A two-sided viewpoint is a viewpoint applied to an object o which takes into account both poles of a given duality. A two-sided viewpoint is opposed to a one-sided viewpoint, where only one pole of a duality is considered. On the other hand, in the two-sided viewpoint, both poles of a duality are taken into account and applied to a given object o. Thus, if the poles A and Ā of a matrix of concepts are taken into account, it will be a two-sided viewpoint.

Franceschi, P., “Éléments d’un contextualisme dialectique” (in english), in Liber Amicorum Pascal Engel, J. Dutant, G. Fassio & A. Meylan (éd.), Université de Genève, 2014, p. 581-608.

The matrix of concepts is a structure that includes six concepts, which is suitable for modeling many common concepts, such as: courage, recklessness, irresolution, eclecticism, superficiality, clemency, instability, selfishness, objectivity, frankness, brusqueness, altruism, etc. Of the six concepts in the matrix:

two are neutral: A^{0} and Ā^{0}

two are positive: A^{+} and Ā^{+}

two are negative: A^{–} and Ā^{–}

These six concepts constitute the canonical poles of the matrix.

The six concepts of the matrix are in particular relationships with each other. Thus:

the neutral concepts A^{0} and Ā^{0} are dual

the positive concept A^{+} and the negative concept Ā^{–} are opposite (or contrary); similarly, the negative concept A^{–} and the positive concept Ā^{+} are opposite (or contrary)

the positive concepts A^{+} and Ā^{+} are complementary

the propensity to take risks and the propensity to avoid risks are dual

audacity and cowardice are contrary, opposite; in the same way, temerity and prudence are contrary, opposite

audacity and prudence are complementary

temerity and cowardice are extreme opposites

Moreover, the three concepts located on the left of the matrix constitute a half-matrix: it is the half-matrix associated with the pole A. In the same way, the three concepts located on the right of the matrix constitute the half-matrix associated to the pole Ā.

Dichotomous analysis is a technique of analysis that tends to consider an object o from the point of view of a given duality, and to distinguish two different situations, each of which corresponds to one of the two poles of the duality considered. Examples of dualities are: Quantitative/Qualitative, Absolute/Relative, Static/Dynamic, Diachronic/Synchronic, Extension/Restriction.

The dichotomous analysis is directly linked to the very statement of the principle of dialectical indifference. Its use responds to the concern not to be vulnerable to one-sidedness bias and to demonstrate, on the contrary, a two-sided viewpoint.

Franceschi, P., “Éléments d’un contextualisme dialectique” (in english), in Liber Amicorum Pascal Engel, J. Dutant, G. Fassio & A. Meylan (éd.), Université de Genève, 2014, p. 581-608.

It is worth mentioning a form of dialectical monism in the ancient Aztec philosophy and especially in the concept of ” Teotl “, which is at the center of Aztec metaphysics and cosmology. Teotl is the expression of an endless alternation of continuous and cyclical oscillation between opposite poles. Teotl is thus characterized by a dual prominent structure, which results from the union of opposites , themselves characterized by complementarity. The dual pairs involved include : the masculine and the feminine, dark and light, order and disorder, hot and cold, life and death, being and non-being etc. The interdependence and higher union of the principles of life and death in Teotl, for example, was represented by Aztec artists of Tlatilco and Oaxaca in masks where one half is alive while the other half died, revealing the skull bones.

James Maffie, Aztec Philosophy, Internet Encyclopedia of Philosophy

The dichotomous analysis as a methodology that can be used to search for solutions to some paradoxes and philosophical problems, results from the statement of the principle of dialectical indifference. 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.

However, every duality is not well-suited to this approach, as for many dualities, the corresponding version of the paradox remains unchanged, regardless of the duality under consideration. 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.

Franceschi, P., “Éléments d’un contextualisme dialectique” (in english), in Liber Amicorum Pascal Engel, J. Dutant, G. Fassio & A. Meylan (éd.), Université de Genève, 2014, p. 581-608.

In the antique Western world, dialectical monism appears not much widespread. But we notably find an elaborate form of dialectical monism in Heraclitus. Several fragments of the philosophy of Heraclitus reflect the expression of this unity that results from the joint presence of two dual principles. For example, the Eigth Fragment:

What opposes unites, and the finest attunement stems from things bearing in opposite directions, and all things come about by strife.

and also the Tenth Fragment:

Things grasped together: things whole, things not whole; being brought together, being separated; consonant, dissonant. Out of all things one thing, out of one thing all things.

Here we find the expression of dialectical monism , through the union of opposites . We see how the dialectic proceeds from the union of opposites : the consonant and dissonant . This dialectical approach that underpins the philosophy of Heraclitus is also illustrated in Fragment 51:

They do not understand how, while differing from it is in agreement with itself. There is a back turning connection, like that of a bow or a lyre.

When only the pole A (respectively the pole Ā) of a given duality is considered, it consists of a one-sided viewpoint. Things are seen exclusively from the perspective of one pole. It contrasts with the two-sided viewpoint, with takes into account both poles of a given duality.

The consequence of taking only into account one pole is that the other pole is ignored, or disqualified. In cognitive psychology, the resulting cognitive distortion applied to the positive/negative duality is termed ‘disqualifying the positive’.

The one-sided bias consists in focusing on a given standpoint when considering a given object, and of neglecting the opposite view. In our framework, such fallacious reasoning consists, when considering an object o and the reference class associated with it, of 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.

Franceschi, P., “Éléments d’un contextualisme dialectique” (in english), in Liber Amicorum Pascal Engel, J. Dutant, G. Fassio & A. Meylan (éd.), Université de Genève, 2014, p. 581-608.

