at the regular interval of one per minute:^{vii}

There are two urns populated with balls as before, but now the balls are not numbered. Suppose you obtain your sample with the following procedure. You are equipped with a stopwatch and a marker. You first choose one of the urns as your subject. It doesn’t matter which urn is chosen. You start the stopwatch. Each minute you reach into the urn and withdraw a ball. The first ball withdrawn you mark with the number one and set aside. The second ball you mark with the number two. In general, the *n*^{th} ball withdrawn you mark with the number *n*. After an arbitrary amount of time has elapsed, you stop the watch and the experiment. In parallel with the original scenario, suppose the last ball withdrawn is marked with a seven. Will there be a probability shift? An examination of the relative likelihoods reveals no.

Thus, under the terms of the viewpoint defended by Eckhardt et al., the human situation corresponding to DA is not analogous with the two-urn case experiment, but with the *numbered balls dispenser*. And this last model leads us to leave the initial probabilities unchanged.

The incremental objection of Eckhardt et al. is based on a disanalogy. Indeed, the human situation corresponding to DA presents a *temporal* nature, for the birth ranks are successively attributed to human beings depending on the time position corresponding to their appearance on Earth. Thus, the corresponding situation takes place, for example, from T_{1} to T_{n}, where 1 and *n* are respectively the birth ranks of the first and of the last humans. However, the two-urn case experiment appears *atemporal*, because when the ball is drawn at random, all the balls are already present within the urn. The two-urn case experiment takes place at a given time T_{0}. It appears thus that the two-urn case experiment is an atemporal model, while the situation corresponding to DA is a temporal model. And this forbids, as Eckhardt et al. underscore, considering the situation corresponding to DA and the two-urn case as isomorphic.^{viii}

At this stage, it appears that the *atemporal-temporal disanalogy* is indeed a reality and it cannot be denied. However, this does not constitute an insurmountable obstacle for DA. As we shall see, it is possible indeed to put in analogy the human situation corresponding to DA, with a temporal variation of the two-urn case. This can be done by considering the following experiment, which can be termed the *incremental two-urn case* (formally, the *two-urn case*^{++}):

*The two-urn case*^{++}. An opaque urn in front of you. You know that it contains either 10 or 1000 numbered balls. A fair coin has been tossed at time T_{0} and if the coin landed tails, then the urn contains only 10 balls, while if the coin landed heads, then the urn contains the same 10 balls plus 990 extra balls, i.e. 1000 balls in total. The balls are numbered 1, 2, 3, …. You formulate then the H_{few} (the box contains only 10 balls) and H_{many} (the box contains 1000 balls) hypotheses with initial probabilities P(H_{few}) = P(H_{many}) = 1/2. At time T_{1}, a device will draw a ball at random, and will eject then every second a numbered ball in increasing order, from the ball #1 until the number of the randomly drawn ball. At that very time, the device will stop.

You are informed of all the foregoing, and the device expels then the ball #1 at T_{1}, the ball #2 at T_{2}, the ball #3 at T_{3}, the ball #4 at T_{4}, and the ball #5 at T_{5}. The device then stops. You wish to estimate the number of balls that were contained at T_{0} in the urn. You conclude then to an upward Bayesian shift in favor of the H_{few} hypothesis.

As we can see, such a variation constitutes a mere adaptation of the original two-urn case, with the addition of an incremental mechanism for the expulsion of the balls. The novelty with this variation^{ix} is that the experience has now a temporal feature, because the random selection is made at T_{1} and the randomly drawn ball is finally ejected, for example at T_{5}.

At this stage, it is also worth analyzing the consequences of the two-urn case^{++} for the analysis developed by Eckhardt et al. Indeed, in the *two-urn case*^{++}, the number of each ball ejected from the device is indexed on the range of its expulsion. For example, I draw the ball #60000000000. But I also know that the previous ball was the ball #59999999999 and that the penultimate ball was the ball #59999999998, and so on. However, this does not prevent me from thinking in the same manner as in the original two-urn case and from concluding to a Bayesian shift in favor of the H_{few} hypothesis. In this context, the *two-urn case*^{++} experiment leads to the following consequence: *the fact of being indexed with regard to time does not mean that the number of the ball is not randomly chosen*. This can now be confronted with the main thesis of the incremental objection raised by Eckhardt et al., i.e. that the birth rank of each human being is not randomly chosen, but is rather indexed on the corresponding time position. Sowers especially believes that the cause of DA is that the number corresponding to the birth rank is time-indexed.^{x} But what the *two-urn case*^{++} experiment and the corresponding analogy demonstrates is that our birth rank can be time-indexed and nevertheless be determined randomly in the context of DA. For this reason, the *numbered balls dispenser* model proposed by Eckhardt and Sowers can not be considered as an alternative model to the two-urn case, within the space of solutions for DA.

**III. ****Success of an alternative model grounded on William Eckhardt’s diachronic objection**

William Eckhardt (1993, 1997) also describes another objection to DA, which we shall call, for the sake of the present discussion, the diachronic objection. This latter objection, as we shall see it, is based on an alternative model to the two-urn case, which is different from the one that corresponds to the incremental objection. Eckhardt highlights the fact that *it is impossible to perform* *a random selection, when* *there exists* *many yet unborn individuals within the corresponding reference class*: “How is it possible in the selection of a random rank to give the appropriate weight to unborn members of the population?” (1997, p. 256).

This second objection is potentially stronger than the incremental objection. In order to assess its scope accurately, it is worth translating now this objection in terms of a probabilistic model. It appears that the model associated with Eckhardt’s diachronic objection can be built from the two-urn case’s structure. The corresponding variation, which can be termed the *diachronic two-urn case*, goes as follows:

*The diachronic two-urn case*. An opaque urn in front of you. You know that it contains either 10 or 1000 numbered balls. A fair coin has been tossed at time T_{0}. If the coin fell tails, 10 balls were then placed in the urn, while if the coin fell heads, 10 balls were also placed in the urn at time T_{0}, but 990 supplementary balls will be also added to the urn at time T_{2}, bringing up the total number of balls finally contained in the urn to 1000. The balls are numbered 1, 2, 3, …. You then formulate H_{few} (the urn finally contains only 10 balls) and H_{many} (the urn finally contains1000 balls) hypotheses with the initial probabilities P (H_{few}) = P (H_{many}) = 1 / 2.

Informed of all the above, you randomly draw at time T_{1} a ball from the urn. You get then the ball #5. You wish to estimate the number of balls that ultimately will be contained in the urn at T_{2}. You conclude then that the initial probabilities remain unchanged.

At this stage, it appears that the protocol described above does justice to Eckhardt’s strong idea that it is impossible to perform a random selection where there are many yet unborn members in the reference class. In the *diachronic two-urn case*, the 990 balls, which are possibly (if the coin falls heads) added in T_{2} account for these members not yet born. In such a situation, it would be quite erroneous to conclude to a Bayesian shift in favor of the H_{few} hypothesis. But what can be inferred rationally in such a case is that the prior probabilities remain unchanged.

We can also see that the structure of the protocol of the *diachronic two-urn case* is quite similar to the original two-urn case experiment (which we shall now term, by contrast, the *synchronic two-urn case*). This will allow now for making easy comparisons. So we see that if the coin lands tails: the situation is the same in both experiments, *synchronic* and *diachronic*. However, the situation is different if the coin lands heads: in the synchronic two-urn case, the 990 balls are already present in the urn at T_{0}; on the other hand, in the model of the diachronic two-urn case, 990 extra balls are added to the urn later, namely at T_{2}. As we can see, the diachronic two-urn case based on Eckhardt’s diachronic objection deserves completely to take a place within the space of solutions for DA.

**IV. ****Construction of the preliminary space of solutions**

In light of the foregoing, we are now in a position to appreciate how much the analogy underlying DA is appropriate. It appears indeed that two alternative models to model the analogy with the human situation corresponding to DA are in competition: on the one hand, the *synchronic* two-urn case advocated by the promoters of DA and, on the other hand, the *diachronic* two-urn case, based on Eckhardt’s diachronic objection. It turns out that these two models share a common structure, which allows for making comparisons.^{xi}

At this step, the question that arises is the following: is the human situation corresponding to DA in analogy with (i) the *synchronic* two-urn case, or (ii) the *diachronic* two-urn case? In response, the next question follows: is there an *objective* criterion that allows one to choose, preferentially, between the two competing models? It appears not. Indeed, neither Leslie nor Eckhardt do provide objective reasons for justifying the choice of their favorite model, *and* for rejecting the alternative model. Leslie, first, defends the analogy of the human situation corresponding to DA with the lottery experiment (here, the synchronic two-urn case). At the same time, Leslie acknowledges that DA is considerably weakened if our universe is of an indeterministic nature, i.e. if the total number of people who will ever exist has not yet been settled.^{xii} But it turns out that such indeterministic situation corresponds completely with the *diachronic two-urn case*. For the protocol of this experiment takes into account the fact that the total number of balls which will ultimately be contained in the urn, is not known at the time when the random drawing is performed. We see it finally, Leslie liberally accepts that the analogy with the *synchronic* *two-urn case *may not prevail in certain indeterministic circumstances, where, as we have seen, the *diachronic two-urn case* would apply.

Otherwise, a weakness in the position defended by Eckhardt is that he rejects the analogy with the lottery experiment (in our terminology, the *synchronic two-urn case*) in all cases. But how can we be certain that an analogy with the *synchronic two-urn case* does not prevail, at least for a given situation? It appears here that we lack the evidence allowing us to reject such an hypothesis with absolute certainty.

To sum now. Within the space of solutions for DA resulting from the foregoing, it follows now that two competing models may also be convenient to model the human situation corresponding to DA: Leslie’s synchronic two-urn case or Eckhardt’s diachronic two-urn case. At this stage, however, it appears that no objective criterion allows for preferring one or the other of these two models. In these circumstances, in the lack of objective evidence to make a choice between the two competing models, we are led to apply a principle of indifference, which leads us to retain both models as roughly equiprobable. We attribute then (Figure 1), applying a principle of indifference, a probability P of 1/2 to the analogy with the *synchronic two-urn case* (associated with a terrifying scenario), and an identical probability of 1/2 to the analogy with the *diachronic two-urn case* (associated with a reassuring scenario).

Case | Model | T_{0} | T_{2} | P | Nature of the scenario |

1 | synchronic two-urn case | ○ | 1/2 | terrifying | |

2 | diachronic two-urn case | ○ | ○ | 1/2 | reassuring |

*Figure 1.*

However, it appears that such an approach is of a preliminary nature, for in order to assign a probability to each specific situation inherent in DA, it is necessary to take into account all the elements underlying DA. But it appears that a key element of DA has not yet been taken into account. It is the notoriously awkward *reference class problem*.

## **V. ****The reference class problem**

Let us begin by recalling the *reference class problem*.^{xiii} Basically, it is the problem of the correct definition of “humans”. More accurately, the problem can be stated as follows: how can the reference class be objectively defined in the context of DA? For a more or less extensive or restrictive definition of the reference class can be used. An extensively defined reference class would include, for example, the somewhat exotic varieties corresponding to a future evolution of humankind, with for example an average IQ equal to 200, a double brain or backward causation abilities. On the other hand, a restrictively defined reference class would only include those humans whose characteristics are exactly those of – for example – our subspecies *Homo sapiens sapiens*. Such a definition would exclude the extinct species such as *Homo sapiens neandertalensis*, as well as a possible future subspecies such as *Homo sapiens supersapiens*. To put this in line with our current taxonomy, the reference class can be set at different levels, which correspond to the *Superhomo* super-genus, the *Homo* genus, the *Homo sapiens* species, the *Homo sapiens sapiens* subspecies, etc. At this stage, it appears that we lack an objective criterion allowing to choose the corresponding level non-arbitrarily.

The solution to the reference class problem proposed by Leslie’s, which is exposed in the response made to Eckhardt (1993) and in *The End of the World* (1996), goes as follows: one can choose the reference class more or less as one wishes, i.e. at any level of extension or of restriction. Once this choice has been made, it suffices to adjust accordingly the initial probabilities, and DA works again. The only reservation mentioned by Leslie is that the reference class should not be chosen at an extreme level of extension or restriction.^{xiv} According to Leslie, the fact that every human being can belong to different classes, depending on whether they are restrictively or extensively defined, is not a problem, because the argument works for each of those classes. In this case, says Leslie, a Bayesian shift follows for whatever class reference, chosen at a reasonable level of extension or of restriction. And Leslie illustrates this point of view by an analogy with a multi-color urn, unlike the one-color urn of the original two-urn case experiment. He considers an urn containing balls of different colors, for example red and green. A red ball is drawn at random from the urn. From a restrictive viewpoint, the ball is a red ball and then there is no difference with the two-urn case. But from a more extensive viewpoint, the ball is also a red-or-green ball.^{xv} According to Leslie, although the initial probabilities are different in each case, a Bayesian shift results in both cases.^{xvi} As we can see, the synchronic two-urn case can be easily adapted to restore the essence of Leslie’s multi-color model. It suffices in effect to replace the red balls of the original synchronic two-urn case with red-or-green balls. The resulting two-color model is then in all respects identical to the original synchronic two-urn case experiment, and leads to a Bayesian shift of the same nature.

At this stage, in order to incorporate properly the reference class problem into the space of solutions for DA, we still need to translate the diachronic two-urn case into a two-color variation.

*A. The two-color diachronic two-urn case*

In the one-color original experiment which corresponds to the diachronic two-urn case, the reference class is that of the red balls. It appears here that one can construct a two-color variation, which is best suited for handling the reference class problem, where the relevant class is that of red-or-green balls. The corresponding two-color variation is in all respects identical with the original diachronic two-urn case, the only difference being that the first 10 balls (#1 to #10) are red and the other 990 balls (#11 to #1000) are green. The corresponding variation runs as follows:

*The two-color diachronic two-urn case*. An opaque urn in front of you. You know it contains either 10 or 1000 numbered balls (consisting of 10 red balls and 990 green balls). The red balls are numbered #1, #2, …, #9, #10 and the green ones #11, #12, .., #999, #1000. A fair coin has been tossed at time T_{0}. If the room fell tails, 10 balls were then placed in the urn, while if the coin fell heads, 10 red balls were also placed in the urn at time T_{0}, but 990 green balls will be then added to the urn at time T_{2}, bringing thus the total number of balls in the urn to 1000. You formulate then the hypotheses H_{few} (the urn contains finally only 10 red-or-green balls) and H_{many} (the box finally contains 1000 red-or-green balls) with the prior probabilities P(H_{few}) = P(H_{many}) = 1/2.

After being informed of all the above, you draw at time T_{1} a ball at random from the urn. You get the red ball #5. You proceed to estimate the number of red-or-green balls which will ultimately be contained in the urn at T_{2}. You conclude that the initial probabilities remain unchanged.

As we can see, the structure of this two-color variation is in all respects similar to that of the one-color version of the diachronic two-urn case. In effect, we can considered here the class of red-or-green balls, instead of the original class of red balls. And in this type of situation, it is rational to conclude in the same manner as in the original one-color version of the diachronic two-urn case experiment that the prior probabilities remain unchanged.

*B. Non-exclusivity of the synchronic one-color model and of the diachronic two-color model*

With the help of the machinery at hand to tackle the reference class problem, we are now in a position to complete the construction of the space of solutions for DA, by incorporating the above elements. On a preliminary basis, we have assigned a probability of 1/2 to each of the one-color two-urn case – synchronic and diachronic – models, by associating them respectively with a terrifying and a reassuring scenario. But what is the situation now, with the presence of two-color models, which are better suited for handling the reference class problem?

Before evaluating the impact of the two-color model on the space of solutions for DA, it is worth defining first how to proceed in putting the two-color models and our present human situation into correspondence. For this, it suffices to assimilate the class of red balls to our current *subspecies* *Homo sapiens sapiens* and the class of red-or-green balls to our current *species* *Homo sapiens*. Similarly, we shall assimilate the class of green balls to the subspecies *Homo sapiens supersapiens*, a subspecies more advanced than our own, which is an evolutionary descendant of *Homo sapiens sapiens*. A situation of this type is very common in the evolutionary process that governs living species. Given these elements, we are now in a position to establish the relationship of the probabilistic models with our present situation.

At this stage it is worth pointing out an important property of the two-color diachronic model. It appears indeed that the latter model is susceptible of being combined with a one-color synchronic two-urn case. Suppose, then, that a one-color synchronic two-urn case prevails: 10 balls or 1000 red balls are placed in the urn at time T_{0}. But this does not preclude green balls from being also added in the urn at time T_{2}. It appears thus that the one-color synchronic model and the diachronic two-color model are not exclusive of one another. For in such a situation, a synchronic one-color two-urn case prevails for the restricted class of red balls, whereas a diachronic two-color model applies to the extended class of red-or-green balls. At this step, it appears that we are on a *third route*, of pluralistic essence. For the fact of matching the human situation corresponding to DA with the synchronic or the (exclusively) diachronic model, are well monist attitudes. In contrast, the recognition of the joint role played by both synchronic and diachronic models, is the expression of a pluralistic point of view. In these circumstances, it is necessary to analyze the impact on the space of solutions for DA of this property of *non-exclusivity* which has just been emphasized.

In light of the foregoing, it appears that four types of situations must now be distinguished, within the space of solutions for DA. Indeed, each of the two initial one-color models – synchronic and diachronic – can be associated with a two-color diachronic two-urn case. Let us begin with the case (1) where the *synchronic one-color* model applies. In this case, one should distinguish between two types of situations: either (1a) nothing happens at T_{2} and no green ball is added to the urn at T_{2}, or (1b) 990 green balls are added in the urn at T_{2}. In the first case (1a) where no green ball is added to the urn at T_{2}, we have a rapid disappearance of the class of red balls. Similarly, we have a disappearance of the corresponding class of red-or-green balls, since it identifies itself here with the class of red balls. In such a case, the rapid extinction of *Homo sapiens sapiens* (the red balls) is not followed by the emergence of *Homo sapiens supersapiens* (the green balls). In such a case, we observe the rapid extinction of the sub-species *Homo sapiens sapiens* and the correlative extinction of the species *Homo sapiens* (the red-or-green balls). Such a scenario, admittedly, corresponds to a form of Doomsday that presents a very frightening nature.

Let us consider now the second case (1b), where we are always in the presence of a synchronic one-color model, but where now green balls are also added in the urn at T_{2}. In this case, 990 green balls are added at T_{2} to the red balls originally placed in the urn at T_{0}. We have then a rapid disappearance of the class of red balls, which accompanies, however, the survival of the class of red-or-green balls given the presence of green balls at T_{2}. In this case (1b), one notices that a *synchronic one-color* model is combined with a *diachronic two-color* model. Both models prove to be compatible, and non-exclusive of one another. If we translate this in terms of the third route, one notices that, according to the pluralistic essence of the latter, the synchronic one-color model applies to the class, narrowly defined, of red balls, while a two-color diachronic model also applies to the class, broadly defined, of red-or-green balls. In this case (1b), the rapid extinction of *Homo sapiens sapiens* (the red balls) is followed by the emergence of the most advanced human subspecies *Homo sapiens supersapiens* (the green balls). In such a situation, the restricted class *Homo sapiens sapiens* goes extinct, while the more extended class *Homo sapiens* (red-or-green balls) survives. While the synchronic one-color model applies to the restricted class *Homo sapiens sapiens*, the diachronic two-color model prevails for the wider class *Homo sapiens*. But such an ambivalent feature has the effect of depriving the original argument of the terror which is initially associated with the one-color synchronic model. And finally, this has the effect of rendering DA innocuous, by depriving it of its originally associated terror. At the same time, this leaves room for the argument to apply to a given class reference, but without its frightening and counter-intuitive consequences .

As we can see, in case (1), the corresponding treatment of the reference class problem is different from that advocated by Leslie. For on Leslie’s view, the synchronic model applies irrespective of the chosen reference class. But the present analysis leads to a differential treatment of the reference class problem. In case (1a), the synchronic model prevails and a Bayesian shift applies, as well as in Leslie’s account, both to the class of red balls and to the class of red-or-green balls. But in case (1b), the situation goes differently. Because if a one-color synchronic model applies to the restricted reference class of red balls and leads to a Bayesian shift, it appears that a diachronic two-color model applies to the extended reference class of red-or-green balls, leaving the initial probability unchanged. In case (1b), as we can see, the third route leads to a pluralistic treatment of the reference class problem.

Let us consider now the second hypothesis (2) where the *diachronic one-color* model prevails. In this case, 10 red balls are placed in the urn at T_{0}, and 990 other red balls are added to the urn at T_{2}. Just as before, we are led to distinguish two situations. Either (2a) no green ball is added to the urn at T_{2}, or (2b) 990 green balls are also added to the urn at T_{2}. In the first case (2a), the diachronic one-color model applies. In such a situation (2a), no appearance of a much-evolved human subspecies such as *Homo sapiens supersapiens* occurs. But the scenario in this case is also very reassuring, since our current subspecies *Homo sapiens sapiens* survives. In the second case (2b), where 990 green balls are added to the urn at T_{2}, a diachronic two-color model adds up to the initial diachronic one-color model. In such a case (2b), it follows the emergence of the most advanced subspecies *Homo sapiens supersapiens*. In this case, the scenario is doubly reassuring, since it leads both to the survival of *Homo sapiens sapiens* and of *Homo sapiens supersapiens*. As we can see, in case (2), it is the diachronic model which remains the basic model, leaving the prior probability unchanged.

At this step, we are in a position to complete the construction of the space of solutions for DA. Indeed, a new application of a principle of indifference leads us here to assign a probability of 1/4 to each of the 4 sub-cases: (1a), (1b), (2a), (2b). The latter are represented in the figure below:

Case | T_{0} | T_{2} | P | |

1 | 1a | ○ | 1/4 | |

1b | ○ | ● | 1/4 | |

2 | 2a | ○ | ○ | 1/4 |

2b | ○ | ● ○ | 1/4 |

*Figure 2.*

It suffices now to determine the nature of the scenario that is associated with each of the four sub-cases just described. As has been discussed above, a worrying scenario is associated with hypothesis (1a), while a reassuring scenario is associated with the hypotheses (1b), (2a) and (2b):

Case | T_{0} | T_{2} | P | Nature of the scenario | P | |

1 | 1a | ○ | 1/4 | terrifying | 1/4 | |

1b | ○ | ● | 1/4 | reassuring | ||

2 | 2a | ○ | ○ | 1/4 | reassuring | 3/4 |

2b | ○ | ● ○ | 1/4 | reassuring |

*Figure 3.*

We see it finally, the foregoing considerations lead to a novel formulation of DA. For it follows from the foregoing that the original scope of DA should be reduced, in two different directions. It should be acknowledged, first, that either the *one-color synchronic model* or the *diachronic one-color model* applies to our current subspecies *Homo sapiens sapiens*. A principle of indifference leads us then to assign a probability of 1/2 to each of these two hypotheses. The result is a weakening of DA, as the Bayesian shift associated with a terrifying assumption no longer concerns but one scenario of the two possible scenarios. A second weakening of DA results from the pluralist treatment of the reference class problem. For in the case where the *one-color synchronic* model (1) applies to our subspecies *Homo sapiens sapiens*, two different situations must be distinguished. Only one of them, (1a) leads to the extinction of both *Homo sapiens sapiens* and *Homo sapiens* and corresponds thus to a frightening Doomsday. In contrast, the other situation (1b) leads to the demise of *Homo sapiens sapiens*, but to the correlative survival of the most advanced human subspecies *Homo sapiens supersapiens*, and constitutes then a quite reassuring scenario. At this stage, a second application of the principle of indifference leads us to assign a probability of 1/2 to each of these two sub-cases (see Figure 3). In total, a frightening scenario is henceforth associated with a probability of no more than 1/4, while a reassuring scenario is associated with a probability of 3/4.

As we can see, given these two sidesteps, a new formulation of DA ensues, which could prove to be more plausible than the original one. Indeed, the present formulation of DA can now be reconciled with our pretheoretical intuition. For the fact of taking into account DA now gives a probability of 3/4 for all reassuring scenarios and a probability of no more than 1/4 for a scenario associated with a frightening Doomsday. Of course, we have not completely eliminated the risk of a frightening Doomsday. And we must, at this stage, accept a certain risk, the scope of which appears however limited. But most importantly, it is no longer necessary now to give up our pretheoretical intuitions.

Finally, the preceding highlights a key facet of DA. For in a narrow sense, it is an argument related to the destiny of humankind. And in a broader sense (the one we have been concerned with so far) it emphasizes the difficulty of applying probabilistic models to everyday situations,^{xvii} a difficulty which is often largely underestimated. This opens the path to a wide field which presents a real practical interest, consisting of a taxonomy of probabilistic models, the philosophical importance of which would have remained hidden without the strong and courageous defense of the Doomsday argument made by John Leslie.^{xviii}

**References**

Bostrom, Nick. 1997. “Investigations into the Doomsday argument.”* preprint*.

———. 2002. *Anthropic Bias: Observation Selection Effects in Science and Philosophy *New York: Routledge.

Chambers, Timothy. 2001. “Do Doomsday’s Proponents Think We Were Born Yesterday?” *Philosophy* 76: 443-450.

Delahaye, Jean-Paul. 1996. “Recherche de modèles pour l’argument de l’apocalypse de Carter-Leslie.” *manuscrit.*

Eckhardt, William. 1993. “Probability Theory and the Doomsday Argument.” *Mind* 102: 483-488.

———. 1997. “A Shooting-Room view of Doomsday.” *Journal of Philosophy* 94: 244-259.

Franceschi, Paul. 1998. “Une solution pour l’argument de l’Apocalypse.” *Canadian Journal of Philosophy* 28: 227-246.

———. 1999. “Comment l’urne de Carter et Leslie se déverse dans celle de Hempel.” *Canadian Journal of Philosophy* 29: 139-156, English translation under the title “The Doomsday Argument and Hempel’s Problem” .

———. 2002. “Une application des n-univers à l’argument de l’Apocalypse et au paradoxe de Goodman.” Corté: University of Corsica, doctoral dissertation.

Hájek, Alan. 2002. “Interpretations of Probability.” *The Stanford Encyclopedia of Philosophy*, E. N. Zalta (ed.), http://plato.stanford.edu/archives/win2002/entries/probability-interpret.

Korb, Kevin. & Oliver, Jonathan. 1998. “A Refutation of the Doomsday Argument.” *Mind* 107: 403-410.

Leslie, John. 1993. “Doom and Probabilities.” *Mind* 102: 489-491.

———. 1996. *The End of the World: the science and ethics of human extinction* London: Routledge.

Sober, Eliott. 2003.**“**An Empirical Critique of Two Versions of the Doomsday Argument – Gott’s Line and Leslie’s Wedge.” *Synthese* 135-3: 415-430.

