# A Brief Introduction to N-universes

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

## A Brief Introduction to N-universes

Paul Franceschi

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

## Presentation of n-universes

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

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

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

– the ΩObj0Time0Loc0: a n-universe comprising a unique object, a temporal constant and a spatial constant

– the ΩObjColTime0Loc: a n-universe comprising multiple objects, a colour variable, a temporal constant and a space variable

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

It is also worth drawing a distinction according to whether the given n-universe comprises either one single (ΩObj0) or multiple objects (ΩObj). To cite a few examples:

– the ΩObj0TimeLoc0: a n-universe comprising a unique object, a temporal variable and a spatial constant

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

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

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

– the ΩObjTime*ObjLoc0: a n-universe with multiple objects comprising a temporal variable and a spatial constant; the objects are also demultiplied with regard to the temporal criterion, so that a given object can thus exemplify several different temporal positions.

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

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

– the ΩObj*Col0Obj*Time0ObjLoc: this type of n-universe with multiple objects comprises a colour constant (Col0), a temporal constant (Time0) and a spatial variable (Loc); moreover, the objects are in a many-one relation with the colour constant, so that all objets share the same colour; in addition, the objects are in a many-one relation with the time constant, so that several objects can exist at a the unique temporal position; on the other hand, the objects are in relation one-one with the space criterion, so that only one object can exist at a given space location (Fig. 1)

– the ΩObjColObj*Time0ObjLoc: this type of n-universe with multiple objects comprises a colour variable (Col), a temporal constant (Time0) and a spatial variable (Loc); moreover, the objects are in a one-one relation with the colour variable, so that all objects can have a different colour; in addition, the objects are in a many-one relation with the time constant, so that several objects can exist at a the unique temporal position; on the other hand, the objects are in relation one-one with the space criterion, so that only one object can exist at a given space location (Fig. 2)

– the ΩObj*ColObj*Time0ObjLoc: this type of n-universe with multiple objects comprises a colour variable (Col), a temporal constant (Time0) and a spatial variable (Loc); moreover, the objects are in a many-one relation with the colour variable, so that several objets can have the same colour; in addition, the objects are in a many-one relation with the time constant, so that several objects can exist at a the unique temporal position; lastly, the objects are in relation one-one with the space criterion, so that only one object can exist at a given space location (Fig. 3)

– the ΩObj*Col0Obj*Time0Obj*Loc: this type of n-universe with multiple objects comprises a colour constant (Col0), a temporal constant (Time0) and a spatial variable (Loc); moreover, the objects are in a many-one relation with the colour constant, so that several objets can share the same colour; in addition, the objects are in a many-one relation with the time constant, so that several objects can exist at a the unique temporal position; on the other hand, the objects are in relation many-one with the space criterion, so that only several objects can occupy the same space location (Fig. 4)

## First steps with n-universes

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

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

Let us proceed now to model the situation corresponding to the Little Puddle/London experiment in terms of n-universes. What are thus the criteria of the corresponding n-universe? It appears first that the corresponding situation characterises itself with the presence of multiple individuals: there are indeed 50 inhabitants in Little Puddle and 10 million in London. Consequently, the corresponding n-universe comprises multiple objects (ΩObj). It also appears that the Little Puddle/London experiment takes place at one single temporal position. Thus, the corresponding n-universe has a constant-time (ΩTime0). Moreover, several inhabitants exist simultaneously at the unique temporal position Time0. Hence, the objects are in relation many-one with the temporal constant (ΩObj*Time0). Moreover, two space locations are explicitly distinguished: Little Puddle (Loc1) and London (Loc2). The corresponding situation can thus be modelled in a n-universe comprising a space variable (ΩLoc) which includes two different locations: Loc1 and Loc2. Furthermore, it proves that each inhabitant is either in Little Puddle or in London, so as a given inhabitant cannot occupy several space locations at the same time. Thus the space criterion is not demultiplied. Lastly, one can observe that several people can be at the same time at a given space location: there are thus 50 inhabitants in Little Puddle and 10 million in London. Consequently, the objects are in relation many-one with the space variable (ΩObj*Loc). Taking into account what precedes, it follows that the situation corresponding to the Little Puddle/London experiment can be modelled in a ΩObj*Time0Obj*Loc, a n-universe with multiple objects, comprising a temporal constant and a space variable, where the objects are in relation many-one with the time-constant and the space variable.

## Conclusion

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

# A Solution to Goodman’s Paradox

English Posprint (with additional illustrations) of a paper published in French in Dialogue Vol. 40, Winter 2001, pp. 99-123 under the title “Une Solution pour le Paradoxe de Goodman”.
In the classical version of Goodman’s paradox, the universe where the problem takes place is ambiguous. The conditions of induction being accurately described, I define then a framework of n-universes, allowing the distinction, among the criteria of a given n-universe, between constants and variables. Within this framework, I distinguish between two versions of the problem, respectively taking place: (i) in an n-universe the variables of which are colour and time; (ii) in an n-universe the variables of which are colour, time and space. Finally, I show that each of these versions admits a specific resolution.

This paper is cited in:

• Alasdair Richmond, The Doomsday Argument, Philosophical Books Vol. 47 No. 2 April 2006, pp. 129–142

# A Solution to Goodman’s Paradox

Paul FRANCESCHI

a paper originally published in Dialogue, winter 2001, vol. 40, pp. 99-123

ABSTRACT: In the classical version of Goodman’s paradox, the universe where the problem takes place is ambiguous. The conditions of induction being accurately described, I define then a framework of n-universes, allowing the distinction, among the criteria of a given n-universe, between constants and variables. Within this framework, I distinguish between two versions of the problem, respectively taking place: (i) in an n-universe the variables of which are colour and time; (ii) in an n-universe the variables of which are colour, time and space. Finally, I show that each of these versions admits a specific resolution.

## 1. The problem

Goodman’s Paradox (thereafter GP) has been described by Nelson Goodman (1946).i Goodman exposes his paradox as follows.ii Consider an urn containing 100 balls. A ball is drawn each day from the urn, during 99 days, until today. At each time, the ball extracted from the urn is red. Intuitively, one expects that the 100th ball drawn from the urn will also be red. This prediction is based on the generalisation according to which all the balls in the urn are red. However, if one considers the property S “drawn before today and red or drawn after today and non-red”, one notes that this property is also satisfied by the 99 instances already observed. But the prediction which now ensue, based on the generalisation according to which all the balls are S, is that the 100th ball will be non-red. And this contradicts the preceding conclusion, which however conforms with our intuition.iii

Goodman expresses GP with the help of an enumerative induction. And one can model GP in terms of the straight rule (SR). If one takes (D) for the definition of the “red” predicate, (I) for the enumeration of the instances, (H) for the ensuing generalisation, and (P) for the corresponding prediction, one has then:

(D) R = red

(I) Rb1·Rb2·Rb3·…·Rb99

(H) Rb1·Rb2·Rb3·…·Rb99·Rb100

∴ (P) Rb100

And also, with the predicate S:

(D*) S = red and drawn before T or non-red and drawn after T

(I*) Sb1·Sb2·Sb3·…·Sb99

(H*) Sb1·Sb2·Sb3·…·Sb99·Sb100 that is equivalent to:

(H’*) Rb1·Rb2·Rb3·…·Rb99·~Rb100

∴ (P*) Sb100 i. e. finally:

∴ (P’*) ~Rb100

The paradox resides here in the fact that the two generalisations (H) and (H*) lead respectively to the predictions (P) and (P’*), which are contradictory. Intuitively, the application of SR to (H*) appears erroneous. Goodman also gives in Fact, Fiction and Forecastiv a slightly different version of the paradox, applied in this case to emeralds.v This form is very well known and based on the predicate “grue” = green and observed before T or non-green and observed after T.

The predicate S used in Goodman (1946) presents with “grue”, a common structure. P and Q being two predicates, this structure corresponds to the following definition: (P and Q) or (~P and ~Q). In what follows, one will designate by grue a predicate having this particular structure, without distinguishing whether the specific form used is that of Goodman (1946) or (1954).

## 2. The unification/differentiation duality

The instances are in front of me. Must I describe them by stressing their differences? Or must I describe them by emphasising their common properties? I can proceed either way. To stress the differences between the instances, is to operate by differentiation. Conversely, to highlight their common properties, is to proceed by unification. Let us consider in turn each of these two modes of proceeding.

Consider the 100 balls composing the urn of Goodman (1946). Consider first the case where my intention is to stress the differences between the instances. There, an option is to apprehend the particular and single moment, where each of them is extracted from the urn. The considered predicates are then: red and drawn on day 1, red and drawn on day 2, …, red and drawn on day 99. There are thus 99 different predicates. But this prohibits applying SR, which requires one single predicate. Thus, what is to distinguish according to the moment when each ball is drawn? It is to stress an essential difference between each ball, based on the criterion of time. Each ball thus is individualised, and many different predicates are resulting from this: drawn at T1, drawn at T2, …, drawn at T99. This indeed prevents then any inductive move by application of SR. In effect, one does not have then a common property to allow induction and to apply SR. Here, the cause of the problem lies in the fact of having carried out an extreme differentiation.

Alternatively, I can also proceed by differentiation by operating an extremely precisevi measurement of the wavelength of the light defining the colour of each ball. I will then obtain a unique measure of the wavelength for each ball of the urn. Thus, I have 100 balls in front of me, and I know with precision the wavelength of the light of 99 of them. The balls respectively have a wavelength of 722,3551 nm, 722,3643 nm, 722,3342 nm, 722,3781 nm, etc. I have consequently 99 distinct predicates P3551, P3643, P3342, P3781, etc. But I have no possibility then to apply SR, which requires one single predicate. Here also, the common properties are missing to allow to implement the inductive process. In the same way as previously, it proves here that I have carried out an extreme differentiation.

What does it occur now if I proceed exclusively by unification? Let us consider the predicate R corresponding to “red or non-red”. One draws 99 red balls before time T. They are all R. One predicts then that the 100th ball will be R after T, i.e. red or non-red. But this form of induction does not bring any information here. The resulting conclusion is empty of information. One will call empty induction this type of situation. In this case, one observes that the process of unification of the instances by the colour was carried out in a radical way, by annihilating in this respect, any step of differentiation. The cause of the problem lies thus in the implementation of a process of extreme unification.

If one considers now the viewpoint of colour, it appears that each case previously considered requires a different taxonomy of colours. Thus, it is made use successively:

– of our usual taxonomy of colours based on 9 predicates: purple, indigo, blue, green, yellow, orange, red, white, black

– of a taxonomy based on a comparison of the wavelengths of the colours with the set of the real numbers (real taxonomy)

– of a taxonomy based on a single predicate (single taxon taxonomy): red or non-red

But it proves that each of these three cases can be replaced in a more general perspective. Indeed, multiple taxonomies of colours are susceptible to be used. And those can be ordered from the coarser (single taxon taxonomy) to the finest (real taxonomy), from the most unified to the most differentiated. We have in particular the following hierarchy of taxonomies:

– TAX1 = {red or non-red} (single taxon taxonomy)

– TAX2 = {red, non-red} (binary taxonomy)

– …

– TAX9 = {purple, indigo, blue, green, yellow, orange, red, white, black} (taxonomy based on the spectral colours, plus white and black)

– …

– TAX16777216 = {(0, 0, 0), …, (255, 255, 255)} (taxonomy used in computer science and distinguishing 256 shades of red/green/blue)

– …

– TAXR = {370, …, 750} (real taxonomy based on the wavelength of the light)

Within this hierarchy, it appears that the use of extreme taxonomies such as the one based on a single taxon, or the real taxonomy, leads to specific problems (respectively extreme unification and extreme differentiation). Thus, the problems mentioned above during the application of an inductive reasoning based on SR occur when the choice in the unification/differentiation duality is carried out too radically. Such problems relate to induction in general. This invites to think that one must rather reason as follows: I should privilege neither unification, nor differentiation. A predicate such as “red”, associated with our usual taxonomy of colours (TAX9)vii, corresponds precisely to such a criterion. It corresponds to a balanced choice in the unification/differentiation duality. This makes it possible to avoid the preceding problems. This does not prevent however the emergence of new problems, since one tries to implement an inductive reasoning, in certain situations. And one of these problems is naturally GP.