A dual pole is one of the components of a pair of concepts that make up a given duality. Among the dualities, we can mention: quantitative/qualitative, static/dynamic, external/internal, unique/multiple, etc. Thus, quantitative, quantitative, static, dynamic,… are dual poles. It should be noted that their nature is neutral, that is, they do not carry a positive or negative connotation. Thus, concepts that have a positive connotation are: audacity, courage, ardor, merit, combativity, etc. Similarly, concepts that have a negative connotation are: cowardice, pessimism, timidity, irresolution, etc.

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

Dualities are pairs of concepts that are dual, antinomic. The pairs of concepts, i.e. the dual poles that make up the dualities are concepts that are neutral in nature. They have no positive or negative connotation, unlike concepts like courage or pessimism, which have a positive or negative nature respectively.

Instances of dualities are the following:

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

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

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.

(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.

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.

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}:

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 componentc {-1, 1}

– a canonical polarityp {-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 degreed 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 positiveincluder ^{+} 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:

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 polarityp Î {-1, 0, 1}

– a neutral canonical principle composed of:

– a duality (or base) A/Ā

– a contrary componentc {-1, 1}

– a neutral polarityq = 0

The positive, neutral, negativecanonical 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

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.

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}

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

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.

1. The paradox

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):

(D1)

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 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:

(C3) S[n, 0] = 1 (base step)

(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:

(D7)

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]

■

■

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:

(D8)

S[1, 0]

S[1, 1]

S[1,s]

■

■

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:

(D10)

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]

■

■

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:

(D11)

S[k, 0]

S[k, 1]

S[n,s]

■

□

S[n-1,s]

■

■

…………

…………

…………

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∆):

(D12)

S[k, 0]

S[k, 1]

S[2,s]

■

□

S[1,s]

■

■

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 :

(D14)

S[1, 0]

S[1, 1]

S[365,s]

■

□

…………

…………

…………

S[1,s]

□

■

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.

6. Conclusion

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.

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.

In On a class of concepts (2002), I described a theory based on the matrices of concepts which aims at constituting an alternative to the classification proposed by Greimas, in the field of paradigmatic analysis. The problem of the determination of the relationships of love/hate/indifference arises in this construction. I state then the problem of the relationships of love/hate/indifference in a detailed way, and several solutions that have been proposed in the literature to solve it. I describe lastly a solution to this problem, based on an extension of the theory of matrices of concepts.

This paper is cited in:

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.

The Problem of the Relationships of Love-Hate-Indifference

Paul Franceschi

I shall be concerned in this paper with presenting a problem related to the proper definition of the relationships of the following concepts: love, hate and indifference. I will describe first the problem in detail and some proposed solutions. Lastly, I will present my own solution to the problem.

1. The problem

The problem is that of the proper definition of the relationships of the concepts love, hate and indifference. Let us call it the LHI problem. What are then the accurate relationships existing between these three concepts? At first sight, the definition of the relation between love and hate is obvious. These concepts are contraries. The definition of such a relation should be consensual. Nevertheless, the problem arises when one considers the relationship of love and indifference, and of hate and indifference. In these latter cases, no obvious response emerges.

However, the issue needs clarifying. In this context, what should we expect of a solution to the LHI problem? In fact, a rigorous solution ought to define precisely the three relations R, S, T such that love R hate, love S indifference and hate T indifference. And the definitions of these relations should be as accurate as possible.

It is worth mentioning that several authors must be credited for having mentioned and investigated the LHI problem. In particular, it is worth stressing that the difficulties presented within propositional calculus by some assertions of the type x loves y, x hates y, or x is indifferent to y have been hinted at by Emile Benzaken (1990)^{1}:

Nevertheless, the difficulty can arise from pairs of words where the one expresses the contrary (negation) of the other; ‘to hate’ can be considered as the strong negation of ‘to love’, whereas ‘to be indifferent’ would be its weak negation.

The author exposes then the problem of the relationships of love/hate/indifference and proposes his own solution: hate is the strong negation of love, and indifferent is the weak negation of love.

However, it turns out that Benzaken’s solution is unsatisfying for a logician, for the following reasons. On the one hand, this way of solving the problem defines the relations between love and hate (strong negation, according to the author) and between love and indifference (weak negation, on the author’s view), but it fails to define accurately the relations existing between indifference and hate. There is a gap, a lack of response at this step. And mentioned above, a satisfying solution should elucidate the nature of the relationships of the three concepts. On the other hand, the difference between weak negation and strong negation is not made fully explicit within the solution provided by Benzaken. For these reasons, Benzaken’s solution to the LHI problem proves to be unsatisfying.

In a very different context, Rick Garlikov (1998) stresses some difficulties of essentially the same nature as those underlined by Benzaken:

In a seminar I attended one time, one of the men came in all excited because he had just come across a quotation he thought very insightful – that it was not hate that was the opposite of love, but that indifference was the opposite of love, because hate was at least still an emotion. I chuckled, and when he asked why I was laughing, I pointed out to him that both hate and indifference were opposites of love, just in different ways, that whether someone hated you or was indifferent toward you, in neither case did they love you.

Garlikov describes in effect the problem of the relationships of love/hate/indifference and implicitly proposes a solution of a similar nature as that provided by Benzaken. For this reason, Galikov’s account suffers from the same defects as those presented by Benzaken’s solution.

In what follows, my concern will be with settling first the relevant machinery, in order to prepare a few steps toward a solution to the LHI problem.

2. The framework

I will sketch here the formal apparatus described in more detail in Franceschi (2002). To begin with, consider a given duality. Let us denote it by A/Ā. At this step, A and Ā are dual concepts. Moreover, A and Ā can be considered as concepts that are characterized by a contrarycomponent c∈ {-1, 1} within a duality A/Ā, such that c[A] = -1 and c[Ā] = 1. Let us also consider that A and Ā are neutral concepts that can be thus denoted by A^{0} and Ā^{0}.