Sowers, George. 2002. “The Demise of the Doomsday Argument.” *Mind* 111: 37-45.

i The present analysis of DA is an extension of Franceschi (2002).

ii Cf. Korb & Oliver (1998).

iii The original description by Bostrom of the two-urn case refers to two separate urns. For the sake of simplicity, we shall refer here equivalently to one single urn (which contains either 10 or 1000 balls).

iv More accurately, Leslie considers an analogy with a *lottery experiment*.

v Cf (2003: 9): “But who or what has the propensity to randomly assign me a temporal location in the duration of the human race? There is no such mechanism.” But Sober is mainly concerned with providing evidence with regard to the assumptions used in the original version of DA and with broadening the scope of the argument by determining the conditions of its application to real-life situations.

vi Cf. (1997: 251).

vii Cf. (2002: 39).

viii I borrow this terminology from Chambers (2001).

ix Other variations of the *two-urn case*^{++} can even be envisaged. In particular, variations of this experiment where the random process is performed diachronically and not synchronically (i.e. at time T_{0}) can even be imagined.

x Cf. Sowers (2002: 40).

xi Both synchronic and diachronic two-urn case experiments can give rise to an incremental variation. The incremental variant of the (synchronic) two-urn case has been mentioned earlier: it consists of the *two-urn case*^{++}. It is also possible to build a similar incremental variation of the diachronic two-urn case, where the ejection of the balls is made at regular time intervals. At this stage it appears that both models can give rise to such incremental variations. Thus, the fact of considering incremental variations of the two competing models – the *synchronic* *two-urn case*^{++} and the *diachronic* *two-urn case*^{++} – does not provide any novel elements with regard to the two original experiments. Similarly, we might consider some variations where the random sampling is done not at T_{0}, but gradually, or some variants where a quantum coin is used, and so on. But in any case, such variations are susceptible to be adapted to each of the two models.

xii Leslie (1993: 490) evokes thus: “(…) the potentially much stronger objection that the number of names in the doomsday argument’s imaginary urn, the number of all humans who will ever have lived, has not yet been firmly settled because the world is indeterministic”.

xiii The *reference class problem* in probability theory is notably mentioned in Hájek (2002: s. 3.3). For a treatment of the reference class problem in the context of DA, see Eckhardt (1993, 1997), Bostrom (1997, 2002: ch. 4 pp. 69-72 & ch. 5), Franceschi (1998, 1999). The point emphasized in Franceschi (1999) can be construed as a treatment of the reference class problem within confirmation theory.

xiv Cf. 1996: 260-261.

xv Cf. Leslie (1996: 259).

xvi Cf. Leslie (1996: 258-259): “The thing to note is that the red ball can be treated either just as a red ball or else as a red-or-green ball. Bayes’s Rule applies in both cases. […] All this evidently continues to apply to when being-red-or-green is replaced by being-red-or-pink, or being-red-or-reddish”.

xvii This important aspect of the argument is also underlined in Delahaye (1996). It is also the main theme of Sober (2003).

xviii I thank Nick Bostrom for useful discussion on the reference class problem, and Daniel Andler, Jean-Paul Delahaye, John Leslie, Claude Panaccio, Elliott Sober, and an anonymous referee for the *Journal of Philosophical Research*, for helpful comments on earlier drafts.

# A Solution to Goodman’s Paradox

*English Posprint (with additional illustrations) of a paper published in French in Dialogue Vol. 40, Winter 2001, pp. 99-123 under the title “Une Solution pour le Paradoxe de Goodman”.*

In the classical version of Goodman’s paradox, the universe where the problem takes place is ambiguous. The conditions of induction being accurately described, I define then a framework of *n*-universes, allowing the distinction, among the criteria of a given *n*-universe, between constants and variables. Within this framework, I distinguish between two versions of the problem, respectively taking place: (i) in an *n*-universe the variables of which are colour and time; (ii) in an *n*-universe the variables of which are colour, time and space. Finally, I show that each of these versions admits a specific resolution.

This paper is cited in:

- Alasdair Richmond, The Doomsday Argument, Philosophical Books Vol. 47 No. 2 April 2006, pp. 129–142

**A Solution to Goodman’s Paradox**

**Paul FRANCESCHI**

a paper originally published in *Dialogue*, winter 2001, vol. 40, pp. 99-123

*ABSTRACT: In the classical version of Goodman’s paradox, the universe where the problem takes place is ambiguous. The conditions of induction being accurately described, I define then a framework of *n*-universes, allowing the distinction, among the criteria of a given *n*-universe, between constants and variables. Within this framework, I distinguish between two versions of the problem, respectively taking place: (i) in an *n*-universe the variables of which are colour and time; (ii) in an *n*-universe the variables of which are colour, time and space. Finally, I show that each of these versions admits a specific resolution.*

**1. The problem**

Goodman’s Paradox (thereafter GP) has been described by Nelson Goodman (1946).^{i} Goodman exposes his paradox as follows.^{ii} Consider an urn containing 100 balls. A ball is drawn each day from the urn, during 99 days, until today. At each time, the ball extracted from the urn is red. Intuitively, one expects that the 100th ball drawn from the urn will also be red. This prediction is based on the generalisation according to which all the balls in the urn are red. However, if one considers the property S “drawn before today and red or drawn after today and non-red”, one notes that this property is also satisfied by the 99 instances already observed. But the prediction which now ensue, based on the generalisation according to which all the balls are S, is that the 100th ball will be non-red. And this contradicts the preceding conclusion, which however conforms with our intuition.^{iii}

Goodman expresses GP with the help of an enumerative induction. And one can model GP in terms of the *straight rule* (SR). If one takes (D) for the definition of the “red” predicate, (I) for the enumeration of the instances, (H) for the ensuing generalisation, and (P) for the corresponding prediction, one has then:

(D) R = red

(I) Rb_{1}·Rb_{2}·Rb_{3}·…·Rb_{99}

(H) Rb_{1}·Rb_{2}·Rb_{3}·…·Rb_{99}·Rb_{100}

∴ (P) Rb_{100}

And also, with the predicate S:

(D*) S = red and drawn before T or non-red and drawn after T

(I*) Sb_{1}·Sb_{2}·Sb_{3}·…·Sb_{99}

(H*) Sb_{1}·Sb_{2}·Sb_{3}·…·Sb_{99}·Sb_{100} that is equivalent to:

(H’*) Rb_{1}·Rb_{2}·Rb_{3}·…·Rb_{99}·~Rb_{100}

∴ (P*) Sb_{100} i. e. finally:

∴ (P’*) ~Rb_{100}

The paradox resides here in the fact that the two generalisations (H) and (H*) lead respectively to the predictions (P) and (P’*), which are contradictory. Intuitively, the application of SR to (H*) appears erroneous. Goodman also gives in *Fact, Fiction and Forecast*^{iv} a slightly different version of the paradox, applied in this case to emeralds.^{v} This form is very well known and based on the predicate “grue” = green and observed before T or non-green and observed after T.

The predicate S used in Goodman (1946) presents with “grue”, a common structure. P and Q being two predicates, this structure corresponds to the following definition: (P and Q) or (~P and ~Q). In what follows, one will designate by *grue* a predicate having this particular structure, without distinguishing whether the specific form used is that of Goodman (1946) or (1954).

**2. The unification/differentiation duality**

The instances are in front of me. Must I describe them by stressing their differences? Or must I describe them by emphasising their common properties? I can proceed either way. To stress the differences between the instances, is to operate by *differentiation*. Conversely, to highlight their common properties, is to proceed by *unification*. Let us consider in turn each of these two modes of proceeding.

Consider the 100 balls composing the urn of Goodman (1946). Consider first the case where my intention is to stress the differences between the instances. There, an option is to apprehend the particular and single moment, where each of them is extracted from the urn. The considered predicates are then: *red and drawn on day 1*, *red and drawn on day 2*, …, *red and drawn on day 99*. There are thus 99 different predicates. But this prohibits applying SR, which requires one single predicate. Thus, what is to distinguish according to the moment when each ball is drawn? It is to stress an essential difference between each ball, based on the criterion of time. Each ball thus is individualised, and many different predicates are resulting from this: drawn at T_{1}, drawn at T_{2}, …, drawn at T_{99}. This indeed prevents then any inductive move by application of SR. In effect, one does not have then a common property to allow induction and to apply SR. Here, the cause of the problem lies in the fact of having carried out an *extreme differentiation*.

Alternatively, I can also proceed by *differentiation* by operating an extremely precise^{vi} measurement of the wavelength of the light defining the colour of each ball. I will then obtain a unique measure of the wavelength for each ball of the urn. Thus, I have 100 balls in front of me, and I know with precision the wavelength of the light of 99 of them. The balls respectively have a wavelength of 722,3551 nm, 722,3643 nm, 722,3342 nm, 722,3781 nm, etc. I have consequently 99 distinct predicates P3551, P3643, P3342, P3781, etc. But I have no possibility then to apply SR, which requires one single predicate. Here also, the common properties are missing to allow to implement the inductive process. In the same way as previously, it proves here that I have carried out an *extreme differentiation*.

What does it occur now if I proceed exclusively by *unification*? Let us consider the predicate R corresponding to “red or non-red”. One draws 99 red balls before time T. They are all R. One predicts then that the 100th ball will be R after T, i.e. red or non-red. But this form of induction does not bring any information here. The resulting conclusion is empty of information. One will call *empty induction* this type of situation. In this case, one observes that the process of unification of the instances by the colour was carried out in a radical way, by annihilating in this respect, any step of differentiation. The cause of the problem lies thus in the implementation of a process of *extreme unification*.

If one considers now the viewpoint of *colour*, it appears that each case previously considered requires a different taxonomy of colours. Thus, it is made use successively:

– of our usual taxonomy of colours based on 9 predicates: purple, indigo, blue, green, yellow, orange, red, white, black

– of a taxonomy based on a comparison of the wavelengths of the colours with the set of the real numbers (*real taxonomy*)

– of a taxonomy based on a single predicate (*single taxon taxonomy*): red or non-red

But it proves that each of these three cases can be replaced in a more general perspective. Indeed, multiple taxonomies of colours are susceptible to be used. And those can be ordered from the coarser (single taxon taxonomy) to the finest (real taxonomy), from the most unified to the most differentiated. We have in particular the following hierarchy of taxonomies:

– TAX_{1} = {red or non-red} (single taxon taxonomy)

– TAX_{2} = {red, non-red} (binary taxonomy)

– …

– TAX_{9} = {purple, indigo, blue, green, yellow, orange, red, white, black} (taxonomy based on the spectral colours, plus white and black)

– …

– TAX_{16777216} = {(0, 0, 0), …, (255, 255, 255)} (taxonomy used in computer science and distinguishing 256 shades of red/green/blue)

– …

– TAX_{R} = {370, …, 750} (real taxonomy based on the wavelength of the light)

Within this hierarchy, it appears that the use of extreme taxonomies such as the one based on a single taxon, or the real taxonomy, leads to specific problems (respectively *extreme unification* and *extreme differentiation*). Thus, the problems mentioned above during the application of an inductive reasoning based on SR occur when the choice in the unification/differentiation duality is carried out too radically. Such problems relate to induction in general. This invites to think that one must rather reason as follows: I should privilege neither *unification*, nor *differentiation*. A predicate such as “red”, associated with our usual taxonomy of colours (TAX_{9})^{vii}, corresponds precisely to such a criterion. It corresponds to a balanced choice in the unification/differentiation duality. This makes it possible to avoid the preceding problems. This does not prevent however the emergence of new problems, since one tries to implement an inductive reasoning, in certain situations. And one of these problems is naturally GP.

Thus, it appears that the stake of the choice in the duality unification/differentiation is essential from the viewpoint of induction, because according to whether I choose one way or the other, I will be able or not to use SR and produce valid inductive inferences. Confronted with several instances, one can implement either a process of differentiation, or a process of unification. But the choice that is made largely conditions the later success of the inductive reasoning carried out on those grounds. I must describe both common properties and differences. From there, a valid inductive reasoning can take place. But at this point, it appears that the role of the unification/differentiation duality proves to be crucial for induction. More precisely, it appears at this stage that a correct choice in the unification/differentiation duality constitutes one of the conditions of induction.

**3. Several problems concerning induction**

The problems which have been just mentioned constitute the illustration of several difficulties inherent to the implementation of the inductive process. However, unlike GP, these problems do not generate a genuine contradiction. From this point of view, they distinguish from GP. Consider now the following situation. I have drawn 99 balls respectively at times T_{1}, T_{2}, …, T_{99}. The 100th ball will be drawn at T_{100}. One observes that the 99 drawn balls are red. They are thus at the same time red and drawn before T_{100}. Let R be the predicate “red” and T the predicate “drawn before T_{100}“. One has then:

(I) RTb_{1}, RTb_{2}, …, RTb_{99}

(H) RTb_{1}, RTb_{2}, …, RTb_{99}, RTb_{100}

∴ (P) RTb_{100}

By direct application of SR, the following prediction ensue: “the 100th ball is red and drawn before T_{100}“. But this is in contradiction with the data of the experiment in virtue of which the 100th ball is drawn in T_{100}. There too, the inductive reasoning is based on a formalisation which is that of SR. And just as for GP, SR leads here to a contradiction. Call 2 this problem, where two predicates are used.

It appears that one can easily build a form of 2 based on one single predicate. A way of doing that is to consider the unique predicate S defined as “red and drawn before T_{100}” in replacement of the predicates R and T used previously. The same contradiction then ensues.

Moreover, it appears that one can highlight another version (1) of this problem comprising only one predicate, without using the “red” property which appears useless here. Let indeed T be the predicate drawn before T_{100}. One has then:

(I) Tb_{1}, Tb_{2}, …, Tb_{99}

(H) Tb_{1}, Tb_{2}, …, Tb_{99}, Tb_{100}

∴ (P) Tb_{100}

Here also, the conclusion according to which the 100th ball is drawn before T_{100} contradicts the data of the experiment according to which the 100th ball is drawn at T_{100}. And one has then a contradictory effect, analogous to that of GP, without the structure of “grue” being implemented. Taking into account the fact that only the criterion of time is used to build this problem, it will be denoted in what follows by 1-time.

It appears here that the problems such as 1-time and 2 lead just as GP to a contradiction. Such is not the case for the other problems related to induction previously mentioned^{viii}, which involve either the impossibility of carrying out induction, or a conclusion empty of information. However, it proves that the contradiction encountered in 1-time is not of the same nature as that observed in GP. Indeed in GP, one has a contradiction between the two concurrent predictions (P) and (P*). On the other hand, in 1-time, the contradiction emerges between on the one hand the conditions of the experiment (T 100) and on the other hand the prediction resulting from generalisation (T < 100).

Anyway, the problems which have been just encountered suggest that the SR formalism does not capture the whole of our intuitions related to induction. Hence, it is worth attempting to define accurately the conditions of induction, and adapting consequently the relevant formalism. However, before carrying out such an analysis, it is necessary to specify in more detail the various elements of the context of GP.

**4. The universe of reference**

Let us consider the law (L1) according to which “diamond scratches the other solids”. A priori, (L1) strikes us as an undeniable truth. Nevertheless, it proves that at a temperature higher than 3550°C, diamond melts. Therefore in last analysis, the law (L1) is satisfied at a normal temperature and in any case, when the temperature is lower than 3550°C. But such a law does not apply beyond 3550°C. This illustrates how the statement of the conditions under which the law (L1) is verified is important, in particular with regard to the conditions of temperature. Thus, when one states (L1), it proves necessary to specify the conditions of temperature in which (L1) finds to apply. This is tantamount to describing the type of universe in which the law is satisfied.

Let also (P1) be the following proposition: “the volume of the visible universe is higher than 1000 times that of the solar system”. Such a proposition strikes us as obvious. But there too, it appears that (P1) is satisfied at modern time, but that it proves to be false at the first moments of the universe. Indeed, when the age of our universe was 10^{-6} second after the big-bang, its volume was approximately equal to that of our solar system. Here also, it thus appears necessary to specify, at the same time as the proposition (P1) the conditions of the universe in which it applies. A nonambiguous formulation of (P1) thus comprises a more restrictive temporal clause, such as: “at our time, the volume of the visible universe is higher than 1000 times that of the solar system”. Thus, generally, one can think that when a generalisation is stated, it is necessary to specify the conditions of the universe in which this generalisation applies. The precise description of the *universe of reference* is fundamental, because according to the conditions of the universe in which one places oneself, the stated law can appear true or false.

One observes in our universe the presence of both constants and variables. There are thus constants, which constitute the fundamental constants of the universe: the speed of light: *c* = 2,998 x10^{8} m/s; Planck’s constant: *h* = 6,626 x 10^{-34} J.s; the electron charge; *e* = 1,602 x 10^{-19} C; etc. There are on the other hand variables. Among those, one can mention in particular: temperature, pressure, altitude, localisation, time, presence of a laser radiation, presence of atoms of titanium, etc.

One often tends, when a generalisation is stated, not to take into account the constants and the variables which are those of our universe envisaged in its totality. Such is the case for example when one considers the situation of our universe on 1 January 2000, at 0h. One places then oneself explicitly in what constitutes a section, a slice of our universe. In effect, time is not regarded then a variable, but well as a constant. Consider also the following: “the dinosaurs had hot blood”^{ix}. Here, one places oneself explicitly in a sub-universe of our where the parameters of time and space have a restricted scope. The temporal variable is reduced to the particular time of the Earth history which knew the appearance of the dinosaurs: the Triassic and the Cretaceous. And similarly, the space parameter is limited to our planet: Earth. Identically, the conditions of temperature are changing within our universe, according to whether one is located at one site or another of it: at the terrestrial equator, the surface of Pluto, the heart of Alpha Centauri, etc. But if one is interested exclusively in the balloon being used for the experimentation within the laboratory of physics, where the temperature is maintained invariably at 12°C, one can then regard valuably the temperature as a constant. For when such generalisations are expressed, one places oneself not in our universe under consideration in his totality, but only in what veritably constitutes a specific part, a restriction of it. One can then assimilate the universe of reference in which one places oneself as a sub-universe of our. It is thus frequent to express generalisations which are only worth for the present time, or for our usual terrestrial conditions. Explicitly or not, the statement of a law comprises a universe of reference. But in the majority of the cases, the variables and the constants of the considered sub-universe are distinct from those allowing to describe our universe in its totality. For the conditions are extremely varied within our universe: the conditions are very different according to whether one places oneself at the 1st second after the big-bang, on Earth at the Precambrian epoch, in our planet in year 2000, inside the particle accelerator of the CERN, in the heart of our Sun, near a white dwarf, or well inside a black hole, etc.

One can also think that it is interesting to be able to model universes the constants of which are different from the fundamental constants of our universe. One can thus wish to study for example a universe where the mass of the electron is equal to 9,325 x10^{-31} kg, or well a universe where the electron charge is equal to 1,598 x 10^{-19} C. And in fact, the toy-universes, which take into account fundamental constants different from those of our familiar universe, are studied by the astrophysicists.

Lastly, when one describes the conditions of a thought experiment, one places oneself, explicitly or not, under the conditions which are related to those of a sub-universe. When one considers for example 100 balls extracted from an urn during 100 consecutive days, one places then oneself in a restriction of our universe where the temporal variable is limited to one period of 100 days and where the spatial location is extremely reduced, corresponding for example to a volume approximately equal to 5 dm^{3}. On the other hand, the number of titanium or zirconium atoms possibly present in the urn, the possible existence of a laser radiation, the presence or the absence of a sound source of 10 db, etc. can be omitted and ignored. In this context, it is not necessary to take into account the existence of such variables. In this situation, it is enough to mention the variables and the constants *actually* used in the thought experiment. For one can think indeed that the number of variables in our universe is so large that it is impossible to enumerate them all. And consequently, it does not appear possible to characterise our universe in function of all its variables, because one can not provide an infinite enumeration of it. It appears sufficient to describe the considered sub-universe, by mentioning only the constants and the variables which play an effective role in the experiment. Thus, in such situations, one will describe the considered sub-universe by mentioning only the effective criteria necessary to the description of the experiment.

What precedes encourages to think that generally, in order to model the context in which the problems such as GP take place, it is convenient to describe a given universe in terms of variables and constants. This leads thus to define a *n*-universe (*n* 0) as a universe the criteria of which comprise *m* constants, and *n* variables, where the *m* constants and *n* variables constitute the *criteria* of the given universe. Within this particular framework, one defines a *temporal 1-universe* (Ω^{1}T) as a universe comprising only one criterion-variable: time. In the same way, one defines a *coloured 1-universe* (Ω^{1}C) as a universe comprising only one criterion-variable: colour. One will define also a *coloured and temporal 2-universe* (Ω^{2}CT) as a universe comprising two criterion-variables: time and colour. Etc. In the same way, a universe where all the objects are red, but are characterised by a different localisation will be modelled by a *localised 1-universe* (Ω^{1}L) a criterion-constant (red) of which is colour.

It should be noted incidentally that the *n*-universe framework makes it possible in particular to model several interesting situations. Thus, a temporal universe can be regarded as a *n*-universe one of the variables of which is a temporal criterion. Moreover, a universe where one single moment T_{0} is considered, deprived of the phenomenon of succession of time, can be regarded as a *n*-universe where time does not constitute one of the variables, but where there is a constant-time. In the same way, an *atemporal* universe corresponds to a *n*-universe no variable of which corresponds to a temporal criterion, and where there is not any time-constant.

In the context which has been just defined, what is it now to be *red*? Here, being “red” corresponds to two different types of situations, according to the type of *n*-universe in which one places oneself. It can be on the one hand a *n*-universe one of the constants of which is colour. In this type of universe, the colour of the objects is not susceptible to change, and all the objects are there invariably red.

The fact of being “red” can correspond, on the second hand, to a *n*-universe one of the criterion-variables of which is constituted by colour. There, an object can be red or non-red. Consider the case of a Ω^{1}C. In such a universe, an object is red or non-red *in the absolute*. No change of colour is possible there, because no other criterion-variable exists, of which can depend such a variation. And in a Ω^{2}CT, being red is being red at time T. Within such a universe, being red is being red *relatively* to time T. Similarly, in a coloured, temporal and localised 3-universe (Ω^{3}CTL), being red is being red at time T and at place L. Etc. In some such universe, being red is being red relatively to other criterion-variables. And the same applies to the *n*-universes which model a universe such as our own.

At this step arises the problem of the *status of the instances* of an object of a given type. What is it thus to be an instance, within this framework? This problem has its importance, because the original versions of GP are based on instances of balls (1946) and emeralds (1954). If one takes into account the case of Goodman (1946), the considered instances are 100 different balls. However, if one considers a unique ball, drawn at times T_{1}, T_{2}, …, T_{100}, one notices that the problem inherent to GP is always present. It suffices indeed to consider a ball whose colour is susceptible to change during the course of time. One has drawn 99 times the ball at times T_{1}, T_{2}, …, T_{99}, and one has noted each time that the ball was red. This leads to the prediction that the ball will be red at T_{100}. However, this last prediction proves to be contradictory with an alternative prediction based on the same observations, and the projection of the predicate S “red and drawn before T_{100} or non-red and drawn at T_{100}“^{x}.

The present framework must be capable of handling the diversity of these situations. Can one thus speak of an instantiated and temporal 1-universe, or well of an instantiated and coloured 1-universe? Here, one must observe that the fact of being instantiated, for a given universe, corresponds to an additional criterion-variable. For, on the contrary, what makes it possible to distinguish between the instances? If no criterion distinguishes them, it is thus only one and the same thing. And if they are distinct, it is thus that a criterion makes it possible to differentiate them. Thus, an instantiated and temporal 1-universe is in fact a 2-universe, whose 2nd criterion, which makes it possible to distinguish the instances between them, is in fact not mentioned nor explicited. By making explicit this second criterion-variable, it is thus clear that one is placed in a 2-universe. In the same way, an instantiated and coloured 1-universe is actually a 2-universe one of the criteria of which is colour and the second criterion exists but is not specified.

Another aspect which deserves mention here, is the question of the reduction of a given *n*-universe to another. Is it not possible indeed, to logically reduce a *n*-universe to a different system of criteria? Consider for example a Ω^{3}CTL. In order to characterise the corresponding universe, one has 3 criterion-variables: colour, time and localisation. It appears that one can reduce this 3-universe to a 2-universe. That can be carried out by reducing two of the criteria of the 3-universe to one single criterion. In particular, one will reduce both criteria of colour and time to a single criterion of tcolour* (shmolor^{xi}). And one will only preserve two taxa of tcolour*: G and ~G. Consider then a criterion of color comprising two taxa (red, non-red) and a criterion of time comprising two taxa (before T, after T). If one associates the taxa of colour and time, one obtains four new predicates: red before T, red after T, non-red before T, non-red after T, which one will denote respectively by RT, R~T, ~RT and ~R~T. Several of these predicates are compatible (RT and R~T, RT and ~R~T, ~RT and R~T, ~RT and ~R~T) whereas others are incompatible (RT and ~RT, R~T and ~R~T). At this stage, one has several manners (16)^{xii} of grouping the compatible predicates, making it possible to obtain two new predicates G and ~G of tcolour*:

0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | |

RT R~T | X | X | X | X | X | X | X | X | ||||||||

RT ~R~T | X | X | X | X | X | X | X | X | ||||||||

~RT R~T | X | X | X | X | X | X | X | X | ||||||||

~RT ~R~T | X | X | X | X | X | X | X | X |

In each of these cases, it results indeed a new single criterion of tcolour* (Z), which substitutes itself to the two preceding criteria of colour and time. One will denote by Z_{i} (0 *i* 15) the taxa of tcolour* thus obtained. If it is clear that Z_{15} leads to the empty induction, it should be observed that several cases corresponding to the situation where the instances are RT lead to the problem inherent to GP. One will note thus that Z_{2}, i.e. *grue*_{2} (by assimilating the Z_{i} to *grue*_{i} and the Z_{15-i} to *bleen*_{i}) is based on the definition: *grue*_{2} = red before T *and* non-red after T. It appears here as a conjunctive interpretation of the definition of “grue”. In the same way, *grue*_{7} corresponds to a definition of “grue” based on an *exclusive disjunction*. Lastly, *grue*_{12} is based on the traditional definition: *grue*_{12} = red before T or non-red after T, where the disjunction is to be interpreted as an *inclusive disjunction*.

Similarly, it also proves that a Ω^{2}CT can be reduced to a tcoloured* 1-universe (Ω^{1}Z). And more generally, a *n*-universe is thus *reducible* to an (*n*-1)-universe (for *n* > 1). Thus, if one considers a given universe, several characterisations in terms of *n*-universe can valuably be used. One can in particular apprehend a same universe like a Ω^{3}CTL, or like a Ω^{2}ZL. In the same way, one can represent a Ω^{2}CT like a Ω^{1}Z. At this stage, none of these views appears fundamentally better than the other. But each of these two characterisations constitute alternative ways to describe a same reality. This shows finally that a *n*-universe constitutes in fact an abstract characterisation of a real or an imaginary universe. A *n*-universe constitutes thus a *system of criteria*, comprising constants and variables. And in order to characterise a same real or imaginary given universe, one can resort valuably to several *n*-universes. Each of them appears finally as a different characterisation of the given universe, simply based on a different set of primitives.

**5. Conditions of induction**

The fact that the SR formalism involves the GP effect suggests that the intuition which governs our concept of induction is not entirely captured by SR. It is thus allowed to think that if the formal approach is necessary and useful to be used as support to induction, it does not constitute however a sufficient step. For it appears also essential to capture the intuition which governs our inductive reasoning. Therefore it proves necessary to supplement the formal approach of induction by a semantic approach. Goodman himself provides us with a definition of induction^{xiii}. He defines induction as the projection of characteristics of the past through the future, or more generally, as the projection of characteristics corresponding to a given aspect of an object through another aspect. This last definition corresponds to our intuition of induction. One can think however that it is necessary to supplement it by taking into account the preceding observations^{xiv} concerning the differentiation/unification duality. In that sense, it has been pointed out that induction consists of an inference from instances presenting both common properties and differences. Let the instances-source (instances-S) be the instances to which relate (I) or (I*) and the instance-destination (instance-D) that which is the subject of (P) or (P*). The common properties relate to the instances-S and the differentiated properties are established between the instances-S and the instance-D. The following definition ensues: induction consists precisely in the fact that the instance-D^{xv} also presents the property that is common to the instances-S, whereas one does vary the criterion (criteria) on which the differences between the instances-S and the instance-D is (are) based. The inductive reasoning is thus based on the constant nature of a property, whereas such other property is variable.

From this definition of induction arise straightforwardly several conditions of induction. I shall examine them in turn. The first two conditions are thus the following ones:

(C1) the instances-S must present some common properties

(C2) the instances-S and the instance-D must present some distinctive properties

This has for consequence that one cannot apply induction in two particular circumstances: firstly (i) when the instances do not present any common property. One will call such a situation a *total differentiation* of the instances. The problems corresponding to this particular circumstance have been mentioned above^{xvi}. And secondly (ii) when the instances do not present any distinctive property. One will call such a situation *total unification*. The problems encountered in this type of situation have also been mentioned previously^{xvii}.

