The matrix of concepts is a structure that includes six concepts, which is suitable for modeling many common concepts, such as: courage, recklessness, irresolution, eclecticism, superficiality, clemency, instability, selfishness, objectivity, frankness, brusqueness, altruism, etc. Of the six concepts in the matrix:
two are neutral: A0 and Ā0
two are positive: A+ and Ā+
two are negative: A– and Ā–
These six concepts constitute the canonical poles of the matrix.
The six concepts of the matrix are in particular relationships with each other. Thus:
the neutral concepts A0 and Ā0 are dual
the positive concept A+ and the negative concept Ā– are opposite (or contrary); similarly, the negative concept A– and the positive concept Ā+ are opposite (or contrary)
the propensity to take risks and the propensity to avoid risks are dual
audacity and cowardice are contrary, opposite; in the same way, temerity and prudence are contrary, opposite
audacity and prudence are complementary
temerity and cowardice are extreme opposites
Moreover, the three concepts located on the left of the matrix constitute a half-matrix: it is the half-matrix associated with the pole A. In the same way, the three concepts located on the right of the matrix constitute the half-matrix associated to the pole Ā.
Dichotomous analysis is a technique of analysis that tends to consider an object o from the point of view of a given duality, and to distinguish two different situations, each of which corresponds to one of the two poles of the duality considered. Examples of dualities are: Quantitative/Qualitative, Absolute/Relative, Static/Dynamic, Diachronic/Synchronic, Extension/Restriction.
The dichotomous analysis is directly linked to the very statement of the principle of dialectical indifference. Its use responds to the concern not to be vulnerable to one-sidedness bias and to demonstrate, on the contrary, a two-sided viewpoint.
Franceschi, P., “Éléments d’un contextualisme dialectique” (in english), in Liber Amicorum Pascal Engel, J. Dutant, G. Fassio & A. Meylan (éd.), Université de Genève, 2014, p. 581-608.
It is worth mentioning a form of dialectical monism in the ancient Aztec philosophy and especially in the concept of ” Teotl “, which is at the center of Aztec metaphysics and cosmology. Teotl is the expression of an endless alternation of continuous and cyclical oscillation between opposite poles. Teotl is thus characterized by a dual prominent structure, which results from the union of opposites , themselves characterized by complementarity. The dual pairs involved include : the masculine and the feminine, dark and light, order and disorder, hot and cold, life and death, being and non-being etc. The interdependence and higher union of the principles of life and death in Teotl, for example, was represented by Aztec artists of Tlatilco and Oaxaca in masks where one half is alive while the other half died, revealing the skull bones.
James Maffie, Aztec Philosophy, Internet Encyclopedia of Philosophy
When only the pole A (respectively the pole Ā) of a given duality is considered, it consists of a one-sided viewpoint. Things are seen exclusively from the perspective of one pole. It contrasts with the two-sided viewpoint, with takes into account both poles of a given duality.
The consequence of taking only into account one pole is that the other pole is ignored, or disqualified. In cognitive psychology, the resulting cognitive distortion applied to the positive/negative duality is termed ‘disqualifying the positive’.
The one-sided bias consists in focusing on a given standpoint when considering a given object, and of neglecting the opposite view. In our framework, such fallacious reasoning consists, when considering an object o and the reference class associated with it, of taking into account the viewpoint of the A pole (respectively Ā), while completely ignoring the viewpoint corresponding to its dual pole Ā (respectively A) to define the reference class. We shall term one-sidedness bias such type of fallacy. The conditions of this type of bias, in violation of the principle of dialectical indifference, needs however to be clarified. Indeed, in this context, we can consider that there are some cases where the two-sidedness with respect to a given duality A/Ā is not required. Such is the case when the elements of the context do not presuppose conditions of objectivity and exhaustiveness of views. Thus, a lawyer who would only emphasise the evidence in defence of his/her client, while completely ignoring the evidence against him/her does not commit the above-mentioned type of error of reasoning. In such a circumstance, in fact, the lawyer would not commit a faulty one-sidedness bias, since it is his/her inherent role. The same would go in a trial for the prosecutor, who conversely, would only focus on the evidence against the same person, by completely ignoring the exculpatory elements. In such a situation also the resulting one-sidedeness bias would not be inappropriate, because it follows well from the context that it consists well of the limited role assigned to the prosecutor. By contrast, a judge who would only take into account the evidence against the accused, or who would commit the opposite error, namely of only considering the exculpatory against the latter, would well commit an inappropriate one-sidedness bias because the mere role of the judge implies that he/she takes into account the two types of elements, and that his/her judgement is the result of the synthesis which is made.