At this point, we are in a position to define the class of the canonical poles. Consider then an extension of the previous class {A^{0}, Ā^{0}}, such that A^{0} and Ā^{0} respectively admit of a positive and a negative correlative concept. Such concepts are intuitively appealing. Let us denote them respectively by {A^{+}, A^{–}} and {Ā^{+}, Ā^{–}}. At this step, for a given duality A/Ā, we get then the following concepts: {A^{+}, A^{0}, A^{–}, Ā^{+}, Ā^{0}, Ā^{–}}. Let us call them canonical poles. It should be noted that one could use alternatively the notation α(A/Ā, c, p) for a canonical pole.^{2} In all cases, the components of a canonical pole are a duality A/Ā, a contrarycomponent c∈ {-1, 1} and a canonicalpolarity p∈ {-1, 0, 1}. This definition of the canonical poles leads to distinguish between the positive (A^{+}, Ā^{+}), neutral (A^{0}, Ā^{0}) and negative (A^{–}, Ā^{–}) canonical poles. Lastly, the class made up by the 6 canonical poles can be termed the canonicalmatrix: {A^{+}, A^{0}, A^{–}, Ā^{+}, Ā^{0}, Ā^{–}}.

Let us investigate now into the nature of the relations existing between the canonical poles of a given matrix. 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: duality, antinomy, complementarity, corollarity, connexity, and anti-connexity. Thus, two canonical poles α_{1}(A/Ā, c_{1}, p_{1}) and α_{2}(A/Ā, c_{2}, p_{2}) of a same matrix are:

(i) dual if their contrary components are opposite and their polarities are neutral^{3}

(ii) contrary (or antinomical) if their contrary components are opposite and their polarities are non-neutral and opposite^{4}

(iii) complementary if their contrary components are opposite and their polarities are non-neutral and equal^{5}

(iv) corollary if their contrary components are equal and their polarities are non-neutral and opposite^{6}

(v) connex if their contrary components are equal and the absolute value of the difference of their polarities equals 1^{7}

(vi) anti-connex if their contrary components are opposite and the absolute value of the difference of their polarities equals 1^{8}

To sum up: {A^{0}, Ā^{0}} are dual, {A^{+}, Ā^{–}} and {A^{–}, Ā^{+}} are contraries, {A^{+}, Ā^{+}} and {A^{–}, Ā^{–}} are complementary, {A^{+}, A^{–}} and {Ā^{+}, Ā^{–}} are corollary, {A^{0}, A^{+}}, {A^{0}, A^{–}}, {Ā^{0}, Ā^{+}} and {Ā^{0}, Ā^{–}} are connex, {A^{0}, Ā^{+}}, {A^{0}, Ā^{–}}, {Ā^{0}, A^{+}} and {Ā^{0}, A^{–}} are anti-connex.

I shall focus now on the types of relations existing, under certain circumstances between the canonical poles of different dualities. Let us define preliminarily the includer relation. Let a concept α be an includer for two other concepts β and γ if and only if α = β∨γ. Such a definition captures the intuition that α is the minimal concept whose semantic content includes that of β and γ. To give an example concerning truth-value, determinate is an includer for {true, false}.

Let now 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^{–}}. These matrices are such that E^{+}, E^{0}, E^{–} are the respective includers for {A^{+}, Ā^{+}}, {A^{0}, Ā^{0}}, {A^{–}, Ā^{–}} i.e. the two matrices are such that E^{+} = A^{+}∨ Ā^{+}, E^{0} = A^{0}∨ Ā^{0} and E^{–} = A^{–}∨ Ā^{–}.^{9}

Let us denote this relation by A < E. One is now in a position to extend the relations previously defined between the canonical poles of a same matrix, to the relations of a same nature between two matrices presenting the properties of A and E, i.e. such that A < E. The relations of 2-duality, 2-antinomy, 2-complementarity, 2-anti-connexity^{10} ensue then straightforwardly. Thus, two canonical poles α_{1}(A/Ā, c_{1}, p_{1}) and α_{2}(E/Ē, c_{2}, p_{2}) of two different matrices are:

(i’) 2-dual (or trichotomic dual) if their polarities are neutral and if the dual of _{2} is an includer for _{1}

(ii’) 2-contrary^{11} (or trichotomic contrary) if their polarities are non-neutral and opposite and if the contrary of α_{2} is an includer for α_{1}

(iii’) 2-complementary (or trichotomic complementary) if their polarities are non-neutral and equal and if the complementary of α_{2} is an includer for α_{1}

(vi’) 2-anti-connex (or trichotomic anti-connex) if the absolute value of the difference of their polarities is equal to 1 and if the anti-connex of α_{2} is an includer for α_{1}

To sum up now: {A^{0}, Ē^{0}} and {Ā^{0}, Ē^{0}} are 2-dual, {A^{+}, Ē^{–}}, {A^{–}, Ē^{+}}, {Ā^{+}, Ē^{–}} and {Ā^{–}, Ē^{+}} are 2-contrary, {A^{+}, Ē^{+}}, {A^{–}, Ē^{–}}, {Ā^{+}, Ē^{+}} and {Ā^{–}, Ē^{–}} are 2-complementary, {A^{0}, Ē^{+}}, {A^{0}, Ē^{–}}, {Ā^{0}, Ē^{+}} and {Ā^{0}, Ē^{–}} are 2-anti-connex.

