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

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

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

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

On a Class of Concepts

Classically, in the discussion relating to polar opposites^{1}, one primarily directs his interest to the common and lexicalized concepts, i.e. for which there exists a corresponding word in the vocabulary inherent to a given language. This way of proceeding tends to generate several disadvantages. One of them resides in the fact (i) that such concepts are likely to vary from one language to another, from one culture to another. Another (ii) of the resulting problems is that certain lexicalized concepts reveal a nuance which is either meliorative or pejorative, with degrees in this type of nuances which prove difficult to appreciate. Finally, another problem (iii) lies in the fact that certain concepts, according to semiotic analysis^{2} are regarded as marked with regard to others concepts which are unmarked, the status of unmarked concept conferring a kind of precedence, of pre-eminence to the concepts in question.

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

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

1. Dualities

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

(i) are different one from the other

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

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

(iv) are predicates

Each of the concepts composing a given duality will be termed a pole. I shall present here a list, which does not pretend to be exhaustive, and could if necessary, be supplemented. Consider then the following enumeration of dualities^{3}:

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

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

It should be noted lastly that several questions^{5} immediately arise with regard to dualities. Do dualities exist (i) in a finite or infinite number? In the same way, does there exist (ii) a logical construction which makes it possible to provide an enumeration of the dualities?

2. Canonical poles

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

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

A capital letter for the first letter of a canonical pole will be used, in order to distinguish it from the corresponding lexicalized concept. If one wishes to refer accurately to a canonical pole whereas the usual language lacks such a concept or well appears ambiguous, one can choose a lexicalized concept, to which the exponent corresponding to the chosen neutral or polarized state will be added. To highlight the fact that one refers explicitly to a canonical pole – positive, neutral or negative – the notations A^{+}, A^{0} et A^{–} will be used. We have thus for example the concepts Unite^{+}, Unite^{0}, Unite^{–} etc. Where Unite^{+} = Solid, Undivided, Coherent and Unite^{–} = Monolithic^{–}. In the same way, Rational^{0} designates the neutral concept corresponding to the term rational of the common language, which reveals a slightly meliorative nuance. In the same way, Irrationnal^{0} designates the corresponding neutral state, whereas the common word irrational reveals a pejorative nuance. One will proceed in the same way, when the corresponding lexicalized word proves ambiguous. One distinctive feature of the present construction is that one begins by constructing the concepts logically, and puts them afterwards in adequacy with the concepts of the usual language, insofar as these latter do exist.

The constituents of a canonical pole are:

– a duality (or base) A/Ā

– a contrary componentc {-1, 1}

– a canonical polarityp {-1, 0, 1}

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

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

– the positive canonical poles: A^{+}, Ā^{+}

– the neutral canonical poles: A^{0}, Ā^{0}

– the negative canonical poles: A^{–}, Ā^{–}

– the canonical matrix consisting of the 6 canonical poles: {A^{+}, A^{0}, A^{–}, Ā^{+}, Ā^{0}, Ā^{–}}. The 6 concepts constituting the canonical matrix can also be denoted under the form of a 3 x 2 matrix.

Let also be a canonical pole, one will denote by ~ its complement, semantically corresponding to non-. We have thus the following complements: ~A^{+}, ~A^{0}, ~A^{–}, ~Ā^{+}, ~Ā^{0}, ~Ā^{–}. The notion of a complement entails the definition of a universe of reference U. Our concern will be thus with the complement of a given canonical pole in regard to the corresponding matrix^{7}. It follows then that: ~A^{+} = {A^{0}, A^{–}, Ā^{+}, Ā^{0}, Ā^{–}}. And a definition of comparable nature for the complements of the other concepts of the matrix ensues.

It should be noted lastly that the following questions arise with regard to canonical poles. The construction of the matrix of the canonical poles of the Positive/Negative duality: {Positive^{+}, Positive^{0}, Positive^{–}, Negative^{+}, Negative^{0}, Negative^{–}} ensues. But do such concepts as Positive^{0}, Negative^{0} and especially Positive^{–}, Negative^{+} exist (i) without contradiction?

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

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

3. Relations between the canonical poles

Among the combinations of relations existing between the 6 canonical poles (A^{+}, A^{0}, A^{–}, Ā^{+}, Ā^{0}, Ā^{–}) of a same duality A/Ā, it is worth emphasizing the following relations (in addition to the identity relation, denoted by I).

Two canonical poles _{1}(A/Ā, c_{1}, p_{1}) and _{2}(A/Ā, c_{2}, p_{2}) of a same duality are dual or antinomical or opposites if their contrary components are opposite and their polarities are opposite^{8}.

Two canonical poles _{1}(A/Ā, c_{1}, p_{1}) and _{2}(A/Ā, c_{2}, p_{2}) of a same duality are complementary if their contrary components are opposite and their polarities are equal^{9}.

Two canonical poles _{1} (A/Ā, c_{1}, p_{1}) et _{2}(A/Ā, c_{2}, p_{2}) of a same duality are corollary if their contrary components are equal and their polarities are opposite^{10}.

Two canonical poles _{1} (A/Ā, c_{1}, p_{1}) and _{2}(A/Ā, c_{2}, p_{2}) of a same duality are connex if their contrary components are equal and the absolute value of the difference in their polarities is equal to 1 ^{11}.

Two canonical poles _{1} (A/Ā, c_{1}, p_{1}) and _{2}(A/Ā, c_{2}, p_{2}) of a same duality are anti-connex if their contrary components are opposite and the absolute value of the difference in their polarities is equal to 1.^{12}^{, }^{13}

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

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

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

4. Degrees of duality

One constructs the class of the degrees of duality, from the intuition that there is a continuous succession of concepts from A^{+} to Ā^{–}, from A^{0} to Ā^{0} and from A^{–} to Ā^{+}. The continuous component of a degree of duality corresponds to a degree in the corresponding dual pair. The approach by degree is underlied by the intuition that there is a continuous and regular succession of degrees, from a canonical pole A^{p} to its contrary Ā^{-p}.^{14} One is thus led to distinguish 3 classes of degrees of duality: (i) from A^{+} to Ā^{–} (ii) from A^{0} to Ā^{0} (iii) from A^{–} to Ā^{+}.

A degree of duality presents the following components:

– a dual pair A^{p}/Ā^{-p} (corresponding to one of the 3 cases: A^{+}/Ā^{–}, A^{0}/Ā^{0} or A^{–}/Ā^{+})

– a degreed Î [-1; 1] in this duality

A degree of duality has thus the form: (A^{+}/Ā^{–}, d), (A^{0}/Ā^{0}, d) or (A^{–}/Ā^{+}, d).

