Posprint in English (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.

Paul Franceschi

p.franceschi@univ-corse.fr

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

Nelson Goodman

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-universe (1T) as a universe comprising only one criterion-variable: time. In the same way, one defines a coloured 1-universe (1C) as a universe comprising only one criterion-variable: colour. One will define also a coloured and temporal 2-universe (2CT) as a universe comprising two criterion-variables: time and colour. Etc. In the same way, a universe where all the objects are red, but are characterised by a different localisation will be modelled by a localised 1-universe (1L) 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 1C. 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 2CT, 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 (3CTL), 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 3CTL. 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*:

 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 RT  R~T X X X X X X X X RT  ~R~T X X X X X X X X ~RT  R~T X X X X X X X X ~RT  ~R~T X X X X X X X X

In each of these cases, it results indeed a new single criterion of tcolour* (Z), which substitutes itself to the two preceding criteria of colour and time. One will denote by 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 2CT can be reduced to a tcoloured* 1-universe (1Z). 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 3CTL, or like a 2ZL. In the same way, one can represent a 2CT like a 1Z. 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 2CT, C°L in a 2CL, Z°L in a 2ZL, etc. Conversely, the following projections are incorrect: T°T in a 1T, Z°Z in a 1Z. In particular, one will note here that the projection T°T in the 1T is that of 1-time. 1-time takes indeed place in a 1T, 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

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 2CT. 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 2CT.

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 2CT 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 minimal2CT, one will consider successively two situations: that of a 2CT, and then that of a 3CT (where denotes a 3rd criterion).

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

Consider first the hypothesis of a 2CT. 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 2CT was reducible to a 1Z. 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 1Z. What is then the situation when one projects “grue” in the 1Z? 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 2CT 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 2CT, 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 1Z, whereas the differentiated properties come from the 2CT. 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 1Z, whereas the use of time is revealing of the fact that one places oneself in a 2CT. 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 1Z with the help of the taxa of the 2CT. Since one knows the code defining the relations between two reducible n-universes – the 1Z and the 2CT – 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 1Z, 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 3CT. This type of universe of reference also corresponds to the definition of a minimal2CT, but it also comprises one 3rd criterion-variablexxv. Let us choose for this last a criterion such as localisationxxvi. Consider then a 3CTL. Consider first (H) in such a 3-universe. To be “red” in the 3CTL, 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 3CTL? It has been observed that the 3CTL could be reduced to a 2ZL. 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 2ZL:

(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 2ZL, 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 3CTL 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 2CT, the 3CTL, the 2ZL 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 2ZL, 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 3CTL 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 3CTL to the 2ZL. 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.

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.

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

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

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

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

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

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

# 1. Cognitive models of delusions

Edvard Munch: The Scream.

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

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

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

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

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

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

# 2. Apophenia

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

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

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

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

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

# 3. Formation and maintenance of delusional ideas

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

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

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

3.1 Formation of delusional ideas

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

Let us begin with delusions of reference. Let us consider the following argument, which leads the patient to conclude that television speaks about him/her, and therefore to delusional ideas of reference (T1 and T2 denote here two successive temporal positions, with a very short time interval between T1 and T2; the symbol denotes the conclusion; and R is taken for reference):

(R1) in T1 I was drinking an aperitif

(R2) in T2 the presenter of the show said: “Stop drinking!”

(R3) in T2 the presenter said: “Stop drinking!” because in T1 I was drinking an aperitif

(R4) in T3 I was upset and anxious

(R5) in T4 the presenter of the show said “Stop stressing”

(R6) in T4 the presenter of the show said “Stop stressing!” because in T3 I was upset and anxious

(R7) in T5 I was smoking a cigarette

(R8) in T6 I heard the presenter saying “That is not good !“

(R9) in T6 the presenter said “That is not good !” because in T5 I was smoking a cigarette

(R10) in T7 I felt fine and lucid and I was relaxed

(R11) in T8 the presenter of the show said: “We are in great form!

(R12) in T8 the presenter said “We are in great form!” because in T7 I felt fine and lucid and I was relaxed

(R…) …

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

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

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

(R3) in T2 the presenter of the show spoke according to what I was doing

(R6) in T4 the presenter of the show spoke according to what I was feeling

(R9) in T6 the presenter of the show spoke according to what I was doing

(R12) in T8 the presenter of the show spoke according to what I felt

(R…) …

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

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

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

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

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

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

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

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

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

(R15) television and the media speak about me

(T16) people can read my thoughts

(I17) I influence other people’s behaviour

(18) I am someone exceptional

(19) I am an extraterrestrial

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

3.2 Maintenance of delusional ideas

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

(R20) in T100 the presenter of the show spoke according to what I was doing

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

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

# 4. Analysis of delusional arguments

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

(T1) in T1 I thought of Michael “What an idiot!”

(T2) in T2 I heard Michael shout

(T3) in T2 I heard Michael shout because in T1 I thought of him “What an idiot!”

It appears here that the two premises (T1) and (T2) constitute genuine facts and therefore turn out to be true. Conversely, the conclusion (T3) that concludes to the existence of a relation of causality between the two consecutive facts F1 (in T1 I thought of Michael “What an idiot!”) and F2 (in T2 I heard Michael shout) is it justified? It appears not. Indeed, both premises are only establishing a relation of anteriority between the two consecutive facts F1 and F2. And the conclusion (T3) which deducts from it a causality relationship turns out therefore to be too strong. The corresponding reasoning presents then a fallacious nature. The corresponding error of reasoning, which concludes to a relation of causality while there is only a simple relation of anteriority, is traditionally called post hoc fallacy, according to the Latin sentence “Post hoc, ergo propter hoc” (thereafter, hence because of it). This is a very common type of fallacious reasoning, which is notably at the root of many superstitions (Martin 1998; Bressan 2002). David Hemsley (1992) notably mentions such type of reasoning in clinical observation: “A patient of the present author, recalling his psychotic experiences noted that the co-occurrence of two events often led immediately to an assumption of a causal relationship between them”. Finally, one can notice that in the context of cognitive distortions, the type of error of reasoning corresponding to post hoc fallacy can be considered a specific case of arbitrary inference.

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

(T22) in T2 Michael spoke according to my thoughts

(T23) in T4 the neighbour spoke according to my thoughts

(T24) in T6 the radio presenter spoke according to my thoughts

(…)

(T25) people act according to my thoughts

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

(T1) in T1 I thought of Michael “What an idiot!”

(26) in T2 Michel was quiet

And an instance of the second form is :

(27) in T1 I didn’t think of Michael

(T2) in T2 I heard Michael shout

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

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

(T28) in T100 my sister spoke according to my thoughts

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

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

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

(T30) people act in function of my thoughts

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

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

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

(R15) television and the media speak about me

(T16) people can read my thoughts

(I17) I influence people’s behaviour

(18) I am someone exceptional

(19) I am an extraterrestrial

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

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

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

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

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

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

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

5. The role of the hallucinations

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

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

(32) in T1 I dropped my pen

(33) in T2 I heard the voice of the presenter of the show saying “Clumsy!”

(34) in T2 the presenter of the show said “Clumsy!” because in T1 I dropped my pen

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

(35) in T1 I was very upset

(36) in T2 I heard the voice of my neighbour saying “Calm down!”

(37) in T2 my neighbour said “Calm down!” because in T1 I was very upset

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

(38) in T2 the TV presenter said “Clumsy!” because in T1 I dropped my pen

(39) in T4 the presenteress said “Calm down!” because in T3 I was upset

(40) in T6 the presenter of the show said “Thank you” because in T5 I thought “I love this presenter”

(…) …

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

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

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

(42) in T50 I thought that people know of my thoughts

(43) in T100 I heard Joseph saying: “I know the slightest of your thoughts”

(44) this confirms that people know of my thoughts

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

(19) in T50 I thought I was an extraterrestrial

(45) in T100 I heard a voice saying : “You come from the planet Mars”

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

# 6. Comparison with other cognitive models of delusions

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

The present analysis, to begin with, is susceptible of inserting itself as part of the adaptation of Albert Ellis’ ABC-analysis described by Chadwick et al. (1996). In this context, the internal and external events with regard to the patient, that are the premises of primary delusional arguments, constitute the As. The primary, secondary, tertiary and quaternary delusional arguments, can also be assimilated to the Bs. Lastly, the negative emotions (anger, anxiety, frustration, etc.) felt by the patient, that result there from the conclusions of delusional tertiary and quaternary arguments, constitute the Cs. As we can see it, the present analysis leads, in comparison with the standard ABC-analysis, to distinguish several steps at the level of the Bs. This distinction is important, since it allows to distinguish several steps whose function is different, within the reasoning that leads to delusional ideas. Thus, the B1s (primary delusional arguments) are instances that lead to the attribution of a causality relationship between internal and external (to the patient) phenomena; the B2s (secondary delusional arguments) result from a generalisation of inductive nature; the B3s (tertiary delusional arguments) correspond to an interpretative step at a monothematic level; finally, the B4s (quaternary delusional arguments) are characteristic of a step of interpretation at a polythematic level, the conclusion of which truly constitutes a global explicative theory of the abnormal phenomena undergone by the patient. On the other hand, we are led there to distinguish between those parts of the patient’s reasoning which are globally valid (the B2s, B3s and B4s) and the part which is invalid (the B1s, based on post hoc fallacy). Such nuanced point of view should be likely to preserve – what constitutes one of the key points of cognitive and behaviour therapy – the therapeutic alliance, i.e., the relation of collaboration between the patient and the therapist aiming at shared objectives in the struggle against the illness. As we can see it, the present analysis leads to especially emphasise post hoc fallacy, which constitutes the weakness in the patient’s reasoning, but the repeated instances of which, triggered by apophenia, truly constitute the building block of the delusional construction.

The present model also has number of affinity with the approaches which are at the root of cognitive therapy of schizophrenia (Kingdon & Turkington 1994; Kingdon & Turkington 2005; Chadwick & al. 1996; Beck & Rector 2000). In this type of approach, the therapist sets out to reduce progressively the patient’s degree of belief in his/her delusional polythematic ideas. To this end, the therapist suggests the patient, in a spirit of dialogue of Socratic inspiration, to elaborate alternative hypotheses; he also teaches the patient the approach which consists in searching elements likely to confirm or to disconfirm his/her own hypotheses, as well as to build out alternative hypotheses. The contribution of the present analysis with regard to cognitive therapy of schizophrenia is likely to manifest itself in several ways. It proves to be useful to specify then, for the clinician, what could be such contribution, and to also provide a specific framework in which the present model will possibly be tested. The distinction of different steps in the development of delusions allows first to distinguish between different hypotheses corresponding to the conclusions of primary, secondary, ternary and quaternary arguments. The degree of belief associated with each of these levels of hypotheses is also susceptible of being evaluated separately, by notably allowing to determinate then at which level resides the strongest degree of conviction. In the same way, each of the conclusions of the primary, secondary, tertiary or quaternary arguments, will possibly be tested (confirmed/infirmed) and give room for the elaboration of alternative hypotheses (David Kingdon, personal communication). For example, at the level of primary delusional arguments, it will be possible to consider the belief according to which the presenter said in T2: “You should not drink!” because the patient was drinking an aperitif in T1; this hypothesis will possibly give rise to a search for evidence, and then confronted with an alternative hypothesis such as: the presenter said in T2: “You should not drink!” because it was scheduled in the script of the television program. At the level of tertiary delusional arguments, the hypothesis according to which television speaks about the patient will also possibly be the object of a search for evidence, etc.