It should also be noted that it is not here a question of intrinsic properties of the instances, but rather of the analysis which is carried out by the one who is on the point of reasoning by induction.

Taking into account the definition of induction which has been given, a third condition can be thus stated:

(C3) a criterion-variable is necessary for the common properties of the instances-S and another criterion-variable for the distinctive properties

This refers to the structure of the considered universe of reference. Consequently, two criterion-variables are at least necessary, in the structure of the corresponding universe of reference. One will call that the *minimal**condition* of induction. Hence, a 2-universe is at least necessary in order that the conditions of induction can be satisfied. Thus, a ^{2}CT will be appropriate. In the same way, a temporal and localised 2-universe (^{2}TL) will also satisfy the conditions which have been just defined, etc^{xviii}.

It should be noted that another way of stating this condition is as follows: the criterion-variable for the common properties and the criterion-variable for the differentiated properties must be distinct. One should not have confusion between the two. One can call that the *condition of separation* of the common properties and the distinctive properties. Such a principle appears as a consequence of the minimal condition for induction: one must have two criteria to perform induction, and these criteria must be different. If one chooses a same criterion for the common properties and the differentiated properties, one is brought back in fact to one single criterion and the context of a 1-universe, itself insufficient to perform induction.

Lastly, a fourth condition of induction results from the preceding definition:

(C4) one must project the common properties of the instances-S (and not the distinctive properties)

The conditions of induction which have been just stated make it possible from now on to handle the problems involved in the use of SR mentioned above^{xix}. It follows indeed that the following projections^{xx} are correct: C°T in a Ω^{2}CT, C°L in a Ω^{2}CL, Z°L in a Ω^{2}ZL, etc. Conversely, the following projections are incorrect: T°T in a Ω^{1}T, Z°Z in a Ω^{1}Z. In particular, one will note here that the projection T°T in the Ω^{1}T is that of 1-time. 1-time takes indeed place in a Ω^{1}T, whereas induction requires at the same time common properties and distinctive properties. Thus, a 2-universe is at least necessary. Usually, the criterion of time is used for differentiation. But here, it is used for unification (“drawn before T”). That can be done, but provided that one uses a distinct criterion for the differentiated properties. However, whereas common properties results here from that, the differentiated properties are missing. It thus misses a second criterion – corresponding to the differentiated properties – in the considered universe, to perform induction validly. Thus 1-time finds its origin in a violation of the *minimal condition* of induction. One can formulate this solution equivalently, with regard to the *condition of separation*. In effect, in 1-time, a same temporal criterion (drawn before T/drawn after T) is used for the common properties and the differentiated properties, whereas two distinct criteria are necessary. It can be thus analysed as a manifest violation of the condition of separation.

Lastly, the conditions of induction defined above lead to adapt the formalism used to describe GP. It proves indeed necessary to distinguish between the common and the distinctive property(ies). One will thus use the following formalism in replacement of the one used above:

(I) RT_{1}·RT_{2}·RT_{3}·…·RT_{99}

(H) RT_{1}·RT_{2}·RT_{3}·…·RT_{99}·RT_{100}

where R denotes the common property and the T_{i} a distinctive property. It should be noted here that it can consist of a single object, or alternatively, of instances which are distinguished by a given criterion (which is not concerned by the inductive process) according to *n*-universe in which one places oneself. Thus, one will use in the case of a single instance , the colour of which is susceptible to change according to time:

(I) RT_{1}·RT_{2}·RT_{3}·…·RT_{99}

or in the case where several instances _{1}, _{2}, …, _{99}, _{100} exist^{xxi}:

(I) RT_{1}_{1}·RT_{2}_{2}·RT_{3}_{3}·…·RT_{99}_{99}

**6. Origin of the paradox**

Given the conditions of induction and the framework of *n*-universes which have been just defined, one is now in a position to proceed to determine the origin of GP. Preliminarily it is worth describing accurately the conditions of the *universe of reference* in which GP takes place. Indeed, in the original version of GP, the choice of the universe of reference is not defined accurately. However one can think that it is essential, in order to avoid any ambiguity, that this last is described precisely.

The universe of reference in which Goodman (1946) places himself is not defined explicitly, but several elements of the statement make it possible to specify its intrinsic nature. Goodman thus mentions the colours “red” and “non-red”. Therefore, colour constitutes one of the criterion-variables of the universe of reference. Moreover, Goodman distinguishes the balls which are drawn at times T_{1}, T_{2}, T_{3}, …, T_{100}. Thus, time is also a criterion-variable of the considered universe. Consequently, one can describe the minimal universe in which Goodman (1946) places himself as a Ω^{2}CT. Similarly, in Goodman (1954), the criterion-variables of colour (green/non-green) and time (drawn before T/drawn after T) are expressly mentioned. In both cases, one thus places oneself implicitly within the *minimal* framework of a Ω^{2}CT.

Goodman in addition mentions *instances* of balls or emeralds. Is it necessary at this stage to resort to an additional criterion-variable making it possible to distinguish between the instances? It appears that not. On the one hand indeed, as we have seen previously^{xxii}, it proves that one has well a version of GP by simply considering a Ω^{2}CT and a single object, the colour of which is susceptible to change during the course of time. On the other hand, it appears that if the criterion which is used to distinguish the instances is not used in the inductive process, it is then neither useful as a common criterion, nor as a differentiated criterion. It follows that one can dispense with this 3rd additional criterion. Thus, it proves that the fact of taking into account one single instance or alternatively, several instances, is not essential in the formulation of GP. In what follows, one will be able thus to consider that the statement applies, indifferently, to a single object or several instances that are distinguished by a criterion which is not used in the inductive process.

At this step, we are in a position to replace GP within the framework of *n*-universes. Taking into account the fact that the context of GP is that of a *minimal*Ω^{2}CT, one will consider successively two situations: that of a Ω^{2}CT, and then that of a Ω^{3}CT (where denotes a 3rd criterion).

*6.1 “Grue” in the coloured and temporal 2-universe*

Consider first the hypothesis of a Ω^{2}CT. In such a universe, being “red” is being red at time T. One has then a criterion of colour for the common properties and a criterion of time for the differentiated properties. Consequently, it appears completely legitimate to project the common property of colour (“red”) into the differentiated time. Such a projection proves to be in conformity with the conditions of induction stated above.

Let us turn now to the projection of “grue”. One has observed previously^{xxiii} that the Ω^{2}CT was reducible to a Ω^{1}Z. Here, the fact of using “grue” (and “bleen”) as primitives, is characteristic of the fact that the system of criteria used is that of a Ω^{1}Z. What is then the situation when one projects “grue” in the Ω^{1}Z? In such a universe of reference, the unique criterion-variable is the tcolour*. An object is there “grue” or “bleen” in the *absolute*. Consequently, if one has well a common criterion (the tcolour*), it appears that the differentiated criterion is missing, in order to perform induction validly. And the situation in which one is placed is that of an extreme differentiation. Thus, such a projection is carried out in violation of the minimal condition of induction. Consequently, it proves that GP cannot take place in the Ω^{2}CT and is then blocked at the stage of the projection of “grue”.

But are these preliminary remarks sufficient to provide, in the context of a Ω^{2}CT, a satisfactory solution to GP? One can think that not, because the paradox also arises in it in another form, which is that of the projection of tcolour* through time. One can formalise this projection Z°T as follows:

(I*) GT_{1}·GT_{2}·GT_{3}·…·GT_{99}

(H*) GT_{1}·GT_{2}·GT_{3}·…·GT_{99}·GT_{100} that is equivalent to:

(H’*) RT_{1}·RT_{2}·RT_{3}·…·RT_{99}·~RT_{100}

(P*) GT_{100} that is equivalent to:

(P’*) ~RT_{100}

where it is manifest that the elements of GP are still present.

Fundamentally in this version, it appears that the common properties are borrowed from the system of criteria of the Ω^{1}Z, whereas the differentiated properties come from the Ω^{2}CT. A first analysis thus reveals that the projection of “grue” under these conditions presents a defect which consists in the choice of a given system of criteria for the common properties (tcolour*) and of a different system of criteria for the differentiated properties (time). For the selection of the tcolour* is characteristic of the choice of a Ω^{1}Z, whereas the use of time is revealing of the fact that one places oneself in a Ω^{2}CT. But one must choose one or the other of the reducible systems of criteria to perform induction. On the hypotheses envisaged previously, the choice of the criteria for the common and differentiated properties was carried out within *the same system of criteria*. But here, the choice of the criteria for the common properties and the differentiated properties is carried out within *two different* (and reducible) *systems of criteria*. Thus, the common and differentiated criteria selected for induction are not genuinely distinct. And this appears as a *violation of the condition of separation*. Consequently, one of the conditions of induction is not respected.

However, the projection Z°T has a certain intuitive support, because it is based on the fact that the notions of “grue before T” and “grue after T” have a certain intuitive meaning. Let us then disregard the violation of the conditions of the induction which has been just mentioned, and consider thus this situation in more detail. In this context, GP is always present, since one observes a contradiction between (P) and (P’*). It is with this contradiction that it is worth from now on being interested. Consider the particular step of the equivalence between (H*) and (H’*). One conceives that “grue before T” is assimilated here to RT, because the fact that the instances-S are red before T results clearly from the conditions of the experiment. On the other hand, it is worth being interested by the step according to which (P*) entails (P’*). According to the classical definition^{xxiv}: “grue” = {RT R~T, RT ~R~T, ~RT ~R~T }. What is it then to be “grue after T”? There, it appears that a “grue” object can be R~T (this corresponds to the case RT R~T) or ~R~T (this correspond to the cases RT ~R~T and ~RT ~R~T). In conclusion, the object can be either R~T or ~R~T. Thus, the fact of knowing that an object is “grue after T” does not make it possible to conclude that this object is ~R~T, because this last can also be R~T. Consequently, the step according to which (P*) involves (P’*) appears finally *false*. From where it ensues that the contradiction between (P) and (P’*) does not have any more a raison d’etre.

One can convince oneself that this analysis does not depend on the choice of the classical definition of “grue” (*grue*_{12}) which is carried out, by considering other definitions. Consider for example the definition based on *grue*_{9}: “grue” = {RT ~R~T, ~RT ~R~T} and “bleen” = {RT R~T, ~RT R~T}. But in this version, one notes that one does not have the emergence of GP, because the instances-S, which are RT, can be at the same time “grue” and ” bleen”. And the same applies if one considers a conjunctive definition (*grue*_{2}) such as “grue” = {RT ~R~T}. In such a case indeed, the instances-S are “grue” only if they are RT but also ~R~T. However this does not correspond to the initial conditions of GP in the ^{2}CT where one ignores if the instances-S are ~R~T.

One could also think that the problem is related to the use of a taxonomy of tcolour* based on two taxa (G and ~G). Consider then a taxonomy of tcolour* based on 4 taxa: Z_{0} = RT R~T, Z_{1} = RT ~R~T, Z_{2} = ~RT R~T, Z_{3} = ~RT ~R~T. But on this hypothesis, it appears clearly that since the instances-S are for example Z_{1}, one finds himself replaced in the preceding situation.

The fact of considering “grue after T”, “grue before T”, “bleen before T”, “bleen after T” can be assimilated with an attempt of expressing “grue” and ” bleen” with the help of our own criteria, and in particular that of time. It can be considered here as a form of anthropocentrism, underlain by the idea to express the Ω^{1}Z with the help of the taxa of the Ω^{2}CT. Since one knows the code defining the relations between two reducible *n*-universes – the Ω^{1}Z and the Ω^{2}CT – and that one has partial data, one can be tempted to elucidate completely the predicates of the foreign *n*-universe. Knowing that the instances are GT, G~T, ~GT, ~G~T, I can deduce that they are respectively {RT, ~RT}, {R~T, ~R~T}, {~RT}, {R~T}. But as we have seen, due to the fact that the instances are GT and RT, I cannot deduce that they will be ~R~T.

The reasoning in this version of GP is based on the apparently inductive idea that what is “grue before T” is also “grue after T”. But in the context which is that of the Ω^{1}Z, when an object is “grue”, it is “grue” in the absolute. For no additional criterion exists which can make its tcolour* vary. Thus, when an object is GT, it is *necessarily* G~T. And from the information according to which an object is GT, one can thus conclude, by *deduction*, that it is also G~T.

From what precedes, it ensues that the version of GP related to the Z°T presents the apparent characters of induction, but it does not constitute an authentic form of this type of reasoning. Z°T thus constitutes a disguised form of induction for two principal reasons: first, it is a projection through the differentiated criterion of time, which constitutes the standard mode of our inductive practice. Second, it is based on the intuitive principle according to which everything that is GT is also G~T. But as we have seen, it consists here in reality of a deductive form of reasoning, whose true nature is masked by an apparent inductive move. And this leads to conclude that the form of GP related to Z°T analyses itself in fact veritably as a *pseudo-induction*.

*6.2 “Grue” in the coloured, temporal and localised 3-universe*

Consider now the case of a Ω^{3}CT. This type of universe of reference also corresponds to the definition of a *minimal* Ω^{2}CT, but it also comprises one 3rd criterion-variable^{xxv}. Let us choose for this last a criterion such as localisation^{xxvi}. Consider then a Ω^{3}CTL. Consider first (H) in such a 3-universe. To be “red” in the Ω^{3}CTL, is to be red at time T and at location L. According to the conditions of GP, colour corresponds to the *common* properties, and time to the *differentiated* properties. One has then the following projection C°TL:

(I) RT_{1}L_{1}·RT_{2}L_{2}·RT_{3}L_{3}·…·RT_{99}L_{99}

(H) RT_{1}L_{1}·RT_{2}L_{2}·RT_{3}L_{3}·…·RT_{99}L_{99}·RT_{100}L_{100}

∴ (P) RT_{100}L_{100}

where taking into account the conditions of induction, it proves to be legitimate to project the common property (“red”) of the instances-S, into differentiated time and location, and to predict that the 100th ball will be red. Such a projection appears completely correct, and proves in all points in conformity with the conditions of induction mentioned above.

What happens now with (H*) in the Ω^{3}CTL? It has been observed that the Ω^{3}CTL could be reduced to a Ω^{2}ZL. In this last *n*-universe, the criterion-variables are tcolour* and localisation. The fact of being “grue” is there relative to location: to be “grue”, is to be “grue” at location L. What is then projected is the tcolour*, i.e. the fact of being “grue” or “bleen”. There is thus a common criterion of tcolour* and a differentiated criterion of localisation. Consequently, if it is considered that the instances-S are “grue”, one can equally well project the property common “grue” into a differentiated criterion of localisation. Consider then the projection Z°L in the Ω^{2}ZL:

(I*) GL_{1}·GL_{2}·GL_{3}·…·GL_{99}

(H*) GL_{1}·GL_{2}·GL_{3}·…·GL_{99}·GL_{100}

∴ (P*) GL_{100}

Such a projection is in conformity with the conditions mentioned above, and constitutes consequently a valid form of induction.

In this context, one can project valuably a predicate having a structure identical to that of “grue”, in the case of emeralds. Consider the definition “grue” = green before T or non-green after T, where T = 10 billion years. It is known that at that time, our Sun will be extinct, and will become gradually a dwarf white. The conditions of our atmosphere will be radically different from what they currently are. And the temperature will rise in particular in considerable proportions, to reach 8000°. Under these conditions, the structure of many minerals will change radically. It should normally thus be the case for our current emeralds, which should see their colour modified, due to the enormous rise in temperature which will follow. Thus, I currently observe an emerald: it is “grue” (for T = 10 billion years). If I project this property through a criterion of *location*, I legitimately conclude from it that the emerald found in the heart of the Amazonian forest will also be “grue”, in the same way as the emerald which has been just extracted from a mine from South Africa.

At this stage, one could wonder whether the projectibility of “grue” is not intrinsically related to the choice of a definition of “grue” based on inclusive disjunction (*grue*_{12})? Nevertheless, one easily checks by using an alternative definition of “grue” that its projection remains valid^{xxvii}.

It should be noticed that one has here the expression of the fact that the taxonomy based on the tcolour* is coarser than that based on time and colour. In effect, the former only comprises 2 taxa (grue/bleen), whereas the latter presents 4 of them. By reducing the criteria of colour and time to a single criterion of tcolor*, one has replaced 4 taxa (RT R~T, RT ~R~T, ~RT R~T, ~RT ~R~T) by 2. Thus, “grue” constitutes from this point of view a predicate coarser than “red”. The universe which is described did not change, but the *n*-universes which are systems of criteria describing these universes are different. With the tcolour* thus defined, one has less predicates at its disposal to describe a same reality. The predicates “grue” and “bleen” are for us not very informative, and are less informative in any case that our predicates “red”, “non-red”, “before T”, etc. But that does not prevent however “grue” and “bleen” to be projectibles.

Whereas the projection of “grue” appears valid in the Ω^{2}ZL, it should be noticed however that one does not observe in this case the contradiction between (P) and (P’*). For here (I*) is indeed equivalent to:

(I’*) RT_{1}L_{1}·RT_{2}L_{2}·RT_{3}L_{3}·…·RT_{99} L_{99}

since, knowing according to the initial data of GP that the instances-S are RT, one valuably replaces the GL_{i} by the RT_{i}L_{i} (*i* < 100). But it appears that on this hypothesis, (P*) does not involve:

(P’*) ~RT_{100}L_{100}

because one does not have an indication relating to the temporality of the 100th instance, due to the fact that only the localisation constitutes here the differentiated criterion. Consequently, one has well in the case of the Ω^{3}CTL a version built with the elements of GP where the projection of “grue” is carried out valuably, but which does not present a paradoxical nature.

**7. Conclusion**

In the solution to GP proposed by Goodman, a predicate is projectible or nonprojectible in the *absolute*. And one has in addition a correspondence between the entrenched^{xxviii}/non-entrenched and the projectible/nonprojectible predicates. Goodman in addition does not provide a justification to this assimilation. In the present approach, there is no such dichotomy, because a given predicate P reveals itself projectible in a given *n*-universe, and nonprojectible in another *n*-universe. Thus, P is projectible *relatively* to such universe of reference. There is thus the *projectible*/*nonprojectible relative to such n-universe* distinction. And this distinction is justified by the conditions of induction, and the fundamental mechanism of induction related to the unification/differentiation duality. There are thus *n*-universes where “green” is projectible and others where it is not. In the same way, “grue” appears here projectible relative to certain *n*-universes. Neither *green* nor *grue* are projectible in the absolute, but only relative to such given universe. Just as of some other predicates, “grue” is projectible in certain universes of reference, but nonprojectible in others^{xxix}.

Thus, it proves that one of the causes of GP resides in the fact that in GP, one classically proceeds to operate a dichotomy between the projectible and the nonprojectible predicates. The solutions classically suggested to GP are respectively based on the distinction temporal/nontemporal, local/non-local, qualitative/nonqualitative, entrenched/non-entrenched, etc. and a one-to-one correspondence with the projectible/nonprojectible distinction. One wonders whether a given predicate P* having the structure of “grue” is projectible, in the *absolute*. This comes from the fact that in GP, one has a contradiction between the two concurrent predictions (P) and (P*). One classically deduces from it that one of the two predictions must be rejected, at the same time as one of the two generalisations (H) or (H*) on which these predictions are respectively based. Conversely, in the present analysis, whether one places himself in the case of the *authentic* projection Z°L or in the case of the *pseudo-projection* Z°T, one does not have a contradiction between (P) and (P’*). Consequently, one is not constrained any more to reject either (H) or (H*). And the distinction between projectible/nonprojectible predicates does not appear indispensable any more^{xxx}.

How is the choice of our usual *n*-universe carried out in this context? *N*-universes such as the Ω^{2}CT, the Ω^{3}CTL, the Ω^{2}ZL etc. are appropriate to perform induction. But we naturally tend to privilege those which are based on criteria structured rather finely to allow a maximum of *combinations* of projections. If one operates from the criteria Z and L in the Ω^{2}ZL, one restricts oneself to a limited number of combinations: Z°L and L°Z. Conversely, if one retains the criteria C, T and L, one places oneself in the Ω^{3}CTL and one has the possibility of projections C°TL, T°CL, L°CT, CT°L^{xxxi}, CL°T, TL°C. One has thus a maximum of combinations. This seems to encourage to prefer the Ω^{3}CTL to the Ω^{2}ZL. Of course, pragmatism seems to have to play a role in the choice of the best alternative of our criteria. But it seems that it is only one of the multiple factors which interact to allow the optimisation of our criteria to carry out the primitive operations of grouping and differentiation, in order to then be able to generalise, classify, order, make assumptions or forecast^{xxxii}. Among these factors, one can in particular mention: pragmatism, simplicity, flexibility of implementation, polyvalence^{xxxiii}, economy in means, power^{xxxiv}, but also the nature of our real universe, the structure of our organs of perception, the state of our scientific knowledge, etc^{xxxv}. Our usual *n*-universes are optimised with regard to these various factors. But this valuably leaves room for the choice of other systems of criteria, according to the variations of one or the other of these parameters^{xxxvi}.

i Nelson Goodman, “A Query On Confirmation”, *Journal of Philosophy*, vol. 43 (1946), p. 383-385; in *Problems and Projects*, Indianapolis, Bobbs-Merrill, 1972, p. 363-366.

ii With some minor adaptations.

iii See Goodman “A Query On Confirmation”, p. 383: “Suppose we had drawn a marble from a certain bowl on each of the ninety-nine days up to and including VE day and each marble drawn was red. We would expect that the marble drawn on the following day would also be red. So far all is well. Our evidence may be expressed by the conjunction “*Ra*_{1}·*Ra*_{2}·…·*Ra*_{99}” which well confirms the prediction *Ra*_{100}.” But increase of credibility, projection, “confirmation” in any intuitive sense, does not occur in the case of every predicate under similar circumstances. Let “S” be the predicate “is drawn by VE day and is red, or is drawn later and is non-red.” The evidence of the same drawings above assumed may be expressed by the conjunction “*Sa*_{1}·*Sa*_{2}·…·*Sa*_{99}“. By the theories of confirmation in question this well confirms the prediction “*Sa*_{100}“; but actually we do not expect that the hundredth marble will be non-red. “*Sa*_{100}” gains no whit of credibility from the evidence offered.”

iv Nelson Goodman, *Fact, Fiction and Forecast*, Cambridge, MA, Harvard University Press, 1954.

v *Ibid*., p. 73-4: “Suppose that all emeralds examined before a certain time *t* are green. At time *t*, then, our observations support the hypothesis that all emeralds are green; and this is in accord with our definition of confirmation. […] Now let me introduce another predicate less familiar than “green”. It is the predicate “grue” and it applies to all things examined before *t* just in case they are green but to other things just in case they are blue. Then at time *t* we have, for each evidence statement asserting that a given emerald is green, a parallel evidence statement asserting that that emerald is grue.”

vi For example with an accuracy of 10^{-4} nm.

vii Or any taxonomy which is similar to it.

viii See §2 above.

ix This assertion is controversial.

x Such a remark also applies to the statement of Goodman, *Fact, Fiction and Forecast*.

xi As J.S. Ullian mentions it, “More one ‘Grue’ and Grue”, *Philosophical Review*, vol. 70 (1961), p. 386-389, in p. 387.

xii I. e. C(0, 4)+C(1, 4)+C(2, 4)+C(3, 4)+C(4, 4) = 2^{4}, where C(*p*, *q*) denotes the number of combinations of *q* elements taken *p* times.

xiii See Goodman, “A Query On Confirmation”, p. 383: “Induction might roughly be described as the projection of characteristics of the past into the future, or more generally of characteristics of one realm of objects into another.”

xiv See §2 above.

xv One can of course alternatively take into account several instances-D.

xvi See §2 above.

xvii *Ibid*.

xviii For the application of this condition, one must take into account the remarks mentioned above concerning the problem of the status of the instances. Thus, one must actually compare an instantiated and temporal 1-universe to a 2-universe one of the criteria of which is temporal, and the second criterion is not explicitly mentioned. Similarly, an instantiated and coloured 1-universe is assimilated in fact to a 2-universe one of the criteria of which is temporal, and the second criterion is not specified.

xix See §3 above.

xx With the notations C (colour), T (time), L (localisation) and Z (tcolour*).

xxi However, since the fact that there exists one or more instances is not essential in the formulation of the given problem, one will obviously be able to abstain from making mention of it.

xxii See §4.

xxiii *Ibid*.

xxiv It is the one based on the inclusive disjunction (*grue*_{12}).

xxv A same solution applies, of course, if one considers a number of criterion-variables higher than 3.

xxvi All other criterion distinct from colour or time, would also be appropriate.

xxvii In particular, it appears that the projection of a conjunctive definition (*grue*_{2}) is in fact familiar for us. In effect, we do not proceed otherwise when we project the predicate “being green before maturity and red after maturity” applicable to tomatoes, through a differentiated criterion of location: this is true of the 99 instance-S observed in Corsica and Provence, and is projected validly to a 100th instance located in Sardinia. One can observe that such a type of projection is in particular regarded as nonproblematic by Jackson (Franck Jackson, “‘Grue'”, *Journal of Philosophy*, vol. 72 (1975), p. 113-131): “There seems no case for regarding ‘grue’ as nonprojectible if it is defined this way. An emerald is grue_{1} just if it is green up to *T* and blue thereafter, and if we discovered that all emeralds so far examined had this property, then, other things being equal, we would probably accept that all emeralds, both examined and unexamined, have this property (…).” If one were to replace such a predicate in the present analysis, one should then consider that the projection is carried out for example through a differentiated criterion of localisation (p. 115).

xxviii Goodman, *Fact, Fiction and Forecast*.

xxix The account presented in J Holland, K Holyoak, R. Nisbett and P. Thagard (*Induction*, Cambridge, MA; London, MIT Press, 1986) appears to me to constitute a variation of Goodman’s solution, directed towards the computer-based processing of data and based on the distinction integrated/non-integrated in the default hierarchy. But Holland’s solution presents the same disadvantages as that of Goodman: what justification if not anthropocentric, does one have for this distinction? See p. 235: “Concepts such as “grue”, which are of no significance to the goals of the learner, will never be generated and hence will not form part of the default hierarchy. (…) Generalization, like other sorts of inference in a processing system, must proceed from the knowledge that the system already has”.

The present analysis also distinguishes from the one presented by Susan Haack (*Evidence and Inquiry*, Oxford; Cambridge, MA, Blackwell, 1993) because the existence of natural kinds does not constitute here a condition for induction. See p. 134: “There *is* a connection between induction and natural kinds. […] the reality of kinds and laws is a necessary condition of successful inductions”. In the present context, the fact that the conditions of induction (a common criterion, a distinct differentiated criterion, etc.) are satisfied is appropriate to perform induction.

xxx A similar remark is made by Franck Jackson in conclusion of his article (“‘Grue'”, p. 131): “[…] the SR can be specified without invoking a partition of predicates, properties or hypotheses into the projectible and the nonprojectible”. For Jackson, all noncontradictory predicates are projectible: “[…] *all* (consistent) predicates are projectible.” (p. 114). Such a conclusion appears however stronger than the one that results from the current analysis. Because for Jackson, all predicates are thus projectible in the *absolute*. However in the present context, there are no projectible or nonprojectible predicates in the absolute. It is only *relative* to a given *n*-universe, that a predicate P reveals projectible or nonprojectible.