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

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

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

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

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

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

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

Does any duality A/Ā admit (ii) a neutral point or trichotomic zero? One can call this question the problem of the general trichotomy. Is it a general rule^{15} or well does there exists some exceptions? It seems a priori that the Abstract/Concrete duality does not admit a neutral point. It appears to be the same for the Finite/Infinite or the Precise/Vague duality. Intuitively, these latter dualities do not admit an intermediate state.

Does the concept corresponding to the neutral point (Neutral^{0}/Polarized^{0}, 0) and responding to the definition: neither neutral nor polarized exist (iii) without contradiction in the present construction?

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

It is worth also considering the relation of includer for the canonical poles. Consider the following pairs of dual canonical poles: A^{+} and Ā^{+}, A^{0} and Ā^{0}, A^{–} and Ā^{–}. We have then the following definitions: a positiveincluder ^{+} is a concept such that it is itself a positive canonical pole and corresponds to the definition ^{+} = A^{+} Ā^{+}. A neutral includer ^{0}is a neutral canonical pole such that ^{0} = A^{0} Ā^{0}. And a negative includer ^{–} is a negative canonical pole such that ^{–} = A^{–} Ā^{–}. Given these definitions, it is clear that one assimilates here the includer to the minimum includer. Examples: Determinate^{0} is an includer for True^{0}/False^{0}. And Determinate^{0} is also a pole for the Determinate^{0}/Indeterminate^{0} duality. In the same way, Polarized^{0}is an includer for Positive^{0}/Negative^{0}.

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

Let us also consider the following derived classes:

– matricial includers: they consist of concepts including the set of the canonical poles of a same duality. They respond to the definition: ^{0} = A^{+} A^{0} A^{–} Ā^{+} Ā^{0} Ā^{–}.

– mixed includers: they consist of concepts responding to the definition _{1} = A^{+} Ā^{–} or well _{2} = A^{–} Ā^{+}

It is worth also considering the types of relations existing between the canonical poles of a different duality. Let A and E be two matrices whose canonical poles are respectively {A^{+}, A^{0}, A^{–}, Ā^{+}, Ā^{0}, Ā^{–}} and {E^{+}, E^{0}, E^{–}, Ē^{+}, Ē^{0}, Ē^{–}} and such that E is an includer for A/Ā i.e. such that E^{+} = A^{+} Ā^{+}, E^{0} = A^{0} Ā^{0} and E^{–} = A^{–} Ā^{–}. One extends then the just-defined relations between the canonical poles of a same matrix, to the relations of comparable nature between two matrices presenting the properties of A and E. We has then the relations of 2-antinomy, 2-complementarity, 2-corollarity, 2-connexity, 2-anti-connexity^{16}. Thus, for example, A^{0} is 2-contrary (or trichotomic contrary) to Ē^{0}, 2-connex (or trichotomic connex) to E^{+} and E^{–} and 2-anti-connex (or trichotomic anti-connex) to Ē^{+} and Ē^{–}. In the same way, A^{+} and Ā^{+} are 2-contrary to Ē^{–}, 2-complementary to Ē^{+}, 2-corollary to E^{–}, 2-connex to E^{0} and 2-anti-connex to Ē^{0}, etc.

Let us consider also the following property of neutral points and includers. Let A and E be two matrices, such that one of the neutral poles of E is an includer for the neutral dual pair of a: E^{0} = A^{0} Ā^{0}. We has then the following property: the canonical pole Ē^{0} for the matrix E is a neutral point for the duality A^{0}/Ā^{0}. Thus, the neutral point for the duality A^{0}/Ā^{0} is the dual of the includer E^{0} of A^{0} and Ā^{0}. Example: Determinate^{0} = True^{0} False^{0}. Here, the neutral point for the True/False duality corresponds to the definition: neither True nor False. And we have then (True^{0}/False^{0}, 0) = (Determinate^{0}/Indeterminate^{0}, -1).

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

One also notes the emergence of a hierarchy, beyond the sole neutral point of a given duality. It consists of the hierarchy of the neutral points of order n, constructed in the following way from the dual canonical poles A_{0} and Ā_{0}:

– A_{0}, Ā_{0}

– A_{1} = neither A_{0} nor Ā_{0}

– A_{21} = neither A_{0} nor A_{1}

– A_{22} = neither Ā_{0} nor A_{1}

– A_{31} = neither A_{0} nor A_{21}

– A_{32} = neither A_{0} nor A_{22}

– A_{33} = neither A_{0} nor A_{21}

– A_{34} = neither Ā_{0} nor A_{22}

– …

One can also consider the emergence of this hierarchy under the following form^{17}:

– A_{0}, Ā_{0}

– A_{1} = neither A_{0} nor Ā_{0}

– A_{2} = neither A_{0} nor Ā_{0} nor A_{1}

– A_{3} = neither A_{0} nor Ā_{0} nor A_{1} nor A_{2}

– A_{4} = neither A_{0} nor Ā_{0} nor A_{1} nor A_{2} nor A_{3}

– A_{5} = neither A_{0} nor Ā_{0} nor A_{1} nor A_{2} nor A_{3} nor A_{4}

– …

Classically, one constructs this infinite hierarchy for True/False by considering I_{1} (Indeterminate), I_{2}, etc. It should be noticed that in this last construction, no mention is made of the includer (Determinate) of True/False. Neither does one make mention of the hierarchy of includers.

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

The following questions finally arise, concerning includers. For certain concepts, does there exist (i) one maximum includer or well does one have an infinite construction for each duality? Concerning the True/False duality in particular, the analysis of the semantic paradoxes has led to the use of a logic based on an infinite number of truth-values^{18}.

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

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

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

6. Canonical principles

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

It should be noticed that a certain number of canonical principles are not lexicalized. The notations A^{+}, A^{0}, A^{–} will be used to denote without ambiguity a canonical principle which is respectively positive, neutral or negative. One could also use the following notation: being a canonical pole, then -ity (or -itude) is a canonical principle. The following notation could then be used: Abstract^{0}–ity, Absolute^{0}–ity, Acessory^{0}–ity, etc. or as above [Abstract^{0}], [Absolute^{0}], etc.

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

It is worth distinguishing finally the following derived classes:

– positive canonical principles

– neutral canonical principles

– negative canonical principles

– polarized canonical principles

with some obvious definitions^{19}.