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

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

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

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

References

 Bayne, Tim & Pacherie, Elisabeth, “In Defense of the Doxastic Conception of Delusions”, Mind and Language, vol. 20-2, 2005, p. 163-88. Bayne, Tim & Pacherie, Elisabeth, “Experience, Belief, and the Interpretive Fold”, Philosophy, Psychiatry, & Psychology, vol. 11-1, 2004, p. 81-86 Beck, Aaron & Rector, Neil, “Cognitive Therapy of Schizophrenia: A New Therapy for the New Millenium”, American Journal of Psychotherapy, vol. 54-3, 2000, p. 291-300. Beck, Aaron, “Delusions: A Cognitive perspective”, Journal of Cognitive Psychotherapy, vol. 16-4, 2002, p. 455-468. Bentall, Richard & Kaney, Sue, “Content specific information processing in persecutory delusions: an investigation using the emotional stroop test”, British Journal of medical Psychology, vol. 62, 1989, p. 355-364. Bentall, Richard, Kinderman, P., & Kaney, S., “Self, attributional processes and abnormal beliefs: Towards a model of persecutory delusions”, Behaviour Research and Therapy, vol. 32, 1994, p. 331-341. Bortolotti, Lisa, “Delusions and the background of rationality”, Mind and Language, 20-2, 2005, p.189-208. Bressan, Paola, “The Connection Between Random Sequences, Everyday Coincidences, and Belief in the Paranormal”, Applied Cognitive Psychology, vol. 16, 2002, p. 17-34. Brugger, Peter, “From Haunted Brain to Haunted Science: A Cognitive Neuroscience View of Paranormal and Pseudoscientific Thought”, in Hauntings and Poltergeists: Multidisciplinary Perspectives, Houran, J., & Lange, R. (Eds), North Carolina: McFarland, 2001. Brundage, B.E., “What I wanted to know but was afraid to ask”, Schizophrenia Bulletin, vol. 9, 1983, p. 583-585. Campbell, John, “Rationality, Meaning, and the Analysis of Delusion”, Philosophy, Psychiatry, & Psychology, vol. 8-2/3, 2001, p. 89-100. Chadwick, Paul, Birchwood, M., & Trower, P., Cognitive Therapy for Delusions, Voices, and Paranoia, Chichester: Wiley, 1996. Chapman, L.J., Chapman, J.P., The genesis of delusions, in T.F. Oltmanns, B.A. Maher (Eds), Delusional beliefs, New York: Wiley, 1988, p. 167-183. Clarke, Isabel, « Madness and Mysticism: clarifying the mystery », Network: The Scientific and Medical Network Review, vol. 72, 2000, p. 11-14. Conrad, K., Die beginnende Schizophrenie. Versuch einer Gestaltanalyse des Wahns, Stuttgart: Thieme, 1958. Davies, M. & Coltheart, M., Langdon, R. & Breen, N., “Monothematic delusions: Towards a two-factor account”, Philosophy, Psychiatry and Psychology, vol. 8-2, 2001, p. 133–158. Davies, Martin & Coltheart, Max, “Introduction: Pathologies of Belief”, Mind and Language, vol. 15, 2000, p.1-46. Ellis, Albert, Reason and emotion in psychotherapy, New York: Lyle Stuart, 1962. Engel, Pascal, “Peut-on parler de croyances délirantes ?”, in J. Chemouni, dir. Clinique de l’intentionnalité, In-Press, Paris, 2001, p. 157-173. Faucher, Luc, “Philosophie psychopathologique : un survol”, Philosophiques, vol. 33-1, 2006, 3-18. Fear, Christopher & Healy David, “Probabilistic reasoning in obsessive-compulsive and delusional disorders”, Psychological Medicine, vol. 27, 1997, p. 199-208. Fowler, David, Garety, P. & Kuipers, E., Cognitive behaviour therapy for psychosis: theory and practice, Chichester: Wiley, 1995. Garety, Philippa, Hemsley, D. & Wessely, S., “Reasoning in deluded schizophrenic and paranoid patients”, Journal of Nervous and Mental Disease, vol. 179, 1991, p. 194-201. Garety, Philippa & Hemsley, David, Delusions: Investigations into the Psychology of Delusional Reasoning, Maudsley Monograph, Psychology Press, 1994. Garety, Philippa, Kuipers, E., Fowler, D., Freeman, D. & Bebbington, P., “A cognitive model of the positive symptoms of psychosis”, Psychological Medicine, vol. 31, 2001, p. 189-195. Hemsley, David, “Disorders of perception and cognition in schizophrenia”, Revue européenne de Psychologie Appliquée, vol. 42-2, 1992, p. 105-114. Jonas E., Schulz-Hardt S., Frey D., & Thelen N. “Confirmation bias in sequential information search after preliminary decisions: an expansion of dissonance theoretical research on selective exposure to information”, Journal of Personality and Social Psychology, vol. 80, 2001, p. 557-571. Kaney, Sue & Bentall, Richard, “Persecutory delusions and attributional style”, British Journal of Medical Psychology, vol. 62, 1989, p. 191-198. Kaney, Sue, Wolfenden, M., Dewey, M.E. & Bentall, R.P, “Persecutory delusions and recall of threatening and non-threatening propositions”, British Journal of Clinical Psychology, vol. 31, 1991, p. 85-87. Kingdon, David, & Turkington, Douglas, Cognitive-behavioural Therapy of Schizophrenia, New York: Guilford, 1994. Kingdon, David, & Turkington, Douglas, Cognitive Therapy of Schizophrenia, New York, London: Guilford, 2005. Maher, Brendan, “Delusional thinking and perceptual disorder”, Journal of Individual Psychology, vol. 30, 1974, p. 98-113. Maher, Brendan, “Anomalous experiences and delusional thinking: the logic of explanations“, in Delusional Beliefs, Oltmanns, T. F., & Maher, B. A. (Eds), New York: Wiley, 1988, p. 15-33. Maher, Brendan, “Anomalous experience in everyday life: Its significance for psychopathology”, The Monist, vol. 82, 1999, p. 547-570. Martin, Bruce, “Coincidences: Remarkable or random”. Skeptical Inquirer, vol. 22-5, 1998, p. 23-27. Nickerson, Raymond, “Confirmation Bias: A Ubiquitous Phenomenon in Many Guises”, Review of General Psychology, vol. 2, 1998, p. 175-220. Pacherie E, Green M, Bayne T. “Phenomenology and delusions: who put the ‘alien’ in alien control?”, Consciousness and Cognition, vol. 15, 2006, p. 566–577. Sass, Louis, The paradoxes of delusion: Wittgenstein, Schreber and the schizophrenic mind, Ithaca, NY: Cornell University Press, 1994. Schneider, Carl, Die Psychologie der Schizophrenen, Leipzig: Thieme, 1930. Spitzer, Manfred, “A neurocomputational approach to delusions”, Comprehensive Psychiatry, vol. 36, 1995, p. 83-105. Stanton, Biba & David, Anthony, “First-person accounts of delusions”, Psychiatric Bulletin, vol. 24, 2000, p. 333-336. Stephens, Lynn & George Graham, “Reconcevoir le délire”, Philosophiques, vol. 33-1, 2006, p. 183-196. Stone, Tony & Young, Andrew, “Delusions and brain injury: the philosophy and psychology of belief”, Mind and Language, vol. 12, 1997, p. 327-364. Young, Andrew , “Delusions”, The Monist, vol. 82, 1999, p. 571-589.

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

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

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

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

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

//

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

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

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## On a Class of Concepts

Classically, in the discussion relating to polar opposites1, one primarily directs his interest to the common and lexicalized concepts, i.e. for which there exists a corresponding word in the vocabulary inherent to a given language. This way of proceeding tends to generate several disadvantages. One of them resides in the fact (i) that such concepts are likely to vary from one language to another, from one culture to another. Another (ii) of the resulting problems is that certain lexicalized concepts reveal a nuance which is either meliorative or pejorative, with degrees in this type of nuances which prove difficult to appreciate. Finally, another problem (iii) lies in the fact that certain concepts, according to semiotic analysis2 are regarded as marked with regard to others concepts which are unmarked, the status of unmarked concept conferring a kind of precedence, of pre-eminence to the concepts in question.

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

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

# 1. Dualities

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

(i) are different one from the other

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

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

(iv) are predicates

Each of the concepts composing a given duality will be termed a pole. I shall present here a list, which does not pretend to be exhaustive, and could if necessary, be supplemented. Consider then the following enumeration of dualities3:

Analytic/Synthetic, Animate/Inanimate, Exceptional/Normal, Antecedent/Consequent, Existent/Inexistent, Absolute/Relative, Abstract/Concrete, Accessory/Principal, Active/Passive, Aleatory/Certain, Discrete/Continuous, Deterministic/Indeterministic, Positive/Negative, True/False, Total/Partial, Neutral/Polarized, Static/Dynamic, Unique/Multiple, Container/Containing, Innate/Acquired (Nature/Nurture), Beautiful/Ugly, Good/Ill, Temporal/Atemporal, Extended/Restricted, Precise/Vague, Finite/Infinite, Simple/Composed, Attracted/Repulsed, Equal/Different, Identical/Contrary, Superior/Inferior, Internal/External, Individual/Collective, Quantitative/Qualitative, Implicit/Explicit4, …

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

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

It should be noted lastly that several questions5 immediately arise with regard to dualities. Do dualities exist (i) in a finite or infinite number? In the same way, does there exist (ii) a logical construction which makes it possible to provide an enumeration of the dualities?

# 2. Canonical poles

The positive canonical poles

Starting from the class of the dualities, we are now in a position to construct the class of the canonical poles. At the origin, the lexicalized concepts corresponding to each pole of a duality reveal a nuance6 which is respectively either meliorative, neutral, or pejorative. The class of the canonical poles corresponds to the intuition that, for each pole of a given duality A/Ā, one can construct 3 concepts: a positive, a neutral and a negative concept. In sum, for a given duality A/Ā, one thus constructs 6 concepts, thus constituting the class of the canonical poles. Intuitively, positive canonical poles respond to the following definition: positive, meliorative form of ; neutral canonical poles correspond to the neutral, i.e. neither meliorative nor pejorative form of ; and negative canonical poles correspond to the negative, pejorative form of . It should be noted that these 6 concepts are exclusively constructed with the help of logical concepts. The only notion which escapes at this step to a logical definition is that of duality or base.

The neutral canonical poles

For a given duality A/Ā, we have thus the following canonical poles: {A+, A0, A, Ā+, Ā0, Ā}, that we can also denote respectively by (A/Ā, 1, 1), (A/Ā, 1, 0) , (A/Ā, 1, -1) , (A/Ā, -1, 1) , (A/Ā, -1, 0) , (A/Ā, -1, -1).

The negative canonical poles

A capital letter for the first letter of a canonical pole will be used, in order to distinguish it from the corresponding lexicalized concept. If one wishes to refer accurately to a canonical pole whereas the usual language lacks such a concept or well appears ambiguous, one can choose a lexicalized concept, to which the exponent corresponding to the chosen neutral or polarized state will be added. To highlight the fact that one refers explicitly to a canonical pole – positive, neutral or negative – the notations A+, A0 et A will be used. We have thus for example the concepts Unite+, Unite0, Unite etc. Where Unite+ = Solid, Undivided, Coherent and Unite = Monolithic. In the same way, Rational0 designates the neutral concept corresponding to the term rational of the common language, which reveals a slightly meliorative nuance. In the same way, Irrationnal0 designates the corresponding neutral state, whereas the common word irrational reveals a pejorative nuance. One will proceed in the same way, when the corresponding lexicalized word proves ambiguous. One distinctive feature of the present construction is that one begins by constructing the concepts logically, and puts them afterwards in adequacy with the concepts of the usual language, insofar as these latter do exist.

The constituents of a canonical pole are:

– a duality (or base) A/Ā

– a contrary component c {-1, 1}

– a canonical polarity p {-1, 0, 1}

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

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

– the positive canonical poles: A+, Ā+

– the neutral canonical poles: A0, Ā0

– the negative canonical poles: A, Ā

– the canonical matrix consisting of the 6 canonical poles: {A+, A0, A, Ā+, Ā0, Ā}. The 6 concepts constituting the canonical matrix can also be denoted under the form of a 3 x 2 matrix.

A canonical matrix

Let also be a canonical pole, one will denote by ~ its complement, semantically corresponding to non. We have thus the following complements: ~A+, ~A0, ~A, ~Ā+, ~Ā0, ~Ā. The notion of a complement entails the definition of a universe of reference U. Our concern will be thus with the complement of a given canonical pole in regard to the corresponding matrix7. It follows then that: ~A+ = {A0, A, Ā+, Ā0, Ā}. And a definition of comparable nature for the complements of the other concepts of the matrix ensues.

It should be noted lastly that the following questions arise with regard to canonical poles. The construction of the matrix of the canonical poles of the Positive/Negative duality: {Positive+, Positive0, Positive, Negative+, Negative0, Negative} ensues. But do such concepts as Positive0, Negative0 and especially Positive, Negative+ exist (i) without contradiction?

In the same way, at the level of the Neutral/Polarized duality, the construction of the matrix {Neutral+, Neutral 0, Neutral, Polarized+, Polarized0, Polarized} ensues. But do Neutral+, Neutral exist (ii) without contradiction? In the same way, does Polarized0 exist without contradiction?

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

# 3. Relations between the canonical poles

Among the combinations of relations existing between the 6 canonical poles (A+, A0, A, Ā+, Ā0, Ā) of a same duality A/Ā, it is worth emphasizing the following relations (in addition to the identity relation, denoted by I).

Two canonical poles 1(A/Ā, c1, p1) and 2(A/Ā, c2, p2) of a same duality are dual or antinomical or opposites if their contrary components are opposite and their polarities are opposite8.

Complementarity

Two canonical poles 1(A/Ā, c1, p1) and 2(A/Ā, c2, p2) of a same duality are complementary if their contrary components are opposite and their polarities are equal9.

Two canonical poles 1 (A/Ā, c1, p1) et 2(A/Ā, c2, p2) of a same duality are corollary if their contrary components are equal and their polarities are opposite10.