Thus, it appears that the stake of the choice in the duality unification/differentiation is essential from the viewpoint of induction, because according to whether I choose one way or the other, I will be able or not to use SR and produce valid inductive inferences. Confronted with several instances, one can implement either a process of differentiation, or a process of unification. But the choice that is made largely conditions the later success of the inductive reasoning carried out on those grounds. I must describe both common properties and differences. From there, a valid inductive reasoning can take place. But at this point, it appears that the role of the unification/differentiation duality proves to be crucial for induction. More precisely, it appears at this stage that a correct choice in the unification/differentiation duality constitutes one of the conditions of induction.

## 3. Several problems concerning induction

The problems which have been just mentioned constitute the illustration of several difficulties inherent to the implementation of the inductive process. However, unlike GP, these problems do not generate a genuine contradiction. From this point of view, they distinguish from GP. Consider now the following situation. I have drawn 99 balls respectively at times T1, T2, …, T99. The 100th ball will be drawn at T100. One observes that the 99 drawn balls are red. They are thus at the same time red and drawn before T100. Let R be the predicate “red” and T the predicate “drawn before T100“. One has then:

(I) RTb1, RTb2, …, RTb99

(H) RTb1, RTb2, …, RTb99, RTb100

∴ (P) RTb100

By direct application of SR, the following prediction ensue: “the 100th ball is red and drawn before T100“. But this is in contradiction with the data of the experiment in virtue of which the 100th ball is drawn in T100. There too, the inductive reasoning is based on a formalisation which is that of SR. And just as for GP, SR leads here to a contradiction. Call 2 this problem, where two predicates are used.

It appears that one can easily build a form of 2 based on one single predicate. A way of doing that is to consider the unique predicate S defined as “red and drawn before T100” in replacement of the predicates R and T used previously. The same contradiction then ensues.

Moreover, it appears that one can highlight another version (1) of this problem comprising only one predicate, without using the “red” property which appears useless here. Let indeed T be the predicate drawn before T100. One has then:

(I) Tb1, Tb2, …, Tb99

(H) Tb1, Tb2, …, Tb99, Tb100

∴ (P) Tb100

Here also, the conclusion according to which the 100th ball is drawn before T100 contradicts the data of the experiment according to which the 100th ball is drawn at T100. And one has then a contradictory effect, analogous to that of GP, without the structure of “grue” being implemented. Taking into account the fact that only the criterion of time is used to build this problem, it will be denoted in what follows by 1-time.

It appears here that the problems such as 1-time and 2 lead just as GP to a contradiction. Such is not the case for the other problems related to induction previously mentionedviii, which involve either the impossibility of carrying out induction, or a conclusion empty of information. However, it proves that the contradiction encountered in 1-time is not of the same nature as that observed in GP. Indeed in GP, one has a contradiction between the two concurrent predictions (P) and (P*). On the other hand, in 1-time, the contradiction emerges between on the one hand the conditions of the experiment (T  100) and on the other hand the prediction resulting from generalisation (T < 100).

Anyway, the problems which have been just encountered suggest that the SR formalism does not capture the whole of our intuitions related to induction. Hence, it is worth attempting to define accurately the conditions of induction, and adapting consequently the relevant formalism. However, before carrying out such an analysis, it is necessary to specify in more detail the various elements of the context of GP.

## 4. The universe of reference

Let us consider the law (L1) according to which “diamond scratches the other solids”. A priori, (L1) strikes us as an undeniable truth. Nevertheless, it proves that at a temperature higher than 3550°C, diamond melts. Therefore in last analysis, the law (L1) is satisfied at a normal temperature and in any case, when the temperature is lower than 3550°C. But such a law does not apply beyond 3550°C. This illustrates how the statement of the conditions under which the law (L1) is verified is important, in particular with regard to the conditions of temperature. Thus, when one states (L1), it proves necessary to specify the conditions of temperature in which (L1) finds to apply. This is tantamount to describing the type of universe in which the law is satisfied.

Let also (P1) be the following proposition: “the volume of the visible universe is higher than 1000 times that of the solar system”. Such a proposition strikes us as obvious. But there too, it appears that (P1) is satisfied at modern time, but that it proves to be false at the first moments of the universe. Indeed, when the age of our universe was 10-6 second after the big-bang, its volume was approximately equal to that of our solar system. Here also, it thus appears necessary to specify, at the same time as the proposition (P1) the conditions of the universe in which it applies. A nonambiguous formulation of (P1) thus comprises a more restrictive temporal clause, such as: “at our time, the volume of the visible universe is higher than 1000 times that of the solar system”. Thus, generally, one can think that when a generalisation is stated, it is necessary to specify the conditions of the universe in which this generalisation applies. The precise description of the universe of reference is fundamental, because according to the conditions of the universe in which one places oneself, the stated law can appear true or false.

One observes in our universe the presence of both constants and variables. There are thus constants, which constitute the fundamental constants of the universe: the speed of light: c = 2,998 x108 m/s; Planck’s constant: h = 6,626 x 10-34 J.s; the electron charge; e = 1,602 x 10-19 C; etc. There are on the other hand variables. Among those, one can mention in particular: temperature, pressure, altitude, localisation, time, presence of a laser radiation, presence of atoms of titanium, etc.

One often tends, when a generalisation is stated, not to take into account the constants and the variables which are those of our universe envisaged in its totality. Such is the case for example when one considers the situation of our universe on 1 January 2000, at 0h. One places then oneself explicitly in what constitutes a section, a slice of our universe. In effect, time is not regarded then a variable, but well as a constant. Consider also the following: “the dinosaurs had hot blood”ix. Here, one places oneself explicitly in a sub-universe of our where the parameters of time and space have a restricted scope. The temporal variable is reduced to the particular time of the Earth history which knew the appearance of the dinosaurs: the Triassic and the Cretaceous. And similarly, the space parameter is limited to our planet: Earth. Identically, the conditions of temperature are changing within our universe, according to whether one is located at one site or another of it: at the terrestrial equator, the surface of Pluto, the heart of Alpha Centauri, etc. But if one is interested exclusively in the balloon being used for the experimentation within the laboratory of physics, where the temperature is maintained invariably at 12°C, one can then regard valuably the temperature as a constant. For when such generalisations are expressed, one places oneself not in our universe under consideration in his totality, but only in what veritably constitutes a specific part, a restriction of it. One can then assimilate the universe of reference in which one places oneself as a sub-universe of our. It is thus frequent to express generalisations which are only worth for the present time, or for our usual terrestrial conditions. Explicitly or not, the statement of a law comprises a universe of reference. But in the majority of the cases, the variables and the constants of the considered sub-universe are distinct from those allowing to describe our universe in its totality. For the conditions are extremely varied within our universe: the conditions are very different according to whether one places oneself at the 1st second after the big-bang, on Earth at the Precambrian epoch, in our planet in year 2000, inside the particle accelerator of the CERN, in the heart of our Sun, near a white dwarf, or well inside a black hole, etc.

One can also think that it is interesting to be able to model universes the constants of which are different from the fundamental constants of our universe. One can thus wish to study for example a universe where the mass of the electron is equal to 9,325 x10-31 kg, or well a universe where the electron charge is equal to 1,598 x 10-19 C. And in fact, the toy-universes, which take into account fundamental constants different from those of our familiar universe, are studied by the astrophysicists.

Lastly, when one describes the conditions of a thought experiment, one places oneself, explicitly or not, under the conditions which are related to those of a sub-universe. When one considers for example 100 balls extracted from an urn during 100 consecutive days, one places then oneself in a restriction of our universe where the temporal variable is limited to one period of 100 days and where the spatial location is extremely reduced, corresponding for example to a volume approximately equal to 5 dm3. On the other hand, the number of titanium or zirconium atoms possibly present in the urn, the possible existence of a laser radiation, the presence or the absence of a sound source of 10 db, etc. can be omitted and ignored. In this context, it is not necessary to take into account the existence of such variables. In this situation, it is enough to mention the variables and the constants actually used in the thought experiment. For one can think indeed that the number of variables in our universe is so large that it is impossible to enumerate them all. And consequently, it does not appear possible to characterise our universe in function of all its variables, because one can not provide an infinite enumeration of it. It appears sufficient to describe the considered sub-universe, by mentioning only the constants and the variables which play an effective role in the experiment. Thus, in such situations, one will describe the considered sub-universe by mentioning only the effective criteria necessary to the description of the experiment.

What precedes encourages to think that generally, in order to model the context in which the problems such as GP take place, it is convenient to describe a given universe in terms of variables and constants. This leads thus to define a n-universe (n 0) as a universe the criteria of which comprise m constants, and n variables, where the m constants and n variables constitute the criteria of the given universe. Within this particular framework, one defines a temporal 1-universe1T) as a universe comprising only one criterion-variable: time. In the same way, one defines a coloured 1-universe1C) as a universe comprising only one criterion-variable: colour. One will define also a coloured and temporal 2-universe2CT) as a universe comprising two criterion-variables: time and colour. Etc. In the same way, a universe where all the objects are red, but are characterised by a different localisation will be modelled by a localised 1-universe1L) a criterion-constant (red) of which is colour.

It should be noted incidentally that the n-universe framework makes it possible in particular to model several interesting situations. Thus, a temporal universe can be regarded as a n-universe one of the variables of which is a temporal criterion. Moreover, a universe where one single moment T0 is considered, deprived of the phenomenon of succession of time, can be regarded as a n-universe where time does not constitute one of the variables, but where there is a constant-time. In the same way, an atemporal universe corresponds to a n-universe no variable of which corresponds to a temporal criterion, and where there is not any time-constant.

In the context which has been just defined, what is it now to be red? Here, being “red” corresponds to two different types of situations, according to the type of n-universe in which one places oneself. It can be on the one hand a n-universe one of the constants of which is colour. In this type of universe, the colour of the objects is not susceptible to change, and all the objects are there invariably red.

The fact of being “red” can correspond, on the second hand, to a n-universe one of the criterion-variables of which is constituted by colour. There, an object can be red or non-red. Consider the case of a Ω1C. In such a universe, an object is red or non-red in the absolute. No change of colour is possible there, because no other criterion-variable exists, of which can depend such a variation. And in a Ω2CT, being red is being red at time T. Within such a universe, being red is being red relatively to time T. Similarly, in a coloured, temporal and localised 3-universe (Ω3CTL), being red is being red at time T and at place L. Etc. In some such universe, being red is being red relatively to other criterion-variables. And the same applies to the n-universes which model a universe such as our own.

At this step arises the problem of the status of the instances of an object of a given type. What is it thus to be an instance, within this framework? This problem has its importance, because the original versions of GP are based on instances of balls (1946) and emeralds (1954). If one takes into account the case of Goodman (1946), the considered instances are 100 different balls. However, if one considers a unique ball, drawn at times T1, T2, …, T100, one notices that the problem inherent to GP is always present. It suffices indeed to consider a ball whose colour is susceptible to change during the course of time. One has drawn 99 times the ball at times T1, T2, …, T99, and one has noted each time that the ball was red. This leads to the prediction that the ball will be red at T100. However, this last prediction proves to be contradictory with an alternative prediction based on the same observations, and the projection of the predicate S “red and drawn before T100 or non-red and drawn at T100x.

The present framework must be capable of handling the diversity of these situations. Can one thus speak of an instantiated and temporal 1-universe, or well of an instantiated and coloured 1-universe? Here, one must observe that the fact of being instantiated, for a given universe, corresponds to an additional criterion-variable. For, on the contrary, what makes it possible to distinguish between the instances? If no criterion distinguishes them, it is thus only one and the same thing. And if they are distinct, it is thus that a criterion makes it possible to differentiate them. Thus, an instantiated and temporal 1-universe is in fact a 2-universe, whose 2nd criterion, which makes it possible to distinguish the instances between them, is in fact not mentioned nor explicited. By making explicit this second criterion-variable, it is thus clear that one is placed in a 2-universe. In the same way, an instantiated and coloured 1-universe is actually a 2-universe one of the criteria of which is colour and the second criterion exists but is not specified.