Lastly, the notion of a complement of a canonical pole also deserves mention. Let α be a canonical pole. Let us denote by ~α its complement, semantically corresponding to non–α. In the present context, the notion of a complement entails the definition of a universe of reference. I shall focus then on the notion of a complement of a canonical pole defined with regard to the corresponding matrix. In this case, the universe of reference is equal to {A^{+}, A^{0}, A^{–}, Ā^{+}, Ā^{0}, Ā^{–}} and then ~α = {A^{+}, A^{0}, A^{–}, Ā^{+}, Ā^{0}, Ā^{–}} – α. One has thus for example ~A^{+} = {A^{0}, A^{–}, Ā^{+}, Ā^{0}, Ā^{–}} and a similar definition for the complements of the other canonical poles of the matrix. Consider now two matrices such that A < E. Under these circumstances, the universe of reference^{12} is equal to {A^{+}, A^{0}, A^{–}, Ā^{+}, Ā^{0}, Ā^{–}, Ē^{+}, Ē^{0}, Ē^{–}}. Call it the 2-matrix of α. It ensues that ~α = {A^{+}, A^{0}, A^{–}, Ā^{+}, Ā^{0}, Ā^{–}, Ē^{+}, Ē^{0}, Ē^{–}} – α. We have then the notion of a 2-complement of a canonical pole α, defined with regard to a universe of reference consisting of the 2-matrix of α. More generally, one has the notion of a n–complement (n > 0) of a canonical pole with regard to the corresponding n-matrix.

3. A solution

With the relevant machinery in place, we are now in a position to present a solution to the LHI problem. Let us now analyze the problem in the light of the above framework. To begin with, let us analyze the relevant concepts in more detail. The concept love has a positive connotation. It is a meliorative concept that can be denoted by love^{+}. Conversely, the concept hate has a negative connotation. It is a pejorative concept that can be rendered by hate^{–}. Similarly, the concept indifference also has a negative connotation. It can be considered a pejorative notion that can be denoted by indifference^{–}.

At this step, a difficulty emerges. In effect, it should be stressed that the three concepts are either meliorative or pejorative at a certain degree. And such a degree might be different from one concept to another. For example hate^{–} might be pejorative at a 0.95 degree, while indifference^{–} might be pejorative at a lesser degree of 0.7. Moreover, it could be said that such a degree might vary from culture to culture, from a given language to another. In sum, the meliorative or pejorative degree of the three concepts, so the objection goes, could be culture-relative.

Nevertheless, such difficulties can be avoided in the present context, since our reasoning will not bear upon the concepts inherent to a specific culture or language, but rather on the canonical concepts described above. Accordingly, we shall replace our usual concepts by the corresponding canonical concepts. There is room for variation in degrees, from culture to culture in the usual concepts of love, hate and indifference. But this point does not affect the current line of reasoning, since it only focuses on canonical concepts. The passage from the non-canonical concepts to the canonical ones goes straightforwardly as follows. Let d[α] be the pejorative or meliorative degree of a concept α. Hence if d[α] ∈ ]0.5; 1] then p[α] = 1 else if d[α]∈ [-1; -0.5[ then p[α] = -1. At this point, one can pose legitimately that p[Love] = 1, p[Hate] = -1 and p[Indifference] = -1^{13}. As a result, the three concepts can be denoted by Love^{+}, Hate^{–}, Indifference^{–}.

As noted from the beginning, the relationship of love/hate is unproblematic and identifies itself with the relation of contrary. This applies straightforwardly to the relationship of the canonical concepts Love^{+}/Hate^{–}. Hence, the corresponding matrix has the following structure: {Love^{+}, A^{0}, A^{–}, Ā^{+}, Ā^{0}, Hate^{–}}. Now the next step is the reconstitution of the complete matrix. This task can be accomplished with the help of the definition of the relations of the canonical poles, namely: A^{–} is corollary to Love^{+}, Ā^{+} is corollary to Hate^{–}, A^{0} is connex to Love^{+} and anti-connex to Hate^{–}, Ā^{0} is connex to Hate^{–} and anti-connex to Love^{+}. Given these elements, we are now in a position to reconstitute the corresponding canonical matrix: {Love^{+}, Attraction^{0}, A^{–}, Defiance^{+}, Repulsion^{0}, Hate^{–}}.^{14}

Let us examine now the case of the concept Indifference^{–}. Such a concept inserts itself into a matrix the structure of which is: {E^{+}, E^{0}, E^{–}, Ē^{+}, Ē^{0}, Indifference^{–}}. Just as before, it is now necessary to reconstitute the complete matrix. This can be done with the help of the corresponding definitions: Ē^{+} is corollary to Indifference^{–}, E^{–} is complementary to Indifference^{–}, E^{+} is contrary to Indifference^{–}, Ē^{0} is connex to Indifference^{–} and to the corollary of Indifference^{–}, E^{0} is anti-connex to Indifference^{–} and to the corollary of Indifference^{–}. The associated matrix is then: {E^{+}, Interest^{0}, E^{–}, Phlegm^{+}, Detachment^{0}, Indifference^{–}}.^{15}

It should be observed now that Interest^{0} = Attraction^{0} Repulsion^{0} i.e. that Interest^{0} is an includer for Attraction^{0} and Repulsion^{0}. At this step, given that {Love^{+}, Attraction^{0}, A^{–}, Repulsion^{+}, Repulsion^{0}, Hate^{–}} {E^{+}, Interest^{0}, E^{–}, Phlegm^{+}, Detachment^{0}, Indifference^{–}}, the relationship of Love^{+}/Indifference^{–} and Hate^{–}/Indifference^{–} now apply straightforwardly. In effect, it ensues from the above definitions that, on the one hand, Love^{+} and Indifference^{– }are trichotomiccontraries and on the other hand, Hate^{–} and Indifference^{– }are trichotomic complementaries. At this point, one is finally in a position to formulate a solution to the LHI problem:

(i) love is contrary to hate

(ii) love is 2-contrary to indifference

(iii) hate is 2-complementary to indifference

Hence, R, S, T identify respectively themselves with contrary, trichotomic contrary, trichotomic complementarity.