Two canonical poles 1 (A/Ā, c1, p1) and 2(A/Ā, c2, p2) of a same duality are connex if their contrary components are equal and the absolute value of the difference in their polarities is equal to 1 11.

Two canonical poles 1 (A/Ā, c1, p1) and 2(A/Ā, c2, p2) of a same duality are anti-connex if their contrary components are opposite and the absolute value of the difference in their polarities is equal to 1.12, 13

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

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

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

# 4. Degrees of duality

One constructs the class of the degrees of duality, from the intuition that there is a continuous succession of concepts from A+ to Ā, from A0 to Ā0 and from A to Ā+. The continuous component of a degree of duality corresponds to a degree in the corresponding dual pair. The approach by degree is underlied by the intuition that there is a continuous and regular succession of degrees, from a canonical pole Ap to its contrary Ā-p.14 One is thus led to distinguish 3 classes of degrees of duality: (i) from A+ to Ā (ii) from A0 to Ā0 (iii) from A to Ā+.

A degree of duality presents the following components:

– a dual pair Ap-p (corresponding to one of the 3 cases: A+, A00 or A+)

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

A degree of duality has thus the form: (A+, d), (A00, d) or (A+, d).

On the other hand, let us call neutral point a concept pertaining to the class of the degrees of duality, whose degree is equal to 0. Let us denote by 0 such a concept, which is thus of the form (Ap-p, 0) with d[0] = 0. Semantically, a neutral point 0 corresponds to a concept which responds to the following definition: neither Ap nor Ā-p. For example, (True/False, 0) corresponds to the definition: neither True nor False. In the same way (Vague/Precise, 0) corresponds the following definition: neither Vague nor Precise. Lastly, when considering the Neutral/Polarized and Positive/Negative dualities, one has then: Neutral0 = (Negative0/Positive0, 0) = (Neutral0/Polarized0, 1).

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

The following property of the neutral points then ensue, for a given duality A/Ā: (A+, 0) = (A00, 0) = (A+, 0).

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

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

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

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

Does any duality A/Ā admit (ii) a neutral point or trichotomic zero? One can call this question the problem of the general trichotomy. Is it a general rule15 or well does there exists some exceptions? It seems a priori that the Abstract/Concrete duality does not admit a neutral point. It appears to be the same for the Finite/Infinite or the Precise/Vague duality. Intuitively, these latter dualities do not admit an intermediate state.

Does the concept corresponding to the neutral point (Neutral0/Polarized0, 0) and responding to the definition: neither neutral nor polarized exist (iii) without contradiction in the present construction?

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

It is worth also considering the relation of includer for the canonical poles. Consider the following pairs of dual canonical poles: A+ and Ā+, A0 and Ā0, A and Ā. We have then the following definitions: a positive includer + is a concept such that it is itself a positive canonical pole and corresponds to the definition + = A+ Ā+. A neutral includer 0 is a neutral canonical pole such that 0 = A0 Ā0. And a negative includer is a negative canonical pole such that = A Ā. Given these definitions, it is clear that one assimilates here the includer to the minimum includer. Examples: Determinate0 is an includer for True0/False0. And Determinate0 is also a pole for the Determinate0/Indeterminate0 duality. In the same way, Polarized0 is an includer for Positive0/Negative0.

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

## – matricial includers: they consist of concepts including the set of the canonical poles of a same duality. They respond to the definition: 0 = A+ A0 A– Ā+ Ā0 Ā–.

mixed includers: they consist of concepts responding to the definition 1 = A+ Ā or well 2 = A Ā+

It is worth also considering the types of relations existing between the canonical poles of a different duality. Let A and E be two matrices whose canonical poles are respectively {A+, A0, A, Ā+, Ā0, Ā} and {E+, E0, E, Ē+, Ē0, Ē} and such that E is an includer for A/Ā i.e. such that E+ = A+ Ā+, E0 = A0 Ā0 and E = A Ā. One extends then the just-defined relations between the canonical poles of a same matrix, to the relations of comparable nature between two matrices presenting the properties of A and E. We has then the relations of 2-antinomy, 2-complementarity, 2-corollarity, 2-connexity, 2-anti-connexity16. Thus, for example, A0 is 2-contrary (or trichotomic contrary) to Ē0, 2-connex (or trichotomic connex) to E+ and E and 2-anti-connex (or trichotomic anti-connex) to Ē+ and Ē. In the same way, A+ and Ā+ are 2-contrary to Ē, 2-complementary to Ē+, 2-corollary to E, 2-connex to E0 and 2-anti-connex to Ē0, etc.

Let us consider also the following property of neutral points and includers. Let A and E be two matrices, such that one of the neutral poles of E is an includer for the neutral dual pair of a: E0 = A0 Ā0. We has then the following property: the canonical pole Ē0 for the matrix E is a neutral point for the duality A00. Thus, the neutral point for the duality A00 is the dual of the includer E0 of A0 and Ā0. Example: Determinate0 = True0 False0. Here, the neutral point for the True/False duality corresponds to the definition: neither True nor False. And we have then (True0/False0, 0) = (Determinate0/Indeterminate0, -1).

This last property can be generalized to a hierarchy of matrices A1, A2, A3, …, An, such that one of the poles 2 of A2 of polarity p is an includer for a dual pair of A1, and that one of the poles 3 of A3 is an includer for a dual pair of A2, …, and that one of the poles n of An is an includer for a dual pair of An-1. It follows then an infinite construction of concepts.

One also notes the emergence of a hierarchy, beyond the sole neutral point of a given duality. It consists of the hierarchy of the neutral points of order n, constructed in the following way from the dual canonical poles A0 and Ā0:

– A0, Ā0

– A1 = neither A0 nor Ā0

– A21 = neither A0 nor A1

– A22 = neither Ā0 nor A1

– A31 = neither A0 nor A21

– A32 = neither A0 nor A22

– A33 = neither A0 nor A21

– A34 = neither Ā0 nor A22

– …

One can also consider the emergence of this hierarchy under the following form17:

– A0, Ā0

– A1 = neither A0 nor Ā0

– A2 = neither A0 nor Ā0 nor A1

– A3 = neither A0 nor Ā0 nor A1 nor A2

– A4 = neither A0 nor Ā0 nor A1 nor A2 nor A3

– A5 = neither A0 nor Ā0 nor A1 nor A2 nor A3 nor A4

– …

Classically, one constructs this infinite hierarchy for True/False by considering I1 (Indeterminate), I2, etc. It should be noticed that in this last construction, no mention is made of the includer (Determinate) of True/False. Neither does one make mention of the hierarchy of includers.

The notion of a complement of a canonical pole corresponds semantically to non. One has the concept of a 2-complement of a canonical pole , defined with regard to a universe of reference U that consists of the 2-matrix of . One has then for example: ~A+ = {A0, A, Ā+, Ā0, Ā, Ē+, Ē0, Ē}. And also, ~A+ = {Ā+, E0, E, Ē+, Ē0, Ē}, etc. More generally, one has then the notion of a n-complement (n > 0) of a canonical pole with regard to the corresponding n-matrix.

The following questions finally arise, concerning includers. For certain concepts, does there exist (i) one maximum includer or well does one have an infinite construction for each duality? Concerning the True/False duality in particular, the analysis of the semantic paradoxes has led to the use of a logic based on an infinite number of truth-values18.

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

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

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

# 6. Canonical principles

Let be a canonical pole. Intuitively, the class of the canonical principles corresponds to the concepts which respond to the following definition: principle corresponding to what is . Examples: Precise Precision; Relative Relativity; Temporal Temporality. The canonical principles can be seen as 0-ary predicates, whereas the canonical poles are n-ary predicates (n > 0). The lexicalized concepts corresponding to canonical principles are often terms for which the suffix –ity (or –itude) has been added to the radical corresponding to a canonical pole. For example: Relativity0, Beauty+, Activity0, Passivity0, Neutrality0, Simplicity0, Temporality0, etc. A list (necessarily non-exhaustive) of the canonical principles is the following:

Analysis0/Synthesis0, [Animate0]/[Inanimate0], [Exceptional0]/Normality0, [Antecedent0]/[Consequent0], Existence0/Inexistence0, Absolute0/Relativity0, Abstraction0/[Concrete], [Accessory0]/[Principal0], Activity0/Passivity0, [Random0]/Certainty0, [Discrete0]/[Continuous0], Determinism0/Indeterminism0, [Positive0]/[Negative0], Truth0/Falsity0, Attraction0/Repulsion0, Neutrality0/Polarization0, [Static0]/Dynamic0, Unicity0/Multiplicity0, Contenance0/[Containing0], Innate0/Acquired0, Beauty+/Ugliness, Good+/Evil, Identity0/Contrary0, Superiority0/Inferiority0, Extension0/Restriction0, Precision0/Vagueness0, Finitude0/Infinitude0, Simplicity0/Complexity0, [Internal0]/[External0], Equality0/Difference0, Whole0/Part0, Temporality0/Atemporality0, Individuality0/Collectivity0, Quantity0/Quality0, [Implicit0]/[Explicit0], …

It should be noticed that a certain number of canonical principles are not lexicalized. The notations A+, A0, A will be used to denote without ambiguity a canonical principle which is respectively positive, neutral or negative. One could also use the following notation: being a canonical pole, then -ity (or -itude) is a canonical principle. The following notation could then be used: Abstract0ity, Absolute0ity, Acessory0ity, etc. or as above [Abstract0], [Absolute0], etc.

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

It is worth distinguishing finally the following derived classes:

positive canonical principles

neutral canonical principles

negative canonical principles

polarized canonical principles

with some obvious definitions19.

# 7. Meta-principles

Let a0 be a neutral canonical principle20. The class of the meta-principles corresponds to a disposition of the mind directed towards what is a0, to an interest with regard to what is a0. Intuitively, a meta-principle corresponds to a point of view, a perspective, an orientation of the human mind. Thus, the attraction for Abstraction0, the interest for Acquired0, the propensity to place oneself from the viewpoint of Unity0, etc. constitute meta-principles. It should be noted that this construction makes it possible in particular to construct some concepts which are not lexicalized. This has the advantage of a better exhaustiveness and leads to a better and richer semantics.

Let a0 be a neutral canonical principle. Let us also denote by p a meta-principle (p {-1, 0, 1}). One denotes thus a positive meta-principle by +, a neutral meta-principle by 0 and a negative meta-principle by . We have then the enumeration of the meta-principles, for a given duality: {A+, A0, A, Ā+, Ā0, Ā}. Moreover, one will be able to denote by aism a meta-principle. Example: Unite Unite-ism. We have then Internalism, Externalism, Relativism, Absolutism, etc. which correspond in particular to dispositions of the mind. A capital letter will preferably be used here to distinguish the meta-principles from the lexicalized concepts, and in particular to differentiate them from the corresponding philosophical doctrines, which often have very different meanings. It will be however possible to make use of the classical terms when they exist to designate the corresponding meta-principle. Thus All-ism corresponds to Holism.

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

The constituents of the meta-principles are:

– a polarity p Î {-1, 0, 1}

– a neutral canonical principle composed of:

– a duality (or base) A/Ā

– a contrary component c {-1, 1}

– a neutral polarity q = 0

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

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

One has lastly the derived classes consisting in:

– the positive meta-principles (p > 0)

– the neutral meta-principles (p = 0)

– the negative meta-principles (p < 0)

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

– the matrix of the canonical meta-principles, consisting of 6 meta-principles applicable to a given duality{A+, A0, A, Ā+, Ā0, Ā}.

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

– the class of the behavioral principles. Intuitively, the class of the behavioral principles constitutes an extension of that of the meta-principles. While the meta-principle constitutes a disposition of the human mind, the concepts concerned here are those which aim to describe, in a more general way, the tendencies of the human behavior21. Among the lexicalized concepts corresponding to the behavioral principles, one can mention: courage, prudence, pessimism, rationality, avarice, fidelity, tendency to analysis, instability, objectivity, pragmatism, etc. A first analysis reveals (i) that a certain number of them reveal a meliorative nuance: courage, objectivity, pragmatism; that (ii) others, by contrast, present a pejorative, unfavorable connotation: cowardice, avarice, instability; and finally (iii) that certain concepts present themselves under a form which is neither meliorative nor pejorative: tendency to analysis22. One has here the same classes as for the meta-principles, and in particular the degrees of behavioral principles. Example: coward is more negative than apprehensive; in the same way, bravery is more positive than courage.

# Conclusion

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

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

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

References

 FINE, Kit (1975). Vagueness, Truth and Logic. Synthese 30: 265-300 GREIMAS, A. J. (1977). Elements of a Narrative Grammar, Diacritics 7: 23-40 JAKOBSON, Roman (1983). Dialogues, Cambridge MA: MIT Press PEACOCKE, C. A. B. (1981). Are Vague Predicates Incoherent?. Synthese 46: 121-141 RESCHER, Nicholas (1969). Many-Valued Logic, New York: McGraw Hill

1 Or polar contraries.

2 Cf. Jakobson (1983).

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

Contrary ° Contrary = Identical.