Another aspect which deserves mention here, is the question of the reduction of a given n-universe to another. Is it not possible indeed, to logically reduce a n-universe to a different system of criteria? Consider for example a Ω3CTL. In order to characterise the corresponding universe, one has 3 criterion-variables: colour, time and localisation. It appears that one can reduce this 3-universe to a 2-universe. That can be carried out by reducing two of the criteria of the 3-universe to one single criterion. In particular, one will reduce both criteria of colour and time to a single criterion of tcolour* (shmolorxi). And one will only preserve two taxa of tcolour*: G and ~G. Consider then a criterion of color comprising two taxa (red, non-red) and a criterion of time comprising two taxa (before T, after T). If one associates the taxa of colour and time, one obtains four new predicates: red before T, red after T, non-red before T, non-red after T, which one will denote respectively by RT, R~T, ~RT and ~R~T. Several of these predicates are compatible (RT and R~T, RT and ~R~T, ~RT and R~T, ~RT and ~R~T) whereas others are incompatible (RT and ~RT, R~T and ~R~T). At this stage, one has several manners (16)xii of grouping the compatible predicates, making it possible to obtain two new predicates G and ~G of tcolour*:

In each of these cases, it results indeed a new single criterion of tcolour* (Z), which substitutes itself to the two preceding criteria of colour and time. One will denote by Zi (0 i 15) the taxa of tcolour* thus obtained. If it is clear that Z15 leads to the empty induction, it should be observed that several cases corresponding to the situation where the instances are RT lead to the problem inherent to GP. One will note thus that Z2, i.e. grue2 (by assimilating the Zi to gruei and the Z15-i to bleeni) is based on the definition: grue2 = red before T and non-red after T. It appears here as a conjunctive interpretation of the definition of “grue”. In the same way, grue7 corresponds to a definition of “grue” based on an exclusive disjunction. Lastly, grue12 is based on the traditional definition: grue12 = red before T or non-red after T, where the disjunction is to be interpreted as an inclusive disjunction.

Similarly, it also proves that a Ω2CT can be reduced to a tcoloured* 1-universe (Ω1Z). And more generally, a n-universe is thus reducible to an (n-1)-universe (for n > 1). Thus, if one considers a given universe, several characterisations in terms of n-universe can valuably be used. One can in particular apprehend a same universe like a Ω3CTL, or like a Ω2ZL. In the same way, one can represent a Ω2CT like a Ω1Z. At this stage, none of these views appears fundamentally better than the other. But each of these two characterisations constitute alternative ways to describe a same reality. This shows finally that a n-universe constitutes in fact an abstract characterisation of a real or an imaginary universe. A n-universe constitutes thus a system of criteria, comprising constants and variables. And in order to characterise a same real or imaginary given universe, one can resort valuably to several n-universes. Each of them appears finally as a different characterisation of the given universe, simply based on a different set of primitives.

## 5. Conditions of induction

The fact that the SR formalism involves the GP effect suggests that the intuition which governs our concept of induction is not entirely captured by SR. It is thus allowed to think that if the formal approach is necessary and useful to be used as support to induction, it does not constitute however a sufficient step. For it appears also essential to capture the intuition which governs our inductive reasoning. Therefore it proves necessary to supplement the formal approach of induction by a semantic approach. Goodman himself provides us with a definition of inductionxiii. He defines induction as the projection of characteristics of the past through the future, or more generally, as the projection of characteristics corresponding to a given aspect of an object through another aspect. This last definition corresponds to our intuition of induction. One can think however that it is necessary to supplement it by taking into account the preceding observationsxiv concerning the differentiation/unification duality. In that sense, it has been pointed out that induction consists of an inference from instances presenting both common properties and differences. Let the instances-source (instances-S) be the instances to which relate (I) or (I*) and the instance-destination (instance-D) that which is the subject of (P) or (P*). The common properties relate to the instances-S and the differentiated properties are established between the instances-S and the instance-D. The following definition ensues: induction consists precisely in the fact that the instance-Dxv also presents the property that is common to the instances-S, whereas one does vary the criterion (criteria) on which the differences between the instances-S and the instance-D is (are) based. The inductive reasoning is thus based on the constant nature of a property, whereas such other property is variable.

From this definition of induction arise straightforwardly several conditions of induction. I shall examine them in turn. The first two conditions are thus the following ones:

(C1) the instances-S must present some common properties

(C2) the instances-S and the instance-D must present some distinctive properties

This has for consequence that one cannot apply induction in two particular circumstances: firstly (i) when the instances do not present any common property. One will call such a situation a total differentiation of the instances. The problems corresponding to this particular circumstance have been mentioned abovexvi. And secondly (ii) when the instances do not present any distinctive property. One will call such a situation total unification. The problems encountered in this type of situation have also been mentioned previouslyxvii.

It should also be noted that it is not here a question of intrinsic properties of the instances, but rather of the analysis which is carried out by the one who is on the point of reasoning by induction.

Taking into account the definition of induction which has been given, a third condition can be thus stated:

(C3) a criterion-variable is necessary for the common properties of the instances-S and another criterion-variable for the distinctive properties

This refers to the structure of the considered universe of reference. Consequently, two criterion-variables are at least necessary, in the structure of the corresponding universe of reference. One will call that the minimalcondition of induction. Hence, a 2-universe is at least necessary in order that the conditions of induction can be satisfied. Thus, a 2CT will be appropriate. In the same way, a temporal and localised 2-universe (2TL) will also satisfy the conditions which have been just defined, etcxviii.

It should be noted that another way of stating this condition is as follows: the criterion-variable for the common properties and the criterion-variable for the differentiated properties must be distinct. One should not have confusion between the two. One can call that the condition of separation of the common properties and the distinctive properties. Such a principle appears as a consequence of the minimal condition for induction: one must have two criteria to perform induction, and these criteria must be different. If one chooses a same criterion for the common properties and the differentiated properties, one is brought back in fact to one single criterion and the context of a 1-universe, itself insufficient to perform induction.

Lastly, a fourth condition of induction results from the preceding definition:

(C4) one must project the common properties of the instances-S (and not the distinctive properties)

The conditions of induction which have been just stated make it possible from now on to handle the problems involved in the use of SR mentioned abovexix. It follows indeed that the following projectionsxx are correct: C°T in a Ω2CT, C°L in a Ω2CL, Z°L in a Ω2ZL, etc. Conversely, the following projections are incorrect: T°T in a Ω1T, Z°Z in a Ω1Z. In particular, one will note here that the projection T°T in the Ω1T is that of 1-time. 1-time takes indeed place in a Ω1T, whereas induction requires at the same time common properties and distinctive properties. Thus, a 2-universe is at least necessary. Usually, the criterion of time is used for differentiation. But here, it is used for unification (“drawn before T”). That can be done, but provided that one uses a distinct criterion for the differentiated properties. However, whereas common properties results here from that, the differentiated properties are missing. It thus misses a second criterion – corresponding to the differentiated properties – in the considered universe, to perform induction validly. Thus 1-time finds its origin in a violation of the minimal condition of induction. One can formulate this solution equivalently, with regard to the condition of separation. In effect, in 1-time, a same temporal criterion (drawn before T/drawn after T) is used for the common properties and the differentiated properties, whereas two distinct criteria are necessary. It can be thus analysed as a manifest violation of the condition of separation.

Lastly, the conditions of induction defined above lead to adapt the formalism used to describe GP. It proves indeed necessary to distinguish between the common and the distinctive property(ies). One will thus use the following formalism in replacement of the one used above:

(I) RT1·RT2·RT3·…·RT99

(H) RT1·RT2·RT3·…·RT99·RT100

where R denotes the common property and the Ti a distinctive property. It should be noted here that it can consist of a single object, or alternatively, of instances which are distinguished by a given criterion (which is not concerned by the inductive process) according to n-universe in which one places oneself. Thus, one will use in the case of a single instance , the colour of which is susceptible to change according to time:

(I) RT1·RT2·RT3·…·RT99

or in the case where several instances 1, 2, …, 99, 100 existxxi:

(I) RT11·RT22·RT33·…·RT9999

## 6. Origin of the paradox

Given the conditions of induction and the framework of n-universes which have been just defined, one is now in a position to proceed to determine the origin of GP. Preliminarily it is worth describing accurately the conditions of the universe of reference in which GP takes place. Indeed, in the original version of GP, the choice of the universe of reference is not defined accurately. However one can think that it is essential, in order to avoid any ambiguity, that this last is described precisely.

The universe of reference in which Goodman (1946) places himself is not defined explicitly, but several elements of the statement make it possible to specify its intrinsic nature. Goodman thus mentions the colours “red” and “non-red”. Therefore, colour constitutes one of the criterion-variables of the universe of reference. Moreover, Goodman distinguishes the balls which are drawn at times T1, T2, T3, …, T100. Thus, time is also a criterion-variable of the considered universe. Consequently, one can describe the minimal universe in which Goodman (1946) places himself as a Ω2CT. Similarly, in Goodman (1954), the criterion-variables of colour (green/non-green) and time (drawn before T/drawn after T) are expressly mentioned. In both cases, one thus places oneself implicitly within the minimal framework of a Ω2CT.

Goodman in addition mentions instances of balls or emeralds. Is it necessary at this stage to resort to an additional criterion-variable making it possible to distinguish between the instances? It appears that not. On the one hand indeed, as we have seen previouslyxxii, it proves that one has well a version of GP by simply considering a Ω2CT and a single object, the colour of which is susceptible to change during the course of time. On the other hand, it appears that if the criterion which is used to distinguish the instances is not used in the inductive process, it is then neither useful as a common criterion, nor as a differentiated criterion. It follows that one can dispense with this 3rd additional criterion. Thus, it proves that the fact of taking into account one single instance or alternatively, several instances, is not essential in the formulation of GP. In what follows, one will be able thus to consider that the statement applies, indifferently, to a single object or several instances that are distinguished by a criterion which is not used in the inductive process.

At this step, we are in a position to replace GP within the framework of n-universes. Taking into account the fact that the context of GP is that of a minimalΩ2CT, one will consider successively two situations: that of a Ω2CT, and then that of a Ω3CT (where  denotes a 3rd criterion).

6.1 “Grue” in the coloured and temporal 2-universe

Consider first the hypothesis of a Ω2CT. In such a universe, being “red” is being red at time T. One has then a criterion of colour for the common properties and a criterion of time for the differentiated properties. Consequently, it appears completely legitimate to project the common property of colour (“red”) into the differentiated time. Such a projection proves to be in conformity with the conditions of induction stated above.

Let us turn now to the projection of “grue”. One has observed previouslyxxiii that the Ω2CT was reducible to a Ω1Z. Here, the fact of using “grue” (and “bleen”) as primitives, is characteristic of the fact that the system of criteria used is that of a Ω1Z. What is then the situation when one projects “grue” in the Ω1Z? In such a universe of reference, the unique criterion-variable is the tcolour*. An object is there “grue” or “bleen” in the absolute. Consequently, if one has well a common criterion (the tcolour*), it appears that the differentiated criterion is missing, in order to perform induction validly. And the situation in which one is placed is that of an extreme differentiation. Thus, such a projection is carried out in violation of the minimal condition of induction. Consequently, it proves that GP cannot take place in the Ω2CT and is then blocked at the stage of the projection of “grue”.

But are these preliminary remarks sufficient to provide, in the context of a Ω2CT, a satisfactory solution to GP? One can think that not, because the paradox also arises in it in another form, which is that of the projection of tcolour* through time. One can formalise this projection Z°T as follows:

(I*) GT1·GT2·GT3·…·GT99

(H*) GT1·GT2·GT3·…·GT99·GT100 that is equivalent to:

(H’*) RT1·RT2·RT3·…·RT99·~RT100

(P*) GT100 that is equivalent to:

(P’*) ~RT100

where it is manifest that the elements of GP are still present.