7. Meta-principles

Let a^{0} be a neutral canonical principle^{20}. The class of the meta-principles corresponds to a disposition of the mind directed towards what is a^{0}, to an interest with regard to what is a^{0}. Intuitively, a meta-principle corresponds to a point of view, a perspective, an orientation of the human mind. Thus, the attraction for Abstraction^{0}, the interest for Acquired^{0}, the propensity to place oneself from the viewpoint of Unity^{0}, etc. constitute meta-principles. It should be noted that this construction makes it possible in particular to construct some concepts which are not lexicalized. This has the advantage of a better exhaustiveness and leads to a better and richer semantics.

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

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

The constituents of the meta-principles are:

– a polarityp Î {-1, 0, 1}

– a neutral canonical principle composed of:

– a duality (or base) A/Ā

– a contrary componentc {-1, 1}

– a neutral polarityq = 0

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

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

One has lastly the derived classes consisting in:

– the positive meta-principles (p > 0)

– the neutral meta-principles (p = 0)

– the negative meta-principles (p < 0)

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

– the matrix of the canonical meta-principles, consisting of 6 meta-principles applicable to a given duality{A^{}^{+}, A^{}^{0}, A^{}^{–}, Ā^{}^{+}, Ā^{}^{0}, Ā^{}^{–}}.

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

– the class of the behavioral principles. Intuitively, the class of the behavioral principles constitutes an extension of that of the meta-principles. While the meta-principle constitutes a disposition of the human mind, the concepts concerned here are those which aim to describe, in a more general way, the tendencies of the human behavior^{21}. Among the lexicalized concepts corresponding to the behavioral principles, one can mention: courage, prudence, pessimism, rationality, avarice, fidelity, tendency to analysis, instability, objectivity, pragmatism, etc. A first analysis reveals (i) that a certain number of them reveal a meliorative nuance: courage, objectivity, pragmatism; that (ii) others, by contrast, present a pejorative, unfavorable connotation: cowardice, avarice, instability; and finally (iii) that certain concepts present themselves under a form which is neither meliorative nor pejorative: tendency to analysis^{22}. One has here the same classes as for the meta-principles, and in particular the degrees of behavioral principles. Example: coward is more negative than apprehensive; in the same way, bravery is more positive than courage.

Conclusion

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

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

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

References

FINE, Kit (1975). Vagueness, Truth and Logic. Synthese 30: 265-300

GREIMAS, A. J. (1977). Elements of a Narrative Grammar, Diacritics 7: 23-40

JAKOBSON, Roman (1983). Dialogues, Cambridge MA: MIT Press

PEACOCKE, C. A. B. (1981). Are Vague Predicates Incoherent?. Synthese 46: 121-141

RESCHER, Nicholas (1969). Many-Valued Logic, New York: McGraw Hill

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

Contrary° Contrary = Identical.

Contrary° Identical = Contrary.

Contrary^{n} = Identical (for n even).

Contrary^{n} = Contrary (for n odd).

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

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

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

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

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

°IcjIIcjccIjjIcjjcI

where for all A Î G, A^{-1} = A, and A ° I = A, I being the neutral element. It should be noted that the group properties make it possible in particular to give straightforwardly a valuation to any propositions of the form: the contrary concept of the complementary of a_{1}is identical to the corollary of the complementary of a_{2}.

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

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

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

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

– A_{2} = A_{21} or A_{22}

– A_{3} = A_{31} or A_{32} or A_{33} or A_{34}

– A_{4} = A_{41} or A_{42} or A_{43} or A_{44} or A_{45} or A_{46} or A_{47} or A_{48}

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

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

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

22 One can consider the following – necessarily partial – enumeration corresponding to the behavioral principles, in the order (A^{+}), (A^{0}), (A^{–}), (Ā^{+}), (Ā^{0}), (Ā^{–}):

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

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

This paper is cited in:

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

The Problem of the Relationships of Love-Hate-Indifference

Paul Franceschi

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

1. The problem

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

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

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

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

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

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

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

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

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

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

2. The framework

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

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

Let us investigate now into the nature of the relations existing between the canonical poles of a given matrix. Among the combinations of relations existing between the 6 canonical poles (A^{+}, A^{0}, A^{–}, Ā^{+}, Ā^{0}, Ā^{–}) of a same duality A/Ā, it is worth emphasizing the following relations: duality, antinomy, complementarity, corollarity, connexity, and anti-connexity. Thus, two canonical poles α_{1}(A/Ā, c_{1}, p_{1}) and α_{2}(A/Ā, c_{2}, p_{2}) of a same matrix are:

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

3. A solution

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

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

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

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

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

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

(i) love is contrary to hate

(ii) love is 2-contrary to indifference

(iii) hate is 2-complementary to indifference

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

4. Concluding remarks

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

References

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Posprint in English (with additional illustrations from wikimedia commons) of a paper published in French in Semiotica, vol. 146(1-4), 2003, 353-367, under the title “Le plan dialectique: pour une alternative au paradigme”. I apply the theory developed in On a Class of Concepts (2002) to the methodology for conceiving a plan. Regarding the dialectical plan, the paradigm is a plan whose structure is thesis-antithesis-synthesis. I describe a new type of matricial dialectical plan, which presents several advantages in regard to the classical dialectical plan and proposes to constitute an alternative to this latter.

The Dialectical Plan: an Alternative to the Paradigm

In Franceschi (2002), I exposed a theory which aims to constitute an alternative to the classification proposed by Greimas in the field of paradigmatic analysis. In the present article, I proceed to draw the consequences of this latter theory by applying it to the technique of conception of a plan. Regarding the dialectic plan, the current paradigm is in effect a plan of the type thesis-antithesis-synthesis. This form of plan is very widespread and its use proves to be consensual. In what follows, I shall propose a novel type of dialectic plan as an alternative to the classical one. It consists of a type of plan which can be qualified as matrix-based, and which presents several advantages with regard to the classical dialectic plan.

The classical dialectic plan

The current paradigm regarding the dialectic plan is a plan of the thesis-antithesis-synthesis type^{1}. This plan finds its origin in the dialectical approach^{2} developed by Hegel. The association of the three concepts thesis-antithesis-synthesis, which is now associated with the dialectical line of reasoning, was elaborated by Hegel and Marx^{3}. The dialectical approach constitutes thus a process of reasoning that proceeds by the statement of two contradictory theses and by their reconciliation at the stage of the synthesis. According to Hegel^{4}, every thesis presents then an inherently incomplete and partial nature, which gives then birth to its contrary, the antithesis. From Hegel’s standpoint, the contraries present, beyond the contradiction underlying them, an indissociable nature. This last property allows thus to make their final union, at a thought level which places itself beyond the one where the contradiction manifests itself. The contraries present thus by essence a genuine unity, from which it is worth grasping the fecund principle, allowing thus to reach, at a higher level, a genuine knowledge. This latter phase constitutes the synthesis, which can thus be considered as the step of reasoning which reconciliates veritably, at a greater level, the contradiction observed between the thesis and the antithesis. The synthesis allows thus to go beyond the conflict raised between the thesis and the antithesis, by further unifying the part of truth simultaneously contained in both of them. However, the process is not limited to that. For the synthesis thus obtained constitutes in turn a novel thesis, which itself yields a novel antithesis and then a novel synthesis, and so on… Within the current language, the dialectical approach designates now the general methodology which allows to go beyond and to solve the contradictions. It is in this dialectical approach that the classical plan of the thesis-antithesis-synthesis type finds its origin.