Contrary ° Identical = Contrary.

Contraryn = Identical (for n even).

Contraryn = Contrary (for n odd).

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

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

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

6 With variable degrees in the nuance.

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

8 Formally c1 = –c2, p1 = – p2 ® 1(A/Ā, c1, p1) = 2(A/Ā, c2, p2).

9 Formally c1 = – c2, p1 = p2 ® 1(A/Ā, c1, p1) = f2(A/Ā, c2, p2).

10 Formally c1 = c2, p1 = – p2 ® 1(A/Ā, c1, p1) = c2(A/Ā, c2, p2).

11 Formally c1 = c2, |p1p2| = 1 ® 1(A/Ā, c1, p1) = g2(A/Ā, c2, p2).

12 Formally c1 = – c2, |p1p2| = 1 ® 1(A/Ā, c1, p1) = b2(A/Ā, c2, p2).

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

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

°IcjIIcjccIjjIcjjcI

where for all A Î G, A-1 = A, and A ° I = A, I being the neutral element. It should be noted that the group properties make it possible in particular to give straightforwardly a valuation to any propositions of the form: the contrary concept of the complementary of a1 is identical to the corollary of the complementary of a2.

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

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

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

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

– A2 = A21 or A22

– A3 = A31 or A32 or A33 or A34

– A4 = A41 or A42 or A43 or A44 or A45 or A46 or A47 or A48

– …

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

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

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

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

22 One can consider the following – necessarily partial – enumeration corresponding to the behavioral principles, in the order (A+), (A0), (A), (Ā+), (Ā0), (Ā):

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

defense, refusal, violence, pacifism, acceptance, weakness

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

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

delicacy, sensitivity, sentimentality, coolness, impassibility, coldness

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

motivation, passion, fanaticism, moderation, reason, tepidity

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

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

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

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

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

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

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

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.

English translation of a paper published in French in the Journal de Thérapie Comportementale et Cognitive, 2009, 19-4, pages 136-140 under the title “Théorie des distorsions cognitives : la sur-généralisation et l’étiquetage”.

In a previous article (Complements to a theory of cognitive distorsions, Journal de Thérapie Comportementale et Cognitive, 2007), we introduced some elements aimed at contributing to a general theory of cognitive distortions. Based on the reference class, the duality and the system of taxa, these elements allow to define the general cognitive distortions as well as the specific cognitive distortions. This model is extended here to the description of two other classical cognitive distortions: over-generalisation and mislabelling. The definition of the two latter cognitive distortions is based on prior differentiation between three levels of reasoning: primary, secondary and ternary pathogenic arguments. The latter analysis also leads to define two other cognitive distortions which insert themselves into this framework: ill-grounded inductive projection and confirmation bias.

## Theory of Cognitive Distortions: Over-generalisation and Labeling

In Franceschi (2007), we set out to introduce several elements aimed at contributing to a general theory of cognitive distortions. These elements are based on three fundamental notions: the reference class, the duality and the system of taxa. With these three elements, we could define within the same conceptual framework the following general cognitive distortions: dichotomous reasoning, disqualification of one pole, minimisation and maximisation, requalification in the other pole and omission of the neutral. We could also describe as specific cognitive distortions: disqualification of the positive, selective abstraction and catastrophism. In the present article, we offer to define and to situate, within the same conceptual framework, two other classical cognitive distortions: over-generalisation and mislabelling.

Over-generalisation and mislabelling constitute two of the twelve traditionally defined cognitive distortions: emotional reasoning; over-generalisation; arbitrary inference; dichotomous reasoning; should statements; divination or mental reading; selective abstraction; disqualification of the positive; maximisation/minimisation; catastrophising; personalisation; mislabelling (Beck 1964, Ellis 1962). Over-generalisation is classically defined as a rough and ill-grounded generalisation, usually including either of the quantifiers “all”, “none”, “never”, “always”. Moreover, it is often described as a cognitive distortion including four subcategories: dichotomous reasoning, selective abstraction, maximisation/minimisation, and disqualification of the positive. Mislabelling is also classically defined as an extreme form of over-generalisation, consisting in the apposition of a label with a strong negative and emotional connotation to oneself or to an external subject.

1. Primary, secondary and ternary pathogenic arguments

Before setting out to define over-generalisation and mislabelling in the present context, it is worth describing preliminarily a structure of pathogenic reasoning (in the etymological sense: engendering suffering), with a general scope, susceptible of being found in some disorders of a very different nature, such as depression, generalised anxiety disorder, body dismorphic disorder, scrupulosity or intermittent explosive disorder. Such structure of reasoning includes several levels of arguments: primary, secondary and ternary. In a simplified way, primary pathogenic arguments are constituted by an enumeration of instances. Secondary pathogenic arguments consist of a generalisation from the latter instances. Lastly, pathogenic ternary arguments are constituted by an interpretation of the latter generalisation. Such reasoning as a whole presents an inductive structure.

At this stage, it is worth mentioning several instances of this type of reasoning. A first instance, susceptible to be found in depression (Beck 1967, 1987), is the following (the symbol denotes the conclusion):

 (11) I gave my ankle a wrench last January premise1 (12) I lost my job last February premise2 (13) Fifteen days ago, I had an influenza with fever premise3 (14) I got into an argument with Lucy last month premise4 (…) (…) (110) Today, my horoscope is not good premise10 (2) ∴ Everything that occurs to me is bad from (11)-(110) (3) ∴ I am a complete failure! from (2)

The patient enumerates first some events of his/her past and present life (11)-(110), that he/she qualifies as negative, through a primary stage which consists of an enumeration of instances. Then he/she performs a generalisation (2) from the previous enumeration, which presents the following structure:

 (2) ∴ All events that occur to me are negative from (11)-(110)

Lastly, the patient interprets (3) the latter conclusion by concluding “I am a complete failure!”. Such instance applies then to the reference class of the present and past events of the patient’s life and to the Positive/Negative duality.

One can also mention a reasoning that presents an identical structure, which is susceptible to be met in body dysmorphic disorder (Veale 2004, Rabinowitz & al. 2007). The patient enumerates then different parts of his/her body, which he/she qualifies as ugly. He/she generalises then by concluding that all parts of his/her body are ugly. Finally, he/she adds: “I am ugly!”. The corresponding reasoning applies then to the Beautiful/Ugly duality and to the reference class of the parts of the patient’s body.

In the same way, in a reasoning of identical structure, susceptible to be met in scrupulosity (Teak & Ulug 2001, Miller & Edges 2007), the patient enumerates several instances corresponding to some acts which he/she made previously or recently, and which he/she considers as morally bad. He/she concludes then: “Everything I do is bad, morally reprehensible”, and he/she further interprets it by concluding: “I am a horrible sinner!”. Such conclusion is likely to trigger an intense feel of guilt and a compulsive practice of religious rituals. The corresponding instance applies here to the duality Good/Evil and to the reference class of the present and past actions of the patient’s life.

Lastly, an instance of this structure of reasoning can contribute to the development of hostility, of a potentially aggressive attitude toward other people. In that case, the patient concludes regarding an external subject: “All acts that he committed toward me are bad”. He/she concludes then: “He is a bastard!”. Such conclusion can then play a role in intermittent explosive disorder (Coccaro & al. 1998, Galovski & al. 2002). In such case, the over-generalisation applies to the Good/Evil duality and to the reference class of the actions of an external subject with regard to the patient.

At this step, it is worth describing in more detail each of the three stages – primary, secondary and ternary – which compose this type of reasoning.

Primary pathogenic arguments

The first step in the aforementioned type of reasoning, consists for the patient to enumerate some instances. The general structure of each instance is as follows:

 (1i) The object xi of the class of reference E has property Ā (in the duality A/Ā) premisei

In the aforementioned example applied to depression, the patient enumerates some events of his/her present and past life, which he/she qualifies as negative, under the form:

 (1i) The event Ei of negative nature occurred to me premisei

Different instances corresponding to this cognitive process can be described under the form of a primary pathogenic argument, the structure of which is the following:

 (1a) The event E1 occurred to me premise (1b) The event E1 was of a negative nature premise (1) ∴ The event E1 of a negative nature occurred to me from (1a), (1b)

By such cognitive process, the patient is led to the conclusion according to which some negative event did occur to him/her.

From a deductive point of view, this type of argument proves to be completely valid (the conclusion is true if the premises are true) since the very event presents well, objectively, a negative nature. However, this type of primary argument can turn out to be fallacious, when the very event presents, objectively, a positive or neutral nature. The flaw in the reasoning resides then in the fact that the premise (1b) turns then out to be false. Such can be case for example if the patient makes use of a specific cognitive distortion such as requalification in the negative. In such case, the patient considers as negative an event the nature of which is objectively positive.

Secondary pathogenic arguments

At the level of the above-mentioned reasoning, secondary pathogenic arguments are constituted by the sequence which proceeds by generalisation, from the instances (11) to (110), according to the following structure:

 (2) ∴ All elements xi of the class of reference E have property Ā from (11)-(110)

Such over-generalisation leads then to the conclusion “All events that occur to me are bad” (depression); “All parts of my body are ugly” (body dysmorphic disorder); “All my acts are morally reprehensible” (scrupulosity); “All acts that he committed toward me are bad” (intermittent explosive disorder).

From a deductive point of view, such generalisation may constitute a completely valid argument. Indeed, the resulting generalisation constitutes a correct deductive reasoning, if the premises (11)-(110) are true. However, it often proves to be that the premises of the argument are false. Such is notably the case when the patient counts among the elements having property Ā, some elements which objectively have the opposite property A. The flaw in the argument resides then in a requalification in the other pole related to some elements and the enumeration of instances includes then some false premises, thus invalidating the resulting generalisation. In such case, secondary pathogenic argument turns out to be ungrounded, because of the falseness of some premises.

In other cases, the secondary pathogenic argument turns out to be fallacious from an inductive standpoint. For some positive (or neutral) events can well have been omitted in the corresponding enumeration of instances. Such omission can result from the use of general cognitive distortions, such as the omission of the neutral or disqualification of the positive. In such case, the elements of the relevant class of reference are only partly taken into account, thus biasing the resulting generalisation. The corresponding reasoning remains then logically valid and sound, but fundamentally incorrect of an inductive point of view, because it does only take partly into account the relevant instances within the reference class. Such feature of over-generalisation – a conclusion resulting from a valid reasoning from a deductive point of view, but inductively wrong – allows to explain how it notably succeeds in deceiving patients whose level of intelligence can otherwise prove to be high.

Ternary pathogenic arguments

It is worth mentioning, lastly, the role played by pathogenic ternary arguments which consist, at the level of the aforementioned reasoning, of the following sequence:

 (2) All events that occur to me are of a negative nature premise (3) ∴ I am a complete failure! from (2)

In such argument, the premise is constituted by the conclusion (2) of the secondary pathogenic argument, of which, in an additional stage (3), the patient aims at making sense by interpreting it. It consists here of a case of mislabelling. At the stage of a ternary pathogenic argument, mislabelling can thus take the following forms: “I am a complete failure!” (depression); “I am ugly!” (bodily dysmorphic disorder); “I am a horrible sinner!” (scrupulosity); “He is a bastard! “ (intermittent explosive disorder). In the present context, mislabelling proves to be an invalid argument, which constitutes a rough and unjustified interpretation of the over-generalisation (2).

2. Over-generalisation

At this stage, we are in a position to give a definition of over-generalisation, by drawing a distinction between general and specific over-generalisations. A general over-generalisation applies to any duality and to any reference class. It can be analysed as the ill-grounded conclusion of a secondary pathogenic argument, the premises of which include some general cognitive distortions: dichotomous reasoning, disqualification of one pole, arbitrary focus, minimisation/maximisation, omission of the neutral or requalification in the other pole. It consists of an ungrounded inductive reasoning, because the resulting generalisation is based on an incorrect counting of the corresponding instances. In the same way, a specific over-generalisation consists of an instance of a general over-generalisation, applied to a given duality and reference class. Thus, the specific over-generalisation “All events which occur to me are of a negative nature” (depression, generalised anxiety disorder) applies to the Positive/Negative duality and to the class of the events of the patient’s life. In the same way, “All parts of my body are ugly” (body dysmorphic disorder) is a specific over-generalisation that applies to the reference class of the parts of the patient’s body and to the Beautiful/Ugly duality.