Fundamentally in this version, it appears that the common properties are borrowed from the system of criteria of the Ω1Z, whereas the differentiated properties come from the Ω2CT. A first analysis thus reveals that the projection of “grue” under these conditions presents a defect which consists in the choice of a given system of criteria for the common properties (tcolour*) and of a different system of criteria for the differentiated properties (time). For the selection of the tcolour* is characteristic of the choice of a Ω1Z, whereas the use of time is revealing of the fact that one places oneself in a Ω2CT. But one must choose one or the other of the reducible systems of criteria to perform induction. On the hypotheses envisaged previously, the choice of the criteria for the common and differentiated properties was carried out within the same system of criteria. But here, the choice of the criteria for the common properties and the differentiated properties is carried out within two different (and reducible) systems of criteria. Thus, the common and differentiated criteria selected for induction are not genuinely distinct. And this appears as a violation of the condition of separation. Consequently, one of the conditions of induction is not respected.

However, the projection Z°T has a certain intuitive support, because it is based on the fact that the notions of “grue before T” and “grue after T” have a certain intuitive meaning. Let us then disregard the violation of the conditions of the induction which has been just mentioned, and consider thus this situation in more detail. In this context, GP is always present, since one observes a contradiction between (P) and (P’*). It is with this contradiction that it is worth from now on being interested. Consider the particular step of the equivalence between (H*) and (H’*). One conceives that “grue before T” is assimilated here to RT, because the fact that the instances-S are red before T results clearly from the conditions of the experiment. On the other hand, it is worth being interested by the step according to which (P*) entails (P’*). According to the classical definitionxxiv: “grue” = {RT  R~T, RT  ~R~T, ~RT  ~R~T }. What is it then to be “grue after T”? There, it appears that a “grue” object can be R~T (this corresponds to the case RT  R~T) or ~R~T (this correspond to the cases RT  ~R~T and ~RT  ~R~T). In conclusion, the object can be either R~T or ~R~T. Thus, the fact of knowing that an object is “grue after T” does not make it possible to conclude that this object is ~R~T, because this last can also be R~T. Consequently, the step according to which (P*) involves (P’*) appears finally false. From where it ensues that the contradiction between (P) and (P’*) does not have any more a raison d’etre.

One can convince oneself that this analysis does not depend on the choice of the classical definition of “grue” (grue12) which is carried out, by considering other definitions. Consider for example the definition based on grue9: “grue” = {RT  ~R~T, ~RT  ~R~T} and “bleen” = {RT  R~T, ~RT  R~T}. But in this version, one notes that one does not have the emergence of GP, because the instances-S, which are RT, can be at the same time “grue” and ” bleen”. And the same applies if one considers a conjunctive definition (grue2) such as “grue” = {RT  ~R~T}. In such a case indeed, the instances-S are “grue” only if they are RT but also ~R~T. However this does not correspond to the initial conditions of GP in the 2CT where one ignores if the instances-S are ~R~T.

One could also think that the problem is related to the use of a taxonomy of tcolour* based on two taxa (G and ~G). Consider then a taxonomy of tcolour* based on 4 taxa: Z0 = RT  R~T, Z1 = RT  ~R~T, Z2 = ~RT  R~T, Z3 = ~RT  ~R~T. But on this hypothesis, it appears clearly that since the instances-S are for example Z1, one finds himself replaced in the preceding situation.

The fact of considering “grue after T”, “grue before T”, “bleen before T”, “bleen after T” can be assimilated with an attempt of expressing “grue” and ” bleen” with the help of our own criteria, and in particular that of time. It can be considered here as a form of anthropocentrism, underlain by the idea to express the Ω1Z with the help of the taxa of the Ω2CT. Since one knows the code defining the relations between two reducible n-universes – the Ω1Z and the Ω2CT – and that one has partial data, one can be tempted to elucidate completely the predicates of the foreign n-universe. Knowing that the instances are GT, G~T, ~GT, ~G~T, I can deduce that they are respectively {RT, ~RT}, {R~T, ~R~T}, {~RT}, {R~T}. But as we have seen, due to the fact that the instances are GT and RT, I cannot deduce that they will be ~R~T.

The reasoning in this version of GP is based on the apparently inductive idea that what is “grue before T” is also “grue after T”. But in the context which is that of the Ω1Z, when an object is “grue”, it is “grue” in the absolute. For no additional criterion exists which can make its tcolour* vary. Thus, when an object is GT, it is necessarily G~T. And from the information according to which an object is GT, one can thus conclude, by deduction, that it is also G~T.

From what precedes, it ensues that the version of GP related to the Z°T presents the apparent characters of induction, but it does not constitute an authentic form of this type of reasoning. Z°T thus constitutes a disguised form of induction for two principal reasons: first, it is a projection through the differentiated criterion of time, which constitutes the standard mode of our inductive practice. Second, it is based on the intuitive principle according to which everything that is GT is also G~T. But as we have seen, it consists here in reality of a deductive form of reasoning, whose true nature is masked by an apparent inductive move. And this leads to conclude that the form of GP related to Z°T analyses itself in fact veritably as a pseudo-induction.

6.2 “Grue” in the coloured, temporal and localised 3-universe

Consider now the case of a Ω3CT. This type of universe of reference also corresponds to the definition of a minimal Ω2CT, but it also comprises one 3rd criterion-variablexxv. Let us choose for this last a criterion such as localisationxxvi. Consider then a Ω3CTL. Consider first (H) in such a 3-universe. To be “red” in the Ω3CTL, is to be red at time T and at location L. According to the conditions of GP, colour corresponds to the common properties, and time to the differentiated properties. One has then the following projection C°TL:

(I) RT1L1·RT2L2·RT3L3·…·RT99L99

(H) RT1L1·RT2L2·RT3L3·…·RT99L99·RT100L100

∴ (P) RT100L100

where taking into account the conditions of induction, it proves to be legitimate to project the common property (“red”) of the instances-S, into differentiated time and location, and to predict that the 100th ball will be red. Such a projection appears completely correct, and proves in all points in conformity with the conditions of induction mentioned above.

What happens now with (H*) in the Ω3CTL? It has been observed that the Ω3CTL could be reduced to a Ω2ZL. In this last n-universe, the criterion-variables are tcolour* and localisation. The fact of being “grue” is there relative to location: to be “grue”, is to be “grue” at location L. What is then projected is the tcolour*, i.e. the fact of being “grue” or “bleen”. There is thus a common criterion of tcolour* and a differentiated criterion of localisation. Consequently, if it is considered that the instances-S are “grue”, one can equally well project the property common “grue” into a differentiated criterion of localisation. Consider then the projection Z°L in the Ω2ZL:

(I*) GL1·GL2·GL3·…·GL99

(H*) GL1·GL2·GL3·…·GL99·GL100

∴ (P*) GL100

Such a projection is in conformity with the conditions mentioned above, and constitutes consequently a valid form of induction.

In this context, one can project valuably a predicate having a structure identical to that of “grue”, in the case of emeralds. Consider the definition “grue” = green before T or non-green after T, where T = 10 billion years. It is known that at that time, our Sun will be extinct, and will become gradually a dwarf white. The conditions of our atmosphere will be radically different from what they currently are. And the temperature will rise in particular in considerable proportions, to reach 8000°. Under these conditions, the structure of many minerals will change radically. It should normally thus be the case for our current emeralds, which should see their colour modified, due to the enormous rise in temperature which will follow. Thus, I currently observe an emerald: it is “grue” (for T = 10 billion years). If I project this property through a criterion of location, I legitimately conclude from it that the emerald found in the heart of the Amazonian forest will also be “grue”, in the same way as the emerald which has been just extracted from a mine from South Africa.

At this stage, one could wonder whether the projectibility of “grue” is not intrinsically related to the choice of a definition of “grue” based on inclusive disjunction (grue12)? Nevertheless, one easily checks by using an alternative definition of “grue” that its projection remains validxxvii.

It should be noticed that one has here the expression of the fact that the taxonomy based on the tcolour* is coarser than that based on time and colour. In effect, the former only comprises 2 taxa (grue/bleen), whereas the latter presents 4 of them. By reducing the criteria of colour and time to a single criterion of tcolor*, one has replaced 4 taxa (RT  R~T, RT  ~R~T, ~RT  R~T, ~RT  ~R~T) by 2. Thus, “grue” constitutes from this point of view a predicate coarser than “red”. The universe which is described did not change, but the n-universes which are systems of criteria describing these universes are different. With the tcolour* thus defined, one has less predicates at its disposal to describe a same reality. The predicates “grue” and “bleen” are for us not very informative, and are less informative in any case that our predicates “red”, “non-red”, “before T”, etc. But that does not prevent however “grue” and “bleen” to be projectibles.

Whereas the projection of “grue” appears valid in the Ω2ZL, it should be noticed however that one does not observe in this case the contradiction between (P) and (P’*). For here (I*) is indeed equivalent to:

(I’*) RT1L1·RT2L2·RT3L3·…·RT99 L99

since, knowing according to the initial data of GP that the instances-S are RT, one valuably replaces the GLi by the RTiLi (i < 100). But it appears that on this hypothesis, (P*) does not involve:

(P’*) ~RT100L100

because one does not have an indication relating to the temporality of the 100th instance, due to the fact that only the localisation constitutes here the differentiated criterion. Consequently, one has well in the case of the Ω3CTL a version built with the elements of GP where the projection of “grue” is carried out valuably, but which does not present a paradoxical nature.

## 7. Conclusion

In the solution to GP proposed by Goodman, a predicate is projectible or nonprojectible in the absolute. And one has in addition a correspondence between the entrenchedxxviii/non-entrenched and the projectible/nonprojectible predicates. Goodman in addition does not provide a justification to this assimilation. In the present approach, there is no such dichotomy, because a given predicate P reveals itself projectible in a given n-universe, and nonprojectible in another n-universe. Thus, P is projectible relatively to such universe of reference. There is thus the projectible/nonprojectible relative to such n-universe distinction. And this distinction is justified by the conditions of induction, and the fundamental mechanism of induction related to the unification/differentiation duality. There are thus n-universes where “green” is projectible and others where it is not. In the same way, “grue” appears here projectible relative to certain n-universes. Neither green nor grue are projectible in the absolute, but only relative to such given universe. Just as of some other predicates, “grue” is projectible in certain universes of reference, but nonprojectible in othersxxix.

Thus, it proves that one of the causes of GP resides in the fact that in GP, one classically proceeds to operate a dichotomy between the projectible and the nonprojectible predicates. The solutions classically suggested to GP are respectively based on the distinction temporal/nontemporal, local/non-local, qualitative/nonqualitative, entrenched/non-entrenched, etc. and a one-to-one correspondence with the projectible/nonprojectible distinction. One wonders whether a given predicate P* having the structure of “grue” is projectible, in the absolute. This comes from the fact that in GP, one has a contradiction between the two concurrent predictions (P) and (P*). One classically deduces from it that one of the two predictions must be rejected, at the same time as one of the two generalisations (H) or (H*) on which these predictions are respectively based. Conversely, in the present analysis, whether one places himself in the case of the authentic projection Z°L or in the case of the pseudo-projection Z°T, one does not have a contradiction between (P) and (P’*). Consequently, one is not constrained any more to reject either (H) or (H*). And the distinction between projectible/nonprojectible predicates does not appear indispensable any morexxx.