4. Concluding remarks

At this point, it is tempting not to consider the above analysis as a solution to the LHI problem per se. In effect, the concepts love, hate and indifference seem to be instances of a wider class of concepts whose relationships are of the same nature. This suggests that the same type of solution should be provided to the general problem of the definition of the relations of three given concepts , , . At first sight, certain concepts such as true, false and indeterminate, fall under the scope of the current analysis. Nevertheless, such a claim should be envisaged with caution. To what extent does the present analysis apply to other concepts? This is another problem that needs to be addressed, but whose resolution goes beyond the scope of the present account.^{16}

References

Benzaken, Claude (1991). “Systèmes formels”. Paris, Masson

Franceschi, Paul (2002). “Une Classe de Concepts”. Semiotica, 139, pp. 211-26, English translation

Garlikov, Rick (1998). “Understanding, Shallow Thinking, and School”. At http://www.garlikov.com/writings.htm

1 My translation. The original text is as follows: ‘La difficulté cependant peut provenir de paires de mots dont l’un exprime le contraire (négation) de l’autre; “haïr” peut être pris comme la négation forte de “aimer” tandis que “être indifférent” en serait la négation faible^{‘}. (p. 63).

2 With the latter notation, the matrix of the canonical poles is rendered as follows: {(A/Ā, -1, 1), (A/Ā, -1, 0), (A/Ā, -1, -1), (A/Ā, 1, 1), (A/Ā, 1, 0), (A/Ā, 1, -1)}.

3 Formally _{1} and _{2} are dual if and only if c[_{1}] = – c[_{2}] and p[_{1}] = p[_{2}] = 0.

4 Formally _{1} and _{2} are antinomical if and only if c[_{1}] = – c[_{2}] and p[_{1}] = – p[_{2}] with p[_{1}], p[_{2}] 0.

5 Formally _{1} and _{2} are complementary if and only if c[_{1}] = – c[_{2}] and p[_{1}] = p[_{2}] with p[_{1}], p[_{2}] 0.

6 Formally _{1} and _{2} are corollary if and only if c[_{1}] = c[_{2}] and p[_{1}] = – p[_{2}] with p[_{1}], p[_{2}] 0.

7 Formally _{1} and _{2} are connex if and only if c[_{1}] = c[_{2}] and │p[_{1}] – p[_{2}]│ = 1.

8 Formally _{1} and _{2} are anti-connex if and only if c[_{1}] = – c[_{2}] and │p[_{1}] – p[_{2}]│ = 1.

9 It should be observed that one of the three conditions is sufficient. In effect, E^{+} = A^{+} Ā^{+} entails E^{0} = A^{0} Ā^{0} and E^{–} = A^{–} Ā^{–}; E^{0} = A^{0} Ā^{0} implies E^{+} = A^{+} Ā^{+} and E^{–} = A^{–} Ā^{–}; E^{–} = A^{–} Ā^{–} entails E^{0} = A^{0} Ā^{0} and E^{+} = A^{+} Ā^{+}.

10 The generalisation to n matrices (n > 1) of the present construction ensues, with the relations of n-duality, n-antinomy, n-complementarity, n-anti-connexity.

12 In this context, E^{+}, E^{0} and E^{–} can be omitted without loss of content, given their nature of includers.

13 The fact of considering alternatively p[indifference] > -0.5 and thus p[Indifference] = 0 also leads to a solution in the present framework. In this last case, the relations S and T both identify themselves with trichotomic anti-connexity.

14 In the process of reconstitution of the complete matrix, some concepts may be missing. The reason is that they are not lexicalized in the corresponding language. This is notably the case for A^{–}. This last concept semantically corresponds to inappropriate, excessive attraction.

15 As far as I can see, the concepts associated with E^{+} and E^{–} are not lexicalized. They respectively correspond to appropriate interest and inappropriate, excessive interest.

16 I thank Professor Claude Panaccio and Rick Garlikov for useful comments on an earlier draft.

The purpose of this study is to describe a conceptual framework for cognitive distortions, which notably allows to specify more accurately their intrinsic relationships. This conceptual framework aims at inserting itself within the apparatus of cognitive therapy and of critical thinking. The present analysis is based on the following fundamental concepts: the reference class, the duality and the system of taxa. With the help of these three notions, each cognitive distortion can be defined. A distinction is also made between, on the one hand, general cognitive distortions and on the other hand, specific cognitive distortions. The present model allows then to define within the same conceptual framework the general cognitive distortions such as dichotomous reasoning, disqualifying a given pole, minimisation and maximisation. It also allows to describe as specific cognitive distortions: disqualifying the positive, selective abstraction and catastrophism. Furthermore, the present model predicts the existence of two other general cognitive distortions: the omission of the neutral and requalifying in the other pole.

This paper is cited in:

Paul Franceschi, Théorie des distorsions cognitives : application à l’anxiété généralisée, Journal de Thérapie Comportementale et Cognitive, Volume 18, Issue 4, December 2008, Pages 127-131, English translation

Paul Franceschi, Théorie des distorsions cognitives : la sur-généralisation et l’étiquetage, Journal de Thérapie Comportementale et Cognitive, Volume 19, Issue 4, December 2009, Pages 136-140, English translation

Pramod Pandey, On the Nature of Reason in the present-day research, in Hasnain, Imtiaz, S. and Chaudhary, S. C. (eds.), Problematizing Language Studies: Cultural, Theoretical and Applied Perspectives- Essays in Honour of of Rama Kant Agnihotri. New Delhi: Aakar Books. Pp. 387-97, 2010.