More generally, the present analysis distinguishes fundamentally from that of Jackson in the sense that the solution suggested to GP does not rest on the counterfactual condition. This last appears indeed too related to the use of certain predicates (*examined*, *sampled*, etc.). On the other hand, in the present context, the problem is considered from a general viewpoint, independently of the particular nature of the predicates constituting the definition of *grue*.

xxxi Such a projection corresponds for example to the generalisation according to which “the anthropomorphic statue-menhirs are of the colour of granite and date from the Age of Bronze”.

xxxii As Ian Hacking underlines it, *Le plus pur nominalisme*, Combas, L’éclat, 1993, p. 9: “Utiliser un nom pour une espèce, c’est (entre autres choses) vouloir réaliser des généralisations et former des anticipations concernant des individus de cette espèce. La classification ne se limite pas au tri : elle sert à prédire. C’est une des leçons de la curieuse “énigme” que Nelson Goodman publia il y a quarante ans.” My translation: “To use a name for a species, it is (among other things) to want to carry out generalisations and to form anticipations concerning the individuals of this species. Classification is not limited to sorting: it is used to predict. It is one of the lessons of the strange “riddle” which Nelson Goodman published forty years ago.”

xxxiii The fact that a same criterion can be used at the same time as a common and a differentiated criterion (while eventually resorting to different taxa).

xxxiv I.e. the number of combinations made possible.

xxxv This enumeration does not pretend to be exhaustive. A thorough study of this question would be of course necessary.

xxxvi I thank the editor of Dialogue and two anonymous referees for very helpful comments on an earlier draft of this paper.

# A Dichotomic Analysis of the Surprise Examination Paradox

*English translation of a paper appeared in French in Philosophiques 2005, vol. 32, pages 399-421 (with minor changes with regard to the published version).*

This paper proposes a new framework to solve the surprise examination paradox. I survey preliminary the main contributions to the literature related to the paradox. I introduce then a distinction between a monist and a dichotomic analysis of the paradox. With the help of a matrix notation, I also present a dichotomy that leads to distinguish two basically and structurally different notions of surprise, which are respectively based on a conjoint and a disjoint structure. I describe then how Quine’s solution and Hall’s reduction apply to the version of the paradox corresponding to the conjoint structure. Lastly, I expose a solution to the version of the paradox based on the disjoint structure.

## A Dichotomic Analysis of the Surprise Examination Paradox

I shall present in what follows a new conceptual framework to solve the *surprise examination paradox* (henceforth, SEP), in the sense that it reorganizes, by adapting them, several elements of solution described in the literature. The solution suggested here rests primarily on the following elements: (i) a distinction between a monist and a dichotomic analysis of the paradox; (ii) the introduction of a matrix definition, which is used as support with several variations of the paradox; (iii) the distinction between a conjoint and a disjoint definition of the cases of surprise and of non-surprise, leading to two structurally different notions of surprise.

In section 1, I proceed to describe the paradox and the main solutions found in the literature. I describe then in section 2, in a simplified way, the solution to the paradox which results from the present approach. I also introduce the distinction between a monist and a dichotomic analysis of the paradox. I present then a dichotomy which makes it possible to distinguish between two basically and structurally different versions of the paradox: on the one hand, a version based on a conjoint structure of the cases of non-surprise and of surprise; in the other hand, a version based on a disjoint structure. In section 3, I describe how Quine’s solution and Hall’s reduction apply to the version of SEP corresponding to the conjoint structure of the cases of non-surprise and of surprise. In section 4, I expose the solution to SEP corresponding to the disjoint structure. Lastly, I describe in section 5, within the framework of the present solution, what should have been the student’s reasoning.

The surprise examination paradox finds its origin in an actual fact. In 1943-1944, the Swedish authorities planned to carry out a civil defence exercise. They diffused then by the radio an announcement according to which a civil defence exercise would take place during the following week. However, in order to perform the latter exercise under optimal conditions, the announcement also specified that nobody could know in advance the date of the exercise. Mathematician Lennart Ekbom understood the subtle problem arising from this announcement of a civil defence exercise and exposed it to his students. A broad diffusion of the paradox throughout the world then ensued.

SEP first appeared in the literature with an article of D. O’ Connor (1948). O’ Connor presents the paradox under the form of the announcement of a military training exercise. Later on, SEP appeared in the literature under other forms, such as the announcement of the appearance of an ace in a set of cards (Scriven 1951) or else of a hanging (Quine 1953). However, the version of the paradox related to the professor’s announcement of a surprise examination has remained the most current form. The traditional version of the paradox is as follows: a professor announces to his/her students that an examination will take place during the next week, but that they will not be able to know in advance the precise day where the examination will occur. The examination will thus occur surprisingly. The students reason as follows. The examination cannot take place on Saturday, they think, for otherwise they would know in advance that the examination would take place on Saturday and thus it could not occur surprisingly. Thus, Saturday is ruled out. Moreover, the examination cannot take place on Friday, for otherwise the students would know in advance that the examination would take place on Friday and thus it could not occur surprisingly. Thus, Friday is also ruled out. By a similar reasoning, the students eliminate successively Thursday, Wednesday, Tuesday and Monday. Finally, all days of the week are then ruled out. However, this does not prevent the examination from finally occurring surprisingly, say, on Wednesday. Thus, the students’ reasoning proved to be fallacious. However, such a reasoning appears intuitively valid. The paradox lies here in the fact that the students’ reasoning seems valid, whereas it finally proves to be in contradiction with the facts, namely that the examination can truly occur surprisingly, in accordance with the announcement made by the professor.

In the literature, several solutions to SEP have been proposed. There does not exist however, at present time, a consensual solution. I will briefly mention the principal solutions which were proposed, as well as the fundamental objections that they raised.

A first attempt at solution appeared with O’ Connor (1948). This author pointed out that the paradox was due to the contradiction which resulted from the professor’s announcement and the implementation of the latter. According to O’ Connor, the professor’s announcement according to which the examination was to occur by surprise was in contradiction with the fact that the details of the implementation of the examination were known. Thus, the statement of SEP was, according to O’ Connor, self-refuting. However, such an analysis proved to be inadequate, because it finally appeared that the examination could truly take place under some conditions where it occurred surprisingly, for example on Wednesday. Thus, the examination could finally occur by surprise, confirming thus and not refuting, the professor’s announcement. This last observation had the effect of making the paradox re-appear.

Quine (1953) also proposed a solution to SEP. Quine considers thus the student’s final conclusion according to which the examination can occur surprisingly on no day of the week. According to Quine, the student’s error lies in the fact of having not considered from the beginning the hypothesis that the examination could not take place on the last day. For the fact of considering precisely that the examination will not take place on the last day makes it finally possible for the examination to occur surprisingly, on the last day. If the student had also taken into account this possibility from the beginning, he would not concluded fallaciously that the examination cannot occur by surprise. However, Quine’s solution has led to criticisms, emanating notably from commentators (Ayer 1973, Janaway 1989 and also Hall 1999) who stressed the fact that Quine’s solution did not make it possible to handle several variations of the paradox. Ayer imagines thus a version of SEP where a given person is informed that the cards of a set will be turned over one by one, but where that person will not know in advance when the ace of Spades will be issued. Nevertheless, the person is authorized to check the presence of the ace of Spades before the set of cards is mixed. The purpose of the objection to Quine’s solution based on such a variation is to highlight a situation where the paradox is quite present but where Quine’s solution does not find to apply any more, because the student knows with certainty, given the initial data of the problem, that the examination will take place as well.

According to another approach, defended in particular by R. Shaw (1958), the structure of the paradox is inherently self-referential. According to Shaw, the fact that the examination must occur by surprise is tantamount to the fact that the date of the examination cannot be deduced in advance. But the fact that the students cannot know in advance, by deduction, the date of the examination constitutes precisely one of the premises. The paradox thus finds its origin, according to Shaw, in the fact that the structure of the professor’s announcement is self-referential. According to the author, the self-reference which results from it constitutes thus the cause of the paradox. However, such an analysis did not prove to be convincing, for it did not make it possible to do justice to the fact that in spite of its self-referential structure, the professor’s announcement was finally confirmed by the fact that the examination could finally occur surprisingly, say on Wednesday.

Another approach, put forth by Richard Montague and David Kaplan (1960) is based on the analysis of the structure of SEP which proves, according to these authors, to be that of the *paradox of the Knower*. The latter paradox constitutes a variation of the *Liar paradox*. What thus ultimately proposes Montague and Kaplan, is a reduction of SEP to the Liar paradox. However, this last approach did not prove to be convincing. Indeed, it was criticized because it did not take account, on the one hand, the fact that the professor’s announcement can be finally confirmed and on the other hand, the fact that one can formulate the paradox in a non-self-referential way.

It is also worth mentioning the analysis developed by Robert Binkley (1968). In his article, Binkley exposes a reduction of SEP to Moore’s paradox. The author makes the point that on the last day, SEP reduces to a variation of the proposition ‘P and I don’t know that P’ which constitutes Moore’s paradox. Binkley extends then his analysis concerning the last day to the other days of the week. However, this approach has led to strong objections, resulting in particular from the analysis of Wright and Sudbury (1977).

Another approach also deserves to be mentioned. It is the one developed by Paul Dietl (1973) and Joseph Smith (1984). According to these authors, the structure of SEP is that of the *sorites paradox*. What then propose Dietl and Smith, is a reduction of SEP to the sorites paradox. However, such an analysis met serious objections, raised in particular by Roy Sorensen (1988).

It is worth lastly mentioning the approach presented by Crispin Wright and Aidan Sudbury (1977). The analysis developed by these authors^{1} results in distinguishing two cases: on the one hand, on the last day, the student is in a situation which is that which results from Moore’s paradox; in addition, on the first day, the student is in a basically different situation where he can validly believe in the professor’s announcement. Thus, the description of these two types of situations leads to the rejection of the *principle of temporal retention*. According to this last principle, what is known at a temporal position T_{0} is also known at a later temporal position T_{1} (with T_{0} < T_{1}). However, the analysis of Wright and Sudbury appeared vulnerable to an argument developed by Sorensen (1982). The latter author presented indeed a version of SEP (the *Designated Student Paradox*) which did not rely on the principle of temporal retention, on which the approach of Wright and Sudbury rested. According to Sorensen’s variation, the paradox was quite present, but without the conditions of its statement requiring to rely on the principle of temporal retention. Sorensen describes thus the following variation of the paradox. Five students, A, B, C, D and E are placed, in this order, one behind the other. The professor then shows to the students four silver stars and one gold star. Then he places a star on the back of each student. Lastly, he announces to them that the one of them who has a gold star in the back has been designated to pass an examination. But, the professor adds, this examination will constitute a surprise, because the students will only know that who was designated when they break their alignment. Under these conditions, it appears that the students can implement a similar reasoning to that which prevails in the original version of SEP. But this last version is diachronic, whereas the variation described by Sorensen appears, by contrast, synchronic. And as such, it is thus not based on whatever principle of temporal retention.

Given the above elements, it appears that the stake and the philosophical implications of SEP are of importance. They are located at several levels and thus relate^{2} to the theory of knowledge, deduction, justification, the semantic paradoxes, self-reference, modal logic, and vague concepts.

2**. Monist or dichotomic analysis of the paradox**

Most analyses classically proposed to solve SEP are based on an overall solution which applies, in a general way, to the situation which is that of SEP. In this type of analysis, a single solution is presented, which is supposed to apply to all variations of SEP. Such type of solution has a unitary nature and appears based on what can be termed a *monist* theory of SEP. Most solutions to SEP proposed in the literature are monist analyses. Characteristic examples of this type of analysis of SEP are the solutions suggested by Quine (1953) or Binkley (1968). In a similar way, the solution envisaged by Dietl (1973) which is based on a reduction of SEP to the sorite paradox also constitutes a monist solution to SEP.

Conversely, a dichotomic analysis of SEP is based on a distinction between two different scenarios of SEP and on the formulation of an *independent* solution for each of the two scenarios. In the literature, the only analysis which has a dichotomic nature, as far as I know, is that of Wright and Sudbury mentioned above. In what follows, I will present a dichotomic solution to SEP. This solution is based on the distinction of two variations of SEP, associated with concepts of surprise that correspond to different structures of the cases of non-surprise and of surprise.

At this step, it proves to be useful to introduce the matrix notation. With the help of this latter, the various cases of non-surprise and of surprise be modelled with the following S[*k*, *s*] table, where *k* denotes the day where the examination takes place and S[*k*, *s*] denotes if the corresponding case of non-surprise (*s* = 0) or of surprise (*s* = 1) is made possible (S[*k*, *s*] = 1) or not (S[*k*, *s*] = 0) by the conditions of the announcement (with 1 *k* *n*).^{3} If one considers for example 7-SEP ^{4}, S[7, 1] = 0 denotes the fact that the surprise is not possible on the 7th day, and conversely, S[7, 1] = 1 denotes the fact that the surprise is possible on the 7th day; in the same way, S[1, 0] = 0 denotes the fact that the non-surprise is not possible on the 1st day by the conditions of the announcement, and conversely, S[1, 0] = 1 denotes the fact that the non-surprise is possible on the 1st day.

The dichotomy on which rests the present solution results directly from the analysis of the structure which makes it possible to describe the concept of surprise corresponding to the statement of SEP. Let us consider first the following matrix, which corresponds to a *maximal* definition, where all cases of non-surprise and of surprise are made possible by the professor’s announcement (with ■ = 1 and □ = 0):

At the level of (D1), as we can see it, all values of the S[*k*, *s*] matrix are equal to 1, which corresponds to the fact that all the cases of non-surprise and of surprise are made possible by the corresponding version of SEP. The associated matrix can be thus defined as a *rectangular* matrix.

At this stage, it appears that one can conceive of some variations of SEP associated with more restrictive matrix structures, where certain cases of non-surprise and of surprise are not authorized by the announcement. In such cases, certain values of the matrix are equal to 0. It is now worth considering the structure of these more restrictive definitions. The latter are such that it exists at least one case of non-surprise or of surprise which is made impossible by the announcement, and where the corresponding value of the matrix S[*k*, *s*] is thus equal to 0. Such a condition leaves place [***room] with a certain number of variations, of which it is now worth studying the characteristics more thoroughly.

One can notice preliminarily that certain types of structures can be discarded from the beginning. It appears indeed that any definition associated with a restriction of (D1) is not adequate. Thus, there are minimal conditions for the emergence of SEP. In this sense, a first condition is that the *base step* be present. This base step is such that the non-surprise must be able to occur on the last day, that is to say S[*n*, 0] = 1. With the previously defined notation, it presents the general form *n***n** and corresponds to 7*7* for 7-SEP. In the lack of this base step, there is no paradoxical effect of SEP. Consequently, a structure of matrix such as S[*n*, 0] = 0 can be discarded from the beginning.

One second condition so that the statement leads to a genuine version of SEP is that the examination can finally occur surprisingly. This renders indeed possible the fact that the professor’s announcement can be finally satisfied. Such a condition – let us call it the *vindication step* – is classically mentioned as a condition for the emergence of the paradox. Thus, a definition which would be such that all the cases of surprise are made impossible by the corresponding statement would also not be appropriate. Thus, the structure corresponding to the following matrix would not correspond either to a licit statement of SEP:

(D2) | S[k, 0] | S[k, 1] | ||

S[7,s] | ■ | □ | ||

S[6,s] | ■ | □ | ||

S[5,s] | ■ | □ | ||

S[4,s] | ■ | □ | ||

S[3,s] | ■ | □ | ||

S[2,s] | ■ | □ | ||

S[1,s] | ■ | □ |

because the surprise is possible here on no day of the week (S[*k*, 1 ] = 0) and the validation step is thus lacking in the corresponding statement.

Taking into account what precedes, one is now in a position to describe accurately the minimal conditions which are those of SEP:

(C4) *k* (1 *k* *n*) such that S[*k*, 1] = 1 (*validation step*)

At this step, it is worth considering the structure of the versions of SEP based on the definitions which satisfy the minimal conditions for the emergence of the paradox which have just been described, i.e. which contain at the same time the basic step and the validation step. It appears here that the structure associated with the cases of non-surprise and of surprise corresponding to a variation with SEP can present two forms of a basically different nature. A first form of SEP is associated with a structure where the possible cases of non-surprise and of surprise are such that it exists during the *n*-period at least one day where the non-surprise and the surprise are simultaneously possible. Such a definition can be called *conjoint*. The following matrix constitutes an example of this type of structure:

(D5) | S[k, 0] | S[k, 1] | ||

S[7,s] | ■ | ■ | ||

S[6,s] | ■ | ■ | ||

S[5,s] | ■ | ■ | ||

S[4,s] | ■ | ■ | ||

S[3,s] | □ | ■ | ||

S[2,s] | □ | ■ | ||

S[1,s] | □ | ■ |

because the non-surprise and the surprise are simultaneously possible here on the 7th, 6th, 5th and 4th days. However, it proves that one can also encounter a second form of SEP the structure of which is basically different, in the sense that for each day of the *n*-period, it is impossible to have simultaneously the surprise and the non-surprise.^{5} A definition of this nature can be called *disjoint*. The following matrix thus constitutes an example of this type of structure:

(D6) | S[k, 0] | S[k, 1] | ||

S[7,s] | ■ | □ | ||

S[6,s] | ■ | □ | ||

S[5,s] | ■ | □ | ||

S[4,s] | □ | ■ | ||

S[3,s] | □ | ■ | ||

S[2,s] | □ | ■ | ||

S[1,s] | □ | ■ |

Consequently, it is worth distinguishing in what follows two structurally distinct versions of SEP: (a) a version based on a *conjoint* structure of the cases of non-surprise and of surprise made possible by the announcement; (b) a version based on a *disjoint* structure of these same cases. The need for making such a dichotomy finds its legitimacy in the fact that in the original version of SEP, the professor does not specify if one must take into account a concept of surprise corresponding to a *disjoint* or a *conjoint* structure of the cases of non-surprise and of surprise. With regard to this particular point, the professor’s announcement of SEP appears ambiguous. Consequently, it is necessary to consider successively two different concepts of surprise, respectively based on a disjoint or conjoint structure of the cases of non-surprise and of surprise, as well as the reasoning which must be associated with them.

3**. The surprise notion corresponding to the conjoint structure**

Let us consider first the case where SEP is based on a concept of surprise corresponding to a conjoint structure of the cases of non-surprise and of surprise. Let SEP(I) be the version associated with such a concept of surprise. Intuitively, this version corresponds to a situation where there exists in the *n*-period at least one day where the non-surprise and the surprise can occur at the same time. Several types of definitions are likely to satisfy this criterion. It is worth considering them in turn.

4*.1 The definition associated with the rectangular matrix and Quine’s solution*

To begin with, it is worth considering the structures which are such that all cases of non-surprise and of surprise are made possible by the statement. The corresponding matrix is a *rectangular* matrix. Let thus SEP(I□) be such a version. The definition associated with such a structure is maximal since all cases of non-surprise and of surprise are authorized. The following matrix corresponds thus to such a general structure:

and the associated professor’s announcement is the following:

(S7) | An examination will occur in the next week but the date of the examination will constitute a surprise. |

At this step, it appears that we also get a version of SEP for *n* = 1 which satisfies this definition. The structure associated with 1-SEP(I□) is as follows:

which corresponds to the following professor’s announcement:

(S8) | An examination will occur on tomorrow but the date of the examination will constitute a surprise. |

Thus, 1-SEP(I□) is the minimal version of SEP which satisfies not only the above condition, but also the base step (C3) according to which the non-surprise must possibly occur on the last day, as well as the validation step (C4) in virtue of which the examination can finally occur by surprise. Moreover, it is a variation which excludes, by its intrinsic structure, the emergence of the version of SEP based on a concept of surprise corresponding to a disjoint structure. For this reason, (D8) can be regarded as the canonical form of SEP(I□). Thus, it is the genuine core of SEP(I□) and in what follows, we will thus endeavour to reason on 1-SEP(I□).

At this stage, it is worth attempting to provide a solution to SEP(I□). For that purpose, let us recall first Quine’s solution. The solution to SEP proposed by Quine (1953) is well-known. Quine highlights the fact that the student eliminates successively the days *n*, *n* -1…, 1, by a reasoning based on *backward-induction* and concludes then that the examination will not take place during the week. The student reasons as follows. On day *n*, I will predict that the examination will take place on day *n*, and consequently the examination cannot take place on day *n*; on day* n* -1, I will predict that the examination will take place on day *n*-1, and consequently the examination cannot take place on day* n* -1; …; on day 1, I will predict that the examination will take place on day 1, and consequently the examination cannot take place on day 1. Finally, the student concludes that the examination will take place on no day of the week. But this last conclusion finally makes it possible to the examination to occur surprisingly, including on day *n*. According to Quine, the error in the student’s reasoning lies precisely in the fact of not having taken into account this possibility since the beginning, which would then have prevented the fallacious reasoning.^{6}

Quine, in addition, directly applies his analysis to the canonical form 1-SEP(I□), where the corresponding statement is that of (S8). In this case, the error of the student lies, according to Quine, in the fact of having considered only the single following assumption: (a) “the examination will take place tomorrow and I will predict that it will take place”. In fact, the student should have also considered three other cases: (b) “the examination will not take place tomorrow and I will predict that it will take place”; (c) “the examination will not take place tomorrow and I will not predict that it will take place”; (d) “the examination will take place tomorrow and I will not predict that it will take place”. And the fact of considering the assumption (a) but also the assumption (d) which is compatible with the professor’s announcement would have prevented the student from concluding that the examination would not finally take place.^{7} Consequently, it is the fact of having taken into account only the hypothesis (a) which can be identified as the cause of the fallacious reasoning. Thus, the student did only take partially into account the whole set of hypotheses resulting from the professor’s announcement. If he had apprehended the totality of the relevant hypotheses compatible with the professor’s announcement, he would not have concluded fallaciously that the examination would not take place during the week.

At this stage, it proves to be useful to describe the student’s reasoning in terms of reconstitution of a matrix. For one can consider that the student’s reasoning classically based on backward-induction leads to reconstitute the matrix corresponding to the concept of surprise in the following way:

(D9) | S[1, 0] | S[1, 1] | ||

S[1,s] | ■ | □ |

In reality, he should have considered that the correct way to reconstitute this latter matrix is the following :

(D8) | S[1, 0] | S[1, 1] | ||

S[1,s] | ■ | ■ |

4*.2 The definition associated with the triangular matrix and Hall’s reduction*

As we have seen, Quine’s solution applies directly to SEP(I□), i.e. to a version of SEP based on a conjoint definition of the surprise and a rectangular matrix. It is now worth being interested in some variations of SEP based on a conjoint definition where the structure of the corresponding matrix is not rectangular, but which satisfies however the conditions for the emergence of the paradox mentioned above, namely the presence of the base step (C3) and the validation step (C4). Such matrices have a structure that can be described as *triangular*. Let thus SEP(I∆) be the corresponding version.

Let us consider first 7-SEP, where the structure of the possible cases of non-surprise and of surprise corresponds to the matrix below:

and to the following announcement of the professor

(S10) | An examination will occur in the next week but the date of the examination will constitute a surprise. Moreover, the fact that the examination will take place constitutes an absolute certainty. |

Such an announcement appears identical to the preceding statement to which the Quine’s solution applies, with however an important difference: the student has from now on the certainty that the examination will occur. And this has the effect of preventing him/her from questioning the fact that the examination can take place, and of making thus impossible the surprise to occur on the last day. For this reason, we note S[7, 1] = 0 in the corresponding matrix. The general structure corresponding to this type of definition is:

And similarly, one can consider the following canonical structure (from where the denomination of *triangular* structure finds its justification), which is that of SEP(I∆) and which corresponds thus to 2-SEP(I∆):

Such a structure corresponds to the following announcement of the professor:

(S12) | An examination will occur on the next two days, but the date of the examination will constitute a surprise. Moreover, the fact that the examination will take place constitutes an absolute certainty. |

As we see it, the additional clause of the statement according to which it is absolutely certain that the examination will occur prevents here the surprise of occurring on the last day. Such a version corresponds in particular to the variation of SEP described by A. J. Ayer. The latter version corresponds to a player, who is authorized to check, before a set of playing cards is mixed, that it contains the ace, the 2, 3…, 7 of Spades. And it is announced that the player that he will not be able to envisage in advance justifiably, when the ace of Spades will be uncovered. Finally the cards, initially hidden, are uncovered one by one. The purpose of such a version is to render impossible, before the 7th card being uncovered, the belief according to which the ace of Spades will not be uncovered. And this has the effect of forbidding to Quine’ solution to apply on the last day.

It is now worth presenting a solution to the versions of SEP associated with the structures corresponding to (D11). Such a solution is based on a reduction recently exposed by Ned Hall, of which it is worth beforehand highlighting the context. In the version of SEP under consideration by Quine (1953), it appears clearly that the fact that the student doubts that the examination will well take place during the week, at a certain stage of the reasoning, is authorized. Quine thus places himself deliberately in a situation where the student has the faculty of doubting that the examination will truly occur during the week. The versions described by Ayer (1973), Janaway (1989) but also Scriven (1951) reveal the intention to prevent this particular step in the student’s reasoning. Such scenarios correspond, in spirit, to SEP(I∆). One can also attach to it the variation of the *Designated Student Paradox* described by Sorensen (1982, 357)^{8}, where five stars – a gold star and four silver stars – are attributed to five students, given that it is indubitable that the gold star is placed on the back of the student who was designated.

However, Ned Hall (1999, 659-660) recently exposed a reduction, which tends to refute the objections classically raised against Quine’s solution. The argumentation developed by Hall is as follows:

We should pause, briefly, to dispense with a bad – though oft-cited – reason for rejecting Quine’s diagnosis. (See for example Ayer 1973 and Janaway 1989). Begin with the perfectly sound observation that the story can be told in such a way that the student *is* justified in believing that, come Friday, he will justifiably believe that an exam is scheduled for the week. Just add a second *Iron Law of the School* : that there must be at least one exam each week. (…) Then the first step of the student’s argument goes through just fine. So Quine’s diagnosis is, evidently, inapplicable.

Perhaps – but in letter only, not in spirit. With the second Iron Law in place, the last disjunct of the professor’s announcement – that E_{5} & J(E_{5}) – is, from the student’s perspective, a *contradiction*. So, from his perspective, the *content* of her announcement is given not by SE_{5} but by SE_{4} : (E_{1} & J_{1}(E_{1})) … (E_{4} & J_{4}(E_{4})). And now Quine’s diagnosis applies straightforwardly : he should simply insist that the student is not justified in believing the announcement and so, come Thursday morning, not justified in believing that crucial part of it which asserts that if the exam is on Friday then it will come as a surprise – which, from the student’s perspective, is tantamount to asserting that the exam is scheduled for one of Monday through Thursday. That is, Quine should insist that the crucial premise that J_{4}(E_{1} E_{2} E_{3} E_{4}) is *false* – which is exactly the diagnosis he gives to an ordinary 4-day surprise exam scenario. Oddly, it seems to have gone entirely unnoticed by those who press this variant of the story against Quine that its only real effect is to convert an *n*-day scenario into an *n*-1 day scenario.

Hall puts then in parallel two types of situations. The first corresponds to the situation in which Quine’s analysis finds classically to apply. The second corresponds to the type of situation under consideration by the opponents to Quine’s solution and in particular by Ayer (1973) and Janaway (1989). On this last hypothesis, a stronger version of SEP is taken into account, where one second *Iron Law of the School* is considered and it is given that the examination will necessarily take place during the week. The argumentation developed by Hall leads to the *reduction* of a version of *n*-SEP of the second type to a version of (*n*-1)-SEP of the quinean type. This equivalence has the effect of annihilating the objections of the opponents to Quine’s solution.^{9} For the effect of this reduction is to make it finally possible to Quine’s solution to apply in the situations described by Ayer and Janaway. In spirit, the scenario under consideration by Ayer and Janaway corresponds thus to a situation where the surprise is not possible on day *n* (i.e. S[*n*, 1] = 0). This has indeed the effect of neutralizing Quine’s solution based on *n*-SEP(I□). But Hall’s reduction then makes it possible to Quine’s solution to apply to (*n*-1)-SEP(I□). The effect of Hall’s reduction is thus of reducing a scenario corresponding to (D11) to a situation based on (D8). Consequently, Hall’s reduction makes it possible to reduce *n*-SEP(I∆) to (*n*-1)-SEP(I□). It results from it that any version of SEP(I∆) for one *n*-period reduces to a version of SEP(I□) for one (*n*-1)-period (formally *n*-SEP(I∆) (*n*-1)-SEP(I□) for *n* > 1). Thus, Hall’s reduction makes it finally possible to apply Quine’s solution to SEP(I∆).^{10}

4**. The surprise notion corresponding to the disjoint structure**

It is worth considering, second, the case where the notion of surprise is based on a *disjoint* structure of the possible cases of non-surprise and of surprise. Let SEP(II) be the corresponding version. Intuitively, such a variation corresponds to a situation where for a given day of the *n*-period, it is not possible to have at the same time the non-surprise and the surprise. The structure of the associated matrix is such that one has exclusively on each day, either the non-surprise or the surprise.