At this step, it is worth considering in turn each component of the thesis-antithesis-synthesis plan. Consider, to begin with, the thesis. This latter constitutes a standpoint expressed by a given author. It consists of the viewpoint on which the discussion is based, and toward which the structure of the plan is oriented. For simplicity, let us assimilate here the thesis to a given proposition. On the other hand, the antithesis is a standpoint which proves to be contrary to that of the thesis. Like the thesis, it is useful to reduce the antithesis, for the sake of simplicity, to a given proposition. At this step, the viewpoints expressed by the thesis and the antithesis are of an antinomical nature. Lastly, the synthesis constitutes the part of the discourse where the antagonist viewpoints developed in the thesis and the antithesis are overcome. The synthesis aims thus classically to go beyond the antinomy existing between the thesis and the antithesis and to encompass it.

In a general way, the advantage of the dialectic plan of the type thesis-antithesis-synthesis is to allow to apprehend the double aspect of a given problem or reality. By placing oneself alternatively from one side and from the other, by considering successively the thesis and then the antithesis, this type of plan allows to avoid a partial or truncated vision of the particular problem raised by the thesis. The aim of the classical dialectic plan is thus to apprehend the two-faceted nature of a given reality and to go beyond the contradiction which results from a preliminary study.

Matrices of concepts

In Franceschi (2002), I described the structure of a matrix of concepts, the scope of which extends to many concepts. For the sake of the present discussion, it is not necessary to describe in a detailed way the structure of concepts put forth in this article. Nevertheless, the type of dialectic plan which will be proposed later derives directly from the notion of a matrix of concepts. It proves then necessary to present the main lines of the basic structure of a matrix of concepts.

Consider first a given duality. Let us denote it by A/Ā. At this step, A and Ā constitute dual concepts. One can then consider that A and Ā are concepts which characterize themselves by a contrarycomponent c {-1, 1} at the level of a given duality A/Ā, such that c[A] = -1 and c[Ā] = 1. One can also consider that A and Ā are neutral concepts which can thus be denoted by A^{0} and Ā^{0}.

At this step, we are in a position to define the class of the canonical poles. It suffices to consider an extension of the preceding class {A^{0}, Ā^{0}}, such that A^{0} and Ā^{0} respectively admit of both a positive and a negative concept which are correlative. Such concepts possess a certain intuitive support. Let us denote them respectively by {A^{+}, A^{–}} and {Ā^{+}, Ā^{–}}. At this step, for a given duality A/Ā, we get the following concepts: {A^{+}, A^{0}, A^{–}, Ā^{+}, Ā^{0}, Ā^{–}}, which constitute the canonical poles. It is worth mentioning here that the notation α(A/Ā, c, p) could be used alternatively, for a given canonical pole^{5}. In all cases, the components of a canonical pole are: a duality A/Ā, a contrary component c {-1, 1} and a canonicalpolarity p {-1, 0, 1}. This definition of the canonical poles leads to distinguish between the positive (A^{+}, Ā^{+}), neutral (A^{0}, Ā^{0}) and negative (A^{–}, Ā^{–}) canonicalpoles. Lastly, the class made up of the six canonical poles of a same matrix can be dubbed the canonical matrix: {A^{+}, A^{0}, A^{–}, Ā^{+}, Ā^{0}, Ā^{–}}.

Let us focus now on the nature of the relationships existing between the canonical poles of a given matrix. Among the combinations of relationships existing between the six canonical poles (A^{+}, A^{0}, A^{–}, Ā^{+}, Ā^{0}, Ā^{–}) of a same duality A/Ā, one will retain the following relations: duality, antinomy, complementarity, corollarity, connexity, anti-connexity. Thus, two canonical poles α_{1}(A/Ā, c_{1}, p_{1}) and α_{2}(A/Ā, c_{2}, p_{2}) of a same matrix are:

(a) dual if their contrary components are opposite and their polarities are neutral^{6}

(b) contrary (or antinomical) if their components are opposite and their polarities are non-neutral and opposite^{7}

(c) complementary if their contrary components are opposite and their polarities are non-neutral and equal^{8}

(d) corollary if their contrary components are equal and their polarities are non-neutral and opposite^{9}

(e) connex if their contrary components are equal and the absolute value of the difference of their polarities equals 1^{10}

(f) anti-connex if their contrary components are opposite and the absolute value of the difference of their polarities equals 1^{11}

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

To fix ideas, let us take the example of the matrix^{12} {eclecticism^{+}, multi-disciplinarity^{0}, dispersion^{–}, expertise^{+}, monodisciplinarity^{0}, compartmentalization^{–}}. One has then the following relationships:

(a’) {multi-disciplinarity^{0}, monodisciplinarity^{0}} are dual

(b’) {eclecticism^{+}, compartmentalization^{–}}, {dispersion^{–}, expertise^{+}} are antinomical

(c’) {eclecticism^{+}, expertise^{+}}, {dispersion^{–}, compartmentalization^{–}} are complementary

(d’) {eclecticism^{+}, dispersion^{–}}, {expertise^{+}, compartmentalization^{–}} are corollary

At this step, it is worth delving more deeply into the internal structure of the thesis to which the plan dialectical applies. I shall draw a distinction here between simple and complex theses.