3. Ungrounded inductive projection

At this step, it proves to be useful to describe another error of reasoning, which is likely to manifest itself at the stage of secondary pathogenic arguments. It consists of an ill-grounded inductive projection. The latter concludes, from the preceding over-generalisation (2), that a new instance will occur in the near future. Such instance is susceptible to be met in depression (Miranda & al. 2008), as well as in generalised anxiety disorder (Franceschi 2008). In the context of depression, such inductive projection presents the following form:

 (2) All events that occur to me are of a negative nature premise (111a) The future event E11 of a negative nature may occur premise (111b) ∴ The future event E11 of a negative nature will occur from (2), (111a)

The corresponding conclusion is susceptible of contributing to depression, notably by triggering the patient’s feeling of despair. Other instances of this type of conclusion are: “My next action will be morally reprehensible” (scrupulosity), or “The next act that he will commit toward me will be bad” (intermittent explosive disorder).

4. Confirmation bias

The cognitive process which has just been described illustrates how over-generalisation contributes to the formation of pathogenic ideas. However, a process of the same nature is also likely to concur to their maintenance. For once the over-generalisation (2) has been established by means of the above reasoning, its maintenance is made as soon as an instance occurs that confirms the generalisation according to which all elements xi of the reference class E have property Ā. This constitutes a confirmation bias, for the patient does only count those elements which present the property Ā, without taking into account those which have the opposite property A, thus disconfirming generalisation (2). Hence, in depression or generalised anxiety disorder, when a new negative event occurs, the patient concludes from it that it confirms that all events which occur to him/her are of a negative nature.

We see it finally, the above developments suggest a classification of cognitive distortions, depending on whether they manifest themselves at the level of primary, secondary or ternary pathogenic arguments. Thus, among the cognitive distortions which arise at the stage of primary pathogenic arguments, one can distinguish: on the one hand, the general cognitive distortions (dichotomous reasoning, disqualification of one pole, minimisation/maximisation, requalification into the other pole, omission of the neutral) and on the other hand, the specific cognitive distortions (disqualification of the positive, requalification into the negative, selective abstraction, catastrophising). Morevoer, among the cognitive distortions which manifest themselves at the stage of secondary pathogenic arguments, one can mention over-generalisation (at the stage of the formation of pathogenic ideas), ill-grounded inductive projection, and confirmation bias (at the stage of the maintenance of pathogenic ideas). Mislabelling, finally, is susceptible to occur at the level of ternary pathogenic arguments.

### References

Beck A. Thinking and depression: Theory and therapy. Archives of General Psychiatry, 1964, 10, 561-571.

Beck, A. Depression: Clinical, experimental, and theoretical aspects, Harper & Row, New York, 1967.

Beck, A. Cognitive models of depression. Journal of Cognitive Psychotherapy, 1, 1987, 5-37.

Coccaro E., Richard J., Kavoussi R., Mitchell E., Berman J., Lish J. Intermittent explosive disorder-revised: Development, reliability, and validity of research criteria. Comprehensive Psychiatry, 39-6, 1998, 368-376.

Eckhardt C., Norlander B., Deffenbacher J., The assessment of anger and hostility: a critical review, Aggression and Violent Behavior, 9-1, 2004, 17-43.

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

Franceschi P. Compléments pour une théorie des distorsions cognitives. Journal de Thérapie Comportementale et Cognitive, 2007, 17-2, 84-88. English translation.

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

Galovski T., Blanchard E., Veazey C. Intermittent explosive disorder and other psychiatric co-morbidity among court-referred and self-referred aggressive drivers. Behaviour Research and Therapy, 40-6, 2002, 641-651.

Miller C., Hedges D. Scrupulosity disorder: An overview and introductory analysis. Journal of Anxiety Disorders, 2007, 22-6, 1042-1048.

Miranda R., Fontes M., Marroquín B. Cognitive content-specificity in future expectancies: Role of hopelessness and intolerance of uncertainty in depression and GAD symptoms. Behaviour Research and Therapy, 46-10, 2008, 1151-1159.

Tek C., Ulug B. Religiosity and religious obsessions in obsessive–compulsive disorder. Psychiatry Research, 2001, 104-2, 99-108.

Rabinowitz D., Neziroglu F., Roberts M. Clinical application of a behavioral model for the treatment of body dysmorphic disorder. Cognitive and Behavioral Practice, 2007, 14-2, 231-237.

Veale D. Advances in a cognitive behavioural model of body dysmorphic disorder. Body Image, 2004, 1, 113-125.

English translation of a paper published in French in the Journal de Thérapie Comportementale et Cognitive, 2010, 20-2, pages 51-55 under the title “Théorie des distorsions cognitives : la personnalisation”.

In a previous paper (Complements to a theory of cognitive distorsionsJournal de Thérapie Comportementale et Cognitive, 2007), we did present some elements aimed at contributing to a general theory of cognitive distortions. Based on the reference class, the duality and the system of taxa, these elements led to distinguish between the general cognitive distortions (dichotomous reasoning, disqualification of one pole, minimisation, maximisation) and the specific cognitive distortions (disqualifying the positive, selective abstraction, catastrophism). By also distinguishing between three levels of reasoning – the instantiation stage, the interpretation stage and the generalisation stage – we did also define two other cognitive distortions: over-generalisation and mislabelling (Théorie des distorsions cognitives : la sur-généralisation et l’étiquetageJournal de Thérapie Comportementale et Cognitive, 2009). We currently extend this model to another classical cognitive distortion: personalisation.

## Theory of Cognitive Distortions: Personalisation

In Franceschi (2007), we set out to introduce several elements intended to contribute to a general theory of cognitive distortions. These elements are based on three fundamental notions: the reference class, the duality and the system of taxa. With the aid of these three elements, we could define within the same conceptual framework the general cognitive distortions such as dichotomous reasoning, disqualification of one pole, minimisation and maximisation, as well as requalification in the other pole and omission of the neutral. In the same way, we could describe as specific cognitive distortions: disqualification of the positive, selective abstraction and catastrophising. In Franceschi (2009), we introduced three levels of reasoning – the instantiation stage, the interpretation stage and the generalisation stage, which allowed to define within the same conceptual framework, two other classical cognitive distortions: over-generalisation and mislabelling. In the present paper, we set out to define and to situate in this conceptual framework another classical cognitive distortion: personalisation.

Personalisation constitutes one of but twelve classically defined cognitive distortions: emotional reasoning; over-generalisation; arbitrary inference; dichotomous reasoning; should statements; divination or mind-reading; selective abstraction; disqualification of the positive; maximisation/minimisation; catastrophising; personalisation; mislabelling (Ellis 1962, Beck 1964). Personalisation is usually defined as the fact of attributing unduly to oneself the cause of an external event. For example, seeing a person who laughs, the patient thinks that it is because of his/her physical appearance. Also, the patient makes himself/herself responsible for a negative event, in an unjustified way. If his/her companion then failed his/her examination, the patient estimates that is due to the fact that he/she is depressed. In what follows, we propose first to clarify the definition of personalisation and to situate it in the context of the theory of cognitive distortions (Franceschi 2007, 2009). Secondly, we set out to clarify the relationships existing between personalisation and several close notions mentioned in the literature: personalising bias (Langdon et al. 2006), ideas of reference (Startup & Startup 2005, Meyer & Lenzenweger 2009) and delusions of reference.

Personalisation and post hoc fallacy

We will set out first to highlight the mere structures of the cases of personalisation. Let us consider the aforementioned example where the patient sees a person who laughs and thinks that this one laughs because of the patient’s physical appearance. This constitutes an instance of personalisation. We can describe more accurately the reasoning which underlies such instance (in what follows, the symbol denotes the conclusion):

 (P11) in T1 I went for a walk premiss1 (P12) in T2 the peasant started to laugh premiss2 (P13) ∴ in T2 the peasant started to laugh because he saw that in T1 I went for a walk conclusion1 (P14) ∴ in T2 the peasant made fun of me conclusion2

The patient puts here in relationship an internal event (“I went for a walk”) with an external event (“the peasant started to laugh”). He/she concludes then that the internal event is the cause of the external event. In this stage, the patient “personalises” an external event, which he/she considers to be the effect of an internal event, while this external event is in reality devoid of any relationship with the patient himself/herself. In a subsequent stage (P14), the patient interprets the previous conclusion (P13) by considering that the peasant made fun of him.

At this stage, it is worth wondering about the specific nature of the patient’s error of reasoning. It appears here that both premises (P11) and (P12) constitute genuine facts and therefore turn out to be true. On the other hand, the conclusion (P13) which concludes to the existence of a relation of causality between two consecutive events E1 (“In T1 I went for a walk”) and E2 (“In T2 the peasant started to laugh”) appears to be unjustified. Indeed, both premises are only establishing a relation of anteriority between the two consecutive facts E1 and E2. And the conclusion (P13) which deducts from it a relation of causality turns out therefore to be too strong. The argument proves here to be invalid and the corresponding reasoning is then fallacious. The corresponding error of reasoning, which concludes to a relation of causality whereas there is only a mere relation of anteriority, is classically termed post hoc fallacy, according to the Latin sentence “Post hoc, ergo propter hoc” (after this therefore because of this). It consists here of a very common error of reasoning, which is notably at the root of many superstitions (Martin 1998, Bressan 2002).

In this context, we can point out that the case of post hoc fallacy which has just been described as an argument of personalisation, also constitutes a case of arbitrary inference, another classically defined cognitive distortion.

Steps of instantiation, of interpretation and of generalisation at the level of the arguments of personalisation

At this step, it proves to be useful to draw a distinction between the levels of arguments that lead to personalisation as cognitive distortion. This leads to differentiate three levels within the arguments of personalisation, among the reasoning’ stages. The latter correspond respectively to three different functions: it consists of the successive stages of instanciation, of interpretation and of generalisation. To this end, it is useful to describe the whole reasoning which underlies the arguments of personalisation and which includes the three aforementioned stages:

 (P11) in T1 I went for a walk premiss11 (P12) in T2 the peasant started to laugh premiss12 (P13) ∴ en T2 the peasant started to laugh because he saw that in T1 I went for a walk conclusion11 (P14) ∴ in T2 the peasant made fun of me conclusion12 (P21) in T3 I was leafing through a magazine in the library premiss21 (P22) in T4, the librarian smirked premiss22 (P23) ∴ en T4 the librarian smirked because in T3 I was leafing through a magazine in the library conclusion21 (P24) ∴ in T4, the librarian made fun of me conclusion22 (P31) in T5 I did enter in the show-room premiss31 (P32) in T6, my colleagues started to laugh premiss32 (P33) ∴ in T6, my colleagues started to laugh because in T5 I did enter in the show-room conclusion31 (P34) ∴ in T6, my colleagues were laughing at me conclusion32 (…) (P105) ∴ people make fun of me from (P14)-(P104)

Here, the instances of the previous arguments (P11)-(P13), (P21)-(P23), (P31)-(P33), etc. constitute primary stages of arguments of personalisation, by which the patient considers that an event related to him/her is the cause of an external event. This type of argument corresponds to the stage of instantiation. As mentioned earlier, such argument is fallacious since it is based on post hoc fallacy. In a subsequent stage the function of which is interpretative, and that is aimed at making sense of the conclusions (P13), (P23), (P33), … of the instances of arguments of the previous type, the patient interprets it by concluding that some people made fun of him. Such conclusions (P14), (P24), (P34) appear to be grounded, inasmuch as the premisses (P13), (P23), (P33) are true. Finally, in a subsequent stage of generalisation, the patient enumerates some instances or circumstances where he/she thinks that people laughed or made fun of him/her ((P14), (P24), (P34), …) and generalises then to the conclusion (P105) according to which people make fun of him/her. This last stage is of an inductive nature, and corresponds to an enumerative induction, the structure of which is the following:

 (P14) in T2 the peasant made fun of me conclusion12 (P24) in T4, the librarian made fun of me conclusion22 (P34) in T6, my colleagues were laughing at me conclusion32 (…) (P105) ∴ people make fun of me from (P14)-(P104)

Given what precedes, we can from now on provide a definition of personalisation. The preceding analysis leads then to distinguish between three stages in arguments of personalisation. At the level of primary arguments of personalisation (stage of instantiation), it consists of the tendency in the patient to establish an unjustified relation of causality between two events, among which one is external and the other one is internal to the patient. The patient personalises then, that is to say puts in relationship with himself/herself, an external event, which proves to be in reality devoid of any relation of causality. The mechanism which underlies such argument consists then of the erroneous attribution of a relation of causality, based on post hoc fallacy. At the level of secondary arguments of personalisation (stage of interpretation), the patient makes sense of the previous conclusion by concluding that at a given time, a person (or several persons) made fun of him, laughed at him, etc. Finally, at the level of arguments of ternary personalisation (stage of generalisation), the patient concludes that, in a general way, people make fun of him.