Paul Franceschi, Traitement cognitif différentiel des délires oolythématiques et du trouble anxieux généralisé, Journal de Thérapie Comportementale et Cognitive, Volume 21, Issue 4, November 2011, Pages 121-125, English translation

Lisa Wake, Karl Nielsen, Nandana Nielsen & Catalin Zaharia, Depression symptom clusters, in “The Clinical Effectiveness of Neurolinguistic Programming”, Routledge 2013, edited by Lisa Wake, Richard M. Gray and Frank S. Bourke

Lizet Fernandez-Jammet, Evaluation longitudinale de l’efficacité d’une prise en charge cognitivo-comportementale de groupe destinée à des patients atteints de fibromyalgie, dissertation doctorale, 2016

Hélène Richard-Lepouriel, Trouble bipolaire, auto-stigmatisation et restructuration cognitive : une première tentative de prise en charge, Journal de Thérapie Comportementale et Cognitive, Volume 27-4, November 2017, Pages 177-183

Valérie Pennequin & Nicolas Combalbert, L’influence des biais cognitifs sur l’anxiété chez des adultes non cliniques, Annales Médico-psychologiques, Volume 175, Issue 2, February 2017, Pages 103-107

Anita Robert, Nicolas Combalbert, Valérie Pennequin, Etude des profils de distorsion cognitive en fonction des états anxieux et dépressifs chez des adultes tout-venant, Annales Médico-Psychologiques 176 (2018) 225–230

Nawal Ouhmad, Nicolas Combalbert, Wissam El-Hage, Cognitive distortions and emotion regulation among post traumatic stress disorder victims, in Psychological Applications and Trends, Ed. by C. Pracana & M. Wang, InScience Press, 2019

Paul Franceschi, For a Typology of Auditory Verbal Hallucinations Based on their Content, Activitas Nervosa Superior, volume 62, pages 104–109, 2020

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

Deperrois Romain & Nicolas Combalbert, Links between cognitive distortions and cognitive emotion regulation strategies in non-clinical young adulthood, in Psychological Applications and Trends, Ed. by C. Pracana & M. Wang, InScience Press, 2021

Ouhmad, Nawal & El-Hage, Wissam & Combalbert, Nicolas. (2022), Maladaptive cognitions and emotional regulation in PTSD, 3-7, Conference: International Psychological Applications Conference and Trend, doi:10.36315/2022.

Complements to a Theory of Cognitive Distortions

Paul FRANCESCHI

The cognitive distortions, introduced by Aaron Beck (1963, 1964) and Albert Ellis (1962) are traditionally defined as fallacious reasoning that plays a crucial role in the emergence of certain mental disorders. The cognitive therapy in particular is based on the identification of these cognitive distortions within the everyday way of thinking of the patient, and their replacement by alternative reasoning. Traditionally, the cognitive distortions are represented as one of the twelve following irrational modes of reasoning: 1. Emotional reasoning 2. Overgeneralization 3. Jumping to conclusions (or arbitrary inference) 4. Dichotomous reasoning 5. Should statements (Ellis 1962) 6. Fortune telling or mind reading 7. Selective abstraction 8. Disqualifying the positive 9. Maximisation and minimisation 10. Catastrophism 11. Personalisation 12. Labelling.

Under their classical form which is that of an enumeration, the cognitive distortions plays a central role within the field of cognitive therapy. Considering also their widespread nature in normal reasoning, one can think however that an accurate understanding of the cognitive distortions proves also to be useful outside the field of psychopathology. In particular, the cognitive distortions can also be seen as part of the apparatus which constitutes critical thinking. For these reasons, it appears that a conceptual framework, notably allowing to define the relationships between the different cognitive distortions, could also turn out to be useful. In what follows, we shall set out to present a general theory of the cognitive distortions, which brings a certain number of supplementary elements in comparison with classical theory.

1. Main notions

The present framework allows to represent several classical cognitive distortions: dichotomous reasoning, disqualification of one of the poles, selective abstraction, minimisation and maximisation. To these, one can add two other cognitive distortions of which the present model allows to predict the existence and which are closely related to the classical cognitive distortions, although they do not appear, to the knowledge of the author, to the number of these last. It consists of the omission of the neutral and the requalification in the other pole.

The cognitive distortions can be constructed, in the present model, from three main notions: the reference class,the duality and the system of taxa. It is necessary, in a preliminary way, to set out to describe these three notions. The reference class, above all, is constituted by a group of phenomena or objects. Several examples can be given here: the class composed of the events and facts of the patient’s life; the class of the future events of the patient’s life; the class constituted by all the parts of the patient’s body; the class which is made up of the patient’s character’s traits.

The notion of duality, second, corresponds to a pair of concepts such as Positive/Negative, Internal/External, Collective/Individual, Nice/Ugly, etc. A duality corresponds then to a criterion under the angle of which the elements of the reference class can be considered or evaluated. Let us denote by A/Ā a given duality, where A and Ā constitute then dual concepts. An enumeration (necessarily partial) of the dualities is as follows: Positive/Negative, Internal/External, Quantitative/Qualitative, Visible/Invisible, Analytical/synthetic, Absolute/Relative, Abstract/Concrete, Static/Dynamic, Unique/Multiple, Aesthetics/Practice, Definite/Vague, Finite/Infinite, Simple/Composite, Individual/Collective, Implicit/Explicit, Intentional/Unintentional.

Finally, thepatient’s system of taxa consists of a taxonomy which allows the patient to evaluate and to classify the elements of the reference class, according to the criterion corresponding to a given duality A/Ā. These taxa can be considered as “what can see” the patient. It consists of a system of values that is inherent to the patient or of a filter through which the patient “sees” the elements of the reference class, i.e. the phenomena or the objects of reality. The figure below represents an optimal system of taxa.