At this step, it appears that a preliminary question can be raised: can’t Quine’s solution apply all the same to SEP(II)? However, the preceding analysis of SEP(I) shows that a necessary condition in order to Quine’s solution to apply is that there exists during the *n*-period at least one day when the non-surprise and the surprise are at the same time possible. However such a property is that of a *conjoint* structure and corresponds to the situation which is that of SEP(I). But in the context of a *disjoint* structure, the associated matrix, in contrast, verifies *k* S[*k*, 0] + S[*k*, 1] = 1. Consequently, this forbids Quine’s solution to apply to SEP(II).

In the same way, one could wonder whether Hall’s reduction wouldn’t also apply to SEP(II). Thus, isn’t there a reduction of SEP(II) for a *n*-period to SEP(I) for a (*n* – 1)-period? It also appears that not. Indeed, as we did see it, Quine’s solution cannot apply to SEP(II). However, the effect of Hall’s reduction is to reduce a given scenario to a situation where Quine’s solution finally finds to apply. But, since Quine’s solution cannot apply in the context of SEP(II), Hall’s reduction is also unable to produce its effect.

Given that Quine’s solution does not apply to SEP(II), it is now worth attempting to provide an adequate solution to the version of SEP corresponding to a concept of surprise associated with a disjoint structure of the cases of non-surprise and of surprise. To this end, it proves to be necessary to describe a version of SEP corresponding to a disjoint structure, as well as the structure corresponding to the canonical version of SEP(II).

In a preliminary way, one can observe that the minimal version corresponding to a disjoint version of SEP is that which is associated with the following structure, i.e. 2-SEP(II):

(D13) | S[1, 0] | S[1, 1] | ||

S[2,s] | ■ | □ | ||

S[1,s] | □ | ■ |

However, for reasons that will become clearer later, the corresponding version of SEP(II) does not have a sufficient degree of realism and of plausibility to constitute a genuine version of SEP, i.e. such that it is susceptible of inducing in error our reasoning.

In order to highlight the canonical version of SEP(II) and the corresponding statement, it is first of all worth mentioning the remark, made by several authors^{11}, according to which the paradox emerges clearly, in the case of SEP(II), when *n* is large. An interesting characteristic of SEP(II) is indeed that the paradox emerges intuitively in a clearer way when great values of *n* are taken into account. A striking illustration of this phenomenon is thus provided to us by the variation of the paradox which corresponds to the following situation, described by Timothy Williamson (2000, 139):

Advance knowledge that there will be a test, fire drill, or the like of which one will not know the time in advance is an everyday fact of social life, but one denied by a surprising proportion of early work on the Surprise Examination. Who has not waited for the telephone to ring, knowing that it will do so within a week and that one will not know a second before it rings that it will ring a second later?

The variation suggested by Williamson corresponds to the announcement made to somebody that he will receive a phone call during the week, without being able however to determine in advance at which precise second the phone call will occur. This variation underlines how the surprise can appear, in a completely plausible way, when the value of *n* is high. The unit of time considered by Williamson is here the second, associated with a period which corresponds to one week. The corresponding value of *n* is here very high and equals 604800 (60 x 60 x 24 x 7) seconds. This illustrates how a great value of *n* makes it possible to the corresponding variation of SEP(II) to take place in both a plausible and realistic way. However, taking into account such large value of *n* is not indeed essential. In effect, a value of *n* which equals, for example, 365, seems appropriate as well. In this context, the professor’s announcement which corresponds to a disjoint structure is then the following:

(S14) | An examination will occur during this year but the date of the examination will constitute a surprise. |

The corresponding definition presents then the following structure :

which is an instance of the following general form :

(D15) | S[1, 0] | S[1, 1] | ||

S[n,s] | ■ | □ | ||

………… | ………… | ………… | ||

S[1,s] | □ | ■ |

This last structure can be considered as corresponding to the canonical version of SEP(II), with *n* large. In the specific situation associated with this version of SEP, the student predicts each day – in a false way but justified by a reasoning based on backward-induction – that the examination will take place on no day of the week. But it appears that at least one case of surprise (for example if the examination occurs on the first day) makes it possible to validate, in a completely realistic way, the professor’s announcement..

The form of SEP(II) which applies to the standard version of SEP is 7-SEP(II), which corresponds to the classical announcement:

(S7) | An examination will occur on the next week but the date of the examination will constitute a surprise. |

but with this difference with the standard version that the context is here exclusively that of a concept of surprised associated with a disjoint structure.

At this stage, we are in a position to determine the fallacious step in the student’s reasoning. For that, it is useful to describe the student’s reasoning in terms of matrix reconstitution. The student’s reasoning indeed leads him/her to attribute a value for S[*k*, 0] and S[*k*, 1]. And when he is informed of the professor’s announcement, the student’s reasoning indeed leads him/her to rebuild the corresponding matrix such that all S[*k*, 0] = 1 and all S[*k*, 1] = 0, in the following way (for *n* = 7):

(D16) | S[k, 0] | S[k, 1] | ||

S[7,s] | ■ | □ | ||

S[6,s] | ■ | □ | ||

S[5,s] | ■ | □ | ||

S[4,s] | ■ | □ | ||

S[3,s] | ■ | □ | ||

S[2,s] | ■ | □ | ||

S[1,s] | ■ | □ |

One can notice here that the order of reconstitution proves to be indifferent. At this stage, we are in a position to identify the flaw which is at the origin of the erroneous conclusion of the student. It appears indeed that the student did not take into account the fact that the surprise corresponds here to a disjoint structure. Indeed, he should have considered here that the last day corresponds to a proper instance of non-surprise and thus that S[*n*, 0] = 1. In the same way, he should have considered that the 1st day^{12} corresponds to a proper instance of surprise and should have thus posed S[1, 1] = 1. The context being that of a disjoint structure, he could have legitimately added, in a second step, that S[*n*, 1] = 0 and S[1, 0] = 0. At this stage, the partially reconstituted matrix would then have been as follows:

(D17) | S[k, 0] | S[k, 1] | ||

S[7,s] | ■ | □ | ||

S[6,s] | ||||

S[5,s] | ||||

S[4,s] | ||||

S[3,s] | ||||

S[2,s] | ||||

S[1,s] | □ | ■ |

The student should then have continued his reasoning as follows. The proper instances of non-surprise and of surprise which are disjoint here do not capture entirely the concept of surprise. In such context, the concept of surprise is not captured exhaustively by the extension and the anti-extension of the surprise. However, such a definition is in conformity with the definition of a vague predicate, which characterizes itself by an extension and an anti-extension which are mutually exclusive and non-exhaustive^{13}. Thus, the surprise notion associated with a disjoint structure is a *vague* one.

What precedes now makes it possible to identify accurately the flaw in the student’s reasoning, when the surprise notion is a vague notion associated with a disjoint structure. For the error which is at the origin of the student’s fallacious reasoning lies in lack of taking into account the fact that the surprise corresponds in the case of a disjoint structure, to a vague concept, and thus comprises the presence of a penumbral zone corresponding to *borderline* cases between the non-surprise and the surprise. There is no need however to have here at our disposal a solution to the sorites paradox. Indeed, whether these borderline cases result from a succession of intermediate degrees, from a precise cut-off between the non-surprise and the surprise whose exact location is impossible for us to know, etc. is of little importance here. For in all cases, the mere fact of taking into account the fact that the concept of surprise is here a concept vague forbids to conclude that S[*k*, 1] = 0, for all values of *k*.

Several ways thus exist to reconstitute the matrix in accordance with what precedes. In fact, there exists as many ways of reconstituting the latter than there are conceptions of vagueness. One in these ways (based on a conception of vagueness based on fuzzy logic) consists in considering that there exists a continuous and gradual succession from the non-surprise to the surprise. The corresponding algorithm to reconstitute the matrix is then the one where the *step* is given by the formula 1/(*n*–*p*) when *p* corresponds to a proper instance of surprise. For *p* = 3, we have here 1/(7-3) = 0,25, with S[3, 1] = 1. And the corresponding matrix is thus the following one:

(D18) | S[k, 0] | S[k, 1] | ||

S[7,s] | 1 | 0 | ||

S[6,s] | 0,75 | 0,25 | ||

S[5,s] | 0,5 | 0,5 | ||

S[4,s] | 0,25 | 0,75 | ||

S[3,s] | 0 | 1 | ||

S[2,s] | 0 | 1 | ||

S[1,s] | 0 | 1 |

where the sum of the values of the matrix associated with a day given is equal to 1. The intuition which governs SEP (II) is here that the non-surprise is total on day *n*, but that there exists intermediate degrees of surprise *s*_{i} (0 < *s*_{i} < 1), such as the more one approaches the last day, the higher the effect of non-surprise. Conversely, the effect of surprise is total on the first days, for example on days 1, 2 and 3.

One can notice here that the definitions corresponding to SEP (II) which have just been described, are such that they present a property of *linearity* (formally, *k* (for 1 < *k* *n*), S[*k*, 0] S[*k*-1, 0]). It appears indeed that a structure corresponding to the possible cases of non-surprise and of surprise which would not present such a property of linearity, would not capture the intuition corresponding to the concept of surprise. For this reason, it appears sufficient to limit the present study to the structures of definitions that satisfy this property of linearity.

An alternative way to reconstitute the corresponding matrix, based on the epistemological conception of vagueness, could also have been used. It consists of the case where the vague nature of the surprise is determined by the existence of a precise cut-off between the cases of non-surprise and of surprise, of which it is however not possible for us to know the exact location. In this case, the matrix could have been reconstituted, for example, as follows:

(D19) | S[k, 0] | S[k, 1] | ||

S[7,s] | ■ | □ | ||

S[6,s] | ■ | □ | ||

S[5,s] | ■ | □ | ||

S[4,s] | □ | ■ | ||

S[3,s] | □ | ■ | ||

S[2,s] | □ | ■ | ||

S[1,s] | □ | ■ |

At this stage, one can wonder whether the version of the paradox associated with SEP(II) cannot be assimilated with the *sorites paradox*. The reduction of SEP to the sorites paradox is indeed the solution which has been proposed by some authors, notably Dietl (1973) and Smith (1984). The latter solutions, based on the assimilation of SEP to the sorites paradox, constitute monist analyses, which do not lead, to the difference of the present solution, to two independent solutions based on two structurally different versions of SEP. In addition, with regard to the analyses suggested by Dietl and Smith, it does not clearly appear whether each step of SEP is fully comparable to the corresponding step of the sorites paradox, as underlined by Sorensen.^{14} But in the context of a conception of surprise corresponding to a disjoint structure, the fact that the last day corresponds to a proper instance of non-surprise can be assimilated here to the base step of the sorites paradox.

Nevertheless, it appears that such a reduction of SEP to the sorites paradox, limited to the notion of surprise corresponding to a disjoint structure, does not prevail here. On the one hand, it does not appear clearly if the statement of SEP can be translated into a variation of the sorites paradox, in particular for what concerns 7-SEP(II). Because the corresponding variation of the sorites paradox would run too fast, as already noted by Sorensen (1988).^{15} It is also noticeable, moreover, as pointed out by Scott Soames (1999), than certain vague predicates are not likely to give rise to a corresponding version of the sorites paradox. Such appears well to be the case for the concept of surprise associated with 7-SEP(II). Because as Soames^{16} points out, the continuum which is semantically associated with the predicates giving rise to the sorites paradox, can be fragmented in units so small that if one of these units is intuitively F, then the following unit is also F. But such is not the case with the variation consisting in 7-SEP(II), where the corresponding units (1 day) are not fine enough with regard to the considered period (7 days).

Lastly and overall, as mentioned above, the preceding solution to SEP(II) applies, *whatever the nature of the solution which will be adopted for the sorites paradox*. For it is the ignorance of the semantic structure of the vague notion of surprise which is at the origin of the student’s fallacious reasoning in the case of SEP(II). And this fact is independent of the solution which should be provided, in a near or far future, to the sorites paradox – whether this approach be of epistemological inspiration, supervaluationnist, based on fuzzy logic…, or of a very different nature.

5**. The solution to the paradox**

The above developments make it possible now to formulate an accurate solution to the surprise examination paradox. The latter solution can be stated by considering what should have been the student’s reasoning. Let us consider indeed, in the light of the present analysis, how the student should have reasoned, after having heard the professor’s announcement:

– The student: Professor, I think that two semantically distinct conceptions of surprise, which are likely to influence the reasoning to hold, can be taken into account. I also observe that you did not specify, at the time of your announcement, to which of these two conceptions you referred. Isn’t it?

– The professor: Yes, it is exact. Continue.

– The student: Since you refer indifferently to one or the other of these conceptions of surprise, it is necessary to consider each one of them successively, as well as the reasoning to be held in each case.

– The professor: Thus let us see that.

– The student: Let us consider, on the one hand, the case where the surprise corresponds to a *conjoint* definition of the cases of non-surprise and of surprise. Such a definition is such that the non-surprise and the surprise are possible at the same time, for example on the last day. Such a situation is likely to arise on the last day, in particular when a student concludes that the examination cannot take place on this same last day, since that would contradict the professor’s announcement. However, this precisely causes to make it possible for the surprise to occur, because this same student then expects that the examination will not take place. And in a completely plausible way, as put forth by Quine, such a situation corresponds then to a case of surprise. In this case, the fact of taking into account the possibility that the examination can occur surprisingly on the last day, prohibits eliminating successively the days *n*, *n*-1, *n*-2, …, 2, and 1. In addition, the concept of surprise associated with a conjoint structure is a concept of total surprise. For one faces on the last day either the non-surprise or the total surprise, without there existing in this case some intermediate situations.

– The professor: I see that. You did mention a second case of surprise…

– The student: Indeed. It is also necessary to consider the case where the surprise corresponds to a *disjoint* definition of the cases of non-surprise and of surprise. Such a definition corresponds to the case where the non-surprise and the surprise are not possible on the same day. The intuition on which such a conception of the surprise rests corresponds to the announcement made to students that they will undergo an examination in the year, while being moreover unaware of the precise day where it will be held. In such a case, it results well from our experience that the examination can truly occur surprisingly, on many days of the year, for example on whatever day of the first three months. It is an actual situation that can be experienced by any student. Of course, in the announcement that you have just made to us, the period is not as long as one year, but corresponds to one week. However, your announcement also leaves place to such a conception of surprise associated with a disjoint structure of the cases of non-surprise and of surprise. Indeed, the examination can indeed occur surprisingly, for example on the 1st day of the week. Thus, the 1st day constitutes a proper instance of surprise. In parallel, the last day constitutes a proper instance of non-surprise, since it results from the announcement that the examination cannot take place surprisingly on this day. At this stage, it also appears that the status of the other days of the corresponding period is not determined. Thus, such a disjoint structure of the cases of non-surprise and of surprise is at the same time disjoint and non-exhaustive. Consequently, the concept of corresponding surprise presents here the criteria of a *vague* notion. And this casts light on the fact that the concept of surprise associated with a conjoint structure is a vague one, and that there is thus a zone of penumbra between the proper instances of non-surprise and of surprise, which corresponds to the existence of borderline cases. And the mere existence of these borderline cases prohibits to eliminate successively, by a reasoning based on backward-induction, the days *n*, *n*-1, *n*-2, …, 2, and then 1. And I finally notice, to the difference of the preceding concept of surprise, that the concept of surprise associated with a conjoint structure leads to the existence of intermediate cases between the non-surprise and the surprise.

– The professor: I see. Conclude now.

– The student: Finally, the fact of considering successively two different concepts of surprise being able to correspond to the announcement which you have just made, resulted in both cases in rejecting the classical reasoning which results in eliminating successively all days of the week. Here, the motivation to reject the traditional reasoning appears different for each of these two concepts of surprise. But in both cases, a convergent conclusion ensues which leads to the rejection of the classical reasoning based on backward-induction.

I shall mention finally that the solution which has been just proposed also applies to the variations of SEP mentioned by Sorensen (1982). Indeed, the structure of the canonical forms of SEP(I□), SEP(I∆) or SEP(II) indicates that whatever the version taken into account, the solution which applies does not require to make use of whatever principle of temporal retention. It is also independent of the order of elimination and can finally apply when the duration of the *n*-period is unknown at the time of the professor’s announcement.

Lastly, it is worth mentioning that the strategy adopted in the present study appears structurally similar to the one used in Franceschi (1999): first, establish a dichotomy which makes it possible to divide the given problem into two distinct classes; second, show that each resulting version admits of a specific resolution.^{17} In a similar way, in the present analysis of SEP, a dichotomy is made and the two resulting categories of problems lead then to an independent resolution. This suggests that the fact that two structurally independent versions are inextricably entangled in philosophical paradoxes could be a more widespread characteristic than one could think at first glance and could also partly explain their intrinsic difficulty.^{18}

**REFERENCES**

AYER, A. J. 1973, “On a Supposed Antinomy”, Mind 82, pp. 125-126. |

BINKLEY, R. 1968, “The Surprise Examination in Modal Logic”, Journal of Philosophy 65, pp. 127-136. |

CHALMERS, D. 2002, “The St. Petersburg two-envelope paradox”, Analysis 62, pp. 155-157. |

CHOW, T. Y. 1998, “The Surprise Examination or Unexpected Hanging Paradox”, The American Mathematical Monthly 105, pp. 41-51. |

DIETL, P. 1973, “The Surprise Examination”, Educational Theory 23, pp. 153-158. |

FRANCESCHI, P. 1999, “Comment l’urne de Carter et Leslie se déverse dans celle de Hempel”, Canadian Journal of Philosophy 29, pp. 139-156. English translation. |

HALL, N. 1999, “How to Set a Surprise Exam”, Mind 108, pp. 647-703. |

HYDE, D. 2002 “Sorites Paradox”, The Stanford Encyclopedia of Philosophy (Fall 2002 Edition), E. N. Zalta (ed.), http ://plato.stanford.edu/archives/fall2002/entries/sorites-paradox. |

JANAWAY, C. 1989, “Knowing About Surprises : A Supposed Antinomy Revisited”, Mind 98, pp. 391-410. |

MONTAGUE, R. & KAPLAN, D. 1960, “A Paradox Regained”, Notre Dame Journal of Formal Logic 3, pp. 79-90. |

O’ CONNOR, D. 1948, “Pragmatic paradoxes”, Mind 57, pp. 358-359. |

QUINE, W. 1953, “On a So-called Paradox”, Mind 62, pp. 65-66. |

SAINSBURY, R. M. 1995, Paradoxes, 2ème édition, Cambridge : Cambridge University Press. |

SCRIVEN, M. 1951, “Paradoxical announcements”, Mind 60, pp. 403-407. |

SHAW, R. 1958, “The Paradox of the Unexpected Examination”, Mind 67, pp. 382-384. |

SMITH, J. W. 1984, “The surprise examination on the paradox of the heap”, Philosophical Papers 13, pp. 43-56. |

SOAMES, S. 1999, Understanding Truth, New York, Oxford : Oxford University Press. |

SORENSEN, R. A. 1982, “Recalcitrant versions of the prediction paradox”, Australasian Journal of Philosophy 69, pp. 355-362. |

SORENSEN, R. A. 1988, Blindspots, Oxford : Clarendon Press. |

WILLIAMSON, T. 2000, Knowledge and its Limits, London & New York : Routledge. |

WRIGHT, C. & SUDBURY, A. 1977, “The Paradox of the Unexpected Examination”, Australasian Journal of Philosophy 55, pp. 41-58. |

1 I simplify here considerably.

2 Without pretending to being exhaustive.

3 In what follows, *n* denotes the last day of the term corresponding to the professor’s announcement.

4 Let 1-SEP, 2-SEP,…, *n*-SEP be the problem for respectively 1 day, 2 days,…, *n* days.

5 The cases where neither the non-surprise nor the surprise are made possible on the same day (i.e. such that S[*k*, 0] + S[*k*, 1] = 0) can be purely and simply ignored.

6 Cf. (1953, 65) : ‘It is notable that *K* acquiesces in the conclusion (wrong, according to the fable of the Thursday hanging) that the decree will not be fulfilled. If this is a conclusion which he is prepared to accept (though wrongly) in the end as a certainty, it is an alternative which he should have been prepared to take into consideration from the beginning as a possibility.’

7 Cf. (1953, 66) : ‘If *K* had reasoned correctly, Sunday afternoon, he would have reasoned as follows : “We must distinguish four cases : first, that I shall be hanged tomorrow noon and I know it now (but I do not) ; second, that I shall be unhanged tomorrow noon and do not know it now (but I do not) ; third, that I shall be unhanged tomorrow noon and know it now ; and fourth, that I shall be hanged tomorrow noon and do not know it now. The latter two alternatives are the open possibilities, and the last of all would fulfill the decree. Rather than charging the judge with self-contradiction, let me suspend judgment and hope for the best.”‘

8 ‘The students are then shown four silver stars and one gold star. One star is put on the back of each student.’.

9 Hall refutes otherwise, but on different grounds, the solution proposed by Quine.

10 Hall’s reduction can be easily generalised. It is then associated with a version of *n*-SEP(I∆) such that the surprise will not possibly occur on the *m* last days of the week. Such a version is associated with a matrix such that (a) *m* (1 *m* < *n*) and S[*n*–*m*, 0] = S[*n*–*m*, 1] = 1 ; (b) *p* > *n*–*m* S[*p*, 0] = 1 and S[*p*, 1] = 0 ; (c) *q* < *n*–*m* S[*q*, 0] = S[*q*, 1] = 1. In this new situation, a *generalised Hall’s reduction *applies to the corresponding version of SEP. In this case, the extended Hall’s reduction leads to : *n*-SEP(I∆) (*n*–*m*)-SEP(I□).

11 Cf. notably Hall (1999, 661), Williamson (2000).

12 It is just an example. Alternatively, one could have chosen here the 2nd or the 3rd day.

13 This definition of a vague predicate is borrowed from Soames. Considering the extension and the anti-extension of a vague predicate, Soames (1999, 210) points out thus: “These two classes are mutually exclusive, though not jointly exhaustive”.

14 Cf. Sorensen (1988, 292-293) : ‘Indeed, no one has simply asserted that the following is just another instance of the sorites.

i. Base step : The audience can know that the exercise will not occur on the last day.

ii. Induction step : If the audience can know that the exercise will not occur on day *n*, then they can also know that the exercise will not occur on day *n* – 1

iii. The audience can know that there is no day on which the exercise will occur.

Why not blame the whole puzzle on the vagueness of ‘can know’? (…) Despite its attractiveness, I have not found any clear examples of this strategy.’

15 Cf. (1988, 324): ‘One immediate qualm about assimilating the prediction paradox to the sorites is that the prediction paradox would be a very ‘fast’ sorites. (…) Yet standard sorites arguments involve a great many borderline cases.’

16 Cf. Soames (1999, 218): ‘A further fact about Sorites predicates is that the continuum semantically associated with such a predicate can be broken down into units fine enough so that once one has characterized one item as F (or not F), it is virtually irresistible to characterize the same item in the same way’.

17 One characteristic example of this type of analysis is also exemplified by the solution to the *two-envelope paradox* described by David Chalmers (2002, 157) : ‘The upshot is a disjunctive diagnosis of the two-envelope paradox. The expected value of the amount in the envelopes is either finite or infinite. If it is finite, then (1) and (2) are false (…). If it is infinite, then the step from (2) to (3) is invalid (…)’.

18 I am grateful toward Timothy Chow, Ned Hall, Claude Panaccio and the anonymous referees for very useful comments concerning previous versions of this paper.

# Theory of Cognitive Distortions: Over-generalisation and Labeling

*English translation of a paper published in French in the Journal de Thérapie Comportementale et Cognitive, 2009, 19-4, pages 136-140 under the title “Théorie des distorsions cognitives : la sur-généralisation et l’étiquetage”.*

In a previous article (Complements to a theory of cognitive distorsions, Journal de Thérapie Comportementale et Cognitive, 2007), we introduced some elements aimed at contributing to a general theory of cognitive distortions. Based on the reference class, the duality and the system of taxa, these elements allow to define the general cognitive distortions as well as the specific cognitive distortions. This model is extended here to the description of two other classical cognitive distortions: over-generalisation and mislabelling. The definition of the two latter cognitive distortions is based on prior differentiation between three levels of reasoning: primary, secondary and ternary pathogenic arguments. The latter analysis also leads to define two other cognitive distortions which insert themselves into this framework: ill-grounded inductive projection and confirmation bias.

This article is cited in:

- Richard-Lepouriel H., Bipolar disorder, self-stigma and cognitive restructuring: A first attempt to support, Journal de Therapie Comportementale et Cognitive, Volume 27, 2017
- Juliette Marty, Et si vous étiez trop perfectionniste ? Eyrolles, 2021

**Theory of Cognitive Distortions: Over-generalisation and Labeling**

In Franceschi (2007), we set out to introduce several elements aimed at contributing to a general theory of cognitive distortions. These elements are based on three fundamental notions: the reference class, the duality and the system of taxa. With these three elements, we could define within the same conceptual framework the following general cognitive distortions: dichotomous reasoning, disqualification of one pole, minimisation and maximisation, requalification in the other pole and omission of the neutral. We could also describe as specific cognitive distortions: disqualification of the positive, selective abstraction and catastrophism. In the present article, we offer to define and to situate, within the same conceptual framework, two other classical cognitive distortions: over-generalisation and mislabelling.

Over-generalisation and mislabelling constitute two of the twelve traditionally defined cognitive distortions: emotional reasoning; over-generalisation; arbitrary inference; dichotomous reasoning; should statements; divination or mental reading; selective abstraction; disqualification of the positive; maximisation/minimisation; catastrophising; personalisation; mislabelling (Beck 1964, Ellis 1962). Over-generalisation is classically defined as a rough and ill-grounded generalisation, usually including either of the quantifiers “all”, “none”, “never”, “always”. Moreover, it is often described as a cognitive distortion including four subcategories: dichotomous reasoning, selective abstraction, maximisation/minimisation, and disqualification of the positive. Mislabelling is also classically defined as an extreme form of over-generalisation, consisting in the apposition of a label with a strong negative and emotional connotation to oneself or to an external subject.

**1. Primary, secondary and ternary pathogenic arguments**

Before setting out to define over-generalisation and mislabelling in the present context, it is worth describing preliminarily a structure of pathogenic reasoning (in the etymological sense: engendering suffering), with a general scope, susceptible of being found in some disorders of a very different nature, such as depression, generalised anxiety disorder, body dismorphic disorder, scrupulosity or intermittent explosive disorder. Such structure of reasoning includes several levels of arguments: primary, secondary and ternary. In a simplified way, primary pathogenic arguments are constituted by an enumeration of instances. Secondary pathogenic arguments consist of a generalisation from the latter instances. Lastly, pathogenic ternary arguments are constituted by an interpretation of the latter generalisation. Such reasoning as a whole presents an *inductive* structure.