How is the choice of our usual n-universe carried out in this context? N-universes such as the Ω2CT, the Ω3CTL, the Ω2ZL etc. are appropriate to perform induction. But we naturally tend to privilege those which are based on criteria structured rather finely to allow a maximum of combinations of projections. If one operates from the criteria Z and L in the Ω2ZL, one restricts oneself to a limited number of combinations: Z°L and L°Z. Conversely, if one retains the criteria C, T and L, one places oneself in the Ω3CTL and one has the possibility of projections C°TL, T°CL, L°CT, CT°Lxxxi, CL°T, TL°C. One has thus a maximum of combinations. This seems to encourage to prefer the Ω3CTL to the Ω2ZL. Of course, pragmatism seems to have to play a role in the choice of the best alternative of our criteria. But it seems that it is only one of the multiple factors which interact to allow the optimisation of our criteria to carry out the primitive operations of grouping and differentiation, in order to then be able to generalise, classify, order, make assumptions or forecastxxxii. Among these factors, one can in particular mention: pragmatism, simplicity, flexibility of implementation, polyvalencexxxiii, economy in means, powerxxxiv, but also the nature of our real universe, the structure of our organs of perception, the state of our scientific knowledge, etcxxxv. Our usual n-universes are optimised with regard to these various factors. But this valuably leaves room for the choice of other systems of criteria, according to the variations of one or the other of these parametersxxxvi.

i Nelson Goodman, “A Query On Confirmation”, Journal of Philosophy, vol. 43 (1946), p. 383-385; in Problems and Projects, Indianapolis, Bobbs-Merrill, 1972, p. 363-366.

ii With some minor adaptations.

iii See Goodman “A Query On Confirmation”, p. 383: “Suppose we had drawn a marble from a certain bowl on each of the ninety-nine days up to and including VE day and each marble drawn was red. We would expect that the marble drawn on the following day would also be red. So far all is well. Our evidence may be expressed by the conjunction “Ra1·Ra2·…·Ra99” which well confirms the prediction Ra100.” But increase of credibility, projection, “confirmation” in any intuitive sense, does not occur in the case of every predicate under similar circumstances. Let “S” be the predicate “is drawn by VE day and is red, or is drawn later and is non-red.” The evidence of the same drawings above assumed may be expressed by the conjunction “Sa1·Sa2·…·Sa99“. By the theories of confirmation in question this well confirms the prediction “Sa100“; but actually we do not expect that the hundredth marble will be non-red. “Sa100” gains no whit of credibility from the evidence offered.”

iv Nelson Goodman, Fact, Fiction and Forecast, Cambridge, MA, Harvard University Press, 1954.

v Ibid., p. 73-4: “Suppose that all emeralds examined before a certain time t are green. At time t, then, our observations support the hypothesis that all emeralds are green; and this is in accord with our definition of confirmation. […] Now let me introduce another predicate less familiar than “green”. It is the predicate “grue” and it applies to all things examined before t just in case they are green but to other things just in case they are blue. Then at time t we have, for each evidence statement asserting that a given emerald is green, a parallel evidence statement asserting that that emerald is grue.”

vi For example with an accuracy of 10-4 nm.

vii Or any taxonomy which is similar to it.

viii See §2 above.

ix This assertion is controversial.

x Such a remark also applies to the statement of Goodman, Fact, Fiction and Forecast.

xi As J.S. Ullian mentions it, “More one ‘Grue’ and Grue”, Philosophical Review, vol. 70 (1961), p. 386-389, in p. 387.

xii I. e. C(0, 4)+C(1, 4)+C(2, 4)+C(3, 4)+C(4, 4) = 24, where C(p, q) denotes the number of combinations of q elements taken p times.

xiii See Goodman, “A Query On Confirmation”, p. 383: “Induction might roughly be described as the projection of characteristics of the past into the future, or more generally of characteristics of one realm of objects into another.”

xiv See §2 above.

xv One can of course alternatively take into account several instances-D.

xvi See §2 above.

xvii Ibid.

xviii For the application of this condition, one must take into account the remarks mentioned above concerning the problem of the status of the instances. Thus, one must actually compare an instantiated and temporal 1-universe to a 2-universe one of the criteria of which is temporal, and the second criterion is not explicitly mentioned. Similarly, an instantiated and coloured 1-universe is assimilated in fact to a 2-universe one of the criteria of which is temporal, and the second criterion is not specified.

xix See §3 above.

xx With the notations C (colour), T (time), L (localisation) and Z (tcolour*).

xxi However, since the fact that there exists one or more instances is not essential in the formulation of the given problem, one will obviously be able to abstain from making mention of it.

xxii See §4.

xxiii Ibid.

xxiv It is the one based on the inclusive disjunction (grue12).

xxv A same solution applies, of course, if one considers a number of criterion-variables higher than 3.

xxvi All other criterion distinct from colour or time, would also be appropriate.

xxvii In particular, it appears that the projection of a conjunctive definition (grue2) is in fact familiar for us. In effect, we do not proceed otherwise when we project the predicate “being green before maturity and red after maturity” applicable to tomatoes, through a differentiated criterion of location: this is true of the 99 instance-S observed in Corsica and Provence, and is projected validly to a 100th instance located in Sardinia. One can observe that such a type of projection is in particular regarded as nonproblematic by Jackson (Franck Jackson, “‘Grue'”, Journal of Philosophy, vol. 72 (1975), p. 113-131): “There seems no case for regarding ‘grue’ as nonprojectible if it is defined this way. An emerald is grue1 just if it is green up to T and blue thereafter, and if we discovered that all emeralds so far examined had this property, then, other things being equal, we would probably accept that all emeralds, both examined and unexamined, have this property (…).” If one were to replace such a predicate in the present analysis, one should then consider that the projection is carried out for example through a differentiated criterion of localisation (p. 115).

xxviii Goodman, Fact, Fiction and Forecast.

xxix The account presented in J Holland, K Holyoak, R. Nisbett and P. Thagard (Induction, Cambridge, MA; London, MIT Press, 1986) appears to me to constitute a variation of Goodman’s solution, directed towards the computer-based processing of data and based on the distinction integrated/non-integrated in the default hierarchy. But Holland’s solution presents the same disadvantages as that of Goodman: what justification if not anthropocentric, does one have for this distinction? See p. 235: “Concepts such as “grue”, which are of no significance to the goals of the learner, will never be generated and hence will not form part of the default hierarchy. (…) Generalization, like other sorts of inference in a processing system, must proceed from the knowledge that the system already has”.

The present analysis also distinguishes from the one presented by Susan Haack (Evidence and Inquiry, Oxford; Cambridge, MA, Blackwell, 1993) because the existence of natural kinds does not constitute here a condition for induction. See p. 134: “There is a connection between induction and natural kinds. […] the reality of kinds and laws is a necessary condition of successful inductions”. In the present context, the fact that the conditions of induction (a common criterion, a distinct differentiated criterion, etc.) are satisfied is appropriate to perform induction.

xxx A similar remark is made by Franck Jackson in conclusion of his article (“‘Grue'”, p. 131): “[…] the SR can be specified without invoking a partition of predicates, properties or hypotheses into the projectible and the nonprojectible”. For Jackson, all noncontradictory predicates are projectible: “[…] all (consistent) predicates are projectible.” (p. 114). Such a conclusion appears however stronger than the one that results from the current analysis. Because for Jackson, all predicates are thus projectible in the absolute. However in the present context, there are no projectible or nonprojectible predicates in the absolute. It is only relative to a given n-universe, that a predicate P reveals projectible or nonprojectible.

More generally, the present analysis distinguishes fundamentally from that of Jackson in the sense that the solution suggested to GP does not rest on the counterfactual condition. This last appears indeed too related to the use of certain predicates (examined, sampled, etc.). On the other hand, in the present context, the problem is considered from a general viewpoint, independently of the particular nature of the predicates constituting the definition of grue.

xxxi Such a projection corresponds for example to the generalisation according to which “the anthropomorphic statue-menhirs are of the colour of granite and date from the Age of Bronze”.

xxxii As Ian Hacking underlines it, Le plus pur nominalisme, Combas, L’éclat, 1993, p. 9: “Utiliser un nom pour une espèce, c’est (entre autres choses) vouloir réaliser des généralisations et former des anticipations concernant des individus de cette espèce. La classification ne se limite pas au tri : elle sert à prédire. C’est une des leçons de la curieuse “énigme” que Nelson Goodman publia il y a quarante ans.” My translation: “To use a name for a species, it is (among other things) to want to carry out generalisations and to form anticipations concerning the individuals of this species. Classification is not limited to sorting: it is used to predict. It is one of the lessons of the strange “riddle” which Nelson Goodman published forty years ago.”

xxxiii The fact that a same criterion can be used at the same time as a common and a differentiated criterion (while eventually resorting to different taxa).

xxxiv I.e. the number of combinations made possible.

xxxv This enumeration does not pretend to be exhaustive. A thorough study of this question would be of course necessary.

xxxvi I thank the editor of Dialogue and two anonymous referees for very helpful comments on an earlier draft of this paper.

# Probabilistic Situations for Goodmanian N-universes

A paper appeared (2006) in French in the Journal of Philosophical Research, vol. 31, pages 123-141, under the title “Situations probabilistes pour n-univers goodmaniens.”

I proceed to describe several applications of the theory of n-universes through several different probabilistic situations. I describe first how n-universes can be used as an extension of the probability spaces used in probability theory. The extended probability spaces thus defined allow for a finer modelling of complex probabilistic situations and fits more intuitively with our intuitions related to our physical universe. I illustrate then the use of n-universes as a methodological tool, with two thought experiments described by John Leslie. Lastly, I model Goodman’s paradox in the framework of n-universes while also showing how these latter appear finally very close to goodmanian worlds.

## Probabilistic Situations for Goodmanian N-universes

The n-universes were introduced in Franceschi (2001, 2002) in the context of the study of the probabilistic situations relating to several paradoxes which are currently the object of intensive studies in the field of analytical philosophy: Goodman’s paradox and the Doomsday Argument. The scope of the present article is twofold: on one hand, to describe how modelling within the n-universes allows to extend the properties of the classical probability spaces used in probability theory, by providing at the same time a finer modelling of some probabilistic situations and a better support for intuition; on the other hand, to show how the use of n-universes allows to simplify considerably the study of complex probabilistic situations such as those which appear in the study of paradoxes.

When one models for example the situation corresponding to the drawing of a ball from an urn, one considers then a restricted temporal space, which limits itself to the few seconds that precede and follow the drawing. Events which took place the day before or one hour before, but also those who will happen for example the day after the drawing, can be purely and simply ignored. A very restricted interval of time, that it is possible to reduce to one or two discrete temporal positions, is then enough for characterising the corresponding situation. It suffices also to consider a restriction of our universe where the space variable is limited to the space occupied by the urn. For it is not useful to take into consideration the space corresponding to the neighbouring room and to the objects which are there. In a similar way, the number of atoms of copper or of molybdenum that are possibly present in the urn, the number of photons which are interacting with the urn at the time of the drawing, or the presence or absence of a sound source of 75 db, etc. can be omitted and ignored. In this context, it is not necessary to take into account the existence of such variables. In such situation, it is enough to mention the variables and constants really used in the corresponding probabilistic situation. For to enumerate all the constants and the variables which describe of our whole universe appears here as an extremely complicated and moreover useless task. In such context, one can legitimately limit oneself to describe a simplified universe, by mentioning only those constants and variables which play a genuine role in the corresponding probabilistic situation.

Let us consider the drawing of a ball from an urn which contains several balls of different colours. To allow the calculation of the likelihood of different events related to the drawing of one or several balls from the urn, probability theory is based on a modelling grounded on probability spaces. The determination of the likelihood of different events is then not based on the modelling of the physical forces which determine the conditions of the drawing, i.e. the mass and the dimensions of the balls, the material of which they are constituted, their initial spatio-temporal position, as well as the characteristics of the forces exercised over the balls to perform a random drawing. The modelling of random phenomena with the help of probability spaces does only retain some very simplified elements of the physical situation which corresponds to the drawing of a ball. These elements are the number and the colour of the balls, as well as their spatio-temporal position. Such methodological approach can be generalised in other probabilistic situations that involve random processes such as the drawing of one or several dices or of one or several cards. Such methodology does not constitute one of the axioms of probability theory, but it consists here of one important tenet of the theory, of which one can suggest that it would be worth being more formalized. It may also be useful to explain in more detail how the elements of our physical world are converted into probability spaces. In what follows, I will set out to show how the probability spaces can be extended, with the help of the theory of n-universes, in order to better restore the structure of the part of our universe which is so modelled.