This last is composed of 11 spheres which represent each a given taxon. The system of taxa is optimal, because all taxa are present. On the other hand, if the patient does not have some taxa, he cannot see nor count the corresponding elements. For example, if he/she lacks the taxa of the duality A/Ā corresponding to pole A, he cannot see the corresponding elements. In the same way, if the patient has no neutral taxon in his/her system of taxa, he cannot see the neutral elements of the reference class. More formally, let us consider then a series of n elements E_{1}, E_{2}, …, E_{n} such that each of them has, in a objective way, a degree d[E_{i}] in a duality A/Ā comprised between -1 and 1 (d Î [-1, +1]). We can consider then a series including 11 elements, E_{1}, E_{2}, …, E_{11}, which present an objective increasing degree (the choice of 11 elements is here arbitrary, and any other number would also do the job). Let us pose then: d[E_{1}] = -1, d[E_{2}] = -4/5, d[E_{3}] = -3/5, d[E_{4}] = -2/5, d[E_{5}] = -1/5, d[E_{6}] = 0, d[E_{7}] = 1/5, d[E_{8}] = 2/5, d[E_{9}] = 3/5, d[E_{10}] = 4/5, d[E_{11}] = 1. Let us also define a subjective degree D[E_{i}] such that it is attributed by the patient to each of the E_{i}. So, E_{1}-E_{5} corresponds to the pole A of duality A/Ā, E_{6} to the neutral taxon and E_{7}-E_{11} corresponds to the pole Ā. Moreover, this optimal system of taxa can be assimilated with one Likert scale with 11 degrees.

At this stage, we are in a position to define the main cognitive distortions, and it is worth considering them in turn. The cognitive distortions can be defined as a type of reasoning which leads to favour, without objective grounds, a subset of the taxa applicable to a given duality A/Ā, in order to qualify a given reference class. It also proves to be useful to draw a distinction, in a preliminary way, between the general cognitive distortions and the specific cognitive distortions. The general cognitive distortions relate to all reference classes and all dualities. By contrast, the specific cognitive distortions are mere instances of the general cognitive distortions which are inherent to a given reference class and to a given duality.

2. The cognitive distortions

2.1 Dichotomous reasoning

In the present context, dichotomous reasoning (or all-or-nothing thinking) can be defined as a general cognitive distortion which leads the patient to consider a given reference class only according to the two extreme taxa which relate to every pole of a given duality. With this type of reasoning, the patient ignores completely the presence of degrees or of intermediate steps. In his/her taxa system, the patient has as well the two extreme taxa corresponding to poles A and Ā. The defect in that way of considering things is that facts or objects corresponding to intermediary taxa are not taken into account. So it results from it a reasoning without nuances nor gradation, which proves to be maladapted to properly apprehend the diversity of human situations. Formally, dichotomous reasoning consists in taking into account only the elements of the reference class such as|d[E_{i}]| = 1, ord[E_{1}] = 1 ord[E_{11}] = -1, by ignoring all the others.

2.2 The disqualification of one of the poles

In the present model, the disqualification of one of the poles is the general cognitive distortion which leads to grant an arbitrary priority in one of the poles of a given duality, in order to qualify the elements of a reference class. It consists then in the fact of attributing more importance to one of the poles rather than to the other one, in the lack of objective motivation. The taxa corresponding to one of the poles of a given duality are lacking in the patient’s system of taxa. So, the patient sees things only through the prism of pole A (respectively Ā), by ignoring completely the viewpoint of the opposed pole Ā (respectively A). Formally, the disqualification of one of the poles leads to consider only the E_{i} such that d[E_{i}] ≤ (respectivelyd[E_{i}] ≥ 0), by ignoring any events such thatd[E_{i}] > 0 (respectivelyd[E_{i}] < 0).

An instance of the disqualification of one of the poles consists in the disqualification of the positive. This last can be analysed, in the present context, as a specific instance of the disqualification of one of the poles, which applies to the Positive/Negative duality and to the reference class including the facts and events of the patient’s life. The patient tends then to ignore positive events, by considering that they do not count, for any reason. Such instance finds to apply in the cognitive therapy of depression.

Another instance of the disqualification of one of the poles also applies to the Positive/Negative duality and to the reference class which comprises the character’s traits of the patient. This one completely ignores his/her positive character’s traits (qualities) and only directs his/her attention to his/her negative character’s traits (defects). This encourages then him/her to conclude that he/she “is worth nothing”, that he/she is “a failure”. Such instance also applies in the cognitive therapy of depression.

2.3 Arbitrary focusing on a given modality

Another type of cognitive general distortion consists in arbitrary focusing on a modality of a given duality. In the present context, this type of general cognitive distortion leads to favour one taxon in the patient’s system of taxa, by ignoring all the others. In arbitrary focusing, the taxon being discussed is present in the patient’s system of taxa, and is affected to an unique element of the reference class. There is eclipsing (in general temporary) of others taxa and other elements of the reference class, so that the patient is haunted by this specific element.

A particular instance of this type of general cognitive distortion, relates to the reference class of the facts of the patient’s life, and to the Positive/Negative duality. It is a specific cognitive distortion, which consists in focusing on a negative event of the patient’s life. It is then one of the classical cognitive distortions, defined as selective abstraction (Mental filter), which consists in the fact of choosing one detail with a negative connotation and to focalise on it. Suchlike, the patient sees only this detail, and his/her vision{view} of reality is darkened because it is entirely tinted with this particular event. Such instance applies in the cognitive therapy of depression.