Franceschi, P., “Éléments d’un contextualisme dialectique” (in english), in Liber Amicorum Pascal Engel, J. Dutant, G. Fassio & A. Meylan (éd.), Université de Genève, 2014, p. 581-608.
A dual pole is one of the components of a pair of concepts that make up a given duality. Among the dualities, we can mention: quantitative/qualitative, static/dynamic, external/internal, unique/multiple, etc. Thus, quantitative, quantitative, static, dynamic,… are dual poles. It should be noted that their nature is neutral, that is, they do not carry a positive or negative connotation. Thus, concepts that have a positive connotation are: audacity, courage, ardor, merit, combativity, etc. Similarly, concepts that have a negative connotation are: cowardice, pessimism, timidity, irresolution, etc.
Franceschi, P. (2002). Une classe de concepts (in english). Semiotica, vol. 139 (1-4), 211-226
Posprint in English (with additional illustrations) of a paper published in French in Semiotica, vol. 139 (1-4), 2002, 211-226, under the title “Une Classe de Concepts”.
This article describes the construction, of philosophical essence, of the class of the matrices of concepts, whose structure and properties present an interest in several fields. The paper emphasises the applications in the field of paradigmatic analysis of the resulting taxonomy and proposes it as an alternative to the semiotic square put forth by Greimas.
Paul Franceschi, Éléments d’un contextualisme dialectique, dans Liber Amicorum Pascal Engel, J. Dutant, G. Fassio & A. Meylan (éd.), Université de Genève, 2014, p. 581-608
Isis Truck, Nesrin Halouani, & Souhail Jebali (2016) Linguistic negation and 2-tuple fuzzy linguistic representation model : a new proposal, pages 81–86, in Uncertainty Modelling in Knowledge Engineering and Decision Making, The 12th International FLINS Conference on Computational Intelligence in Decision and Control, Eds. Xianyi Zeng, Jie Lu, Etienne E Kerre, Luis Martinez, Ludovic Koehl, 2016, Singapore: World Scientific Publishing.
On a Class of Concepts
Classically, in the discussion relating to polar 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:
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 componentc {-1, 1}
– a canonical polarityp {-1, 0, 1}
A canonical pole presents the form: (A/Ā, c, p).
Furthermore, it is worth distinguishing, at the level of each duality A/Ā, the following derived classes:
– the positive canonical poles: A+, Ā+
– the neutral canonical poles: 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/Ā-p (corresponding to one of the 3 cases: A+/Ā–, A0/Ā0 or A–/Ā+)
– a degreed Î [-1; 1] in this duality
A degree of duality has thus the form: (A+/Ā–, d), (A0/Ā0, d) or (A–/Ā+, d).
On the other hand, let us call neutral point a concept pertaining to the class of the degrees of duality, whose degree is equal to 0. Let us denote by 0 such a concept, which is thus of the form (Ap/Ā-p, 0) with d[0] = 0. Semantically, a neutral point 0 corresponds to a concept which responds to the following definition: neither Apnor 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 Apnor Ā-p. It will appear on the contrary that several concepts and even hierarchies of concepts can correspond to this latter definition.
The following property of the neutral points then ensue, for a given duality A/Ā: (A+/Ā–, 0) = (A0/Ā0, 0) = (A–/Ā+, 0).
At this point, it is worth also taking into account the following derived classes:
– a discrete and truncated class, built from the degrees of duality, including only those concepts whose degree of duality is such that d {-1, -0.5, 0, 0.5, 1}.
– the class of the degrees of complementarity, the degrees of corollarity, etc. The class of the degrees of duality corresponds to the relation of antinomy. But it is worth considering, in a general way, as many classes as there exists relations between the canonical poles of a same duality. This leads to as many classes of comparable nature for the other relations, corresponding respectively to degrees of complementarity, corollarity, connexity and anti-connexity.
It is worth noting finally the following questions, with regard to degrees of duality and neutral points. Does there exist (i) one (or several) canonical pole which is its own neutral point? A priori, it is only possible for a neutral pole.
Does any duality A/Ā admit (ii) a neutral point or trichotomic zero? One can call this question the problem of the general trichotomy. Is it a general 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 positiveincluder + is a concept such that it is itself a positive canonical pole and corresponds to the definition + = A+ Ā+. A neutral includer 0is 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, Polarized0is 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.