Personalisation and personalising bias

At this step, it proves to be useful to distinguish personalisation as cognitive distortion from personalising bias. The latter is defined as an attribution bias (“personalising attributional bias”), by whom the patient attributes to other persons rather than to circumstances the cause of a negative event (McKay & al. 2005, Langdon & al. 2006). Personalising bias is often related to polythematic delusions (Kinderman & Bentall 1997, Garety & Freeman 1999, McKay & al. 2005) met in schizophrenia.

Considering this definition, the difference between the two notions can be thus underlined: in personalisation as cognitive distortion, the patient attributes the cause of an external event to an event which concerns the patient himself/herself; on the other hand, in personalising bias the patient attributes the cause of an internal event to external persons. This allows to highlight several fundamental differences between the two notions. Firstly, in personalisation as cognitive distortion, the “person” is the patient himself/herself, while in personalising bias, it consist of external “persons”. Secondly, in the structure of personalisation, an internal event precedes an external event; by contrast, in the scheme of personalising bias, it is an external event which precedes an internal event. Finally, in personalisation as cognitive distortion, the internal event is indifferently of a positive, neutral or negative nature, whereas in personalising bias, the internal event is of a negative type. Hence, it finally proves to be that both notions appear fundamentally distinct.

Personalisation and ideas of reference

It appears also useful, for the sake of clarity, to specify the relationships between personalisation and ideas of reference. It is worth preliminary mentioning that one usually distinguishes between ideas of reference and delusions of reference (Dziegielewski 2002, p. 266). Ideas of reference characterise themselves by the fact that a patient considers that insignificant events relate to himself/herself, while is not the case in reality. For example, the patient hears several persons laugh, and considers, in an unjustified way, that the latter make fun of him/her. In parallel, delusions of reference constitute one of the most salient symptoms noticed in schizophrenia, and leads the patient to be persuaded that the media, television, or the radio speak about him/her or issue messages concerning him/her. Several criteria allow to draw a distinction between ideas of reference and delusions of reference. First, ideas of reference have much less impact on the patient’s life than reference delusions of reference. Second, the degree of conviction which is associated with ideas of reference is far lesser than with delusions of reference. Lastly, ideas of reference (“the neighbour made fun of me”) are related with beliefs the degree of plausibility of which is much stronger than the one which is inherent to delusions of reference (“newspapers speak about me”).

In this context, the aforementioned arguments of personalisation (P11)-(P14), (P21)-(P24), and (P31)-(P34), by whom the patient concludes that some people make fun of him, corresponds completely to the definition of ideas of reference. It appears then that personalisation, such as it was defined above as cognitive distortion, identifies itself with ideas of reference.

Personalisation and delusion of reference

One traditionally distinguishes at the level of polythematic delusions met in schizophrenia between: delusions of reference, delusions of influence, delusions of control, telepathy-like delusions, delusions of grandeur, and delusions of persecution. Delusions of reference leads for example the patient to believe with a very strong conviction that the media, the newspapers, the television speak about him/her.

It is worth describing here a mechanism which is susceptible to lead to the formation of delusions of reference. Such mechanism appears to be grounded on a reasoning (Franceschi 2008) which includes, as well as the above-mentioned primary instances of personalisation, a post hoc fallacy:

 (DR11) in T1 I was drinking an appetizer premiss11 (DR12) in T2 the presenter of the show said: “Stop drinking!” premiss12 (DR13) ∴ in T2 the presenter of the show said: “Stop drinking!” because in T1 I was drinking an appetizer conclusion11 (DR14) ∴ in T2 the presenter of the show spoke about me conclusion12

Consider also this second instance :

 (DR21) in T3 I hardly got out of bed premiss21 (DR22) in T4 the radio presenter said: “Be forceful:” premiss22 (DR23) ∴ in T4 the radio presenter said: “Be forceful:” because in T3 I hardly got out of bed conclusion21 (DR24) ∴ in T4 the radio presenter spoke about me conclusion22

At the level of the instantial step (DR11)-(DR13), (DR21)-(DR23), … the patient concludes here that an internal event is the cause of an external event. In a further interpretative stage, he/she interprets the conclusions (DR13), (DR23), … of the preceding arguments by considering that the presenters of radio or of television speak about him/her. Finally, in a generalisation step, of inductive nature, the patient enumerates the conclusions (DR14), (DR24), … of secondary arguments (interpretation stage) and generalises thus:

 (DR14) ∴ in T2 , the presenter of the show spoke about me (DR24) ∴ in T4, the radio presenter spoke about me (…) (DR105) ∴ the media speak about me conclusion

It proves then that the structure of the mechanism which leads to the formation of delusions of reference thus described, is identical to that of the reasoning which leads to ideas of reference which is associated with personalisation as cognitive distortion.

Finally, it appears that the preceding developments allow to provide a definition of personalisation and to situate it in the context of cognitive distortions (Franceschi 2007, 2009). Personalisation is then likely to manifest itself at the level of primary, secondary or ternary pathogenic arguments, which correspond respectively to the stages of instantiation, of interpretation, and of generalisation. At the level of primary pathogenic arguments, corresponding to a function of instantiation, it consists of instances, the conclusions of which lead the patient to conclude in an unjustified way that some external events are caused by some of his/her actions. At the level of secondary pathogenic arguments, which correspond to a function of interpretation, personalisation takes the form of a reasoning by which the patient interprets the conclusion of primary pathogenic argument by concluding for example that people make fun of him/her. Finally, at the level of ternary pathogenic arguments, associated with a function of generalisation, the patient generalises from the conclusions of several secondary pathogenic arguments and concludes that, in a general way, people make fun of him/her.

Lastly, it appears that the previous definition of personalisation as cognitive distortion allows to describe precisely the relationships between personalisation and close notions such as personalising bias, ideas of reference and delusions of reference.

### References

Beck A. Thinking and depression: Theory and therapy. Archives of General Psychiatry 1964; 10:561-571.

Bressan, P. The Connection Between Random Sequences, Everyday Coincidences, and Belief in the Paranormal. Applied Cognitive Psychology, 2002, 16, 17-34.

Dziegielewski, S. F. DSM-IV-TR in action, Wiley, New York, 2002.

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

Franceschi P. Compléments pour une théorie des distorsions cognitives. Journal de Thérapie Comportementale et Cognitive, 2007, 17-2, 84-88. English translation: http://cogprints.org/5261/

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

Franceschi P. Théorie des distorsions cognitives : la sur-généralisation et l’étiquetage. Journal de Thérapie Comportementale et Cognitive, 2009, 19-4. English translation.

Garety, P., Freeman, D., 1999. Cognitive approaches to delusions: a critical review of theories and evidence. British Journal of Clinical Psychology 38, 113-154.

Langdon R., Cornera T., McLarena J., Wardb P. & Coltheart M., 2006, Externalizing and personalizing biases in persecutory delusions: The relationship with poor insight and theory-of-mind, Behaviour Research and Therapy, 44:699-713

Kinderman, P., Bentall, R., 1997. Causal attributions in paranoia and depression: internal, personal, and situational attributions for negative events. Journal of Abnormal Psychology 106 (2), 341- 345.

Martin, B. Coincidences: Remarkable or random. Skeptical Inquirer, 1998, 22-5, 23-27.

McKay, R., Langdon, R. & Coltheart, 2005. M. Paranoia, persecutory delusions and attributional biases, Psychiatry Research, 136, 233–245

Meyer, E., Lenzenweger, M., 2009. The specificity of referential thinking: A comparison of schizotypy and social anxiety, Psychiatry Research, 165, 78-87.

Startup, M., Startup, S., 2005. On two kinds of delusion of reference, Psychiatry Research ,137, 87-92.

English translation and postprint (with additional illustrations) of a paper published in French under the title “Théorie des distorsions cognitives : application à l’anxiété généralisée” in the Journal de Thérapie Comportementale et Cognitive, 2008, 18, pp. 127-131.
This article follows the contribution to the general theory of cognitive distortions exposed in “Complements to a theory of cognitive distorsions” (Journal de Thérapie Comportementale et Cognitive, 2007). The elements described, namely the reference class, the duality and the system of taxa, are applied here to generalised anxiety disorder. On the one hand, these elements allow to describe the cognitive distortions which are specific to generalised anxiety disorder, consistent with recent work emphasising the role played uncertain situations relative to future events. On the second hand, they allow to define a type of structured reasoning, of inductive nature, which leads to the formation and maintenance of anxious ideas.

## Theory of Cognitive Distortions: Application to Generalised Anxiety Disorder

In Franceschi (2007), we set out to introduce several elements intended to contribute to a general theory of cognitive distortions. These elements are based on three basic notions: the reference class, the duality and the system of taxa. With the help of these three elements, we could define within the same conceptual framework the general cognitive distortions such as dichotomous reasoning, the disqualification of one pole, minimisation and maximisation, as well as the requalification in the other pole and the omission of the neutral. In addition, we could describe as specific cognitive distortions: the disqualification of the positive, selective abstraction and catastrophism.

In what follows, we offer to extend this work by applying it in a specific way to generalised anxiety disorder (GAD), in order to allow their use within cognitive therapy. The present study inserts itself in the context of recent work (Butler & Mathews 1983, 1987, Dalgleish et al. 1997), which notably underlined the major role played, in the context of GAD, by indeterminate situations, and especially by uncertain situations relating to future events. Recent developments, emphasising especially the intolerance with regard to indeterminate future situations, echoed this (Dugas et al. 2004, Canterbury et al. 2004, Carleton et al. 2007).

We shall be interested successively in two main forms of reasoning likely to occur in the context of GAD: on the one hand, the cognitive distortions which are specific to GAD; on the other hand, a structured argument relating to GAD and grounded on inductive logic, which is likely to include one or several of the aforementioned cognitive distortions.

Cognitive distortions in the context of generalized anxiety disorder

The optimal system of taxa

The conceptual framework defined in Franceschi (2007) is based on three fundamental elements: the duality, the reference class, and the system of taxa, which allow to define the general cognitive distortions. These three notions also allow to describe the specific cognitive distortions which are applicable to GAD. In this context, as we will see it, the reference class for the latter specific cognitive distortions identifies itself with the class of future events of the patient’s life. Moreover, the duality assimilates itself to the Positive/Negative duality. Finally, for the sake of the present discussion, we shall make use of the system of taxa (its choice is more or less arbitrary) described in Franceschi (2007), which includes 11 taxa, denoted by E1 to E11, where E6 denotes the neutral taxon. Such conceptual framework allows then to define the specific cognitive distortions in the context of GAD. We offer to examine them in turn.

Dichotomous reasoning

Dichotomous reasoning

An instance of dichotomous reasoning related to GAD consists for the patient to only consider future events from the viewpoint of the extreme taxa corresponding to each pole of the Positive/Negative duality. Hence, the patient only considers future events which present either a very positive, or a very negative nature. All other events, being either neutral, positive or negative to a lesser degree, are thus ignored. This type of reasoning can be analysed as an instance of dichotomous reasoning, applied to the class of the events of the patient’s future life and to the Positive/Negative duality.

Disqualification of one pole

The disqualification of one pole

An instance of the disqualification of one pole related to GAD consists for the patient to only envisage, among future events likely to occur, those which present a negative nature. The patient tends then to be unaware of positive future events that could happen, by considering that they do not count, for this or that reason. In the present context, this type of reasoning can be analysed as an instance of disqualification of one pole, applied to the reference class of the events of the patient’s future life and to the Positive/Negative duality, i.e. disqualification of the positive.

Arbitrary focus on a given modality

Arbitrary focus

In GAD, a typical instance of arbitrary focus, consists for the patient to focus on a possible future event, the nature of which turns out to be negative. This can be analysed as focusing on one of the taxa of the Positive/Negative duality, at the level of the class of the future events of the patient’s life.

Omission of the neutral

Omission of the neutral

A specific instance for GAD consists for the patient to be completely unaware of possible future events the nature of which is neutral, i.e. those which are neither positive nor negative.

Requalification into the other pole

Requalifcation into the other pole

In the context of GAD, the corresponding cognitive distortion consists in requalifying as negative a possible future event, whereas it should be considered objectively as positive. Such cognitive distortion consists of a requalification in the other pole applied to the reference class of the future events of the patient’s life and to the Positive/Negative duality, i.e. requalification in the negative.

Minimisation or maximisation

Maximisation and minimisation

A specific instance of minimisation applied to GAD consists for the patient to consider some possible future events as less positive than they truly are in reality. With maximisation, the patient considers some possible future events as more negative than they objectively are.