1. Introduction to n-universes

It is worth describing preliminarily the basic principles underlying the n-universes. N-universes constitute a simplified model of the physical world which is studied in a probabilistic situation. Making use of Ockam’s razor, we set out then to model a physical situation with the help of the simplest universe’s model, in a compatible way however with the preservation of the inherent structure of the corresponding physical situation. At this stage, it proves to be necessary to highlight several important features of n-universes.

1.1. Constant-criteria and variable-criteria

The criteria of a given n-universe include both constants and variables. Although n-universes allow to model situations which do not correspond to our physical world, our concern will be here exclusively with the n-universes which correspond to common probabilistic situations, in adequacy with the fundamental characteristics of our physical universe. The corresponding n-universes include then at the very least one temporal constant or variable, as well as one constant or variable of location. One distinguishes then among n-universes: a T0L0 (a n-universe including a temporal constant and a location constant), a T0L (a temporal constant and a location variable), a TL0 (a temporal variable and a location constant), a TL (a temporal variable and a location variable). Other n-universes also include a constant or a variable of colour, of direction, etc.

1.2. N-universes with a unique object or with multiple objects

Every n-universe includes one or several objects. One distinguishes then, for example: a 0TL0 (n-world including a unique object, a temporal variable and a constant of location), a TL0 (multiple objects, a temporal variable and a location constant).

1.3. Demultiplication with regard to a variable-criterion

It is worth highlighting the property of demultiplication of a given object with regard to a variable-criterion of a given n-universe. In what follows, we shall denote a variable-criterion  with demultiplication by *. Whatever variable-criterion of a given n-universe can so be demultiplicated. The fact for a given object to be demultiplicated with regard to a criterion  is the property for this object to exemplify several taxa of criterion . Let us take the example of the time criterion. The fact for a given object to be demultiplicated with regard to time resides in the fact of exemplifying several temporal positions. In our physical world, an object 0 can exist at several (successive) temporal positions and finds then itself demultiplicated with regard to the time criterion. Our common objects have then a property of temporal persistence, which constitutes a special case of temporal demultiplication. So, in our universe of which one of the variable-criteria is time, it is common to note that a given object 0 which exists at T1 also exists at T2, …, Tn. Such object has a life span which covers the period T1-Tn. The corresponding n-universe presents then the structure 0T*L0 (T* with simplified notation).

1.4. Relation one/many of the multiple objects with a given criterion

At this stage, it proves to be necessary to draw an important distinction. It is worth indeed distinguishing between two types of situations. An object can thus exemplify, as we did just see it, several taxa of a given variable-criterion. This corresponds to the case of demultiplication which has just been described with regard to a given variable-criterion. But it is also worth taking into account another type of situation, which concerns only those n-universes with multiple objects. Indeed, several objects can instantiate the same taxon of a given criterion. Let us consider first the temporal criterion. Let us place ourselves, for example, in a n-universe with multiple objects including at the same time a temporal variable and a location constant L0. This can correspond to two types of different n-universes. In the first type of n-universe, there is one single object by temporal position. At some point in time, it is therefore only possible to have a unique object in L0 in the corresponding n-universe. We can consider in that case that every object of this n-universe is in relation one with the time taxa. We denote by T*L0 (with simplified notation T) such n-universe. Let us consider now a n-universe with multiple objects including a temporal variable and a location constant, but where several objects 1, 2, 3 can exist at the same time. In that case, the multiple objects are at a given temporal position in L0. The situation then differs fundamentally from the T*L0, because several objects can now occupy the same given temporal position. In other words, the objects can co-exist at a given time. In that case, one can consider that the objects are in relation many with the temporal taxa. We denote then by *T*L0 such n-universe (with simplified notation *T) .

Let us place ourselves now from the point of view of the location criterion. Let us consider a n-universe with multiple objects including at the same time a temporal variable and a variable of location, and where the objects are in relation many with the temporal criterion. It is also worth distinguishing here between two types of n-universes. In the first, a single object can find itself at a given taxon of the location criterion at the same time. There is then one single object by space position at a given time. This allows for example to model the situation which is that of the pieces of a chess game. Let us denote by *TL such n-universe (with simplified notation *TL). In that case, the objects are in relation one with the location criterion. On the other hand, in the second type of n-universe, several objects can find themselves in the same taxon of a location criterion at the same time. Thus, for example, the objects 1, 2, 3 are in L1 at T1. Such situation corresponds for example to an urn (which is thus assimilated with a given taxon of location) where there are several balls at a given time. We denote by *T*L such n-universe, where the objects are in relation many with the location taxa.

One can notice lastly that such differentiation is also worth for the variable-criterion of colour. One can then draw a distinction between: (a) a *T0*L0C (with simplified notation C) where several objects which can co-exist at the same time in a given space position present all necessarily a different colour, because the objects are in relation one with the colour criterion there; (b) a *T0*L0*C (with simplified notation *C) where several objects which can co-exist at the same time at a given space position can present the same colour, because the objects are in relation many with the colour criterion there.

1.5. Notation

At this stage, it is worth highlighting an important point which concerns the used notation. It was indeed made use in what precedes of an extended and of a simplified notation. The extended notation includes the explicit specification of all criteria of the considered n-universe, including at the same time the variable-criteria and the constant-criteria. By contrast, the simplified notation includes only the explicit specification of the variable-criteria of the considered n-universe. For constant-criteria of time and of location of the considered n-universe can be merely deduced from variable-criteria of the latter. This is made possible by the fact that the studied n-universes include, in a systematic way, one or several objects, but also a variable-criterion or a constant-criterion of time and of location.

Let us illustrate what precedes by an example. Consider first the case where we situate ourselves in a n-universe including multiple objects, a constant-criterion of time and a constant-criterion of location. In that case, it appears that the multiple objects exist necessarily at T0. As a result, in the considered n-universe, the multiple objects are in relation many with the constant-criterion of time. And also, there exist necessarily multiple objects at L0. So, the multiple objects are also in relation many with the constant-criterion of location. We place ourselves then in the situation which is that of a *T0*L0. But for the reasons which have just been mentioned, such n-universe can be denoted, in a simplified way, by .

The preceding remarks suggest then a simplification, in a general way, at the level of the used notation. Indeed, since a n-universe includes multiple objects and since it includes a constant-criterion of time, the multiple objects are necessarily in relation many with the constant-criterion of time. The n-universe is then a *T0. But it is possible to simplify the corresponding notation into . If a n-universe also includes multiple objects and a constant-criterion of location, the multiple objects are necessarily in relation many with the constant-criterion of location. The given n-universe is then a *L0, and it is possible to simplify the notation of the considered n-universe in . As a result, it is possible to simplify the notations *L0*T0 into , *L0T into T, *L0*T into *T, *L0*T* into *T*, etc.

2. Modelling random events with n-universes

The situations traditionally implemented in probability theory involve dices, coins, card games or else some urns that contain balls. It is worth setting out to describe how such objects can be modelled within the n-universes. It also proves to be necessary to model the notion of a “toss” in the probability spaces extended to n-universes. One can make use of the modellings that follow:1

2.1. Throwing a dice

How can we model a toss such as the result of the throwing of the dice is “5 “? We model here the dice as a unique object that finds itself at a space location L0 and which is susceptible of presenting at time T0 one discrete modality of space direction among {1,2,3,4,5,6}. The corresponding n-universe includes then a unique object, a variable of direction and a temporal constant. The unique object can only present one single direction at time T0 and is not with demultiplication with regard to the criterion of direction. The n-universe is a O (with extended notation 0T0L0O). Traditionally, we have the sample space  = {1,2,6} and the event {5}. The drawing of “5 ” consists here for the unique object to have direction 5 among {1,2,6} at time T0 and at location L0. We denote then the sample space by 0T0L0O{1,2,…,6} and the event by 0T0L0O{5}.2

How can we model two successive throws of the same dice, such as the result is “5” and then “1”? Traditionally, we have the sample space  = {1,2,…,6}2 and the event {5,1}. Here, it corresponds to the fact that the dice 0 has direction 5 and 1 respectively at T1 and T2. In the corresponding n-universe, we have now a time variable, including two positions: T1 and T2. Moreover, the time variable is with demultiplication because the unique object exists at different temporal positions. The considered n-universe is therefore a T*O (with extended notation 0T*L0O). We denote then the sample space by 0T*{1,2}L0O{1,2,…,6} and the event by {0T*{1}L0O{5}, 0T*{2}L0O{1}}.

2.2. Throwing a coin

How can we model the toss, for example of Tails, resulting from the flipping of a coin? We model here the coin as a unique object presenting 2 different modalities of direction among {P,F}. The corresponding n-universe is identical to the one which allows to model the dice, with the sole difference that the direction criterion includes only two taxa: {P,F}. The corresponding n-universe is therefore a O (with extended notation 0T0L0O). Classically, we have:  = {P,F} and {P}. Here, the Tails-toss is assimilated with the fact for the unique object to take direction {P} among {P,F} at time T0 and at location L0. The sample space is then denoted by 0T0L0O{P,F} and the event by 0T0L0O{P}.

How can we model two successive tosses of the same coin, such as the result is “Heads” and then “Tails”? Classically, we have the sample space  = {P,F}2 and the event {F,P}. As well as for the modelling of the successive throws of the same dice, the corresponding n-universe is here a T*O (with extended notation 0T*L0O). The sample space is then denoted by by 0T*{1,2}L0O{P,F} and the event by {0T*{1}L0O{F}, 0T*{2}L0O{P}}.

2.3. Throwing several discernible dices

How can we model the throwing of two discernible dices at the same time, for example the simultaneous toss of one “3” and of one “5”? The discernible dices are modelled here as multiple objects being each at a given space position and susceptible of presenting at time T0 one modality of space direction among {1,2,3,4,5,6}. The multiple objects co-exist at the same temporal position, so that the objects are in relation many with the temporal constant. In addition, the multiple objects can only present one single direction at time T0 and are not therefore with demultiplication with regard to the criterion of direction. The fact that both dices could have the same direction corresponds to the fact that objects are in relation many with the criterion of direction. There exists also a location variable, each of the dices 1 and 2 being at one distinct space position. We consider then that the latter property renders the dices discernible. The objects are here in relation one with the location criterion. In addition, the objects can only occupy one single space position at time T0 and are not therefore with demultiplication with regard to the location criterion. The n-universe is then a L*O (with extended notation *T0L*O). Classically, one has:  = {1,2,3,4,5,6}2 and {3,5}. Here, it corresponds to the fact that the dices 1 and 2 are to be found respectively at L1 and L2 and present a given direction among {1,2,6} at time T0. We denote then the sample space by {1,2}*T0L{1,2}*O{1,2,…,6} and the event by {{1}*T0L{1}*O{3}, {2}*T0L{2}*O{5}}.

2.4. Throwing several indiscernible dices

How can we model the throwing of two indiscernible dices, for example the toss of one “3” and one “5” at the same time? Both indiscernible dices are modelled as multiple objects being at space position L0 and susceptible of presenting at time T0 one modality of space direction among {1,2,3,4,5,6} at a given location. The multiple objects co-exist at the same temporal position, so that the objects are in relation many with the temporal constant. The multiple objects can only present one single direction at time T0 and are not therefore with demultiplication with regard to the criterion of direction. The fact that both dices are susceptible of having the same direction corresponds to the fact that the objects are in relation many with the criterion of direction. Both dices 1 and 2 are at the same location L0, what makes them indiscernible. In addition, the multiple objects are in relation many with the constant-criterion of location. Lastly, the objects can only be at one single space position at time T0 and are not therefore with demultiplication with regard to the location criterion. The corresponding n-universe is then a *O (with extended notation *T0*L0*O). Classically, we have:  = (i, j) with 1  ij  6 and {3,5}. Here, it corresponds to the fact that the dices 1 and 2 are both in L0 and present a given direction among {1,2,…,6} at T0. The sample space is then denoted by {1,2}*T0*L0*O{1,…,6} and the event by {{1}*T0*L0*O{3}, {2}*T0*L0*O{5}}.