Let us also consider the following derived classes:
– matricial includers: they consist of concepts including the set of the canonical poles of a same duality. They respond to the definition: 0 = A+ 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 A0/Ā0. Thus, the neutral point for the duality A0/Ā0 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:
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: Abstract0–ity, Absolute0–ity, Acessory0–ity, 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:
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 a-ism a meta-principle. Example: Unite Unite-ism. We have then Internalism, Externalism, Relativism, Absolutism, etc. which correspond in particular to dispositions of the mind. A capital letter will preferably be used here to distinguish the meta-principles from the lexicalized concepts, and in particular to differentiate them from the corresponding philosophical doctrines, which often have very different meanings. It will be however possible to make use of the classical terms when they exist to designate the corresponding meta-principle. Thus All-ism corresponds to Holism.
One can term Ultra-a-ism or Hyper-a-ism the concept corresponding to –. This latter form corresponds to an exclusive, excessive, exaggerated use of the viewpoint corresponding to a given principle. One has thus for example: Externalism– = Hyper-externalism.
The constituents of the meta-principles are:
– a polarityp Î {-1, 0, 1}
– a neutral canonical principle composed of:
– a duality (or base) A/Ā
– a contrary componentc {-1, 1}
– a neutral polarityq = 0
The positive, neutral, negativecanonical meta-principles are respectively of the form ((A/Ā, c, 0), 1), ((A/Ā, c, 0), 0), ((A/Ā, c, 0), -1).
Between the canonical meta-principles of a same duality, one has the same relations as for the canonical poles.
One has lastly the derived classes consisting in:
– the positive meta-principles (p > 0)
– the neutral meta-principles (p = 0)
– the negative meta-principles (p < 0)
– the polarized meta-principles which include the positive and negative meta-principles
– the matrix of the canonical meta-principles, consisting of 6 meta-principles applicable to a given duality{A+, 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
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.
13 We have then the following properties, with regard to the above-mentioned relations. The relation of identity constitutes a relation of equivalence. Antinomy, complementarity and corollarity are symmetrical, anti-reflexive, non-associative, involutive.
The operation of composition on the relations {identity, corollarity, antinomy, complementarity} defines an abelian group of order 4. With G = {I, c, , j}:
°IcjIIcjccIjjIcjjcI
where for all A Î G, A-1 = A, and A ° I = A, I being the neutral element. It should be noted that the group properties make it possible in particular to give straightforwardly a valuation to any propositions of the form: the contrary concept of the complementary of a1is 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
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
In On a class of concepts (2002), I described a theory based on the matrices of concepts which aims at constituting an alternative to the classification proposed by Greimas, in the field of paradigmatic analysis. The problem of the determination of the relationships of love/hate/indifference arises in this construction. I state then the problem of the relationships of love/hate/indifference in a detailed way, and several solutions that have been proposed in the literature to solve it. I describe lastly a solution to this problem, based on an extension of the theory of matrices of concepts.
This paper is cited in:
Isis Truck, Nesrin Halouani, & Souhail Jebali (2016) Linguistic negation and 2-tuple fuzzy linguistic representation model : a new proposal, pages 81–86, in Uncertainty Modelling in Knowledge Engineering and Decision Making, The 12th International FLINS Conference on Computational Intelligence in Decision and Control, Eds. Xianyi Zeng, Jie Lu, Etienne E Kerre, Luis Martinez, Ludovic Koehl, 2016, Singapore: World Scientific Publishing.
The Problem of the Relationships of Love-Hate-Indifference
Paul Franceschi
I shall be concerned in this paper with presenting a problem related to the proper definition of the relationships of the following concepts: love, hate and indifference. I will describe first the problem in detail and some proposed solutions. Lastly, I will present my own solution to the problem.
1. The problem
The problem is that of the proper definition of the relationships of the concepts love, hate and indifference. Let us call it the LHI problem. What are then the accurate relationships existing between these three concepts? At first sight, the definition of the relation between love and hate is obvious. These concepts are contraries. The definition of such a relation should be consensual. Nevertheless, the problem arises when one considers the relationship of love and indifference, and of hate and indifference. In these latter cases, no obvious response emerges.
However, the issue needs clarifying. In this context, what should we expect of a solution to the LHI problem? In fact, a rigorous solution ought to define precisely the three relations R, S, T such that love R hate, love S indifference and hate T indifference. And the definitions of these relations should be as accurate as possible.