At this stage, it is worth mentioning several instances of this type of reasoning. A first instance, susceptible to be found in depression (Beck 1967, 1987), is the following (the symbol denotes the conclusion):

(1_{1}) | I gave my ankle a wrench last January | premise_{1} |

(1_{2}) | I lost my job last February | premise_{2} |

(1_{3}) | Fifteen days ago, I had an influenza with fever | premise_{3} |

(1_{4}) | I got into an argument with Lucy last month | premise_{4} |

(…) | (…) | |

(1_{10}) | Today, my horoscope is not good | premise_{10} |

(2) | ∴ Everything that occurs to me is bad | from (1_{1})-(1_{10}) |

(3) | ∴ I am a complete failure! | from (2) |

The patient enumerates first some events of his/her past and present life (1_{1})-(1_{10}), that he/she qualifies as negative, through a primary stage which consists of an enumeration of instances. Then he/she performs a generalisation (2) from the previous enumeration, which presents the following structure:

(2) | ∴ All events that occur to me are negative | from (1_{1})-(1_{10}) |

Lastly, the patient interprets (3) the latter conclusion by concluding “I am a complete failure!”. Such instance applies then to the reference class of the present and past events of the patient’s life and to the Positive/Negative duality.

One can also mention a reasoning that presents an identical structure, which is susceptible to be met in body dysmorphic disorder (Veale 2004, Rabinowitz & al. 2007). The patient enumerates then different parts of his/her body, which he/she qualifies as ugly. He/she generalises then by concluding that all parts of his/her body are ugly. Finally, he/she adds: “I am ugly!”. The corresponding reasoning applies then to the Beautiful/Ugly duality and to the reference class of the parts of the patient’s body.

In the same way, in a reasoning of identical structure, susceptible to be met in scrupulosity (Teak & Ulug 2001, Miller & Edges 2007), the patient enumerates several instances corresponding to some acts which he/she made previously or recently, and which he/she considers as morally bad. He/she concludes then: “Everything I do is bad, morally reprehensible”, and he/she further interprets it by concluding: “I am a horrible sinner!”. Such conclusion is likely to trigger an intense feel of guilt and a compulsive practice of religious rituals. The corresponding instance applies here to the duality Good/Evil and to the reference class of the present and past actions of the patient’s life.

Lastly, an instance of this structure of reasoning can contribute to the development of hostility, of a potentially aggressive attitude toward other people. In that case, the patient concludes regarding an external subject: “All acts that he committed toward me are bad”. He/she concludes then: “He is a bastard!”. Such conclusion can then play a role in intermittent explosive disorder (Coccaro & al. 1998, Galovski & al. 2002). In such case, the over-generalisation applies to the Good/Evil duality and to the reference class of the actions of an external subject with regard to the patient.

At this step, it is worth describing in more detail each of the three stages – primary, secondary and ternary – which compose this type of reasoning.

**Primary pathogenic arguments**

The first step in the aforementioned type of reasoning, consists for the patient to enumerate some instances. The general structure of each instance is as follows:

(1_{i}) | The object x_{i} of the class of reference E has property Ā (in the duality A/Ā) | premise_{i} |

In the aforementioned example applied to depression, the patient enumerates some events of his/her present and past life, which he/she qualifies as negative, under the form:

(1_{i}) | The event E_{i} of negative nature occurred to me | premise_{i} |

Different instances corresponding to this cognitive process can be described under the form of a *primary pathogenic argument*, the structure of which is the following:

(1a) | The event E_{1} occurred to me | premise |

(1b) | The event E_{1} was of a negative nature | premise |

(1) | ∴ The event E_{1} of a negative nature occurred to me | from (1a), (1b) |

By such cognitive process, the patient is led to the conclusion according to which some negative event did occur to him/her.

From a *deductive* point of view, this type of argument proves to be completely valid (the conclusion is true if the premises are true) since the very event presents well, objectively, a negative nature. However, this type of primary argument can turn out to be fallacious, when the very event presents, objectively, a positive or neutral nature. The flaw in the reasoning resides then in the fact that the premise (1b) turns then out to be false. Such can be case for example if the patient makes use of a specific cognitive distortion such as requalification in the negative. In such case, the patient considers as negative an event the nature of which is objectively positive.

**Secondary pathogenic arguments**

At the level of the above-mentioned reasoning, secondary pathogenic arguments are constituted by the sequence which proceeds by *generalisation*, from the instances (1_{1}) to (1_{10}), according to the following structure:

(2) | ∴ All elements x_{i} of the class of reference E have property Ā | from (1_{1})-(1_{10}) |

Such over-generalisation leads then to the conclusion “All events that occur to me are bad” (depression); “All parts of my body are ugly” (body dysmorphic disorder); “All my acts are morally reprehensible” (scrupulosity); “All acts that he committed toward me are bad” (intermittent explosive disorder).

From a *deductive* point of view, such generalisation may constitute a completely *valid* argument. Indeed, the resulting generalisation constitutes a correct deductive reasoning, if the premises (1_{1})-(1_{10}) are true. However, it often proves to be that the premises of the argument are false. Such is notably the case when the patient counts among the elements having property Ā, some elements which objectively have the opposite property A. The flaw in the argument resides then in a *requalification in the other pole* related to some elements and the enumeration of instances includes then some false premises, thus invalidating the resulting generalisation. In such case, secondary pathogenic argument turns out to be ungrounded, because of the falseness of some premises.

In other cases, the secondary pathogenic argument turns out to be fallacious from an inductive standpoint. For some positive (or neutral) events can well have been omitted in the corresponding enumeration of instances. Such omission can result from the use of general cognitive distortions, such as the omission of the neutral or disqualification of the positive. In such case, the elements of the relevant class of reference are only partly taken into account, thus biasing the resulting generalisation. The corresponding reasoning remains then logically valid and sound, but fundamentally incorrect of an inductive point of view, because it does only take partly into account the relevant instances within the reference class. Such feature of over-generalisation – a conclusion resulting from a valid reasoning from a deductive point of view, but inductively wrong – allows to explain how it notably succeeds in deceiving patients whose level of intelligence can otherwise prove to be high.

**Ternary pathogenic arguments**

It is worth mentioning, lastly, the role played by pathogenic ternary arguments which consist, at the level of the aforementioned reasoning, of the following sequence:

(2) | All events that occur to me are of a negative nature | premise |

(3) | ∴ I am a complete failure! | from (2) |

In such argument, the premise is constituted by the conclusion (2) of the secondary pathogenic argument, of which, in an additional stage (3), the patient aims at making sense by interpreting it. It consists here of a case of *mislabelling*. At the stage of a ternary pathogenic argument, mislabelling can thus take the following forms: “I am a complete failure!” (depression); “I am ugly!” (bodily dysmorphic disorder); “I am a horrible sinner!” (scrupulosity); “He is a bastard! “ (intermittent explosive disorder). In the present context, mislabelling proves to be an invalid argument, which constitutes a rough and unjustified interpretation of the over-generalisation (2).

**2. Over-generalisation**

At this stage, we are in a position to give a definition of over-generalisation, by drawing a distinction between general and specific over-generalisations. A general over-generalisation applies to any duality and to any reference class. It can be analysed as the ill-grounded conclusion of a secondary pathogenic argument, the premises of which include some general cognitive distortions: dichotomous reasoning, disqualification of one pole, arbitrary focus, minimisation/maximisation, omission of the neutral or requalification in the other pole. It consists of an ungrounded inductive reasoning, because the resulting generalisation is based on an incorrect counting of the corresponding instances. In the same way, a specific over-generalisation consists of an instance of a general over-generalisation, applied to a given duality and reference class. Thus, the specific over-generalisation “All events which occur to me are of a negative nature” (depression, generalised anxiety disorder) applies to the Positive/Negative duality and to the class of the events of the patient’s life. In the same way, “All parts of my body are ugly” (body dysmorphic disorder) is a specific over-generalisation that applies to the reference class of the parts of the patient’s body and to the Beautiful/Ugly duality.

**3. Ungrounded inductive projection**

At this step, it proves to be useful to describe another error of reasoning, which is likely to manifest itself at the stage of secondary pathogenic arguments. It consists of an ill-grounded inductive projection. The latter concludes, from the preceding over-generalisation (2), that a new instance will occur in the near future. Such instance is susceptible to be met in depression (Miranda & al. 2008), as well as in generalised anxiety disorder (Franceschi 2008). In the context of depression, such inductive projection presents the following form:

(2) | All events that occur to me are of a negative nature | premise |

(1_{11a}) | The future event E_{11} of a negative nature may occur | premise |

(1_{11b}) | ∴ The future event E_{11} of a negative nature will occur | from (2), (1_{11a}) |

The corresponding conclusion is susceptible of contributing to depression, notably by triggering the patient’s feeling of despair. Other instances of this type of conclusion are: “My next action will be morally reprehensible” (scrupulosity), or “The next act that he will commit toward me will be bad” (intermittent explosive disorder).

**4. Confirmation bias**

The cognitive process which has just been described illustrates how over-generalisation contributes to the *formation* of pathogenic ideas. However, a process of the same nature is also likely to concur to their *maintenance*. For once the over-generalisation (2) has been established by means of the above reasoning, its maintenance is made as soon as an instance occurs that *confirms* the generalisation according to which all elements *x*_{i} of the reference class E have property Ā. This constitutes a *confirmation bias*, for the patient does only count those elements which present the property Ā, without taking into account those which have the opposite property A, thus disconfirming generalisation (2). Hence, in depression or generalised anxiety disorder, when a new negative event occurs, the patient concludes from it that it confirms that all events which occur to him/her are of a negative nature.

We see it finally, the above developments suggest a classification of cognitive distortions, depending on whether they manifest themselves at the level of primary, secondary or ternary pathogenic arguments. Thus, among the cognitive distortions which arise at the stage of *primary* pathogenic arguments, one can distinguish: on the one hand, the general cognitive distortions (dichotomous reasoning, disqualification of one pole, minimisation/maximisation, requalification into the other pole, omission of the neutral) and on the other hand, the specific cognitive distortions (disqualification of the positive, requalification into the negative, selective abstraction, catastrophising). Morevoer, among the cognitive distortions which manifest themselves at the stage of *secondary* pathogenic arguments, one can mention over-generalisation (at the stage of the *formation* of pathogenic ideas), ill-grounded inductive projection, and confirmation bias (at the stage of the *maintenance* of pathogenic ideas). Mislabelling, finally, is susceptible to occur at the level of *ternary* pathogenic arguments.

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# Theory of Cognitive Distortions: Personalisation

*English translation of a paper published in French in the Journal de Thérapie Comportementale et Cognitive, 2010, 20-2, pages 51-55 under the title “Théorie des distorsions cognitives : la personnalisation”.*

In a previous paper (Complements to a theory of cognitive distorsions, *Journal de Thérapie Comportementale et Cognitive*, 2007), we did present some elements aimed at contributing to a general theory of cognitive distortions. Based on the reference class, the duality and the system of taxa, these elements led to distinguish between the general cognitive distortions (dichotomous reasoning, disqualification of one pole, minimisation, maximisation) and the specific cognitive distortions (disqualifying the positive, selective abstraction, catastrophism). By also distinguishing between three levels of reasoning – the instantiation stage, the interpretation stage and the generalisation stage – we did also define two other cognitive distortions: over-generalisation and mislabelling (Théorie des distorsions cognitives : la sur-généralisation et l’étiquetage, *Journal de Thérapie Comportementale et Cognitive*, 2009). We currently extend this model to another classical cognitive distortion: personalisation.

This article is cited in:

- Juliette Marty, Et si vous étiez trop perfectionniste ? Eyrolles, 2021

**Theory of Cognitive Distortions: Personalisation**

In Franceschi (2007), we set out to introduce several elements intended to contribute to a general theory of cognitive distortions. These elements are based on three fundamental notions: the reference class, the duality and the system of taxa. With the aid of these three elements, we could define within the same conceptual framework the *general* cognitive distortions such as dichotomous reasoning, disqualification of one pole, minimisation and maximisation, as well as requalification in the other pole and omission of the neutral. In the same way, we could describe as *specific* cognitive distortions: disqualification of the positive, selective abstraction and catastrophising. In Franceschi (2009), we introduced three levels of reasoning – the instantiation stage, the interpretation stage and the generalisation stage, which allowed to define within the same conceptual framework, two other classical cognitive distortions: *over-generalisation* and *mislabelling*. In the present paper, we set out to define and to situate in this conceptual framework another classical cognitive distortion: *personalisation*.

Personalisation constitutes one of but twelve classically defined cognitive distortions: emotional reasoning; over-generalisation; arbitrary inference; dichotomous reasoning; should statements; divination or mind-reading; selective abstraction; disqualification of the positive; maximisation/minimisation; catastrophising; personalisation; mislabelling (Ellis 1962, Beck 1964). Personalisation is usually defined as the fact of attributing unduly to oneself the cause of an external event. For example, seeing a person who laughs, the patient thinks that it is because of his/her physical appearance. Also, the patient makes himself/herself responsible for a negative event, in an unjustified way. If his/her companion then failed his/her examination, the patient estimates that is due to the fact that he/she is depressed. In what follows, we propose first to clarify the definition of personalisation and to situate it in the context of the theory of cognitive distortions (Franceschi 2007, 2009). Secondly, we set out to clarify the relationships existing between personalisation and several close notions mentioned in the literature: personalising bias (Langdon et al. 2006), ideas of reference (Startup & Startup 2005, Meyer & Lenzenweger 2009) and delusions of reference.

**Personalisation and post hoc fallacy**

We will set out first to highlight the mere structures of the cases of personalisation. Let us consider the aforementioned example where the patient sees a person who laughs and thinks that this one laughs because of the patient’s physical appearance. This constitutes an instance of personalisation. We can describe more accurately the reasoning which underlies such instance (in what follows, the symbol denotes the conclusion):

(P1_{1}) | in T_{1} I went for a walk | premiss_{1} |

(P1_{2}) | in T_{2} the peasant started to laugh | premiss_{2} |

(P1_{3}) | ∴ in T_{2} the peasant started to laugh because he saw that in T_{1} I went for a walk | conclusion_{1} |

(P1_{4}) | ∴ in T_{2} the peasant made fun of me | conclusion_{2} |

The patient puts here in relationship an internal event (“I went for a walk”) with an external event (“the peasant started to laugh”). He/she concludes then that the internal event is the *cause* of the external event. In this stage, the patient “personalises” an external event, which he/she considers to be the effect of an internal event, while this external event is in reality devoid of any relationship with the patient himself/herself. In a subsequent stage (P1_{4}), the patient interprets the previous conclusion (P1_{3}) by considering that the peasant made fun of him.

At this stage, it is worth wondering about the specific nature of the patient’s error of reasoning. It appears here that both premises (P1_{1}) and (P1_{2}) constitute genuine facts and therefore turn out to be true. On the other hand, the conclusion (P1_{3}) which concludes to the existence of a relation of causality between two consecutive events E_{1} (“In T_{1} I went for a walk”) and E_{2} (“In T_{2} the peasant started to laugh”) appears to be unjustified. Indeed, both premises are only establishing a relation of anteriority between the two consecutive facts E_{1} and E_{2}. And the conclusion (P1_{3}) which deducts from it a relation of causality turns out therefore to be too strong. The argument proves here to be invalid and the corresponding reasoning is then fallacious. The corresponding error of reasoning, which concludes to a relation of causality whereas there is only a mere relation of anteriority, is classically termed *post hoc fallacy*, according to the Latin sentence “Post hoc, ergo propter hoc” (after this therefore because of this). It consists here of a very common error of reasoning, which is notably at the root of many superstitions (Martin 1998, Bressan 2002).

In this context, we can point out that the case of post hoc fallacy which has just been described as an argument of personalisation, also constitutes a case of *arbitrary inference*, another classically defined cognitive distortion.

**Steps of instantiation, of interpretation and of generalisation at the level of the arguments of personalisation**

At this step, it proves to be useful to draw a distinction between the levels of arguments that lead to personalisation as cognitive distortion. This leads to differentiate three levels within the arguments of personalisation, among the reasoning’ stages. The latter correspond respectively to three different functions: it consists of the successive stages of *instanciation*, of *interpretation* and of *generalisation*. To this end, it is useful to describe the whole reasoning which underlies the arguments of personalisation and which includes the three aforementioned stages:

(P1_{1}) | in T_{1} I went for a walk | premiss_{11} |

(P1_{2}) | in T_{2} the peasant started to laugh | premiss_{12} |

(P1_{3}) | ∴ en T_{2} the peasant started to laugh because he saw that in T_{1} I went for a walk | conclusion_{11} |

(P1_{4}) | ∴ in T_{2} the peasant made fun of me | conclusion_{12} |

(P2_{1}) | in T_{3} I was leafing through a magazine in the library | premiss_{21} |

(P2_{2}) | in T_{4}, the librarian smirked | premiss_{22} |

(P2_{3}) | ∴ en T_{4} the librarian smirked because in T_{3} I was leafing through a magazine in the library | conclusion_{21} |

(P2_{4}) | ∴ in T_{4}, the librarian made fun of me | conclusion_{22} |

(P3_{1}) | in T_{5} I did enter in the show-room | premiss_{31} |

(P3_{2}) | in T_{6}, my colleagues started to laugh | premiss_{32} |

(P3_{3}) | ∴ in T_{6}, my colleagues started to laugh because in T_{5} I did enter in the show-room | conclusion_{31} |

(P3_{4}) | ∴ in T_{6}, my colleagues were laughing at me | conclusion_{32} |

(…) | ||

(P10_{5}) | ∴ people make fun of me | from (P1_{4})-(P10_{4}) |

Here, the instances of the previous arguments (P1_{1})-(P1_{3}), (P2_{1})-(P2_{3}), (P3_{1})-(P3_{3}), etc. constitute primary stages of arguments of personalisation, by which the patient considers that an event related to him/her is the cause of an external event. This type of argument corresponds to the stage of instantiation. As mentioned earlier, such argument is fallacious since it is based on post hoc fallacy. In a subsequent stage the function of which is interpretative, and that is aimed at making sense of the conclusions (P1_{3}), (P2_{3}), (P3_{3}), … of the instances of arguments of the previous type, the patient interprets it by concluding that some people made fun of him. Such conclusions (P1_{4}), (P2_{4}), (P3_{4}) appear to be grounded, *inasmuch* as the premisses (P1_{3}), (P2_{3}), (P3_{3}) are true. Finally, in a subsequent stage of generalisation, the patient enumerates some instances or circumstances where he/she thinks that people laughed or made fun of him/her ((P1_{4}), (P2_{4}), (P3_{4}), …) and generalises then to the conclusion (P10_{5}) according to which people make fun of him/her. This last stage is of an inductive nature, and corresponds to an enumerative induction, the structure of which is the following:

(P1_{4}) | in T_{2} the peasant made fun of me | conclusion_{12} |

(P2_{4}) | in T_{4}, the librarian made fun of me | conclusion_{22} |

(P3_{4}) | in T_{6}, my colleagues were laughing at me | conclusion_{32} |

(…) | ||

(P10_{5}) | ∴ people make fun of me | from (P1_{4})-(P10_{4}) |

Given what precedes, we can from now on provide a definition of personalisation. The preceding analysis leads then to distinguish between three stages in arguments of personalisation. At the level of primary arguments of personalisation (stage of instantiation), it consists of the tendency in the patient to establish an unjustified relation of causality between two events, among which one is external and the other one is internal to the patient. The patient personalises then, that is to say puts in relationship with himself/herself, an external event, which proves to be in reality devoid of any relation of causality. The mechanism which underlies such argument consists then of the erroneous attribution of a relation of causality, based on post hoc fallacy. At the level of secondary arguments of personalisation (stage of interpretation), the patient makes sense of the previous conclusion by concluding that *at a given time*, a person (or several persons) made fun of him, laughed at him, etc. Finally, at the level of arguments of ternary personalisation (stage of generalisation), the patient concludes that, *in a general way*, people make fun of him.

**Personalisation and personalising bias**

At this step, it proves to be useful to distinguish personalisation as cognitive distortion from *personalising bias*. The latter is defined as an attribution bias (“personalising attributional bias”), by whom the patient attributes to other persons rather than to circumstances the cause of a negative event (McKay & al. 2005, Langdon & al. 2006). Personalising bias is often related to polythematic delusions (Kinderman & Bentall 1997, Garety & Freeman 1999, McKay & al. 2005) met in schizophrenia.

Considering this definition, the difference between the two notions can be thus underlined: in personalisation as cognitive distortion, the patient attributes the cause of an external event to an event which concerns the patient himself/herself; on the other hand, in personalising bias the patient attributes the cause of an internal event to external persons. This allows to highlight several fundamental differences between the two notions. Firstly, in personalisation as cognitive distortion, the “person” is the patient himself/herself, while in personalising bias, it consist of external “persons”. Secondly, in the structure of personalisation, an internal event precedes an external event; by contrast, in the scheme of personalising bias, it is an external event which precedes an internal event. Finally, in personalisation as cognitive distortion, the internal event is indifferently of a positive, neutral or negative nature, whereas in personalising bias, the internal event is of a negative type. Hence, it finally proves to be that both notions appear fundamentally distinct.

**Personalisation and ideas of reference**

It appears also useful, for the sake of clarity, to specify the relationships between personalisation and *ideas of reference*. It is worth preliminary mentioning that one usually distinguishes between ideas of reference and delusions of reference (Dziegielewski 2002, p. 266). Ideas of reference characterise themselves by the fact that a patient considers that insignificant events relate to himself/herself, while is not the case in reality. For example, the patient hears several persons laugh, and considers, in an unjustified way, that the latter make fun of him/her. In parallel, delusions of reference constitute one of the most salient symptoms noticed in schizophrenia, and leads the patient to be persuaded that the media, television, or the radio speak about him/her or issue messages concerning him/her. Several criteria allow to draw a distinction between ideas of reference and delusions of reference. First, ideas of reference have much less impact on the patient’s life than reference delusions of reference. Second, the degree of conviction which is associated with ideas of reference is far lesser than with delusions of reference. Lastly, ideas of reference (“the neighbour made fun of me”) are related with beliefs the degree of plausibility of which is much stronger than the one which is inherent to delusions of reference (“newspapers speak about me”).

In this context, the aforementioned arguments of personalisation (P1_{1})-(P1_{4}), (P2_{1})-(P2_{4}), and (P3_{1})-(P3_{4}), by whom the patient concludes that some people make fun of him, corresponds completely to the definition of ideas of reference. It appears then that personalisation, such as it was defined above as cognitive distortion, identifies itself with ideas of reference.

**Personalisation and delusion of reference**

One traditionally distinguishes at the level of polythematic delusions met in schizophrenia between: delusions of reference, delusions of influence, delusions of control, telepathy-like delusions, delusions of grandeur, and delusions of persecution. Delusions of reference leads for example the patient to believe with a very strong conviction that the media, the newspapers, the television speak about him/her.

It is worth describing here a mechanism which is susceptible to lead to the formation of delusions of reference. Such mechanism appears to be grounded on a reasoning (Franceschi 2008) which includes, as well as the above-mentioned primary instances of personalisation, a post hoc fallacy:

(DR1_{1}) | in T_{1} I was drinking an appetizer | premiss_{11} |

(DR1_{2}) | in T_{2} the presenter of the show said: “Stop drinking!” | premiss_{12} |

(DR1_{3}) | ∴ in T_{2} the presenter of the show said: “Stop drinking!” because in T_{1} I was drinking an appetizer | conclusion_{11} |

(DR1_{4}) | ∴ in T_{2} the presenter of the show spoke about me | conclusion_{12} |

Consider also this second instance :

(DR2_{1}) | in T_{3} I hardly got out of bed | premiss_{21} |

(DR2_{2}) | in T_{4} the radio presenter said: “Be forceful:” | premiss_{22} |

(DR2_{3}) | ∴ in T_{4} the radio presenter said: “Be forceful:” because in T_{3} I hardly got out of bed | conclusion_{21} |

(DR2_{4}) | ∴ in T_{4} the radio presenter spoke about me | conclusion_{22} |

At the level of the instantial step (DR1_{1})-(DR1_{3}), (DR2_{1})-(DR2_{3}), … the patient concludes here that an internal event is the *cause* of an external event. In a further interpretative stage, he/she interprets the conclusions (DR1_{3}), (DR2_{3}), … of the preceding arguments by considering that the presenters of radio or of television speak about him/her. Finally, in a generalisation step, of inductive nature, the patient enumerates the conclusions (DR1_{4}), (DR2_{4}), … of secondary arguments (interpretation stage) and generalises thus:

(DR1_{4}) | ∴ in T_{2} , the presenter of the show spoke about me | |

(DR2_{4}) | ∴ in T_{4}, the radio presenter spoke about me | |

(…) | ||

(DR10_{5}) | ∴ the media speak about me | conclusion |

It proves then that the structure of the mechanism which leads to the formation of delusions of reference thus described, is identical to that of the reasoning which leads to ideas of reference which is associated with personalisation as cognitive distortion.

Finally, it appears that the preceding developments allow to provide a definition of personalisation and to situate it in the context of cognitive distortions (Franceschi 2007, 2009). Personalisation is then likely to manifest itself at the level of primary, secondary or ternary pathogenic arguments, which correspond respectively to the stages of instantiation, of interpretation, and of generalisation. At the level of primary pathogenic arguments, corresponding to a function of instantiation, it consists of instances, the conclusions of which lead the patient to conclude in an unjustified way that some external events are caused by some of his/her actions. At the level of secondary pathogenic arguments, which correspond to a function of interpretation, personalisation takes the form of a reasoning by which the patient interprets the conclusion of primary pathogenic argument by concluding for example that people make fun of him/her. Finally, at the level of ternary pathogenic arguments, associated with a function of generalisation, the patient generalises from the conclusions of several secondary pathogenic arguments and concludes that, in a general way, people make fun of him/her.

Lastly, it appears that the previous definition of personalisation as cognitive distortion allows to describe precisely the relationships between personalisation and close notions such as personalising bias, ideas of reference and delusions of reference.

### References

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Franceschi P. Une défense logique du modèle de Maher pour les délires polythématiques. *Philosophiques*, 2008, 35-2, 451-475. English translation.

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# Theory of Cognitive Distortions: Application to Generalised Anxiety Disorder

*English translation and postprint (with additional illustrations) of a paper published in French under the title “Théorie des distorsions cognitives : application à l’anxiété généralisée” in the Journal de Thérapie Comportementale et Cognitive, 2008, 18, pp. 127-131.*

This article follows the contribution to the general theory of cognitive distortions exposed in “Complements to a theory of cognitive distorsions” (Journal de Thérapie Comportementale et Cognitive, 2007). The elements described, namely the reference class, the duality and the system of taxa, are applied here to generalised anxiety disorder. On the one hand, these elements allow to describe the cognitive distortions which are specific to generalised anxiety disorder, consistent with recent work emphasising the role played uncertain situations relative to future events. On the second hand, they allow to define a type of structured reasoning, of inductive nature, which leads to the formation and maintenance of anxious ideas.

This paper is cited in:

- Étude des profils de distorsion cognitive en fonction des états anxieux et dépressifs chez des adultes tout-venant, Anita Robert, Nicolas Combalbert, Valérie Pennequin, Annales Médico-psychologiques, revue psychiatrique, Volume 176-3, 2018, pages 225-230
- Deperrois Romain & Nicolas Combalbert, Links between cognitive distortions and cognitive emotion regulation strategies in non-clinical young adulthood, in Psychological Applications and Trends, Eb. by C. Pracana & M. Wang, InScience Press, 2019
- Nawal Ouhmad, Nicolas Combalbert, Wissam El-Hage, Cognitive distortions and emotion regulation among post traumatic stress disorder victims, in Psychological Applications and Trends, Eb. by C. Pracana & M. Wang, InScience Press, 2019
- A.Robert, N.Combalbert, V.Pennequin, R.Deperrois, N.Ouhmad, Création de l’Échelle de Distorsions Cognitives pour adultes (EDC-A) : étude des propriétés psychométriques en population générale et association avec l’anxiété et la dépression, Psychologie Française, 2021

**Theory of Cognitive Distortions: Application to Generalised Anxiety Disorder**

In Franceschi (2007), we set out to introduce several elements intended to contribute to a general theory of cognitive distortions. These elements are based on three basic notions: the reference class, the duality and the system of taxa. With the help of these three elements, we could define within the same conceptual framework the general cognitive distortions such as dichotomous reasoning, the disqualification of one pole, minimisation and maximisation, as well as the requalification in the other pole and the omission of the neutral. In addition, we could describe as specific cognitive distortions: the disqualification of the positive, selective abstraction and catastrophism.