One can also mention another instance of arbitrary focusing, which also applies to the Positive/Negative duality, but relates to the class of reference composed of the hypothetical future events of the patient’s life. In that case, the patient focalises on the possible happening of a very negative event. Such instance finds to apply in the cognitive therapy of generalised anxiety disorder.

Another specific instance of arbitrary focusing applies to the Nice/Ugly duality and to a reference class which identifies itself with all the parts of the patient’s body. The patient focalises then on a detail of his/her anatomy which he considers to be ugliness. The patient has well, in his/her system of taxa the Ugly taxon in question. Moreover, he/she affects this taxon to an unique part of his/her body, while all the others taxa are temporarily eclipsed. Such specific cognitive distortion finds to apply in the cognitive therapy of body dysmorphic disorder (Neziroglu and Yaryura-Tobias 1993, Veale and Riley 2001, Veale 2004).

2.4 The omission of the neutral

The present model also leads to predict the existence of another type of general cognitive distortion, which consists in the omission of the neutral. This latter cognitive distortion results from the absence, in the patient’s system of taxa, of the neutral taxon. It follows that the elements of the reference class which can objectively be defined as neutral with regard to a given duality A/Ā, are not taken into account by the patient. Formally, the patient omits to consider the E_{i} such that d[E_{i}] = 0. The omission of the neutral sometimes plays an important role, notably when there is a gaussian distribution of the elements of the reference class, where the elements objectively corresponding to the neutral taxon are precisely those which are the most numerous.

2.5 The requalification in the other pole

The present model also leads to predict the existence of another type of general cognitive distortion. That is the reasoning which consists in re-qualifying an event belonging to a given duality A, in the other duality Ā. Formally, the subjective degree attributed by the patient to a given event E is the opposite of its objective degree, so that: D[E] = (-1) x d[E].

A characteristic instance of requalification in the other pole consists in the specific cognitive distortion which applies to the class of the events of the patient’s life and to the Positive/Negative duality. This consists typically in re-describing as negative an event which should be objectively considered as positive. By requalifying positive events in a negative way, the patient can reach the conclusion that all events of his/her life are of a negative nature. For instance, by considering the past events of his/her life, the patient notes that he/she made no act of violence. He/she considers this to be “suspect”. This type of instance also finds to apply within the cognitive therapy of depression.

Another instance of requalification in the other pole consists in the specific cognitive distortion which applies to the class of the parts of the patient’s body and to the Nice/Ugly duality. Typically, the patient re-qualifies as “ugly” a part of his/her body which is objectively “nice”. Such specific cognitive distortion relates to the cognitive therapy of body dysmorphic disorder.

2.6 Minimisation and maximisation

This general cognitive distortion consists in attributing to an element of the reference class, a taxon according to the criterion of a duality A/Ā which proves to be lower (minimisation) or greater (maximisation) than its objective value. It consists here of a classical cognitive distortion. The subjective degree D[E] which is attributed by the patient to an event E differs significantly from its objective degree d[E]. In minimisation, this subjective degree is distinctly less than, so that |D[E]| < |d[E]|. In maximisation, by contrast, the subjective degree is distinctly greater, such that |D[E]| > |d[E]|.

A specific instance of minimisation relates to the class of the facts of the patient’s life and to the Positive/Negative duality. The patient tends to consider certain facts of his/her existence as less positive than they in reality are. In maximisation, he/she considers certain facts of his/her life as more negative than they really are. In the present context, the classical cognitive distortion of catastrophism (or dramatisation) can be considered as a specific cognitive distortion, which consists of a maximisation applied to the negative pole of the Positive/Negative duality. The patient attributes then a subjective degree D[E] in the Positive/Negative duality to an event, while the absolute value of its objective degree d[E] is very distinctly lesser. Such instance applies to the cognitive therapy of depression.

3. Conclusion

As we see it, the present theory provides several elements, in comparison with classical theory, that allow to define and to classify the classical cognitive distortions, within the same conceptual framework. These last are considered, either as general cognitive distortions, or as specific cognitive distortions, i.e. as instances of the general cognitive distortions which relate to a given reference class and duality. Thus, dichotomous reasoning, maximisation and minimisation constitute general cognitive distortions. In addition, disqualification of the positive, selective abstraction, selective negative focus and catastrophism constitute then specific cognitive distortions. Besides, the present analysis allowed to describe two additional general cognitive distortions: the omission of the neutral and the requalification in the other pole.

References

Beck AT.: 1963, Thinking and depression: Idiosyncratic content and cognitive distortions. Archives of General Psychiatry9, 324-333.

Beck AT.: 1964, Thinking and depression: Theory and therapy, Archives of General Psychiatry10, 561-571.

Ellis A.: 1962, Reason and Emotion in Psychotherapy, Lyle Stuart, New York.

Neziroglu FA et JA. Yaryura-Tobias: 1993, Exposure, response prevention, and cognitive therapy in the treatment of body dysmorphic disorder. Behav Ther 24, 431-438.

Veale D et S. Riley: 2001, Mirror, mirror on the wall, who is the ugliest of them all? The psychopathology of mirror gazing in previous termbody dysmorphic disorder.next term. Behaviour Research and Therapy39, 1381-1393.

Veale D.: 2004, Advances in a cognitive behavioural model of body dysmorphic disorder. Body Image1, 113-125.

Viewpoints relative to a given duality are viewpoints that concerne a given duality A/Ā.

For example, one can place oneself under the angle of the Static/Dynamic duality. Or, one can place oneself according to the point of view of the Absolute/Relative duality.

In On a Class of Concepts (2002), I presented a theory based on matrices of concepts which aims to constitute an alternative to the classification proposed by Greimas, in the field of paradigmatic analysis. I proceed here to apply the matrices de concepts to the analysis of a corpus made up of Corsican proverbs.