2.5. Drawing a card

How can we model the drawing of a card, for example the card #13, in a set of 52 cards? Cards are modelled here as multiple objects presenting each a different colour among {1,2,…,52}. The cards’ numbers are assimilated here with taxa of colour, numbered from 1 to 52. Every object can have only one single colour at a given time. As a result, the multiple objects are not with demultiplication with regard to the colour criterion. In addition, a given card can only present one single colour at the same time. Hence, the objects are in relation one with the colour criterion. Moreover, the multiple objects can be at a given time at the same space location (to fix ideas, on the table). The objects are then in relation many with the location criterion. Lastly, the objects can co-exist at the same given temporal position. Thus, they are in relation many with the time criterion. The corresponding n-universe is then a C (with extended notation *T0*L0C). How can we model the drawing of a card? Classically, we have the sample space  = {1,2,…,52} and the event {13}. Here, the drawing of the card #13 is assimilated with the fact that the object the colour of which is #13 is at T0 at location L0. The sample space is then denoted by {1,2,…,52}*T0*L0C{1,2,…,52} and the event by {1}*T0*L0C{13}.

The drawing of two cards at the same time or the successive drawing of two cards are then modelled in the same way.

2.6 Drawing of a ball from an urn containing red and blue balls

How can we model the drawing of, for example, a red bowl, from an urn containing 10 balls among which 3 red balls and 7 blue balls? The balls are modelled here as multiple objects presenting each one colour among {R,B}. There exists then a colour variable in the corresponding n-universe. In addition, several objects can present the same colour. The objects are then in relation many with the variable-criterion of colour. Moreover, the objects are in relation many with regard to the constant-criteria of time and location. The corresponding n-universe is therefore a *T0**L0*C (with simplified notation *C). Classically, we have the sample space  = {R,R,R,B,B,B,B,B,B,B} and the event {R}. The sample space is then denoted by {1,2,…,10}*T0**L0*C{R,B} and the event by {{1}*T0**L0*C{R}}.

The drawing of two balls at the same time or the successive drawing of two balls are modelled in the same way.

3. Dimorphisms and isomorphisms

The comparison of the structures of the extended (to n-universes) sample spaces corresponding to two given probabilistic situations allows to determine if these situations are, from a probabilistic viewpoint, isomorphic or not. The examination of the structures of the sample spaces allows to determine easily the isomorphisms or, on the contrary, the dimorphisms. Let us give some examples.

Consider a first type of application where one wonders whether two probabilistic situations are of comparable nature. To this end, we model the two distinct probabilistic situations within the n-universes. The first situation is thus modelled in a *T0*L0*C (with simplified notation *C), and the second one in a *T0*L0C (with simplified notation C). One notices then a dimorphism between the n-universes that make it possible to model respectively the two probabilistic situations. Indeed, in the first situation, the multiple objects are in relation many with the colour criterion, corresponding thus to the fact that several objects can have an identical colour at a given moment and location. On the other hand, in the second situation, the multiple objects are in relation one with the colour criterion, what corresponds to the fact that each object has a different colour at a given time and location. The dimorphism observed at the level of the demultiplication of the variable-criterion of colour in the two corresponding n-universes makes it possible to conclude that the two probabilistic situations are not of a comparable nature.

It is worth considering now a second type of application. The throwing of two discernible dice is modelled, as we did see it, in a {1,2}T0*L{1,2}*O{1,…,6}. Now let us consider a headlight which can take at a given time one colour of 6 colours numbered from 1 to 6. If one considers now two headlights of this type, it appears that the corresponding situation can be modelled in a {1,2}T0*L{1,2}*C{1,…, 6}. In this last case, it appears that the variable-criterion of colour replaces the criterion of orientation. At this stage, it proves that the structure of such n-universe (with simplified notation L*C) is isomorphic to that of the n-universe in which the throwing of two discernible dice was modelled (with simplified notation L*O). This makes it possible to conclude that the two probabilistic situations are of a comparable nature.

Let us consider now a concrete example. John Leslie (1996, 20) describes in the following terms the Emerald case:

Imagine an experiment planned as follows. At some point in time, three humans would each be given an emerald. Several centuries afterwards, when a completely different set of humans was alive, five thousands humans would again each be given an emerald in the experiment. You have no knowledge, however, of whether your century is the earlier century in which just three people were to be in this situation, or the later century in which five thousand were to be in it. Do you say to yourself that if yours were the earlier century then the five thousand people wouldn’t be alive yet, and that therefore you’d have no chance of being among them? On this basis, do you conclude that you might just as well bet that you lived in the earlier century?

Leslie thus puts in parallel a real situation related to some emeralds and a probabilistic model concerning some balls in a urn. Let us proceed then to model the real, concrete, situation, described by Leslie, in terms of n-universes. It appears first that the corresponding situation is characterized by the presence of multiple objects: the emeralds. We find then ourselves in a n-universe with multiple objects. On the second hand, one can consider that the emeralds are situated at one single place: the Earth. Thus, the corresponding n-universe has a location constant (L0). Leslie also distinguishes two discrete temporal positions in the experiment: the one corresponding to a given time and the other being situated several centuries later. The corresponding n-universe comprises then a time variable with two taxa: T1 and T2. Moreover, it proves to be that the emeralds existing in T1 do not exist in T2 (and reciprocally). Consequently, the n-universe corresponding to the emerald case is a n-universe which is not with temporal demultiplication. Moreover, one can observe that several emeralds can be at the same given temporal position Ti: three emeralds exist thus in T1 and five thousand in T2. Thus, the objects are in relation many with the time variable. Lastly, several emeralds can coexist in L0 and the objects are thus in relation many with the location constant. Taking into account what precedes, it appears thus that the Emerald case takes place in a *T (with extended notation *T*L0), a n-universe with multiple objects, comprising a location constant and a time variable with which the objects are in relation many.

Compare now with the situation of the Little Puddle/London experiment, also described by Leslie (1996, 191):

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

The latter experiment is based on a real, concrete, situation, to be put in relation with an implicit probabilistic model. It appears first that the corresponding situation characterises itself by the presence of multiple inhabitants: 50 in Little Puddle and 10 million in London. The corresponding n-universe is then a n-universe with multiple objects. It appears, second, that this experiment takes place at one single time: the corresponding n-universe has then one time constant (T0). Moreover, two space positions – Little Puddle and London – are distinguished, so that we can model the corresponding situation with the help of a n-universe comprising two space positions: L1 and L2. Moreover, each inhabitant is either in Little Puddle or in London, but but no one can be at the two places at the same time. The corresponding n-universe is then not with local demultiplication. Lastly, one can notice that several people can find themselves at a given space position Li: there are thus 50 inhabitants at Little Puddle (L1) and 10 million in London (L2). The objects are thus in a relation many with the space variable. And in a similar way, several inhabitants can be simultaneously either in Little Puddle, or in London, at time T0. Thus, the objects are in relation many with the time constant. Taking into account what precedes, it appears that the situation of the Little Puddle/London experiment takes place in a *L (with extended notation *T0*L), a n-universe with multiple objects, comprising a time constant and a location variable, with which the objects are in relation many.

As we can see it, the emerald case takes place in a *T, whereas the Little Puddle/London experiment situates itself in a *L. This makes it possible to highlight the isomorphic structure of the two n-universes in which the two experiments are respectively modelled. This allows first to conclude that the probabilistic model which applies to the one, is also worth for the other one. Moreover, it appears that both the *T and the *L are isomorphic with the *C. This makes it possible to determine straightforwardly the corresponding probabilistic model. Thus, the situation corresponding to both the emerald case and the Little Puddle/London experiment can be modelled by the drawing of a ball from an urn comprising red and blue balls. In the emerald case, it consists of an urn comprising 3 red balls and 5000 green balls. In the Little Puddle/London experiment, the urn includes thus 50 red balls and 107 green balls.

4. Goodman’s paradox

Another interest of the n-universes as a methodological tool resides in their use to clarify complex situations such as those which are faced in the study of paradoxes. I will illustrate in what follows the contribution of the n-universes in such circumstances through the analysis of Goodman’s paradox.3

Goodman’s paradox was described in Fact, Fiction and Forecast (1954, 74-75). Goodman explains then his paradox as follows. Every emeralds which were until now observed turned out to be green. Intuitively, we foresee therefore that the next emerald that will be observed will also be green. Such prediction is based on the generalisation according to which all emeralds are green. However, if one considers the property grue, that is to say “observed before today and green, or observed after today and not-green”,4 we can notice that this property is also satisfied by all instances of emeralds observed before. But the prediction which results from it now, based on the generalisation according to which all emeralds are grue, is that the next emerald to be observed will be not-green. And this contradicts the previous conclusion, which is conforms however with our intuition. The paradox comes here from the fact that the application of an enumerative induction to the same instances, with the two predicates green and grue, leads to predictions which turn out to be contradictory. This contradiction constitutes the heart of the paradox. One of the inductive inferences must then be fallacious. And intuitively, the conclusion according to which the next observed emerald will be not-green appears erroneous.

Let us set out now to model the Goodman’s experiment in terms of n-universes. It is necessary for it to describe accurately the conditions of the universe of reference in which the paradox takes place. Goodman makes thus mention of properties green and not-green applicable to emeralds. Colour constitutes then one of the variable-criteria of the n-universe in which the paradox takes place. Moreover, Goodman draws a distinction between emeralds observed before T and those which will be observed after T. Thus, the corresponding n-universe also includes a variable-criterion of time. As a result, we are in a position to describe the minimal universe in which Goodman (1954) situates himself as a coloured and temporal n-universe, i.e. a CT.

Moreover, Goodman makes mention of several instances of emeralds. It could then be natural to model the paradox in a n-universe with multiple objects, coloured and temporal. However, it does not appear necessary to make use of a n-universe including multiple objects. Considering the methodological objective which aims at avoiding a combinatorial explosion of cases, it is indeed preferable to model the paradox in the simplest type of n-universe, i.e. a n-universe with a unique object. We observe then the emergence of a version of the paradox based on one unique emerald the colour of which is likely to vary in the course of time. This version is the following. The emerald which I currently observe was green all times when I did observe it before. I conclude therefore, by induction, that it will be also green the next time when I will observe it. However, the same type of inductive reasoning also leads me to conclude that it will be grue, and therefore not-green. As we can see, such variation always leads to the emergence of the paradox. The latter version takes p lace in a n-universe including a unique object and a variable of colour and of time, i.e. a CT. At this step, given that the original statement of the paradox turns out to be ambiguous in this respect, and that the minimal context is that of a CT, we will be led to distinguish between two situations: the one which situates itself in a CT, and the one which takes place in a CT (where  denotes a third variable-criterion).

Let us place ourselves first in the context of a coloured and temporal n-universe, i;e. a CT. In such universe, to be green, is to be green at time T. In this context, it appears completely legitimate to project the shared property of colour (green) of the instances through time. The corresponding projection can be denoted by C°T. The emerald was green every time where I observed it before, and the inductive projection leads me to conclude that it will be also green next time when I will observe it. This can be formalized as follows (V denoting green):

The previous reasoning appears completely correct and conforms to our inductive practice. But are we thus entitled to conclude from it that the green predicate is projectible without restriction in the CT? It appears not. For the preceding inductive enumeration applies indeed to a n-universe where the temporal variable corresponds to our present time, for example the period of 100 years surrounding our present epoch, that is to say the interval [-100, +100] years. But what would it be if the temporal variable extended much more far, by including for example the period of 10 thousand million years around our current time, that is to say the interval [-1010, +1010] years. In that case, the emerald is observed in 10 thousand million years. At that time, our sun is burned out, and becomes progressively a white dwarf. The temperature on our planet then warmed itself up in significant proportions to the point of attaining 8000°: the observation reveals then that the emerald – as most mineral – suffered important transformations and proves to be now not-green. Why is the projection of green correct in the CT where the temporal variable is defined by restriction in comparison with our present time, and incorrect if the temporal variable assimilates itself by extension to the interval of 10 thousand million years before or after our present time? In the first case, the projection is correct because the different instances of emeralds are representative of the reference class on which the projection applies. An excellent way of getting representative instances of a given reference class is then to choose the latter by means of a random toss. On the other hand, the projection is not correct in the second case, for the different instances are not representative of the considered reference class. Indeed, the 99 observations of emeralds come from our modern time while the 100th concerns an extremely distant time. So, the generalisation (H2) results from 99 instances which are not representative of the CT[-1010, +1010] and does not allow to be legitimately of use as support for induction. Thus green is projectible in the CT[-102, +102] and not projectible in the CT[-1010, +1010]. At this stage, it already appears that green is not projectible in the absolute but turns out to be projectible or not projectible relative to this or that n-universe.