It is worth mentioning that several authors must be credited for having mentioned and investigated the LHI problem. In particular, it is worth stressing that the difficulties presented within propositional calculus by some assertions of the type x loves y, x hates y, or x is indifferent to y have been hinted at by Emile Benzaken (1990)1:
Nevertheless, the difficulty can arise from pairs of words where the one expresses the contrary (negation) of the other; ‘to hate’ can be considered as the strong negation of ‘to love’, whereas ‘to be indifferent’ would be its weak negation.
The author exposes then the problem of the relationships of love/hate/indifference and proposes his own solution: hate is the strong negation of love, and indifferent is the weak negation of love.
However, it turns out that Benzaken’s solution is unsatisfying for a logician, for the following reasons. On the one hand, this way of solving the problem defines the relations between love and hate (strong negation, according to the author) and between love and indifference (weak negation, on the author’s view), but it fails to define accurately the relations existing between indifference and hate. There is a gap, a lack of response at this step. And mentioned above, a satisfying solution should elucidate the nature of the relationships of the three concepts. On the other hand, the difference between weak negation and strong negation is not made fully explicit within the solution provided by Benzaken. For these reasons, Benzaken’s solution to the LHI problem proves to be unsatisfying.
In a very different context, Rick Garlikov (1998) stresses some difficulties of essentially the same nature as those underlined by Benzaken:
In a seminar I attended one time, one of the men came in all excited because he had just come across a quotation he thought very insightful – that it was not hate that was the opposite of love, but that indifference was the opposite of love, because hate was at least still an emotion. I chuckled, and when he asked why I was laughing, I pointed out to him that both hate and indifference were opposites of love, just in different ways, that whether someone hated you or was indifferent toward you, in neither case did they love you.
Garlikov describes in effect the problem of the relationships of love/hate/indifference and implicitly proposes a solution of a similar nature as that provided by Benzaken. For this reason, Galikov’s account suffers from the same defects as those presented by Benzaken’s solution.
In what follows, my concern will be with settling first the relevant machinery, in order to prepare a few steps toward a solution to the LHI problem.
2. The framework
I will sketch here the formal apparatus described in more detail in Franceschi (2002). To begin with, consider a given duality. Let us denote it by A/Ā. At this step, A and Ā are dual concepts. Moreover, A and Ā can be considered as concepts that are characterized by a contrarycomponent c∈ {-1, 1} within a duality A/Ā, such that c[A] = -1 and c[Ā] = 1. Let us also consider that A and Ā are neutral concepts that can be thus denoted by A0 and Ā0.
Figure 1
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 α(A/Ā, c, p) for a canonical pole.2 In all cases, the components of a canonical pole are a duality A/Ā, a contrarycomponent c∈ {-1, 1} and a canonicalpolarity p∈ {-1, 0, 1}. This definition of the canonical poles leads to distinguish between the positive (A+, Ā+), neutral (A0, Ā0) and negative (A–, Ā–) canonical poles. Lastly, the class made up by the 6 canonical poles can be termed the canonicalmatrix: {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 α1(A/Ā, c1, p1) and α2(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 α be an includer for two other concepts β and γ if and only if α = β∨γ. Such a definition captures the intuition that α is the minimal concept whose semantic content includes that of β and γ. To give an example concerning truth-value, determinate is an includer for {true, false}.
Let now A and E be two matrices whose canonical poles are respectively {A+, 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 α1(A/Ā, c1, p1) and α2(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 α2 is an includer for α1
(iii’) 2-complementary (or trichotomic complementary) if their polarities are non-neutral and equal and if the complementary of α2 is an includer for α1
(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 α2 is an includer for α1
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 α be a canonical pole. Let us denote by ~α its complement, semantically corresponding to non–α. In the present context, the notion of a complement entails the definition of a universe of reference. I shall focus then on the notion of a complement of a canonical pole defined with regard to the corresponding matrix. In this case, the universe of reference is equal to {A+, A0, A–, Ā+, Ā0, Ā–} and then ~α = {A+, A0, A–, Ā+, Ā0, Ā–} – α. One 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 α. It ensues that ~α = {A+, A0, A–, Ā+, Ā0, Ā–, Ē+, Ē0, Ē–} – α. We have then the notion of a 2-complement of a canonical pole α, defined with regard to a universe of reference consisting of the 2-matrix of α. More generally, one has the notion of a n–complement (n > 0) of a canonical pole with regard to the corresponding n-matrix.