Primary, secondary and tertiary anxiogenous arguments

At this stage, it is worth also considering a certain type of reasoning, likely to be met in GAD, which can include several instances of the aforementioned cognitive distortions. This type of reasoning presents an anxiogenous nature, because it leads the patient to predict that a future event of negative nature is going to occur. Such reasoning is underlain by a structure which presents an inductive nature. Before analysing in detail the different steps of the corresponding reasoning, it is worth describing preliminarily its internal structure. The latter is the following (in what follows, the symbol denotes the conclusion):

 (1) the event E1 of negative nature did occur to me premiss (2) the event E2 of negative nature did occur to me premiss (3) the event E3 of negative nature did occur to me premiss (…) premiss (10) the event E10 of negative nature did occur to me premiss (11) all events that occur to me are of negative nature from (1)-(10) (12) « I am always unlucky », « I am ill-fated » from (11) (13) the future event E11 of negative nature may occur premiss (14)  the future event E11 of negative nature will occur from (11), (13)

The essence of such reasoning is of a logically inductive nature. The patient enumerates then some events of his/her past or present life, the nature of which he/she considers as negative. He/she reaches then by generalisation the conclusion according to which all events that which occur to him/her are negative. From this generalisation, he/she infers a prediction relating to a future event, likely to happen, which he/she considers as negative. The patient is thus led to the anxiogenous conclusion that an event of negative nature is going to occur.

In such reasoning, it is worth pointing out that the reference class identifies itself with the class of past, present and future events, of the patient’s life. Typically, in this type of reasoning, the generalisation is grounded on present or past events, while a future event is the object of the corresponding inductive prediction. This is different from the reference class applicable to the cognitive distortions mentioned above, where the reference class identifies itself exclusively with the future events of the patient’s life.

At this stage, it proves to be necessary to identify the fallacious steps in the patient’s reasoning, to allow their use in cognitive therapy of GAD. To this end, we can differentiate several steps in the structure of the corresponding reasoning. It proves indeed that some steps are valid arguments (an argument is valid when its conclusion is true if its premises are true), while others are invalid. For this purpose, it is worth drawing within this type of reasoning, a distinction between primary, secondary or ternary anxiogenous arguments.

Primary anxiogenous arguments

The first step in the type of aforementioned reasoning, consists for the patient to think to a past negative event, in the following way:

 (1) the event E1 of negative nature did occur to me

It is however possible to describe more accurately the corresponding cognitive process, under the form of an argument that we can term a primary anxiogenous argument, the structure of which is the following:

 (1a) the event E1 did occur to me (1b) the event E1 was of negative nature (1)  the event E1 of negative nature did occur to me from (1a), (1b)

By such cognitive process, the patient is led to the conclusion that some negative event did occur to him/her. This type of argument proves to be entirely valid inasmuch as the event in question presents well, objectively, a negative nature. However, it can also turn out to be invalid, if the event in question presents, objectively, a positive (or neutral) nature. What is then defective in this type of reasoning, is the fact that premise (1b) turns then out to be false. Such can notably be the case for example if the patient makes use of a cognitive distortion such as requalification in the negative. In such case, the patient considers then as negative an event the nature of which is objectively positive.

Secondary anxiogenous arguments

Anxiogenous secondary arguments are constituted, at the level of the above-mentioned reasoning, of the part that takes into account the instances (1)-(10) and proceeds then by generalisation. The patient counts thus some instances of events that did occur to him/her, the nature of which he/she considers as negative, and concludes that all events that did occur to him/her were negative, in the following way:

 (1) the event E1 of negative nature did occur to me (2) the event E2 of negative nature did occur to me (…) (10) the event E10 of negative nature did occur to me (11) all events that occur to me are of negative nature from (1)-(10)

Such generalisation may constitute a fully valid argument. For the resulting generalisation constitutes a fully correct inductive reasoning, if the premises (1)-(10) are true. However, such type of reasoning is most often defective from two different viewpoints, thus distorting the conclusion which results from it. Above all, as we have did just see it, some past events of positive nature can have been counted among the number of negative events, by the effect of a requalification in the negative. In that case, the enumeration of instances includes then some false premises, thus invalidating the resulting generalisation. Secondly, some past or present positive (or neutral) events can have been omitted in the corresponding enumeration. Such omission can result from the use of some cognitive distortions, such as disqualification of the positive. In such case, the relevant class of reference consisting in present and past events of the patient’s life is only taken into account in a partial or erroneous way. The corresponding reasoning remains then logically valid, but proves to be incorrect, since it takes into account only partly the relevant instances within the reference class, that of the present and past events of the patient’s life.

As we see it finally, the patient proceeds then to a reconstruction of the relevant reference class which proves to be erroneous, due to the use of the following specific cognitive distortions: requalification in the negative and disqualification of the positive (and possibly, omission of the neutral). The corresponding type of reasoning is illustrated on the figure below:

 A series of events of the patient’s life, seen (objectively) from the optimal system of taxa After omission of the neutral After requalification in the negative After disqualification of the positive Conclusion: «All events that occur to me are negative »

Figure 1. Incorrect construction of the reference class for induction, after omission of the neutral, requalification in the negative, and then disqualification of the positive

Such mechanism, as we did see it, illustrates how the formation of anxious ideas is made. However; a mechanism of the same nature is also likely to contribute to their maintenance. For once the generalisation (11) according to which all events which occur to the patient are of negative nature, has been established by means of the above reasoning, its maintenance is made as soon as an event occurs which confirms this latter generalisation. When a new negative event indeed happens, the patient concludes from it that it confirms generalisation (11). Such mechanism, at the stage of the maintenance of anxious ideas, constitutes a confirmation bias. For the patient only counts those events of negative nature related to him/her that confirm indeed the generalisation (11), but without taking into account those events of positive nature which occur to him/her and that would then disconfirm the idea according to which all events which occur to him/her are of negative nature.

Ternary anxiogenous arguments

Lastly, it is worth mentioning the role played by ternary anxiogenous arguments which consist, at the level of the aforementioned reasoning, in the following sequence:

(11) all events that occur to me are of negative nature

(12) « I am always unlucky », « I am ill-fated »

It consists here of an argument which follows the conclusion of the secondary anxiogenous argument (11), and which, by an additional step (12), aims at interpreting it, at making sense of it. The patient interprets here the fact that the events which occur to him/her are negative, due to the fact that he/she is unlucky, ill-fated.

As we did see it, the interest of drawing a distinction between three types of arguments resides in the fact that each of them has a specific function: the primary stage proceeds by enumerating the instances, the secondary stage operates by generalisation, and the ternary stage, lastly, proceeds by interpretation (Franceschi 2008).

The present study, as we see it, extends recent work (Butler and Mathews on 1987, Dalgleish et al. 1997) emphasising the role played, in GAD, by anticipations concerning indeterminate situations related to future events. In this context, the specific cognitive distortions as well as a reasoning of an inductive structure, contribute then to the vicious circle (Sgard et al. 2006), which results from the process of formation and maintenance of the anxious state.

### References

Butler G & Matews A. Cognitive processes in anxiety. Advances in Behaviour Research and Therapy 1983 ; 5 : 51-62.

Butler G & Matews A. Anticipatory anxiety and risk perception. Cognitive Therapy and Research 1987 ; 11 : 551-565.

Carleton R, Norton M & Aslundson G. Fearing the unknown: A short version of the Intolerance of Uncertainty Scale. Journal of Anxiety Disorders 2007 ; 21-1 : 105-117.

Canterbury R, Golden A, Taghavi R, Neshat-Doost H, Moradi A & Yule W. Anxiety and judgements about emotional events in children and adolescents. Personality and Individual Differences 2004 ; 36 : 695-704.

Dalgleish T, Taghavi R, Neshat-Doost H, Moradi A, Yule W & Canterbury R. Information processing in clinically depressed and anxious children and adolescents. Journal of Child Psychology and Psychiatry 1997 ; 38 : 535-541.

Dugas M, Buhr K & Ladouceur R. The role of intolerance of uncertainty in etiology and maintenance. In R. Heimberg, C. Turk, & D. Mennin (Eds.). Generalized anxiety disorder: Advances in research and practice. Guilford, New York, 2004 (143-163).

Franceschi P. Compléments pour une théorie des distorsions cognitives. Journal de Thérapie Comportementale et Cognitive 2007 ; 17-2 : 84-88

Franceschi P. Une défense logique du modèle de Maher pour les délires polythématiques. Philosophiques 2008 ; 35-2 : 451-475

Sgard F, Rusinek S, Hautekeete M & Graziani P. Biais anxieux de perception des risques. Journal de Thérapie Comportementale et Cognitive 2006 ; 16-1 : 12-15.

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 *TL such n-universe (with simplified notation *TL). In that case, the objects are in relation one with the location criterion. On the other hand, in the second type of n-universe, several objects can find themselves in the same taxon of a location criterion at the same time. Thus, for example, the objects 1, 2, 3 are in 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*L0C (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 , *L0T 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 *T0L*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}*T0L{1,2}*O{1,2,…,6} and the event by {{1}*T0L{1}*O{3}, {2}*T0L{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*L0C). 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*L0C{1,2,…,52} and the event by {1}*T0*L0C{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*L0C (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.

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 CTL and ZTL do not require a separate analysis. Indeed, these two projections present the same structure as those of the projections CT and ZT which have just been studied, except for an additional differentiated criterion of location. The conditions under which the paradox dissolves when one compares the projections CT and ZT apply therefore identically to the variation of the paradox which emerges when one relates the projections CTL and ZTL .

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

 (I7) 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}*T0L{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/. —. 2002. Une application des n-univers à l’argument de l’Apocalypse et au paradoxe de Goodman. Doctoral dissertation, Corté: University of Corsica. [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).

Review of John Leslie, Infinite Minds, Oxford, Oxford University Press, 2001, 234 pages.1

Paul Franceschi

Post-publication of the review appeared in Philosophiques, Volume 30, number 2, Autumn 2003

Infinite Minds is the fourth book of John Leslie, which follows Value and Existence (1979), Universes (1989) and The End of the World (1996). Infinite Minds presents a very rich content, and covers a number of particularly varied subjects. Among these latter, one can notably mention: omniscience, the problem of Evil, the fine-tuning argument, observational selection effects, the identity of indiscernables, time, infiniteness, the nature of consciousness.

The book places itself clearly within the field of speculative philosophy. And Leslie is primarily concerned here with considerations not of rigorous demonstration, but rather of plausibility and of coherence. He thus does not hesitate sometimes to attribute a rather weak probability to certain assertions.

Some readers may be rebutted from the beginning by the counter-intuitive assertion that galaxies, planets, animals, but also each of us and our surrounding objects, are mere structures among divine thoughts. One can think that such an assertion has motivated the commentary placed on the book’s cover by a reader from Oxford University Press, according to which it may be difficult to believe that the universe is such that the author describes it. This was also my primary reaction. But if certain readers were to draw from that a hasty conclusion, they would miss then, I think, what constitutes the hidden treasure of the book. Because Infinite Minds resembles a sumptuous temple, whose access however is dissimulated by a gate which looks poorly attractive. Those who will not cross the door, rebutted by the aspect of this latter, will not have the occasion to contemplate the hidden treasures that the book contains. Because the book presents an overall deep structure and coherence, based on the consistency of the author’s pantheist conception of the universe with our current most advanced scientific views with regard to cosmology, physics, as well as with the solutions to several contemporary philosophical problems. To show synthetically how a pantheist vision of the world can cohere with our most recent views with regard to multiple universes, physics and quantum computer science, inasmuch as with relativity theory and recent discussions relating to omniscience, the problem of Evil, the fine-tuning argument, observational selection effects, etc. appears both an immense and deeply original task.

It should be observed here that Leslie is familiar with this type of wide-scale work. It suffices for that to consider his whole work relating to the Doomsday Argument. It is worth evaluating here the immense task which consists in defending point by point the Doomsday Argument against a good hundred different objections. But this vigorous defense of the Doomsday argument has stimulated in return the development of a rich literature, which continuously enlightens a number of fields hitherto ignored.

The variety of pantheism described by Leslie, inspired by Spinoza, characterizes itself by the fact that each of us is nothing but a structure of divine thoughts. Because the divine mind only exists. The galaxies, the planets, the mountains, the human beings that we are, the animals, the flowers are nothing but structures within divine minds. As Leslie points out, this is coherent with the way physicists themselves describe physical objects, by specifying their intrinsic properties. Nevertheless, Leslie is not committed to a conception of panpsychism where all beings and objects which are part of our universe, have mental properties. For according to the author, physical objects such as trees, rocks, sand, exist as structures within the divine mind, but without being equipped themselves with conscience or thought. Here, all things are not equipped with conscience, but are such however that a conscience of these latter things exists.

Moreover, universes in infinite number can exist as structures in the divine mind. The author’s theory appears thus compatible with recent cosmological theories based on the existence of multiple universes. One of these universes is thus our own, which presents such characteristics and an accurate tuning of its parameters (the ratio of the respective masses of the electron and proton, the electron charge, the gravitational constant, Planck’s constant, etc.), that it allows the emergence of an intelligent life.

Furthermore, Leslie suggests the existence not of a single divine mind, but as well of an infinity of divine minds. Each of them is absolutely identical to the others, but has however an autonomous conscience of its own existence.