In the light of what precedes, we are from now on in a position to highlight what proved to be fallacious in the projection of generalisation according to which “all swans are white”. In 1690, such hypothesis resulted from the observation of a big number of instances of swans in Europe, in America, in Asia and in Africa. The n-universe in which such projection did take place was a n-universe with multiple objects, including a variable of colour and of location. To simplify, we can consider that all instances had being picked at constant time T0. The corresponding inductive projection C°L led to the conclusion that the next observed swan would be white. However, such prediction turned out to be false, when occurred the discovery in 1697 by the Dutch explorer Willem de Vlamingh of black swans in Australia. In the n-universe in which such projection did take place, the location criterion was implicitly assimilating itself to our whole planet. However, the generalisation according to which “all swans are white” was founded on the observation of instances of swans which came only from one part of the n-universe of reference. The sample turned out therefore to be biased and not representative of the reference class, thus yielding the falseness of the generalisation and of the corresponding inductive conclusion.

Let us consider now the projection of grue. The use of the grue property, which constitutes (with bleen) a taxon of tcolour*, is revealing of the fact that the used system of criteria comes from the Z. The n-universe in which takes place the projection of grue is then a Z, a n-universe to which the CT reduces. For the fact that there exists two taxa of colour (green, not-green) and two taxa of time (before T, after T) in the CT determines four different states: green before T, not-green before T, green after T, not-green after T. By contrast, the Z only determines two states: grue and bleen. The reduction of the CT to the Z is made by transforming the taxa of colour and of time into taxa of tcolour*. The classical definition of grue (green before T or not-green after T) allows for that. In this context, it appears that the paradox is still present. It comes indeed under the following form: the emerald was grue every time that I did observe it before, and I conclude inductively that the emerald will also be grue and thus not-green the next time when I will observe it. The corresponding projection Z°T can then be formalized (G denoting grue):

What is it then that leads to deceive our intuition in this specific variation of the paradox? It appears here that the projection of grue comes under a form which is likely to create an illusion. Indeed, the projection Z°T which results from it is that of the tcolor* through time. The general idea which underlies inductive reasoning is that the instances are grue before T and therefore also grue after T. But it should be noticed here that the corresponding n-universe is a Z. And in a Z, the only variable-criterion is tcolor*. In such n-universe, an object is grue or bleen in the absolute. By contrast, an object is green or not-green in the CT relative to a given temporal position. But in the Z where the projection of grue takes place, an additional variable-criterion is missing so that the projection of grue could be legitimately made. Due to the fact that an object is grue or bleen in the absolute in a Z, when it is grue before T, it is also necessarily grue after T. And from the information according to which an object is grue before T, it is therefore possible to conclude, by deduction, that it is also grue after T. As we can see it, the variation of the paradox corresponding to the projection Z°T presents a structure which gives it the appearance of an enumerative generalisation but that constitutes indeed a genuine deductive reasoning. The reasoning that ensues from it constitutes then a disguised form of induction, a pseudo-induction.

Let us envisage now the case of a coloured, temporal n-universe, but including an additional variable-criterion , i.e. a CT. A n-universe including variable-criteria of colour, of time and location,5 i.e. a CTL, will be suited for that. To be green in a CTL, is to be green at time T and at location L. Moreover, the CTL reduces to a ZL, a n-universe the variable-criteria of which are tcolor* and location. The taxa of tcolor* are grue and bleen. And to be grue in the ZL, is to be grue at location L.

In a preliminary way, one can point out here that the projections CTL and ZTL do not require a separate analysis. Indeed, these two projections present the same structure as those of the projections CT and ZT which have just been studied, except for an additional differentiated criterion of location. The conditions under which the paradox dissolves when one compares the projections CT and ZT apply therefore identically to the variation of the paradox which emerges when one relates the projections CTL and ZTL .

On the other hand, it appears here opportune to relate the projections CT°L and Z°L which respectively take place in the CTL and the ZL. Let us begin with the projection CT°L. The shared criteria of colour and of time are projected here through a differentiated criterion of location. The taxa of time are here before T and after T. In this context, the projection of green comes under the following form. The emerald was green before T in every place where I did observe it before, and I conclude from it that it will be also green before T in the next place where it will be observed. The corresponding projection C°TL can then be formalized as follows:

At this step, it seems completely legitimate to project the green and before T shared by the instances, through a differentiated criterion of location, and to predict that the next emerald which will be observed at location L will present the same properties.

What is it now of the projection of grue in the CTL? The use of grue conveys the fact that we place ourselves in a ZL, a n-universe to which reduces the CTL and the variable-criteria of which are tcolour* and location. The fact of being grue is relative to the variable-criterion of location. In the ZL, to be grue is to be grue at location L. The projection relates then to a taxon of tcolour* ( grue or bleen) which is shared by the instances, through a differentiated criterion of location. Consider then the classical definition of grue (green before T or non-grue after T). Thus, the emerald was grue in every place where I did observe it before, and I predict that it will also be grue in the next place where it will be observed. If we take T = in 1010 years, the projection Z°L in the ZL appears then as a completely valid form of induction (V~T denoting green after T):

As pointed out by Franck Jackson (1975, 115), such type of projection applies legitimately to all objects which colour changes in the course of time, such as tomatoes or cherries. More still, one can notice that if we consider a very long period of time, which extends as in the example of emeralds until 10 thousand million years, such property applies virtually to all concrete objects. Finally, one can notice here that the contradiction between both concurrent predictions (P9) and (P12’*) has now disappeared since the emerald turns out to be green before T in L100 (VTL100) in both cases.

As we can see, in the present analysis, a predicate turns out to be projectible or not projectible in relative to this or that universe of reference. As well as green, grue is not projectible in the absolute but turns out to be projectible in some n-universes and not projectible in others. It consists here of a difference with several classical solutions offered to solve the Goodman’s paradox, according to which a predicate turns out to be projectible or not projectible in the absolute. Such solutions lead to the definition of a criterion allowing to distinguish the projectible predicates from the unprojectible ones, based on the differentiation temporal/non-temporal, local/non-local, qualitative/non-qualitative, etc. Goodman himself puts then in correspondence the distinction projectible/ unprojectible with the distinction entrenchedi/unentrenched, etc. However, further reflexions of Goodman, formulated in Ways of Worldmakingii, emphasize more the unabsolute nature of projectibility of green or of grue: “Grue cannot be a relevant kind for induction in the same world as green, for that would preclude some of the decisions, right or wrong, that constitute inductive inference”. As a result, grue can turn out to be projectible in a goodmanian world and not projectible in some other one. For green and grue belong for Goodman to different worlds which present different structures of categories.6 In this sense, it appears that the present solution is based on a form of relativism the nature of which is essentially goodmanian.

5. Conclusion

From what precedes and from Goodman’s paradox analysis in particular, one can think that the n-universes are of a fundamentally goodmanian essence. From this viewpoint, the essence of n-universes turns out to be pluralist, thus allowing numerous descriptions, with the help of different systems of criteria, of a same reality. A characteristic example, as we did see it, is the reduction of the criteria of colour and time in a CTL into a unique criterion of tcolour* in a ZL. In this sense, one can consider the n-universes as an implementation of the programme defined by Goodman in Ways of Worldmaking. Goodman offers indeed to construct worlds by composition, by emphasis, by ordering or by deletion of some elements. The n-universes allow in this sense to represent our concrete world with the help of different systems of criteria, which correspond each to a relevant point of view, a way of seeing or of considering a same reality. In this sense, to privilege this or that system of criteria, to the detriment of others, leads to a truncated view of this same reality. And the exclusive choice, without objective motivation, of such or such n-universe leads to engender a biased point of view.

However, the genuine nature of the n-universes turns out to be inherently ambivalent. For the similarity of the n-universes with the goodmanian worlds does not prove to be exclusive of a purely ontological approach. Alternatively, it is indeed possible to consider the n-universes from the only ontological point of view, as a methodological tool allowing to model directly this or that concrete situation. The n-universes constitute then so much universes with different properties, according to combinations resulting from the presence of a unique object or multiple objects, in relation one or many, with demultiplication or not, with regard to the criteria of time, location, colour, etc. In a goodmanian sense also, the n-universes allow then to build so much universes with different structures, which sometimes correspond to the properties of our real world, but which have sometimes some exotic properties. To name only the simplest of the latter, the L* is then a n-universe which includes only one ubiquitous object, presenting the property of being at several locations at the same time.7

At this stage, it is worth mentioning several advantages which would result from the use of the n-universes for modelling probabilistic situations. One of these advantages would be first to allow a better intuitive apprehension of a given probabilistic situation, by emphasising its essential elements and by suppressing its superfluous elements. By differentiating for example depending on whether the situation to model presents a constant or a time variable, a constant or a space variable, a unique object or several objects, etc. the modelling of concrete situations in the n-universes provides a better support to intuition. On the other hand, the distinction according to whether the objects are or not with demultiplication or in relation one/many with regard to the different criteria allows for a precise classification of the different probabilistic situations which are encountered.

One can notice, second, that the use of the notation of the probability spaces extended to the n-universes would allow to withdraw the ambiguity which is sometimes associated with classical notation. As we did see it, we sometimes face an ambiguity. Indeed, it proves to be that {1,2,…,6}2 denotes at the same time the sample space of a simultaneous throwing of two discernible dices in T0 and that of two successive throwing of the same dice in T1 and then in T2. With the use of the notation extended to n-universes, the ambiguity disappears. In effect, the sample space of the simultaneous throwing of two discernible dices in T0 is a {1,2}*T0L{1,2}*O{1,2,…,6}, whilst that of two successive throwing of the same dice in T1 and then in T2 is a 0T*{1,2}L0O{1,2,…,6}.

Finally, an important advantage, as we have just seen it, which would result from a modelling of probabilistic situations extended to n-universe is the easiness with which it allows comparisons between several probabilistic models and it highlights the isomorphisms and the corresponding dimorphisms. But the main advantage of the use of the n-universes as a methodological tool, as we did see it through Goodman’s paradox, would reside in the clarification of the complex situations which appear during the study of paradoxes.8

References

1 Il convient de noter que ces différentes modélisations ne constituent pas une manière unique de modéliser les objets correspondants dans les n-univers. Cependant, elles correspondent à l’intuition globale que l’on a de ces objets.

2 De manière alternative, on pourrait utiliser la notation 0T0L0O5 en lieu et place de 0T0L0O{5}. Cette dernière notation est toutefois préférée ici, car elle se révèle davantage compatible avec la notation classique des événements.

3 Cette analyse du paradoxe de Goodman correspond, de manière simplifiée et avec plusieurs adaptations, à celle initalement décrite dans Franceschi (2001). La variation du paradoxe qui est considérée ici est celle de Goodman (1954), mais avec une émeraude unique.

4 P and Q being two predicates, grue presents the following structure: (P and Q) or (~P and ~Q).

5 Tout autre critère différent de la couleur et du temps tel que la masse, la température, l’orientation, etc. conviendrait également.

6 Cf. Goodman (1978, 11): “(…) a green emerald and a grue one, even if the same emerald (…) belong to worlds organized into different kinds”.

7 Les n-univers aux propriétés non standard nécessitent une étude plus détaillée, qui dépasse le cadre de la présente étude.

8 Je suis reconnaissant envers Jean-Paul Delahaye pour la suggestion de l’utilisation des n-univers en tant qu’espaces de probabilité étendus. Je remercie également Claude Panaccio et un expert anonyme pour le Journal of Philosophical Research pour des discussions et des commentaires très utiles.

i Entrenched.

ii Cf. Goodman (1978, 11).