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.

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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, …, T_n. Such object has a life span which covers the period T₁-T_n. 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  i  j  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 T_i: 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 L_i: 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):

(I1)	VT1·VT2·VT3·…·VT99	instances
(H2)	VT1·VT2·VT3·…·VT99·VT100	generalisation
(P3)	 VT100	from (H2)

The previous reasoning appears completely correct and conforms to our inductive practice. But are we thus entitled to conclude from it that the green predicate is projectible without restriction in the CT? It appears not. For the preceding inductive enumeration applies indeed to a n-universe where the temporal variable corresponds to our present time, for example the period of 100 years surrounding our present epoch, that is to say the interval [-100, +100] years. But what would it be if the temporal variable extended much more far, by including for example the period of 10 thousand million years around our current time, that is to say the interval [-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):

(I4*)	GT1·GT2·GT3·…·GT99	instances
(H5*)	GT1·GT2·GT3·…·GT99·GT100	generalisation
(H5’*)	VT1·VT2·VT3·…·VT99·~VT100	from (H5*), definition
(P6*)	 GT100	prediction
(P6’*)	 ~VT100	from (P6*), definition

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:

(I7)	VTL1·VTL2·VTL3·…·VTL99	instances
(H8)	VTL1·VTL2·VTL3·…·VTL99·VTL100	generalisation
(P9)	 VTL100	prediction

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

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

(I10*)	GL1·GL2·GL3·…·GL99	instances
(H11*)	GL1·GL2·GL3·…·GL99·GL100	generalisation
(H11’*)	VT~V~TL1·VT~V~TL2·VT~V~TL3·…·VT~V~TL99·VT~V~TL100	from (H11*), definition
(P12*)	 GL100	prediction
(P12’*)	 VT~V~TL100	from (P12*), definition

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

Franceschi, Paul. 2001. Une solution pour le paradoxe de Goodman. Dialogue 40: 99-123, English translation under the title The Doomsday Argument and Hempel’s Problem, http://cogprints.org/2172/. English translation

—. 2002. Une application des n-univers à l’argument de l’Apocalypse et au paradoxe de Goodman. Doctoral dissertation, Corté: University of Corsica. <http://www.univ-corse.fr/~franceschi/index-fr.htm> [retrievec Dec.29, 2003]

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

—. 1978. Ways of Worldmaking. Indianapolis: Hackett Publishing Company.

Jackson, Franck. 1975. “Grue”. The Journal of Philosophy 72: 113-131.

Leslie, John. 1996. The End of the World: The Science and Ethics of Human Extinction. London: Routledge.

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

2 De manière alternative, on pourrait utiliser la notation 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).