Simples theses

In general, a simple thesis presents a structure which is that of an appreciation – negative, neutral or positive – relative to a given concept. Let α be such a concept; one denotes then by ^{p}(α) such structure of thesis, where p denotes a negative polarity, neutral or positive such that respectively p {-1, 0, 1}. The negative appreciation can be assimilated to a blame and the positive appreciation to a praise. The blame of a given concept α is thus denoted by ^{–}(α), the neutralappreciation by ^{0}(α) and the praise by ^{+}(α). In a general way, the propositions corresponding to the simple theses present the following structure: ^{p}(α), with p {-1, 0, 1} and α {A^{+}, A^{0}, A^{–}, Ā^{+}, Ā^{0}, Ā^{–}}. By referring to the matrix notion, one notes that the different theoretical cases are the following, with regard to the six concepts of a given matrix: {^{–}(A^{+}), ^{–}(A^{0}), ^{–}(A^{–}), ^{–}(Ā^{+}), ^{–}(Ā^{0}), ^{–}(Ā^{–}), ^{0}(A^{+}), ^{0}(A^{0}), ^{0}(A^{–}), ^{0}(Ā^{+}), ^{0}(Ā^{0}), ^{0}(Ā^{–}), ^{+}(A^{+}), ^{+}(A^{0}), ^{+}(A^{–}), ^{+}(Ā^{+}), ^{+}(Ā^{0}), ^{+}(Ā^{–})}. At this step, it appears that the neutralappreciation is somewhat rarely found. Thus, for the sake of simplicity, we shall be mainly concerned here with describing more accurately the theses which present the structure of a blame or of a praise.

Let us begin with the blame. A number of theses are thus composed of a depreciative appreciation, related to a behavior, a way of doing or apprehending things, a given situation. Such statements correspond to propositions that present the structure of a blame. Such propositions can be denoted by ^{–}(s) where s designates a way of apprehending or of doing things.

Let us take, to fix ideas, a few examples. Consider the following thesis:

(1) In the contempt of ambition is to be found one of the essential principles of happiness on earth.(Edgar Poe, The Domain of Arnheim)

The author considers here the “contempt of ambition” as a fundamental principle allowing to reach happiness. Such a viewpoint can be analyzed as a negative, depreciative judgment toward ambition. This latter concept can be considered as a neutral notion^{13}. Hence, such a simple thesis presents the structure which is that of the blame of ambition^{0} and can be thus denoted by ^{–}(ambition^{0}).

Consider also this other thesis:

(2) Love, the scourge of the world, atrocious folly. (Alfred of Musset, Premières poésies)

The content of this latter thesis can be analyzed as a very pejorative appreciation formulated with regard to love^{+}. Here also, such thesis presents a structure that can be analyzed as a blame of love^{+}, that one can thus denote by ^{–}(love^{+}).

Conversely, one also frequently encounters some theses which are composed of a flattering appreciation with regard to a given behavior, a propensity to act, a situation or a way of apprehending things. The structure of the corresponding proposition is then that of a praise. One denotes such propositions by ^{+}(s) where s designates a way of considering things or a given behavior.

Consider then a few examples. To begin with, the following viewpoint illustrates this type of structure:

(3) Nothing of great importance came true in the world without passion. (Hegel, Introduction to the Philosophy of History)

The author formulates here a praise related to the passion, considering thus that “nothing of great importance” ever came true without this latter. One can consider here the passion as a neutral notion^{14}. Such a viewpoint presents thus the structure of a praise of passion^{0}, i.e. formally ^{+}(passion^{0}).

One also encounters an identical type of structure, regarding the following affirmation:

(4) Passion is an illness that abhors all medication. (Kant)

which can be analyzed as a blame of passion^{0}, i.e. formally ^{–}( passion^{0}).

Lastly, the following simple thesis:

(5) The worst vice of the fanatic is his sincerity. (Oscar Wilde)

constitutes an example of praise of the negative concept of fanaticism, i.e. formally ^{+}(fanaticism^{–}).

At this step, we are in a position to determine the truth value of the simple theses. The truth value of each type of praise, of neutralappreciation or of blame indicates if the considered affirmation is plausible and coherent or not, given that the praise of a positiveconcept is true, in the same way as the neutralappreciation of a neutral concept and the blame of a negative concept. Conversely, the praise of a non-positive concept^{15}, the neutralappreciation of a non-neutral concept or well the blame of a non-negative concept^{16} are false. Formally, the truth value [v] of propositions of the type P = ^{p}(α^{q}), with p, q {-1, 0, 1} and α {A^{+}, A^{0}, A^{–}, Ā^{+}, Ā^{0}, Ā^{–}} can be calculated as follows: [v] = 1 (true) if p = q and [v] = -1 (false) if p q.^{17} Hence, among the different cases which have just been enumerated, those whose truth value is true are: {^{–}(A^{–}), ^{–}(Ā^{–}), ^{0}(A^{0}), ^{0}(Ā^{0}), ^{+}(A^{+}), ^{+}(Ā^{+})}. And those whose truth value is false are: {^{–}(A^{+}), ^{–}(A^{0}), ^{–}(Ā^{+}), ^{–}(Ā^{0}), ^{0}(A^{+}), ^{0}(A^{–}), ^{0}(Ā^{+}), ^{0}(Ā^{–}), ^{+}(A^{0}), ^{+}(A^{–}), ^{+}(Ā^{0}), ^{+}(Ā^{–})}.

Complex theses

Whereas simple theses contain a judgment related to one single concept belonging to a given matrix, complextheses are composed of appreciations relative to several concepts of a same matrix. A complex thesis can thus be defined in a general way as the conjunction of several simple theses. A complex thesis can thus be composed of appreciations relative to two, three, …, n different concepts. One will use accordingly the term of n-complex thesis. Under these circumstances, the combinations prove to be numerous, without it being nevertheless necessary to enumerate them exhaustively. A given proposition P constituting a complex thesis presents thus the following structure: P = Q_{1} Q_{2} … Q_{n}, for n > 1, and Q_{i} = ^{pi}(α^{qi}), with p_{i}, q_{i} {-1, 0, 1} and α {A^{+}, A^{0}, A^{–}, Ā^{+}, Ā^{0}, Ā^{–}}. We have then the 2-complex, 3-complex, …, n-complex theses.

At this step, it appears necessary to consider first the 2-complex theses, which constitute, among the complex theses, the most common case. The 2-complex theses are composed of some appreciations relative to two concepts of a same matrix. They present the structure: ^{p}(α_{1}(A/Ā, c_{1}, q)) ^{r}(α_{2}(A/Ā, c_{2}, s)). The following appreciation constitutes thus an example of 2-complex thesis:

(6) All theory is gray, but the golden tree of life is green. (Goethe)

This 2-complex thesis is in effect composed of both the blame of theory (“all theory is gray”) and the praise of pragmatism (“the golden tree of life is green”). It proves here that the concepts of interest for theory and of pragmatism belong to the following matrix: {capacity of abstraction^{+}, interest for theory^{0}, dogmatism^{–}, pragmatism^{+}, interest for practise^{0}, prosaicness^{–}}. The structure of the thesis is thus ^{–}(interest for theory^{0}) ^{+}(pragmatism^{+}) i.e. ^{–}(A^{0}) ^{+}(Ā^{+}).