3. A solution
With the relevant machinery in place, we are now in a position to present a solution to the LHI problem. Let us now analyze the problem in the light of the above framework. To begin with, let us analyze the relevant concepts in more detail. The concept love has a positive connotation. It is a meliorative concept that can be denoted by love+. Conversely, the concept hate has a negative connotation. It is a pejorative concept that can be rendered by hate–. Similarly, the concept indifference also has a negative connotation. It can be considered a pejorative notion that can be denoted by indifference–.
At this step, a difficulty emerges. In effect, it should be stressed that the three concepts are either meliorative or pejorative at a certain degree. And such a degree might be different from one concept to another. For example hate– might be pejorative at a 0.95 degree, while indifference– might be pejorative at a lesser degree of 0.7. Moreover, it could be said that such a degree might vary from culture to culture, from a given language to another. In sum, the meliorative or pejorative degree of the three concepts, so the objection goes, could be culture-relative.
Nevertheless, such difficulties can be avoided in the present context, since our reasoning will not bear upon the concepts inherent to a specific culture or language, but rather on the canonical concepts described above. Accordingly, we shall replace our usual concepts by the corresponding canonical concepts. There is room for variation in degrees, from culture to culture in the usual concepts of love, hate and indifference. But this point does not affect the current line of reasoning, since it only focuses on canonical concepts. The passage from the non-canonical concepts to the canonical ones goes straightforwardly as follows. Let d[α] be the pejorative or meliorative degree of a concept α. Hence if d[α] ∈ ]0.5; 1] then p[α] = 1 else if d[α]∈ [-1; -0.5[ then p[α] = -1. At this point, one can pose legitimately that p[Love] = 1, p[Hate] = -1 and p[Indifference] = -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 Indifference– are trichotomiccontraries and on the other hand, Hate– and Indifference– are trichotomic complementaries. At this point, one is finally in a position to formulate a solution to the LHI problem:
(i) love is contrary to hate
(ii) love is 2-contrary to indifference
(iii) hate is 2-complementary to indifference
Hence, R, S, T identify respectively themselves with contrary, trichotomic contrary, trichotomic complementarity.
4. Concluding remarks
At this point, it is tempting not to consider the above analysis as a solution to the LHI problem per se. In effect, the concepts love, hate and indifference seem to be instances of a wider class of concepts whose relationships are of the same nature. This suggests that the same type of solution should be provided to the general problem of the definition of the relations of three given concepts , , . At first sight, certain concepts such as true, false and indeterminate, fall under the scope of the current analysis. Nevertheless, such a claim should be envisaged with caution. To what extent does the present analysis apply to other concepts? This is another problem that needs to be addressed, but whose resolution goes beyond the scope of the present account.16
References
Benzaken, Claude (1991). “Systèmes formels”. Paris, Masson
Franceschi, Paul (2002). “Une Classe de Concepts”. Semiotica, 139, pp. 211-26, English translation
Garlikov, Rick (1998). “Understanding, Shallow Thinking, and School”. At http://www.garlikov.com/writings.htm
1 My translation. The original text is as follows: ‘La difficulté cependant peut provenir de paires de mots dont l’un exprime le contraire (négation) de l’autre; “haïr” peut être pris comme la négation forte de “aimer” tandis que “être indifférent” en serait la négation faible‘. (p. 63).
2 With the latter notation, the matrix of the canonical poles is rendered as follows: {(A/Ā, -1, 1), (A/Ā, -1, 0), (A/Ā, -1, -1), (A/Ā, 1, 1), (A/Ā, 1, 0), (A/Ā, 1, -1)}.
3 Formally 1 and 2 are dual if and only if c[1] = – c[2] and p[1] = p[2] = 0.
4 Formally 1 and 2 are antinomical if and only if c[1] = – c[2] and p[1] = – p[2] with p[1], p[2] 0.
5 Formally 1 and 2 are complementary if and only if c[1] = – c[2] and p[1] = p[2] with p[1], p[2] 0.
6 Formally 1 and 2 are corollary if and only if c[1] = c[2] and p[1] = – p[2] with p[1], p[2] 0.
7 Formally 1 and 2 are connex if and only if c[1] = c[2] and │p[1] – p[2]│ = 1.
8 Formally 1 and 2 are anti-connex if and only if c[1] = – c[2] and │p[1] – p[2]│ = 1.
9 It should be observed that one of the three conditions is sufficient. In effect, E+ = A+ Ā+ entails 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.
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.