What is then the status of abstract objects, such as natural integers, in this context? According to certain philosophers, abstract objects also constitute divine thoughts. Such is in particular the viewpoint put forth by Alvin Plantinga, according to which natural integers constitute divine thoughts. But Leslie adopts a different line of thought. Abstract objects such as natural integers have in Infinite Minds’ ontology a completely original status, which is not prima facie obvious, and which deserves a detailed mention. Abstract objects such as natural integers, the idea of an apple, or the idea that “2 + 2 = 4”, are of Platonic essence. And Leslie points out that such abstract objects do not result from our brains, which themselves constitute thoughts in the divine mind. Neither do such objects of Platonic nature result from the divine mind itself. The natural integers, the idea of an apple, or the idea that “2 + 2 = 4”, constitute eternal realities, which are independent of our existence as human beings, of our thoughts and of our language. Leslie explains clearly how the idea that “2 + 2 = 4”, i.e. the fact that “IF two sets of two apples exist, THEN four apples exist” (p. 160) constitutes a Platonic reality, independent of the thoughts of the divine mind and of the human beings that we are.

Leslie also develops the topic of omniscience. According to Leslie, God simply knows all that is worth knowning: (“God knows everything worth knowing”). This seems probably more plausible than the idea according to which God has any knowledge, which notably conflicts with the logically impossible existence, already noted by Patrick Grim, of the set which contains absolutely all truths. Our pretheoretical conception of an omniscient God could well appear naïve, as the author points out, because a many unimportant facts could appear undesirable knowledge there.

Lastly, Leslie develops the point of view according to which God exists by ethical need (“because of its ethical requiredness”). The existence of God and of the cosmos in his entirety is ethically necessary, from all eternity. This argument could well appear more convincing than certain ontological arguments. Because such an ethical need has, according to Leslie, an inherently creative power. But such creative capacity, of Platonic essence, does not proceed of any external cause. It is simply inherent by nature to the ethical necessity.

Leslie’s book also constitutes the courageous expression of a viewpoint. For such pantheist conception does not constitute a widespread opinion within contemporary analytical philosophy. Moreover, Leslie’s variety of pantheism also constitutes a variation of panpsychism. But the attitude of the author appears eminently constructive, because it constrains us to consider more attentively some doctrines than we would tend to reject too easily. One will or not adhere to the pantheist and panpsychist theory exposed in Infinite Minds. But for the majority of readers for whom we can suppose that they will not adhere to the variation of panpsychism thus described, Leslie’s work constitutes nevertheless an admirable and highly original synthesis, showing how an astonishing construction can be elaborated around the pantheist model, while bringing answers to many contemporary philosophical problems. The work will provide new arguments to the defenders of panpsychism. But Infinite Minds will be also prove to be essential to the detractors of panpsychism, who will find there a particularly strong and structured defense.

1This review only differs from the version published in Philosophiques with regard to the status of abstract objects. I thank John Leslie for very useful discussion on this topic.

English translation of a paper published in French in Semiotica, vol. 150(1-4), 2004 under the title “Le problème des relations amour-haine-indifférence”.

This paper is cited in:

• Isis Truck, Nesrin Halouani, & Souhail Jebali (2016) Linguistic negation and 2-tuple fuzzy linguistic representation model : a new proposal, pages 81–86, in Uncertainty Modelling in Knowledge Engineering and Decision Making, The 12th International FLINS Conference on Computational Intelligence in Decision and Control, Eds. Xianyi Zeng, Jie Lu, Etienne E Kerre, Luis Martinez, Ludovic Koehl, 2016, Singapore: World Scientific Publishing.

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

## The Problem of the Relationships of Love-Hate-Indifference

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

1. The problem

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

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

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

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

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

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

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

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

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

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

2. The framework

I will sketch here the formal apparatus described in more detail in Franceschi (2002). To begin with, consider a given duality. Let us denote it by A/Ā. At this step, A and Ā are dual concepts. Moreover, A and Ā can be considered as concepts that are characterized by a contrary component c {-1, 1} within a duality A/Ā, such that c[A] = -1 and c[Ā] = 1. Let us also consider that A and Ā are neutral concepts that can be thus denoted by A0 and Ā0.

Figure 1: The canonical matrix

At this point, we are in a position to define the class of the canonical poles. Consider then an extension of the previous class {A0, Ā0}, such that A0 and Ā0 respectively admit of a positive and a negative correlative concept. Such concepts are intuitively appealing. Let us denote them respectively by {A+, A} and {Ā+, Ā}. At this step, for a given duality A/Ā, we get then the following concepts: {A+, A0, A, Ā+, Ā0, Ā}. Let us call them canonical poles. It should be noted that one could use alternatively the notation (A/Ā, c, p) for a canonical pole.2 In all cases, the components of a canonical pole are a duality A/Ā, a contrary component c {-1, 1} and a canonical polarity p {-1, 0, 1}. This definition of the canonical poles leads to distinguish between the positive (A+, Ā+), neutral (A0, Ā0) and negative (A, Ā) canonical poles. Lastly, the class made up by the 6 canonical poles can be termed the canonical matrix: {A+, A0, A, Ā+, Ā0, Ā}.

Let us investigate now into the nature of the relations existing between the canonical poles of a given matrix. Among the combinations of relations existing between the 6 canonical poles (A+, A0, A, Ā+, Ā0, Ā) of a same duality A/Ā, it is worth emphasizing the following relations: duality, antinomy, complementarity, corollarity, connexity, and anti-connexity. Thus, two canonical poles 1(A/Ā, c1, p1) and 2(A/Ā, c2, p2) of a same matrix are:

(i) dual if their contrary components are opposite and their polarities are neutral3

(ii) contrary (or antinomical) if their contrary components are opposite and their polarities are non-neutral and opposite4

(iii) complementary if their contrary components are opposite and their polarities are non-neutral and equal5

(iv) corollary if their contrary components are equal and their polarities are non-neutral and opposite6

(v) connex if their contrary components are equal and the absolute value of the difference of their polarities equals 17

(vi) anti-connex if their contrary components are opposite and the absolute value of the difference of their polarities equals 18

To sum up: {A0, Ā0} are dual, {A+, Ā} and {A, Ā+} are contraries, {A+, Ā+} and {A, Ā} are complementary, {A+, A} and {Ā+, Ā} are corollary, {A0, A+}, {A0, A}, {Ā0, Ā+} and {Ā0, Ā} are connex, {A0, Ā+}, {A0, Ā}, {Ā0, A+} and {Ā0, A} are anti-connex.

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

Let now A and E be two matrices whose canonical poles are respectively {A+, A0, A, Ā+, Ā0, Ā} and {E+, E0, E, Ē+, Ē0, Ē}. These matrices are such that E+, E0, E are the respective includers for {A+, Ā+}, {A0, Ā0}, {A, Ā} i.e. the two matrices are such that E+ = A+ Ā+, E0 = A0 Ā0 and E = A Ā.9

Figure 2

Let us denote this relation by A E. One is now in a position to extend the relations previously defined between the canonical poles of a same matrix, to the relations of a same nature between two matrices presenting the properties of A and E, i.e. such that A E. The relations of 2-duality, 2-antinomy, 2-complementarity, 2-anti-connexity10 ensue then straightforwardly. Thus, two canonical poles 1(A/Ā, c1, p1) and 2(E/Ē, c2, p2) of two different matrices are:

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

(ii’) 2-contrary11 (or trichotomic contrary) if their polarities are non-neutral and opposite and if the contrary of 2 is an includer for 1

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

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

To sum up now: {A0, Ē0} and {Ā0, Ē0} are 2-dual, {A+, Ē}, {A, Ē+}, {Ā+, Ē} and {Ā, Ē+} are 2-contrary, {A+, Ē+}, {A, Ē}, {Ā+, Ē+} and {Ā, Ē} are 2-complementary, {A0, Ē+}, {A0, Ē}, {Ā0, Ē+} and {Ā0, Ē} are 2-anti-connex.

Lastly, the notion of a complement of a canonical pole also deserves mention. Let be a canonical pole. Let us denote by ~ its complement, semantically corresponding to non. In the present context, the notion of a complement entails the definition of a universe of reference. I shall focus then on the notion of a complement of a canonical pole defined with regard to the corresponding matrix. In this case, the universe of reference is equal to {A+, A0, A, Ā+, Ā0, Ā} and then ~ = {A+, A0, A, Ā+, Ā0, Ā} – . On has thus for example ~A+ = {A0, A, Ā+, Ā0, Ā} and a similar definition for the complements of the other canonical poles of the matrix. Consider now two matrices such that A E. Under these circumstances, the universe of reference12 is equal to {A+, A0, A, Ā+, Ā0, Ā, Ē+, Ē0, Ē}. Call it the 2-matrix of . It ensues that ~ = {A+, A0, A, Ā+, Ā0, Ā, Ē+, Ē0, Ē} – . We have then the notion of a 2-complement of a canonical pole , defined with regard to a universe of reference consisting of the 2-matrix of . More generally, one has the notion of a ncomplement (n > 0) of a canonical pole with regard to the corresponding n-matrix.

3. A solution

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

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

Nevertheless, such difficulties can be avoided in the present context, since our reasoning will not bear upon the concepts inherent to a specific culture or language, but rather on the canonical concepts described above. Accordingly, we shall replace our usual concepts by the corresponding canonical concepts. There is room for variation in degrees, from culture to culture in the usual concepts of love, hate and indifference. But this point does not affect the current line of reasoning, since it only focuses on canonical concepts. The passage from the non-canonical concepts to the canonical ones goes straightforwardly as follows. Let d[] be the pejorative or meliorative degree of a concept . Hence if d[] ]0.5; 1] then p[] = 1 else if d[] [-1; -0.5[ then p[] = -1. At this point, one can pose legitimately that p[Love] = 1, p[Hate] = -1 and p[Indifference] = -113. As a result, the three concepts can be denoted by Love+, Hate, Indifference.

Figure 3

As noted from the beginning, the relationship of love/hate is unproblematic and identifies itself with the relation of contrary. This applies straightforwardly to the relationship of the canonical concepts Love+/Hate. Hence, the corresponding matrix has the following structure: {Love+, A0, A, Ā+, Ā0, Hate}. Now the next step is the reconstitution of the complete matrix. This task can be accomplished with the help of the definition of the relations of the canonical poles, namely: A is corollary to Love+, Ā+ is corollary to Hate, A0 is connex to Love+ and anti-connex to Hate, Ā0 is connex to Hate and anti-connex to Love+. Given these elements, we are now in a position to reconstitute the corresponding canonical matrix: {Love+, Attraction0, A, Defiance+, Repulsion0, Hate}.14

Let us examine now the case of the concept Indifference. Such a concept inserts itself into a matrix the structure of which is: {E+, E0, E, Ē+, Ē0, Indifference}. Just as before, it is now necessary to reconstitute the complete matrix. This can be done with the help of the corresponding definitions: Ē+ is corollary to Indifference, E is complementary to Indifference, E+ is contrary to Indifference, Ē0 is connex to Indifference and to the corollary of Indifference, E0 is anti-connex to Indifference and to the corollary of Indifference. The associated matrix is then: {E+, Interest0, E, Phlegm+, Detachment0, Indifference}.15

Figure 4

It should be observed now that Interest0 = Attraction0 Repulsion0 i.e. that Interest0 is an includer for Attraction0 and Repulsion0. At this step, given that {Love+, Attraction0, A, Repulsion+, Repulsion0, Hate} {E+, Interest0, E, Phlegm+, Detachment0, Indifference}, the relationship of Love+/Indifference and Hate/Indifference now apply straightforwardly. In effect, it ensues from the above definitions that, on the one hand, Love+ and Indifferenceare trichotomic contraries and on the other hand, Hate and Indifferenceare trichotomic complementaries. At this point, one is finally in a position to formulate a solution to the LHI problem:

(i) love is contrary to hate

(ii) love is 2-contrary to indifference

(iii) hate is 2-complementary to indifference

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

4. Concluding remarks

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

References

 Benzaken, Claude (1991). “Systèmes formels”. Paris, Masson Franceschi, Paul (2002). “Une Classe de Concepts”. Semiotica, 139, pp. 211-26, English translation Garlikov, Rick (1998). “Understanding, Shallow Thinking, and School”. At http://www.garlikov.com/writings.htm

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

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

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

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

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

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

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

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

9 It should be observed that one of the three conditions is sufficient. In effect, E+ = A+ Ā+ entails E0 = A0 Ā0 and E = A Ā; E0 = A0 Ā0 implies E+ = A+ Ā+ and E = A Ā; E = A Ā entails E0 = A0 Ā0 and E+ = A+ Ā+.

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

11 Or 2-antinomical.

12 In this context, E+, E0 and E can be omitted without loss of content, given their nature of includers.

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

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

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

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

//