In the same way, the following appreciation constitutes a case of 2-complex thesis:

(7) The art of being sometimes very audacious, sometimes very cautious is the art of success. (Napoleon Bonaparte)

This 2-complex thesis is composed of both the praise of boldness (“The art of being (…) very audacious (…) is the art of success”) and the praise of the cautiousness (“the art of being (…) very cautious is the art of success”). It appears that these latter concepts belong to the following matrix: {boldness^{+}, propensity to take risk^{0}, temerity^{–}, cautiousness^{+}, propensity to avoid the risk^{0}, cowardice^{–}}. The thesis is thus composed here of the praise of two complementary positive concepts of a same matrix. The particular structure of this type of complex thesis is thus composed of the praise of A^{+} and the praise of Ā^{+}, i.e. formally ^{+}(boldness^{+}) ^{+}(cautiousness^{+}).

Consider lastly the following thesis, which also constitutes a case of 2-complex thesis:

(8) Two excesses: to exclude reason, and to admit nothing else than reason.(Pascal, Thoughts)

This last thesis is in effect composed of both the blame of irrationality (“exclude the reason”) and the blame of hyper-rationalism (“to admit nothing else than reason”). The corresponding reconstituted matrix is the following: {imagination^{+}, inspiration^{0}, irrationality^{–}, rationality^{+}, reason^{0}, hyper-rationalism^{–}}. As we see it, we face here a 2-complex thesis whose structure is ^{–}(irrationality^{–}) ^{–}(hyper-rationalism^{–}) i.e. ^{–}(A^{–}) ^{–}(Ā^{–}).

Lastly, the following 2-complex thesis:

(9) How can we tolerate that passion be placed on the same level than reason? (Sénèque, De Ira)

can be analyzed as a blame of passion^{0} and a praise of reason^{0}, i.e. formally ^{–}(passion^{0}) ^{+}(reason^{0}), i.e. ^{–}(A^{0}) ^{+}(Ā^{0}) at the level of the matrix {motivation^{+}, passion^{0}, fanaticism^{–}, level-headedness^{+}, reason^{0}, lukewarmness^{–}}.

It is worth noting here that this last type of 2-complex thesis corresponds to a common case, for motives of internal coherence. It is in effect logical when one criticizes or depreciates such or such value or concept, of flattering its contrary. To blame such or such thing amounts naturally to praising its opposite, and conversely. For that reason, the 2-complex theses whose particular structure is ^{–}(A^{–}) ^{+}(Ā^{+}) or well ^{+}(A^{+}) ^{–}(Ā^{–}) also constitute, among all possible combinations of 2-complex theses, a common case.

For what concerns the truth value of the 2-complex theses, it can be determined in the same way as for the simple theses. Let thus P Q be a 2-complex thesis, such that P = ^{p}(α^{q}) and Q = ^{r}(β^{s}), with p, q, r, s {-1, 0, 1} and α, β {A^{+}, A^{0}, A^{–}, Ā^{+}, Ā^{0}, Ā^{–}}. Formally, the truth value [v] of a 2-complex thesis P Q is true if v[P] = v[Q] = true, and false in other cases^{18}. It is worth noting that the most common types of 2-complex theses are those whose truth value are true. Such is the case when the truth-value of each of the two propositions included within the complex thesis is true. Under this hypothesis, the two propositions reinforce themselves. It consists thus of the cases corresponding to: {^{+}(A^{+}) ^{–}(A^{–}), ^{+}(A^{+}) ^{+}(Ā^{+}), ^{+}(A^{+}) ^{–}(Ā^{–}), ^{–}(A^{–}) ^{+}(Ā^{+}), ^{–}(A^{–}) ^{–}(Ā^{–}), ^{+}(Ā^{+}) ^{–}(Ā^{–})}.

Dual theses

At this step, it is worth focusing on the notion of a dual thesis of a given thesis. This last notion applies both to the simple theses and to the complex ones. The dual thesis constitutes here an element of the dialectical discussion, which proves to be important since it is the basis of the discussion related to the thesis under consideration.

Let us focus, to begin with, on dual theses of simple theses. Let us begin by giving a general definition. Formally, a simple thesis ^{p}(α_{1}(A/Ā, c, q)) admits of a dual thesis that corresponds to the following definition: ^{p}(α_{2}(A/Ā, –c, q)). Thus, a dual thesis of a simple thesis presents the following characteristics: (i) the polarities of the appreciation of the dual thesis and of the simple thesis are identical; (ii) the contrary components of the concepts on which bear the appreciations of the dual thesis and of the simple thesis are opposite; (iii) the polarities of the concepts on which bear the appreciations of the dual thesis and of the simple thesis are identical.

Let us consider first the dual theses of the true simple theses. The types of true simple theses can be thus enumerated as follows: {^{+}(A^{+}), ^{0}(A^{0}), ^{–}(A^{–}), ^{+}(Ā^{+}), ^{0}(Ā^{0}), ^{–}(Ā^{–})}. Formally, a true simple thesis ^{p}(α_{1}(A/Ā, c, p)) presents a dual thesis which responds to the following definition: ^{p}(α_{2}(A/Ā, –c, p)). Thus, the dual theses of the true simple theses are respectively: {^{+}(Ā^{+}), ^{0}(Ā^{0}), ^{–}(Ā^{–}), ^{+}(A^{+}), ^{0}(A^{0}), ^{–}(A^{–})}.

To take an example, consider the following true simple thesis:

(10) What you can do, or dream you can do, begin it. Boldness has genius, power and magic in it. (Goethe)

which presents the structure ^{+}(boldness^{+}) i.e. ^{+}(A^{+}) at the level of the matrix {boldness^{+}, propensity to take risk^{0}, temerity^{–}, cautiousness^{+}, propensity to avoid risk^{0}, cowardice^{–}}. The thesis below whose structure is ^{+}(cautiousness^{+}) i.e. ^{+}(Ā^{+}) constitutes thus its dual thesis:

(11) Cautiousness is as much superior to the other virtues as sight is to the other senses. (Bion of Phlossa)

Consider also the dual theses of the false simple theses. The types of false simple theses are: {^{–}(A^{+}), ^{–}(A^{0}), ^{–}(Ā^{+}), ^{–}(Ā^{0}), ^{0}(A^{+}), ^{0}(A^{–}), ^{0}(Ā^{+}), ^{0}(Ā^{–}), ^{+}(A^{0}), ^{+}(A^{–}), ^{+}(Ā^{0}), ^{+}(Ā^{–})}. And the dual theses of the false simple theses are respectively: {^{–}(Ā^{+}), ^{–}(Ā^{0}), ^{–}(A^{+}), ^{–}(A^{0}), ^{0}(Ā^{+}), ^{0}(Ā^{–}), ^{0}(A^{+}), ^{0}(A^{–}), ^{+}(Ā^{0}), ^{+}(Ā^{–}), ^{+}(A^{0}), ^{+}(A^{–})}.