In what follows, we offer to extend this work by applying it in a specific way to generalised anxiety disorder (GAD), in order to allow their use within cognitive therapy. The present study inserts itself in the context of recent work (Butler & Mathews 1983, 1987, Dalgleish et al. 1997), which notably underlined the major role played, in the context of GAD, by indeterminate situations, and especially by uncertain situations relating to future events. Recent developments, emphasising especially the intolerance with regard to indeterminate future situations, echoed this (Dugas et al. 2004, Canterbury et al. 2004, Carleton et al. 2007).

We shall be interested successively in two main forms of reasoning likely to occur in the context of GAD: on the one hand, the cognitive distortions which are specific to GAD; on the other hand, a structured argument relating to GAD and grounded on inductive logic, which is likely to include one or several of the aforementioned cognitive distortions.

**Cognitive distortions in the context of generalized anxiety disorder**

The conceptual framework defined in Franceschi (2007) is based on three fundamental elements: the duality, the reference class, and the system of taxa, which allow to define the general cognitive distortions. These three notions also allow to describe the specific cognitive distortions which are applicable to GAD. In this context, as we will see it, the reference class for the latter specific cognitive distortions identifies itself with the class of *future events of the patient’s life*. Moreover, the duality assimilates itself to the *Positive/Negative* duality. Finally, for the sake of the present discussion, we shall make use of the system of taxa (its choice is more or less arbitrary) described in Franceschi (2007), which includes 11 taxa, denoted by E_{1} to E_{11}, where E_{6} denotes the neutral taxon. Such conceptual framework allows then to define the specific cognitive distortions in the context of GAD. We offer to examine them in turn.

**Dichotomous reasoning**

An instance of dichotomous reasoning related to GAD consists for the patient to only consider future events from the viewpoint of the extreme taxa corresponding to each pole of the Positive/Negative duality. Hence, the patient only considers future events which present either a very positive, or a very negative nature. All other events, being either neutral, positive or negative to a lesser degree, are thus ignored. This type of reasoning can be analysed as an instance of dichotomous reasoning, applied to the class of the events of the patient’s future life and to the Positive/Negative duality.

**Disqualification of one pole**

An instance of the disqualification of one pole related to GAD consists for the patient to only envisage, among future events likely to occur, those which present a negative nature. The patient tends then to be unaware of positive future events that could happen, by considering that they do not count, for this or that reason. In the present context, this type of reasoning can be analysed as an instance of disqualification of one pole, applied to the reference class of the events of the patient’s future life and to the Positive/Negative duality, i.e. disqualification of the positive.

**Arbitrary focus on a given modality**

In GAD, a typical instance of arbitrary focus, consists for the patient to focus on a possible future event, the nature of which turns out to be negative. This can be analysed as focusing on one of the taxa of the Positive/Negative duality, at the level of the class of the future events of the patient’s life.

**Omission of the neutral**

A specific instance for GAD consists for the patient to be completely unaware of possible future events the nature of which is neutral, i.e. those which are neither positive nor negative.

**Requalification into the other pole**

In the context of GAD, the corresponding cognitive distortion consists in requalifying as negative a possible future event, whereas it should be considered objectively as positive. Such cognitive distortion consists of a requalification in the other pole applied to the reference class of the future events of the patient’s life and to the Positive/Negative duality, i.e. requalification in the negative.

**Minimisation or maximisation**

A specific instance of minimisation applied to GAD consists for the patient to consider some possible future events as less positive than they truly are in reality. With maximisation, the patient considers some possible future events as more negative than they objectively are.

**Primary, secondary and tertiary anxiogenous arguments**

At this stage, it is worth also considering a certain type of reasoning, likely to be met in GAD, which can include several instances of the aforementioned cognitive distortions. This type of reasoning presents an anxiogenous nature, because it leads the patient to predict that a future event of negative nature is going to occur. Such reasoning is underlain by a structure which presents an *inductive* nature. Before analysing in detail the different steps of the corresponding reasoning, it is worth describing preliminarily its internal structure. The latter is the following (in what follows, the symbol ∴ denotes the conclusion):

(1) the event E_{1} of negative nature did occur to me | premiss |

(2) the event E_{2} of negative nature did occur to me | premiss |

(3) the event E_{3} of negative nature did occur to me | premiss |

(…) | premiss |

(10) the event E_{10} of negative nature did occur to me | premiss |

(11) all events that occur to me are of negative nature | from (1)-(10) |

(12) « I am always unlucky », « I am ill-fated » | from (11) |

(13) the future event E_{11} of negative nature may occur | premiss |

(14) ∴ the future event E_{11} of negative nature will occur | from (11), (13) |

The essence of such reasoning is of a logically inductive nature. The patient enumerates then some events of his/her past or present life, the nature of which he/she considers as negative. He/she reaches then by generalisation the conclusion according to which all events that which occur to him/her are negative. From this generalisation, he/she infers a prediction relating to a future event, likely to happen, which he/she considers as negative. The patient is thus led to the anxiogenous conclusion that an event of negative nature is going to occur.

In such reasoning, it is worth pointing out that the reference class identifies itself with the class of *past, present and future events*, of the patient’s life. Typically, in this type of reasoning, the generalisation is grounded on present or past events, while a future event is the object of the corresponding inductive prediction. This is different from the reference class applicable to the cognitive distortions mentioned above, where the reference class identifies itself exclusively with the *future events* of the patient’s life.

At this stage, it proves to be necessary to identify the fallacious steps in the patient’s reasoning, to allow their use in cognitive therapy of GAD. To this end, we can differentiate several steps in the structure of the corresponding reasoning. It proves indeed that some steps are valid arguments (an argument is valid when its conclusion is true if its premises are true), while others are invalid. For this purpose, it is worth drawing within this type of reasoning, a distinction between primary, secondary or ternary anxiogenous arguments.

**Primary anxiogenous arguments**

The first step in the type of aforementioned reasoning, consists for the patient to think to a past negative event, in the following way:

(1) the event E_{1} of negative nature did occur to me |

It is however possible to describe more accurately the corresponding cognitive process, under the form of an argument that we can term a primary anxiogenous argument, the structure of which is the following:

(1a) the event E_{1} did occur to me | |

(1b) the event E_{1} was of negative nature | |

(1) ∴ the event E_{1} of negative nature did occur to me | from (1a), (1b) |

By such cognitive process, the patient is led to the conclusion that some negative event did occur to him/her. This type of argument proves to be entirely valid inasmuch as the event in question presents well, objectively, a negative nature. However, it can also turn out to be invalid, if the event in question presents, objectively, a positive (or neutral) nature. What is then defective in this type of reasoning, is the fact that premise (1b) turns then out to be false. Such can notably be the case for example if the patient makes use of a cognitive distortion such as requalification in the negative. In such case, the patient considers then as negative an event the nature of which is objectively positive.

**Secondary anxiogenous arguments**

Anxiogenous secondary arguments are constituted, at the level of the above-mentioned reasoning, of the part that takes into account the instances (1)-(10) and proceeds then by *generalisation*. The patient counts thus some instances of events that did occur to him/her, the nature of which he/she considers as negative, and concludes that all events that did occur to him/her were negative, in the following way:

(1) the event E_{1} of negative nature did occur to me | |

(2) the event E_{2} of negative nature did occur to me | |

(…) | |

(10) the event E_{10} of negative nature did occur to me | |

(11) ∴all events that occur to me are of negative nature | from (1)-(10) |

Such generalisation *may* constitute a fully *valid* argument. For the resulting generalisation constitutes a fully correct inductive reasoning, if the premises (1)-(10) are true. However, such type of reasoning is most often defective from two different viewpoints, thus distorting the conclusion which results from it. Above all, as we have did just see it, some past events of positive nature can have been counted among the number of negative events, by the effect of a *requalification in the negative*. In that case, the enumeration of instances includes then some false premises, thus invalidating the resulting generalisation. Secondly, some past or present positive (or neutral) events can have been omitted in the corresponding enumeration. Such omission can result from the use of some cognitive distortions, such as *disqualification of the positive*. In such case, the relevant class of reference consisting in present and past events of the patient’s life is only taken into account in a partial or erroneous way. The corresponding reasoning remains then logically valid, but proves to be incorrect, since it takes into account only partly the relevant instances within the reference class, that of the present and past events of the patient’s life.

As we see it finally, the patient proceeds then to a reconstruction of the relevant reference class which proves to be erroneous, due to the use of the following specific cognitive distortions: *requalification in the negative* and *disqualification of the positive* (and possibly, *omission of the neutral*). The corresponding type of reasoning is illustrated on the figure below:

*omission of the neutral*,

*requalification in the negative*, and then

*disqualification of the positive*

Such mechanism, as we did see it, illustrates how the *formation* of anxious ideas is made. However; a mechanism of the same nature is also likely to contribute to their *maintenance*. For once the generalisation (11) according to which all events which occur to the patient are of negative nature, has been established by means of the above reasoning, its maintenance is made as soon as an event occurs which confirms this latter generalisation. When a new negative event indeed happens, the patient concludes from it that it confirms generalisation (11). Such mechanism, at the stage of the maintenance of anxious ideas, constitutes a *confirmation bias*. For the patient only counts those events of negative nature related to him/her that *confirm* indeed the generalisation (11), but without taking into account those events of positive nature which occur to him/her and that would then *disconfirm* the idea according to which all events which occur to him/her are of negative nature.

**Ternary anxiogenous arguments**

Lastly, it is worth mentioning the role played by ternary anxiogenous arguments which consist, at the level of the aforementioned reasoning, in the following sequence:

(11) ∴all events that occur to me are of negative nature

(12) ∴« I am always unlucky », « I am ill-fated »

It consists here of an argument which follows the conclusion of the secondary anxiogenous argument (11), and which, by an additional step (12), aims at interpreting it, at making sense of it. The patient interprets here the fact that the events which occur to him/her are negative, due to the fact that he/she is unlucky, ill-fated.

As we did see it, the interest of drawing a distinction between three types of arguments resides in the fact that each of them has a specific function: the primary stage proceeds by enumerating the instances, the secondary stage operates by generalisation, and the ternary stage, lastly, proceeds by interpretation (Franceschi 2008).

The present study, as we see it, extends recent work (Butler and Mathews on 1987, Dalgleish et al. 1997) emphasising the role played, in GAD, by anticipations concerning indeterminate situations related to future events. In this context, the specific cognitive distortions as well as a reasoning of an inductive structure, contribute then to the vicious circle (Sgard et al. 2006), which results from the process of formation and maintenance of the anxious state.

**References**

Butler G & Matews A. Cognitive processes in anxiety. *Advances in Behaviour Research and* *Therapy* 1983 ; 5 : 51-62.

Butler G & Matews A. Anticipatory anxiety and risk perception. *Cognitive Therapy and Research* 1987 ; 11 : 551-565.

Carleton R, Norton M & Aslundson G. Fearing the unknown: A short version of the Intolerance of Uncertainty Scale. *Journal of Anxiety Disorders* 2007 ; 21-1 : 105-117.

Canterbury R, Golden A, Taghavi R, Neshat-Doost H, Moradi A & Yule W. Anxiety and judgements about emotional events in children and adolescents. *Personality and Individual Differences* 2004 ; 36 : 695-704.

Dalgleish T, Taghavi R, Neshat-Doost H, Moradi A, Yule W & Canterbury R. Information processing in clinically depressed and anxious children and adolescents. *Journal of Child Psychology and Psychiatry* 1997 ; 38 : 535-541.

Dugas M, Buhr K & Ladouceur R. The role of intolerance of uncertainty in etiology and maintenance. In R. Heimberg, C. Turk, & D. Mennin (Eds.). Generalized anxiety disorder: Advances in research and practice. Guilford, New York, 2004(143-163).

Franceschi P. Compléments pour une théorie des distorsions cognitives. *Journal de Thérapie Comportementale et Cognitive* 2007 ; 17-2 : 84-88. English translation.

Franceschi P. Une défense logique du modèle de Maher pour les délires polythématiques. *Philosophiques* 2008 ; 35-2 : 451-475. English translation.

Sgard F, Rusinek S, Hautekeete M & Graziani P. Biais anxieux de perception des risques. *Journal de Thérapie Comportementale et Cognitive* 2006 ; 16-1 : 12-15.

# Probabilistic Situations for Goodmanian N-universes

*A paper appeared (2006) in French in the Journal of Philosophical Research, vol. 31, pages 123-141, under the title “Situations probabilistes pour n-univers goodmaniens.”*

I proceed to describe several applications of the theory of n-universes through several different probabilistic situations. I describe first how n-universes can be used as an extension of the probability spaces used in probability theory. The extended probability spaces thus defined allow for a finer modelling of complex probabilistic situations and fits more intuitively with our intuitions related to our physical universe. I illustrate then the use of n-universes as a methodological tool, with two thought experiments described by John Leslie. Lastly, I model Goodman’s paradox in the framework of n-universes while also showing how these latter appear finally very close to goodmanian worlds.

**Probabilistic Situations for Goodmanian N-universes**

The *n-universes* were introduced in Franceschi (2001, 2002) in the context of the study of the probabilistic situations relating to several paradoxes which are currently the object of intensive studies in the field of analytical philosophy: Goodman’s paradox and the Doomsday Argument. The scope of the present article is twofold: on one hand, to describe how modelling within the n-universes allows to extend the properties of the classical probability spaces used in probability theory, by providing at the same time a finer modelling of some probabilistic situations and a better support for intuition; on the other hand, to show how the use of n-universes allows to simplify considerably the study of complex probabilistic situations such as those which appear in the study of paradoxes.

When one models for example the situation corresponding to the drawing of a ball from an urn, one considers then a restricted temporal space, which limits itself to the few seconds that precede and follow the drawing. Events which took place the day before or one hour before, but also those who will happen for example the day after the drawing, can be purely and simply ignored. A very restricted interval of time, that it is possible to reduce to one or two discrete temporal positions, is then enough for characterising the corresponding situation. It suffices also to consider a restriction of our universe where the space variable is limited to the space occupied by the urn. For it is not useful to take into consideration the space corresponding to the neighbouring room and to the objects which are there. In a similar way, the number of atoms of copper or of molybdenum that are possibly present in the urn, the number of photons which are interacting with the urn at the time of the drawing, or the presence or absence of a sound source of 75 db, etc. can be omitted and ignored. In this context, it is not necessary to take into account the existence of such variables. In such situation, it is enough to mention the variables and constants really used in the corresponding probabilistic situation. For to enumerate all the constants and the variables which describe of our whole universe appears here as an extremely complicated and moreover useless task. In such context, one can legitimately limit oneself to describe a simplified universe, by mentioning only those constants and variables which play a genuine role in the corresponding probabilistic situation.

Let us consider the drawing of a ball from an urn which contains several balls of different colours. To allow the calculation of the likelihood of different events related to the drawing of one or several balls from the urn, probability theory is based on a modelling grounded on probability spaces. The determination of the likelihood of different events is then not based on the modelling of the physical forces which determine the conditions of the drawing, i.e. the mass and the dimensions of the balls, the material of which they are constituted, their initial spatio-temporal position, as well as the characteristics of the forces exercised over the balls to perform a random drawing. The modelling of random phenomena with the help of probability spaces does only retain some very simplified elements of the physical situation which corresponds to the drawing of a ball. These elements are the number and the colour of the balls, as well as their spatio-temporal position. Such methodological approach can be generalised in other probabilistic situations that involve random processes such as the drawing of one or several dices or of one or several cards. Such methodology does not constitute one of the axioms of probability theory, but it consists here of one important tenet of the theory, of which one can suggest that it would be worth being more formalized. It may also be useful to explain in more detail how the elements of our physical world are converted into probability spaces. In what follows, I will set out to show how the probability spaces can be extended, with the help of the theory of n-universes, in order to better restore the structure of the part of our universe which is so modelled.

1**. Introduction to n-universes**

It is worth describing preliminarily the basic principles underlying the n-universes. N-universes constitute a simplified model of the physical world which is studied in a probabilistic situation. Making use of Ockam’s razor, we set out then to model a physical situation with the help of the simplest universe’s model, in a compatible way however with the preservation of the inherent structure of the corresponding physical situation. At this stage, it proves to be necessary to highlight several important features of n-universes.

**1.1. Constant-criteria and variable-criteria**

The criteria of a given n-universe include both constants and variables. Although n-universes allow to model situations which do not correspond to our physical world, our concern will be here exclusively with the n-universes which correspond to common probabilistic situations, in adequacy with the fundamental characteristics of our physical universe. The corresponding n-universes include then at the very least one temporal constant or variable, as well as one constant or variable of location. One distinguishes then among n-universes: a T_{0}L_{0} (a n-universe including a temporal constant and a location constant), a T_{0}L (a temporal constant and a location variable), a TL_{0} (a temporal variable and a location constant), a TL (a temporal variable and a location variable). Other n-universes also include a constant or a variable of colour, of direction, etc.

**1.2. N-universes with a unique object or with multiple objects**

Every n-universe includes one or several objects. One distinguishes then, for example: a _{0}TL_{0} (n-world including a unique object, a temporal variable and a constant of location), a TL_{0} (multiple objects, a temporal variable and a location constant).

**1.3. Demultiplication with regard to a variable-criterion**

It is worth highlighting the property of *demultiplication* of a given object with regard to a variable-criterion of a given n-universe. In what follows, we shall denote a variable-criterion with demultiplication by *. Whatever variable-criterion of a given n-universe can so be demultiplicated. The fact for a given object to be demultiplicated with regard to a criterion is the property for this object to exemplify several taxa of criterion . Let us take the example of the time criterion. The fact for a given object to be demultiplicated with regard to time resides in the fact of exemplifying several temporal positions. In our physical world, an object _{0} can exist at several (successive) temporal positions and finds then itself demultiplicated with regard to the time criterion. Our common objects have then a property of temporal persistence, which constitutes a special case of temporal demultiplication. So, in our universe of which one of the variable-criteria is time, it is common to note that a given object _{0} which exists at T_{1} also exists at T_{2}, …, T_{n}. Such object has a life span which covers the period T_{1}-T_{n}. The corresponding n-universe presents then the structure _{0}T*L_{0} (T* with simplified notation).

**1.4. Relation one/many of the multiple objects with a given criterion**

At this stage, it proves to be necessary to draw an important distinction. It is worth indeed distinguishing between two types of situations. An object can thus exemplify, as we did just see it, several taxa of a given variable-criterion. This corresponds to the case of *demultiplication* which has just been described with regard to a given variable-criterion. But it is also worth taking into account another type of situation, which concerns only those n-universes with multiple objects. Indeed, several objects can instantiate the same taxon of a given criterion. Let us consider first the temporal criterion. Let us place ourselves, for example, in a n-universe with multiple objects including at the same time a temporal variable and a location constant L_{0}. This can correspond to two types of different n-universes. In the first type of n-universe, there is one single object by temporal position. At some point in time, it is therefore only possible to have a unique object in L_{0} in the corresponding n-universe. We can consider in that case that every object of this n-universe is in relation *one* with the time taxa. We denote by T*L_{0} (with simplified notation T) such n-universe. Let us consider now a n-universe with multiple objects including a temporal variable and a location constant, but where several objects _{1}, _{2}, _{3} can exist at the same time. In that case, the multiple objects are at a given temporal position in L_{0}. The situation then differs fundamentally from the T*L_{0}, because several objects can now occupy the same given temporal position. In other words, the objects can co-exist at a given time. In that case, one can consider that the objects are in relation *many* with the temporal taxa. We denote then by *T*L_{0} such n-universe (with simplified notation *T) .

Let us place ourselves now from the point of view of the location criterion. Let us consider a n-universe with multiple objects including at the same time a temporal variable and a variable of location, and where the objects are in relation *many* with the temporal criterion. It is also worth distinguishing here between two types of n-universes. In the first, a single object can find itself at a given taxon of the location criterion at the same time. There is then one single object by space position at a given time. This allows for example to model the situation which is that of the pieces of a chess game. Let us denote by *TL such n-universe (with simplified notation *TL). In that case, the objects are in relation *one* with the location criterion. On the other hand, in the second type of n-universe, several objects can find themselves in the same taxon of a location criterion at the same time. Thus, for example, the objects _{1}, _{2}, _{3} are in L_{1} at T_{1}. Such situation corresponds for example to an urn (which is thus assimilated with a given taxon of location) where there are several balls at a given time. We denote by *T*L such n-universe, where the objects are in relation *many* with the location taxa.

One can notice lastly that such differentiation is also worth for the variable-criterion of colour. One can then draw a distinction between: (a) a *T_{0}*L_{0}C (with simplified notation C) where several objects which can co-exist at the same time in a given space position present all necessarily a different colour, because the objects are in relation *one* with the colour criterion there; (b) a *T_{0}*L_{0}*C (with simplified notation *C) where several objects which can co-exist at the same time at a given space position can present the same colour, because the objects are in relation *many* with the colour criterion there.

**1.5. Notation**

At this stage, it is worth highlighting an important point which concerns the used notation. It was indeed made use in what precedes of an extended and of a simplified notation. The extended notation includes the explicit specification of all criteria of the considered n-universe, including at the same time the variable-criteria and the constant-criteria. By contrast, the simplified notation includes only the explicit specification of the variable-criteria of the considered n-universe. For constant-criteria of time and of location of the considered n-universe can be merely deduced from variable-criteria of the latter. This is made possible by the fact that the studied n-universes include, in a systematic way, one or several objects, but also a variable-criterion or a constant-criterion of time and of location.

Let us illustrate what precedes by an example. Consider first the case where we situate ourselves in a n-universe including multiple objects, a constant-criterion of time and a constant-criterion of location. In that case, it appears that the multiple objects exist necessarily at T_{0}. As a result, in the considered n-universe, the multiple objects are in relation *many* with the constant-criterion of time. And also, there exist necessarily multiple objects at L_{0}. So, the multiple objects are also in relation *many* with the constant-criterion of location. We place ourselves then in the situation which is that of a *T_{0}*L_{0}. But for the reasons which have just been mentioned, such n-universe can be denoted, in a simplified way, by .

The preceding remarks suggest then a simplification, in a general way, at the level of the used notation. Indeed, since a n-universe includes multiple objects and since it includes a constant-criterion of time, the multiple objects are necessarily in relation *many* with the constant-criterion of time. The n-universe is then a *T_{0}. But it is possible to simplify the corresponding notation into . If a n-universe also includes multiple objects and a constant-criterion of location, the multiple objects are necessarily in relation *many* with the constant-criterion of location. The given n-universe is then a *L_{0}, and it is possible to simplify the notation of the considered n-universe in . As a result, it is possible to simplify the notations *L_{0}*T_{0} into , *L_{0}T into T, *L_{0}*T into *T, *L_{0}*T* into *T*, etc.

2**. ****Modelling random events with n-universes**

The situations traditionally implemented in probability theory involve dices, coins, card games or else some urns that contain balls. It is worth setting out to describe how such objects can be modelled within the n-universes. It also proves to be necessary to model the notion of a “toss” in the probability spaces extended to n-universes. One can make use of the modellings that follow:^{1}

**2.1. Throwing a dice**

How can we model a toss such as the result of the throwing of the dice is “5 “? We model here the dice as a unique object that finds itself at a space location L_{0} and which is susceptible of presenting at time T_{0} one discrete modality of space direction among {1,2,3,4,5,6}. The corresponding n-universe includes then a unique object, a variable of direction and a temporal constant. The unique object can only present one single direction at time T_{0} and is not with demultiplication with regard to the criterion of direction. The n-universe is a O (with extended notation _{0}T_{0}L_{0}O). Traditionally, we have the sample space = {1,2,6} and the event {5}. The drawing of “5 ” consists here for the unique object to have direction 5 among {1,2,6} at time T_{0} and at location L_{0}. We denote then the sample space by _{0}T_{0}L_{0}O{1,2,…,6} and the event by _{0}T_{0}L_{0}O{5}.^{2}

How can we model two successive throws of the same dice, such as the result is “5” and then “1”? Traditionally, we have the sample space = {1,2,…,6}^{2} and the event {5,1}. Here, it corresponds to the fact that the dice _{0} has direction 5 and 1 respectively at T_{1} and T_{2}. In the corresponding n-universe, we have now a time variable, including two positions: T_{1} and T_{2}. Moreover, the time variable is with demultiplication because the unique object exists at different temporal positions. The considered n-universe is therefore a T*O (with extended notation _{0}T*L_{0}O). We denote then the sample space by _{0}T*{1,2}L_{0}O{1,2,…,6} and the event by {_{0}T*{1}L_{0}O{5}, _{0}T*{2}L_{0}O{1}}.

**2.2. Throwing a coin**

How can we model the toss, for example of Tails, resulting from the flipping of a coin? We model here the coin as a unique object presenting 2 different modalities of direction among {P,F}. The corresponding n-universe is identical to the one which allows to model the dice, with the sole difference that the direction criterion includes only two taxa: {P,F}. The corresponding n-universe is therefore a O (with extended notation _{0}T_{0}L_{0}O). Classically, we have: = {P,F} and {P}. Here, the Tails-toss is assimilated with the fact for the unique object to take direction {P} among {P,F} at time T_{0} and at location L_{0}. The sample space is then denoted by _{0}T_{0}L_{0}O{P,F} and the event by _{0}T_{0}L_{0}O{P}.

How can we model two successive tosses of the same coin, such as the result is “Heads” and then “Tails”? Classically, we have the sample space = {P,F}^{2} and the event {F,P}. As well as for the modelling of the successive throws of the same dice, the corresponding n-universe is here a T*O (with extended notation _{0}T*L_{0}O). The sample space is then denoted by by _{0}T*{1,2}L_{0}O{P,F} and the event by {_{0}T*{1}L_{0}O{F}, _{0}T*{2}L_{0}O{P}}.

**2.3. ****Throwing several discernible dices**

How can we model the throwing of two discernible dices at the same time, for example the simultaneous toss of one “3” and of one “5”? The discernible dices are modelled here as multiple objects being each at a given space position and susceptible of presenting at time T_{0} one modality of space direction among {1,2,3,4,5,6}. The multiple objects co-exist at the same temporal position, so that the objects are in relation *many* with the temporal constant. In addition, the multiple objects can only present one single direction at time T_{0} and are not therefore with demultiplication with regard to the criterion of direction. The fact that both dices could have the same direction corresponds to the fact that objects are in relation *many* with the criterion of direction. There exists also a location variable, each of the dices _{1} and _{2} being at one distinct space position. We consider then that the latter property renders the dices discernible. The objects are here in relation *one* with the location criterion. In addition, the objects can only occupy one single space position at time T_{0} and are not therefore with demultiplication with regard to the location criterion. The n-universe is then a L*O (with extended notation *T_{0}L*O). Classically, one has: = {1,2,3,4,5,6}^{2} and {3,5}. Here, it corresponds to the fact that the dices _{1} and _{2} are to be found respectively at L_{1} and L_{2} and present a given direction among {1,2,6} at time T_{0}. We denote then the sample space by {1,2}*T_{0}L{1,2}*O{1,2,…,6} and the event by {{1}*T_{0}L{1}*O{3}, {2}*T_{0}L{2}*O{5}}.

**2.4. ****Throwing several indiscernible dices**

How can we model the throwing of two indiscernible dices, for example the toss of one “3” and one “5” at the same time? Both indiscernible dices are modelled as multiple objects being at space position L_{0} and susceptible of presenting at time T_{0} one modality of space direction among {1,2,3,4,5,6} at a given location. The multiple objects co-exist at the same temporal position, so that the objects are in relation *many* with the temporal constant. The multiple objects can only present one single direction at time T_{0} and are not therefore with demultiplication with regard to the criterion of direction. The fact that both dices are susceptible of having the same direction corresponds to the fact that the objects are in relation *many* with the criterion of direction. Both dices _{1} and _{2} are at the same location L_{0}, what makes them indiscernible. In addition, the multiple objects are in relation *many* with the constant-criterion of location. Lastly, the objects can only be at one single space position at time T_{0} and are not therefore with demultiplication with regard to the location criterion. The corresponding n-universe is then a *O (with extended notation *T_{0}*L_{0}*O). Classically, we have: = (*i*, *j*) with 1 *i* *j* 6 and {3,5}. Here, it corresponds to the fact that the dices _{1} and _{2} are both in L_{0} and present a given direction among {1,2,…,6} at T_{0}. The sample space is then denoted by {1,2}*T_{0}*L_{0}*O{1,…,6} and the event by {{1}*T_{0}*L_{0}*O{3}, {2}*T_{0}*L_{0}*O{5}}.

**2.5. Drawing a card**

How can we model the drawing of a card, for example the card #13, in a set of 52 cards? Cards are modelled here as multiple objects presenting each a different colour among {1,2,…,52}. The cards’ numbers are assimilated here with taxa of colour, numbered from 1 to 52. Every object can have only one single colour at a given time. As a result, the multiple objects are not with demultiplication with regard to the colour criterion. In addition, a given card can only present one single colour at the same time. Hence, the objects are in relation *one* with the colour criterion. Moreover, the multiple objects can be at a given time at the same space location (to fix ideas, on the table). The objects are then in relation *many* with the location criterion. Lastly, the objects can co-exist at the same given temporal position. Thus, they are in relation *many* with the time criterion. The corresponding n-universe is then a C (with extended notation *T_{0}*L_{0}C). How can we model the drawing of a card? Classically, we have the sample space = {1,2,…,52} and the event {13}. Here, the drawing of the card #13 is assimilated with the fact that the object the colour of which is #13 is at T_{0} at location L_{0}. The sample space is then denoted by {1,2,…,52}*T_{0}*L_{0}C{1,2,…,52} and the event by {1}*T_{0}*L_{0}C{13}.

The drawing of two cards at the same time or the successive drawing of two cards are then modelled in the same way.

**2.6 Drawing of a ball from an urn containing red and blue balls**

How can we model the drawing of, for example, a red bowl, from an urn containing 10 balls among which 3 red balls and 7 blue balls? The balls are modelled here as multiple objects presenting each one colour among {R,B}. There exists then a colour variable in the corresponding n-universe. In addition, several objects can present the same colour. The objects are then in relation *many* with the variable-criterion of colour. Moreover, the objects are in relation *many* with regard to the constant-criteria of time and location. The corresponding n-universe is therefore a *T_{0}**L_{0}*C (with simplified notation *C). Classically, we have the sample space = {R,R,R,B,B,B,B,B,B,B} and the event {R}. The sample space is then denoted by {1,2,…,10}*T_{0}**L_{0}*C{R,B} and the event by {{1}*T_{0}**L_{0}*C{R}}.

The drawing of two balls at the same time or the successive drawing of two balls are modelled in the same way.

3**. Dimorphisms and isomorphisms**

The comparison of the structures of the extended (to n-universes) sample spaces corresponding to two given probabilistic situations allows to determine if these situations are, from a probabilistic viewpoint, isomorphic or not. The examination of the structures of the sample spaces allows to determine easily the isomorphisms or, on the contrary, the dimorphisms. Let us give some examples.

Consider a first type of application where one wonders whether two probabilistic situations are of comparable nature. To this end, we model the two distinct probabilistic situations within the n-universes. The first situation is thus modelled in a *T_{0}*L_{0}*C (with simplified notation *C), and the second one in a *T_{0}*L_{0}C (with simplified notation C). One notices then a dimorphism between the n-universes that make it possible to model respectively the two probabilistic situations. Indeed, in the first situation, the multiple objects are in relation *many* with the colour criterion, corresponding thus to the fact that several objects can have an identical colour at a given moment and location. On the other hand, in the second situation, the multiple objects are in relation *one* with the colour criterion, what corresponds to the fact that each object has a different colour at a given time and location. The dimorphism observed at the level of the demultiplication of the variable-criterion of colour in the two corresponding n-universes makes it possible to conclude that the two probabilistic situations are not of a comparable nature.

It is worth considering now a second type of application. The throwing of two discernible dice is modelled, as we did see it, in a {1,2}T_{0}*L{1,2}*O{1,…,6}. Now let us consider a headlight which can take at a given time one colour of 6 colours numbered from 1 to 6. If one considers now two headlights of this type, it appears that the corresponding situation can be modelled in a {1,2}T_{0}*L{1,2}*C{1,…, 6}. In this last case, it appears that the variable-criterion of colour replaces the criterion of orientation. At this stage, it proves that the structure of such n-universe (with simplified notation L*C) is isomorphic to that of the n-universe in which the throwing of two discernible dice was modelled (with simplified notation L*O). This makes it possible to conclude that the two probabilistic situations are of a comparable nature.

Let us consider now a concrete example. John Leslie (1996, 20) describes in the following terms the *Emerald case*:

Imagine an experiment planned as follows. At some point in time, three humans would each be given an emerald. Several centuries afterwards, when a completely different set of humans was alive, five thousands humans would again each be given an emerald in the experiment. You have no knowledge, however, of whether your century is the earlier century in which just three people were to be in this situation, or the later century in which five thousand were to be in it. Do you say to yourself that if yours were the earlier century then the five thousand people wouldn’t be alive yet, and that therefore you’d have no chance of being among them? On this basis, do you conclude that you might just as well bet that you lived in the earlier century?

Leslie thus puts in parallel a real situation related to some emeralds and a probabilistic model concerning some balls in a urn. Let us proceed then to model the real, concrete, situation, described by Leslie, in terms of n-universes. It appears first that the corresponding situation is characterized by the presence of multiple objects: the emeralds. We find then ourselves in a n-universe with multiple objects. On the second hand, one can consider that the emeralds are situated at one single place: the Earth. Thus, the corresponding n-universe has a location constant (L_{0}). Leslie also distinguishes two discrete temporal positions in the experiment: the one corresponding to a given time and the other being situated several centuries later. The corresponding n-universe comprises then a time variable with two taxa: T_{1} and T_{2}. Moreover, it proves to be that the emeralds existing in T_{1} do not exist in T_{2} (and reciprocally). Consequently, the n-universe corresponding to the *emerald case* is a n-universe which is not with temporal demultiplication. Moreover, one can observe that several emeralds can be at the same given temporal position T_{i}: three emeralds exist thus in T_{1} and five thousand in T_{2}. Thus, the objects are in relation *many* with the time variable. Lastly, several emeralds can coexist in L_{0} and the objects are thus in relation *many* with the location constant. Taking into account what precedes, it appears thus that the *Emerald case* takes place in a *T (with extended notation *T*L_{0}), a n-universe with multiple objects, comprising a location constant and a time variable with which the objects are in relation *many*.

Compare now with the situation of the Little Puddle/London experiment, also described by Leslie (1996, 191):

Compare the case of geographical position. You develop amnesia in a windowless room. Where should you think yourself more likely to be: in Little Puddle with a tiny situation, or in London? Suppose you remember that Little Puddle’s population is fifty while London’s is ten million, and suppose you have nothing but those figures to guide you. (…) Then you should prefer to think yourself in London. For what if you instead saw no reason for favouring the belief that you were in the larger of the two places? Forced to bet on the one or on the other, suppose you betted you were in Little Puddle. If everybody in the two places developed amnesia and betted as you had done, there would be ten million losers and only fifty winners. So, it would seem, betting on London is far more rational. The right estimate of your chances of being there rather than in Little Puddle, on the evidence on your possession, could well be reckoned as ten million to fifty.

The latter experiment is based on a real, concrete, situation, to be put in relation with an implicit probabilistic model. It appears first that the corresponding situation characterises itself by the presence of multiple inhabitants: 50 in Little Puddle and 10 million in London. The corresponding n-universe is then a n-universe with multiple objects. It appears, second, that this experiment takes place at one single time: the corresponding n-universe has then one time constant (T_{0}). Moreover, two space positions – Little Puddle and London – are distinguished, so that we can model the corresponding situation with the help of a n-universe comprising two space positions: L_{1} and L_{2}. Moreover, each inhabitant is either in Little Puddle or in London, but but no one can be at the two places at the same time. The corresponding n-universe is then not with local demultiplication. Lastly, one can notice that several people can find themselves at a given space position L_{i}: there are thus 50 inhabitants at Little Puddle (L_{1}) and 10 million in London (L_{2}). The objects are thus in a relation *many* with the space variable. And in a similar way, several inhabitants can be simultaneously either in Little Puddle, or in London, at time T_{0}. Thus, the objects are in relation *many* with the time constant. Taking into account what precedes, it appears that the situation of the Little Puddle/London experiment takes place in a *L (with extended notation *T_{0}*L), a n-universe with multiple objects, comprising a time constant and a location variable, with which the objects are in relation *many*.

As we can see it, the emerald case takes place in a *T, whereas the Little Puddle/London experiment situates itself in a *L. This makes it possible to highlight the isomorphic structure of the two n-universes in which the two experiments are respectively modelled. This allows first to conclude that the probabilistic model which applies to the one, is also worth for the other one. Moreover, it appears that both the *T and the *L are isomorphic with the *C. This makes it possible to determine straightforwardly the corresponding probabilistic model. Thus, the situation corresponding to both the emerald case and the Little Puddle/London experiment can be modelled by the drawing of a ball from an urn comprising red and blue balls. In the emerald case, it consists of an urn comprising 3 red balls and 5000 green balls. In the Little Puddle/London experiment, the urn includes thus 50 red balls and 10^{7} green balls.

4**. Goodman’s paradox**

Another interest of the n-universes as a methodological tool resides in their use to clarify complex situations such as those which are faced in the study of paradoxes. I will illustrate in what follows the contribution of the n-universes in such circumstances through the analysis of Goodman’s paradox.^{3}

Goodman’s paradox was described in *Fact, Fiction and Forecast* (1954, 74-75). Goodman explains then his paradox as follows. Every emeralds which were until now observed turned out to be green. Intuitively, we foresee therefore that the next emerald that will be observed will also be green. Such prediction is based on the generalisation according to which all emeralds are green. However, if one considers the property *grue*, that is to say “observed before today and green, or observed after today and not-green”,^{4} we can notice that this property is also satisfied by all instances of emeralds observed before. But the prediction which results from it now, based on the generalisation according to which all emeralds are *grue*, is that the next emerald to be observed will be not-green. And this contradicts the previous conclusion, which is conforms however with our intuition. The paradox comes here from the fact that the application of an enumerative induction to the same instances, with the two predicates *green* and *grue*, leads to predictions which turn out to be contradictory. This contradiction constitutes the heart of the paradox. One of the inductive inferences must then be fallacious. And intuitively, the conclusion according to which the next observed emerald will be *not-green* appears erroneous.

Let us set out now to model the Goodman’s experiment in terms of n-universes. It is necessary for it to describe accurately the conditions of the *universe of reference* in which the paradox takes place. Goodman makes thus mention of properties *green* and *not-green* applicable to emeralds. Colour constitutes then one of the variable-criteria of the n-universe in which the paradox takes place. Moreover, Goodman draws a distinction between emeralds observed *before T* and those which will be observed *after T*. Thus, the corresponding n-universe also includes a variable-criterion of time. As a result, we are in a position to describe the minimal universe in which Goodman (1954) situates himself as a coloured and temporal n-universe, i.e. a CT.

Moreover, Goodman makes mention of several *instances* of emeralds. It could then be natural to model the paradox in a n-universe with multiple objects, coloured and temporal. However, it does not appear necessary to make use of a n-universe including multiple objects. Considering the methodological objective which aims at avoiding a combinatorial explosion of cases, it is indeed preferable to model the paradox in the simplest type of n-universe, i.e. a n-universe with a unique object. We observe then the emergence of a version of the paradox based on one unique emerald the colour of which is likely to vary in the course of time. This version is the following. The emerald which I currently observe was green all times when I did observe it before. I conclude therefore, by induction, that it will be also green the next time when I will observe it. However, the same type of inductive reasoning also leads me to conclude that it will be *grue*, and therefore not-green. As we can see, such variation always leads to the emergence of the paradox. The latter version takes p lace in a n-universe including a unique object and a variable of colour and of time, i.e. a CT. At this step, given that the original statement of the paradox turns out to be ambiguous in this respect, and that the minimal context is that of a CT, we will be led to distinguish between two situations: the one which situates itself in a CT, and the one which takes place in a CT (where denotes a third variable-criterion).

Let us place ourselves first in the context of a coloured and temporal n-universe, i;e. a CT. In such universe, to be *green*, is to be *green* at time T. In this context, it appears completely legitimate to project the shared property of colour (*green*) of the instances through time. The corresponding projection can be denoted by C°T. The emerald was green every time where I observed it before, and the inductive projection leads me to conclude that it will be also green next time when I will observe it. This can be formalized as follows (V denoting *green*):

(I1) | VT_{1}·VT_{2}·VT_{3}·…·VT_{99} | instances |

(H2) | VT_{1}·VT_{2}·VT_{3}·…·VT_{99}·VT_{100} | generalisation |

(P3) | VT_{100} | from (H2) |

The previous reasoning appears completely correct and conforms to our inductive practice. But are we thus entitled to conclude from it that the *green* predicate is projectible without restriction in the CT? It appears not. For the preceding inductive enumeration applies indeed to a n-universe where the temporal variable corresponds to our present time, for example the period of 100 years surrounding our present epoch, that is to say the interval [-100, +100] years. But what would it be if the temporal variable extended much more far, by including for example the period of 10 thousand million years around our current time, that is to say the interval [-10^{10}, +10^{10}] years. In that case, the emerald is observed in 10 thousand million years. At that time, our sun is burned out, and becomes progressively a white dwarf. The temperature on our planet then warmed itself up in significant proportions to the point of attaining 8000°: the observation reveals then that the emerald – as most mineral – suffered important transformations and proves to be now *not-green*. Why is the projection of *green* correct in the CT where the temporal variable is defined by *restriction* in comparison with our present time, and incorrect if the temporal variable assimilates itself by *extension* to the interval of 10 thousand million years before or after our present time? In the first case, the projection is correct because the different instances of emeralds are representative of the reference class on which the projection applies. An excellent way of getting representative instances of a given reference class is then to choose the latter by means of a random toss. On the other hand, the projection is not correct in the second case, for the different instances are not representative of the considered reference class. Indeed, the 99 observations of emeralds come from our modern time while the 100th concerns an extremely distant time. So, the generalisation (H2) results from 99 instances which are not representative of the CT[-10^{10}, +10^{10}] and does not allow to be legitimately of use as support for induction. Thus *green* is projectible in the CT[-10^{2}, +10^{2}] and not projectible in the CT[-10^{10}, +10^{10}]. At this stage, it already appears that *green* is not projectible in the absolute but turns out to be projectible or not projectible *relative* to this or that n-universe.

In the light of what precedes, we are from now on in a position to highlight what proved to be fallacious in the projection of generalisation according to which “all swans are white”. In 1690, such hypothesis resulted from the observation of a big number of instances of swans in Europe, in America, in Asia and in Africa. The n-universe in which such projection did take place was a n-universe with multiple objects, including a variable of colour and of location. To simplify, we can consider that all instances had being picked at constant time T_{0}. The corresponding inductive projection C°L led to the conclusion that the next observed swan would be white. However, such prediction turned out to be false, when occurred the discovery in 1697 by the Dutch explorer Willem de Vlamingh of black swans in Australia. In the n-universe in which such projection did take place, the location criterion was implicitly assimilating itself to our whole planet. However, the generalisation according to which “all swans are white” was founded on the observation of instances of swans which came only from one part of the n-universe of reference. The sample turned out therefore to be biased and not representative of the reference class, thus yielding the falseness of the generalisation and of the corresponding inductive conclusion.

Let us consider now the projection of *grue*. The use of the *grue* property, which constitutes (with *bleen*) a taxon of tcolour*, is revealing of the fact that the used system of criteria comes from the Z. The n-universe in which takes place the projection of *grue* is then a Z, a n-universe to which the CT reduces. For the fact that there exists two taxa of colour (*green*, *not-green*) and two taxa of time (*before T*, *after T*) in the CT determines four different states: *green before T*, *not-green before T*, *green after T*, *not-green after T*. By contrast, the Z only determines two states: *grue* and *bleen*. The reduction of the CT to the Z is made by transforming the taxa of colour and of time into taxa of tcolour*. The classical definition of *grue* (*green before T* or *not-green after T*) allows for that. In this context, it appears that the paradox is still present. It comes indeed under the following form: the emerald was *grue* every time that I did observe it before, and I conclude inductively that the emerald will also be *grue* and thus *not-green* the next time when I will observe it. The corresponding projection Z°T can then be formalized (G denoting *grue*):

What is it then that leads to deceive our intuition in this specific variation of the paradox? It appears here that the projection of *grue* comes under a form which is likely to create an illusion. Indeed, the projection Z°T which results from it is that of the tcolor* through time. The general idea which underlies inductive reasoning is that the instances are *grue before T* and therefore also *grue after T*. But it should be noticed here that the corresponding n-universe is a Z. And in a Z, the only variable-criterion is tcolor*. In such n-universe, an object is *grue* or *bleen* in the *absolute*. By contrast, an object is *green* or *not-green* in the CT *relative* to a given temporal position. But in the Z where the projection of *grue* takes place, an additional variable-criterion is missing so that the projection of *grue* could be legitimately made. Due to the fact that an object is *grue* or *bleen* in the absolute in a Z, when it is *grue before T*, it is also necessarily *grue after T*. And from the information according to which an object is *grue before T*, it is therefore possible to conclude, by *deduction*, that it is also *grue after T*. As we can see it, the variation of the paradox corresponding to the projection Z°T presents a structure which gives it the appearance of an enumerative generalisation but that constitutes indeed a genuine deductive reasoning. The reasoning that ensues from it constitutes then a disguised form of induction, a *pseudo-induction*.

Let us envisage now the case of a coloured, temporal n-universe, but including an additional variable-criterion , i.e. a CT. A n-universe including variable-criteria of colour, of time and location,^{5} i.e. a CTL, will be suited for that. To be *green* in a CTL, is to be *green* at time T and at location L. Moreover, the CTL reduces to a ZL, a n-universe the variable-criteria of which are tcolor* and location. The taxa of tcolor* are *grue* and *bleen*. And to be *grue* in the ZL, is to be *grue* at location L.

In a preliminary way, one can point out here that the projections CTL and ZTL do not require a separate analysis. Indeed, these two projections present the same structure as those of the projections CT and ZT which have just been studied, except for an additional differentiated criterion of location. The conditions under which the paradox dissolves when one compares the projections CT and ZT apply therefore identically to the variation of the paradox which emerges when one relates the projections CTL and ZTL .

On the other hand, it appears here opportune to relate the projections CT°L and Z°L which respectively take place in the CTL and the ZL. Let us begin with the projection CT°L. The shared criteria of colour and of time are projected here through a differentiated criterion of location. The taxa of time are here *before T* and *after T*. In this context, the projection of *green* comes under the following form. The emerald was *green before T* in every place where I did observe it before, and I conclude from it that it will be also *green before T* in the next place where it will be observed. The corresponding projection C°TL can then be formalized as follows:

(I7) | VTL_{1}·VTL_{2}·VTL_{3}·…·VTL_{99} | instances |

(H8) | VTL_{1}·VTL_{2}·VTL_{3}·…·VTL_{99}·VTL_{100} | generalisation |

(P9) | VTL_{100} | prediction |

At this step, it seems completely legitimate to project the *green* and *before T* shared by the instances, through a differentiated criterion of location, and to predict that the next emerald which will be observed at location L will present the same properties.

What is it now of the projection of *grue* in the CTL? The use of *grue* conveys the fact that we place ourselves in a ZL, a n-universe to which reduces the CTL and the variable-criteria of which are tcolour* and location. The fact of being *grue* is relative to the variable-criterion of location. In the ZL, to be *grue* is to be *grue* at location L. The projection relates then to a taxon of tcolour* ( *grue* or *bleen*) which is shared by the instances, through a differentiated criterion of location. Consider then the classical definition of *grue* (*green before T or non-grue after T*). Thus, the emerald was *grue* in every place where I did observe it before, and I predict that it will also be *grue* in the next place where it will be observed. If we take T = in 10^{10} years, the projection Z°L in the ZL appears then as a completely valid form of induction (V~T denoting *green after T*):

As pointed out by Franck Jackson (1975, 115), such type of projection applies legitimately to all objects which colour changes in the course of time, such as tomatoes or cherries. More still, one can notice that if we consider a very long period of time, which extends as in the example of emeralds until 10 thousand million years, such property applies virtually to all concrete objects. Finally, one can notice here that the contradiction between both concurrent predictions (P9) and (P12’*) has now disappeared since the emerald turns out to be green before T in L_{100} (VTL_{100}) in both cases.

As we can see, in the present analysis, a predicate turns out to be projectible or not projectible in *relative* to this or that universe of reference. As well as *green*, *grue* is not projectible in the absolute but turns out to be projectible in some n-universes and not projectible in others. It consists here of a difference with several classical solutions offered to solve the Goodman’s paradox, according to which a predicate turns out to be projectible or not projectible in the *absolute*. Such solutions lead to the definition of a criterion allowing to distinguish the projectible predicates from the unprojectible ones, based on the differentiation temporal/non-temporal, local/non-local, qualitative/non-qualitative, etc. Goodman himself puts then in correspondence the distinction projectible/ unprojectible with the distinction entrenched^{i}/unentrenched, etc. However, further reflexions of Goodman, formulated in *Ways of Worldmaking*^{ii}, emphasize more the unabsolute nature of projectibility of *green* or of *grue*: “Grue cannot be a relevant kind for induction in the same world as green, for that would preclude some of the decisions, right or wrong, that constitute inductive inference”. As a result, *grue* can turn out to be projectible in a goodmanian world and not projectible in some other one. For *green* and *grue* belong for Goodman to different worlds which present different structures of categories.^{6} In this sense, it appears that the present solution is based on a form of relativism the nature of which is essentially goodmanian.

5**. Conclusion**

From what precedes and from Goodman’s paradox analysis in particular, one can think that the n-universes are of a fundamentally goodmanian essence. From this viewpoint, the essence of n-universes turns out to be pluralist, thus allowing numerous descriptions, with the help of different systems of criteria, of a same reality. A characteristic example, as we did see it, is the reduction of the criteria of colour and time in a CTL into a unique criterion of tcolour* in a ZL. In this sense, one can consider the n-universes as an implementation of the programme defined by Goodman in *Ways of Worldmaking*. Goodman offers indeed to construct worlds by composition, by emphasis, by ordering or by deletion of some elements. The n-universes allow in this sense to represent our concrete world with the help of different systems of criteria, which correspond each to a relevant point of view, a way of seeing or of considering a same reality. In this sense, to privilege this or that system of criteria, to the detriment of others, leads to a truncated view of this same reality. And the exclusive choice, without objective motivation, of such or such n-universe leads to engender a biased point of view.

However, the genuine nature of the n-universes turns out to be inherently ambivalent. For the similarity of the n-universes with the goodmanian worlds does not prove to be exclusive of a purely ontological approach. Alternatively, it is indeed possible to consider the n-universes from the only ontological point of view, as a methodological tool allowing to model directly this or that concrete situation. The n-universes constitute then so much universes with different properties, according to combinations resulting from the presence of a unique object or multiple objects, in relation *one* or *many*, with demultiplication or not, with regard to the criteria of time, location, colour, etc. In a goodmanian sense also, the n-universes allow then to build so much universes with different structures, which sometimes correspond to the properties of our real world, but which have sometimes some exotic properties. To name only the simplest of the latter, the L* is then a n-universe which includes only one ubiquitous object, presenting the property of being at several locations at the same time.^{7}

At this stage, it is worth mentioning several advantages which would result from the use of the n-universes for modelling probabilistic situations. One of these advantages would be first to allow a better intuitive apprehension of a given probabilistic situation, by *emphasising* its essential elements and by *suppressing* its superfluous elements. By differentiating for example depending on whether the situation to model presents a constant or a time variable, a constant or a space variable, a unique object or several objects, etc. the modelling of concrete situations in the n-universes provides a better support to intuition. On the other hand, the distinction according to whether the objects are or not with demultiplication or in relation one/many with regard to the different criteria allows for a precise classification of the different probabilistic situations which are encountered.

One can notice, second, that the use of the notation of the probability spaces extended to the n-universes would allow to withdraw the ambiguity which is sometimes associated with classical notation. As we did see it, we sometimes face an ambiguity. Indeed, it proves to be that {1,2,…,6}^{2} denotes at the same time the sample space of a simultaneous throwing of two discernible dices in T_{0} and that of two successive throwing of the same dice in T_{1} and then in T_{2}. With the use of the notation extended to n-universes, the ambiguity disappears. In effect, the sample space of the simultaneous throwing of two discernible dices in T_{0} is a {1,2}*T_{0}L{1,2}*O{1,2,…,6}, whilst that of two successive throwing of the same dice in T_{1} and then in T_{2} is a _{0}T*{1,2}L_{0}O{1,2,…,6}.

Finally, an important advantage, as we have just seen it, which would result from a modelling of probabilistic situations extended to n-universe is the easiness with which it allows comparisons between several probabilistic models and it highlights the isomorphisms and the corresponding dimorphisms. But the main advantage of the use of the n-universes as a methodological tool, as we did see it through Goodman’s paradox, would reside in the clarification of the complex situations which appear during the study of paradoxes.^{8}

**References**

Franceschi, Paul. 2001. Une solution pour le paradoxe de Goodman. Dialogue 40: 99-123, English translation under the title The Doomsday Argument and Hempel’s Problem, http://cogprints.org/2172/. English translation |

—. 2002. Une application des n-univers à l’argument de l’Apocalypse et au paradoxe de Goodman. Doctoral dissertation, Corté: University of Corsica. <http://www.univ-corse.fr/~franceschi/index-fr.htm> [retrievec Dec.29, 2003] |

Goodman, Nelson. 1954. Fact, Fiction and Forecast. Cambridge, MA: Harvard University Press. |

—. 1978. Ways of Worldmaking. Indianapolis: Hackett Publishing Company. |

Jackson, Franck. 1975. “Grue”. The Journal of Philosophy 72: 113-131. |

Leslie, John. 1996. The End of the World: The Science and Ethics of Human Extinction. London: Routledge. |

1 Il convient de noter que ces différentes modélisations ne constituent pas une manière unique de modéliser les objets correspondants dans les n-univers. Cependant, elles correspondent à l’intuition globale que l’on a de ces objets.

2 De manière alternative, on pourrait utiliser la notation _{0}T_{0}L_{0}O_{5} en lieu et place de _{0}T_{0}L_{0}O{5}. Cette dernière notation est toutefois préférée ici, car elle se révèle davantage compatible avec la notation classique des événements.

3 Cette analyse du paradoxe de Goodman correspond, de manière simplifiée et avec plusieurs adaptations, à celle initalement décrite dans Franceschi (2001). La variation du paradoxe qui est considérée ici est celle de Goodman (1954), mais avec une émeraude unique.

4 P and Q being two predicates, grue presents the following structure: (P and Q) or (~P and ~Q).

5 Tout autre critère différent de la couleur et du temps tel que la masse, la température, l’orientation, etc. conviendrait également.

6 Cf. Goodman (1978, 11): “(…) a green emerald and a grue one, even if the same emerald (…) belong to worlds organized into different kinds”.

7 Les n-univers aux propriétés non standard nécessitent une étude plus détaillée, qui dépasse le cadre de la présente étude.

8 Je suis reconnaissant envers Jean-Paul Delahaye pour la suggestion de l’utilisation des n-univers en tant qu’espaces de probabilité étendus. Je remercie également Claude Panaccio et un expert anonyme pour le *Journal of Philosophical Research* pour des discussions et des commentaires très utiles.

i *Entrenched*.

ii Cf. Goodman (1978, 11).