To take an example, the following false simple thesis:

(4) Passion is an illness that abhors all medication. (Kant)

presents the structure ^{–}(passion^{0}) i.e. ^{–}(A^{0}) at the level of the matrix {motivation^{+}, passion^{0}, fanaticism^{–}, level-headedness^{+}, reason^{0}, lukewarmness^{–}}. The following thesis whose structure is ^{–}(reason^{0}) i.e. ^{–}(Ā^{0}) constitutes thus its dual thesis:

(12) If reason dominated on the earth, nothing would happen there. (Bernard Fontenelle)

It is worth considering now, on the other hand, the dual theses of the complex theses. These latter are such that the contrary components of the concepts on which bear the appreciations of the two simple theses, which are part of the dual thesis and of the considered thesis, are opposite^{19}. Consider then the true 2-complex theses. Thus, the dual thesis of ^{+}(A^{+}) ^{–}(Ā^{–}) is ^{+}(Ā^{+}) ^{–}(A^{–}). And also, the dual thesis of ^{0}(A^{0}) ^{+}(A^{+}) is ^{0}(Ā^{0}) ^{+}(Ā^{+}). It is worth noting here in particular that the dual thesis of ^{0}(A^{0}) ^{0}(Ā^{0}) is ^{0}(Ā^{0}) ^{0}(A^{0}), that the dual thesis of ^{+}(A^{+}) ^{+}(Ā^{+}) is ^{+}(Ā^{ +}) ^{+}(A^{+}) and that the dual thesis ^{–}(A^{–}) ^{–}(Ā^{–}) is ^{–}(A^{–}) ^{–}(Ā^{–}).

Let us also give a few examples. Thus, the true 2-complex thesis corresponding to the following proposition:

(6) All theory is gray, but the golden tree of life is green. (Goethe)

presents the structure ^{–}(A^{0}) ^{+}(Ā^{+}) i.e. ^{–}(interest for theory^{0}) ^{+}(pragmatism^{+}) at the level of the matrix {capacity of abstraction^{+}, interest for theory^{0}, dogmatism^{–}, pragmatism^{+}, interest for practice^{0}, prosaicness^{–}}. The following thesis whose structure is ^{–}(Ā^{0}) ^{+}(A^{+}) i.e. ^{–}(interest for practice^{0}) ^{+}(capacity of abstraction^{+}) constitutes thus its dual thesis:

(13) All practice is vile, but fecund and elevated is the quest of the genuine abstraction.

Similarly, the following proposition:

(8) Two excesses: to exclude reason, and to admit nothing else than reason.(Pascal, Thoughts)

constitutes a true 2-complex thesis whose structure is ^{–}(irrationality^{–}) ^{–}(hyper-rationalism^{–}) i.e. ^{–}(A^{–}) ^{–}(Ā^{–}) at the level of the matrix: {imagination^{+}, inspiration^{0}, irrationality^{–}, rationality^{+}, reason^{0}, hyper-rationalism^{–}}. The thesis below whose structure is ^{+}(imagination^{+}) ^{+}(rationality^{+}) i.e. ^{+}(A^{+}) ^{+}(Ā^{+}) constitutes thus its dual thesis:

(14) The art of being sometimes very imaginative, sometimes very rational is the art of success.

Lastly, it is worth noting that one has also analogous definitions for 3-complex, 4-complex, etc. theses. To take then an example, the dual thesis of the 3-complex thesis ^{+}(A^{+}) ^{0}(A^{0}) ^{0}(Ā^{0}) is ^{+}(Ā^{+}) ^{0}(Ā^{0}) ^{0}(A^{0}). In the same way, the dual thesis of the 3-complex thesis ^{+}(A^{+}) ^{0}(A^{0}) ^{–}(A^{–}) is ^{+}(Ā^{+}) ^{0}(Ā^{0}) ^{–}(Ā^{–}).

The matrix-based dialectic plan

The preceding developments allow now to describe the steps of the dialectical reasoning applicable to the analysis of a given particular thesis, from the above-mentioned principles. The first step consists thus in the accurate determination of the structure of the thesis under consideration. The second step, which results directly from it, is the attribution of a truth-value to this latter thesis. The following step consists then in the reconstitution of the whole matrix applicable to the concept(s) which are the object of the thesis. One is then in a position to determine the dual thesis of the considered thesis in the same way as the true simple theses other than the considered thesis and its dual thesis. Lastly, the final step is the synthesis which consists in the conjunction of the true simple theses relative to each of the 6 concepts of the considered matrix: ^{+}(A^{+}) ^{0}(A^{0}) ^{–}(A^{–}) ^{+}(Ā^{+}) ^{0}(Ā^{0}) ^{–}(Ā^{–}). Such a synthesis allows to encompass a threefold antinomy: the one existing between A^{+} and Ā^{–}, A^{0} and Ā^{0}, and A^{–} and Ā^{+}. It should be observed here that one can eventually retain from the synthesis but a simplified form consisting of the conjunction of the true simple theses constituting a praise or a blame: ^{+}(A^{+}) ^{–}(A^{–}) ^{+}(Ā^{+}) ^{–}(Ā^{–}). In the same way, one may sometimes limit oneself to a truncated form of synthesis consisting in ^{+}(A^{+}) ^{+}(Ā^{+}), which emphasizes the complementarity between A^{+} and Ā^{+}.^{20}

At this step, we are in a position to present the matrix-based dialectic plan. Such a plan results directly from the structure of matrix of concepts which has been just described. The corresponding matrix-based dialectic plan presents thus the following structure:^{21}

(15) 1. From the viewpoint of A^{0}

1.1 Praise of A^{+}

1.2 Blame of A^{–}

2. From the viewpoint of Ā^{0}

2.1 Praise of Ā^{+}

2.2 Blame of Ā^{–}

3. Complementarity between A^{+} and Ā^{+}^{22}

Consider then, to take an example the following true simple thesis:

(16) Success was always a child of audacity. (Prosper Crebillon, Catilina)

whose structure is ^{+}(boldness^{+}) i.e. ^{+}(A^{+}) at the level of the matrix {boldness^{+}, propensity to take risk^{0}, temerity^{–}, cautiousness^{+}, propensity to avoid risk^{0}, cowardice^{–}}. It results then the following matrix-based plan:

(17) 1. From the viewpoint of risk taking^{0}

1.1 The necessity of boldness^{+}

1.2 The dangers of temerity^{–}

2. From the viewpoint of risk avoidance^{0}

2.1 The advantages of the cautiousness^{+}

2.2 The risk of cowardice^{–}

3. The necessary complementarity between boldness^{+} and cautiousness^{+}

Consider also the following false simple thesis:

(12) If reasondominated on the earth, nothing would happen there. (Bernard Fontenelle)

whose structure is ^{–}(reason^{0}). The corresponding matrix is: {level-headedness^{+}, reason^{0}, lukewarmness^{–}, motivation^{+}, passion^{0}, fanaticism^{–}}. And the following matrix-based plan then ensues:

(18) Introduction: (i) structure of the thesis; (ii) truth value; (iii) matrix

1. From the viewpoint of reason^{0}

1.1 The pitfall of lukewarmness^{–}

1.2 The necessity of level-headedness^{+}

2. From the viewpoint of passion^{0}

2.1 The dangers of fanaticism^{–}

2.2 The necessity of motivation^{+}

3. The necessary complementarity between level-headedness^{+} and motivation^{+}

Lastly, such a type of plan also proves to be adapted to a true 2-complex thesis such as the following:

(19) In the first place comes your profession, because doing just one thing well will procure a higher development for you than doing one hundred by halves. (Goethe)

This latter thesis can be analyzed as a 2-complex thesis whose structure is ^{+}(expertise^{+}) ^{–}(superficiality^{–}) i.e. ^{+}(A^{+}) ^{–}(Ā^{–}) at the level of the matrix: {expertise^{+}, monodisciplinarity^{0}, compartmentalization^{–}, eclecticism^{+}, multi-disciplinarity^{0}, superficiality^{–}}. And the following matrix-based plan^{23} then ensues:

(20) 1. From the viewpoint of monodisciplinarity^{0}

1.1 The advantages of expertise^{+}

1.2 The risk of compartmentalization^{–}

2. From the viewpoint of multi-disciplinarity^{0}

2.1 The necessity of eclecticism^{+}

2.2 The dangers of superficiality^{–}

3. The necessary complementarity between expertise^{+} and eclecticism^{+}

Conclusion

From the above developments, it should be noted that the matrix-based dialectic plan presents a number of advantages with regard to the classical dialectic plan. First, the dialectical approach which has just been described performs first an analysis of the structure of the thesis under consideration, which leads then to assign a truth value to it, on objective grounds.

Second, it appears that the matrix-based dialectic plan replaces the thesis or the main proposition in a context that comprises a greater number of concepts than the classical dialectic plan. In effect, the classical dialectic plan usually places the thesis in an environment comprising in general two, or even three concepts. By contrast, the matrix-based dialectic plan replaces the thesis in a context comprising six concepts which are related to this latter.

Third, one of the advantages of the matrix-based dialectic plan is that it also allows to take into account some concepts which are not lexicalized. In effect, a matrix of concepts describes six canonical concepts. But it is rare that the totality of these latter concepts are lexicalized. In effect, the most common situation is that only some concepts – in general two or three – among the six described by the corresponding matrix, are lexicalized. Here also, the advantage of the matrix-based dialectic plan is to allow to take into account exhaustively the six concepts of a same matrix and to incorporate them in the corresponding discussion.

It should also be noted that the step of the antithesis at the level of the classical dialectic plan is replaced here by the determination of the dual thesis, which presents an identical structure to that of the initial thesis. The dual thesis, which serves here as a basis for dialectical reasoning, presents by its simple or well n-complex structure a more elaborated nature than the traditional antithesis.

Lastly, it proves that the classical dialectic plan allows to overcome an antinomy existing between two concepts, which serve respectively as a support to the thesis and to the antithesis. It consists most often of A^{+} and Ā^{–}, of A^{0} and Ā^{0}, or well of A^{–} and Ā^{+}. Most of the time, it consists of a dual or antinomical pair of concepts which present the property of being lexicalized. Conversely, the matrix-based dialectic plan constitutes the expression of a dialectical move of the thought which allows to go beyond a threefold antinomy: the one existing at the same time between A^{+} and Ā^{–}, A^{0} and Ā^{0}, and finally A^{–} and Ā^{+}, whether these concepts are lexicalized or not.

References

Franceschi, Paul (2002). Une classe de concepts. Semiotica 139 (1-4), 211-226. English translation.

Hegel, Georg Wilhelm Friedrich (1812-1816). Wissenschaft der Logik. Science de la logique, trad. Bourgeois, Paris, Aubier Montaigne, 1972.

Hegel, Georg Wilhelm Friedrich (1817). Die Encyclopädie der philosophischen Wissenschaften im Grundrisse. Précis de l’encyclopédie des sciences philosophiques, trad. J. Gibelin. Vrin, Paris, 1978

1 One also finds the antithesis-thesis-synthesis variant.

2 Platon envisaged dialectic under the form of a dialogue between two persons, based on alternate questions and responses. One also finds a dialectical approach in Kant, but also in Fichte and Schelling.

3 In the context of dialectical materialism, the dialectic finds its expression on the social terrain, through the conflict or the struggle, which are viewed as the manifestation, at a material level, of the contradiction. Historical progress and social advances ensue once this conflict has been overcome. For Marx also, the dialectical objective situates itself veritably at the level of the reality, finding thus its expression in the facts and the phenomena. Conversely, the dialectical move observed at the level of human thought only constitutes the subjective reflect of the essential dialectic, a simple transposition of the latter at the level of the humain brain.

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

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

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

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

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

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

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

12 For a more comprehensive list of matrices of concepts, see Franceschi (2002).

13 Personal ambition could be fruitful (ambition^{+}) or well excessive, or even immoderate (ambition^{–}).

14 A passion could be positive (passion^{+}) or well excessive, destructive (passion^{–}).

17 One could as well distinguish here degrees of truth value, by making use of degrees of appreciation, with p [-1, 1]. An approach by degree of the truth value ensues, by calculating thus this latter with regard to the absolute value of the difference between p and q: [v] = 1- |(p – q)/2|.

18 Such a definition generalizes to the determination of the truth values of the 3-composed theses, …, n-composed.

19 Formally, let thus P Q be a 2-composed thesis, such that P = ^{p1}(α_{1}(A/Ā, c1, q1)) and Q = ^{P2}(α_{2}(A/Ā, c2, q2), with p1, p2, q1, q2 {-1, 0, 1}, c1, c2 {-1, 1} and α, β {A^{+}, A^{0}, A^{–}, Ā^{+}, Ā^{0}, Ā^{–}}; then the dual thesis of P Q is of the form: ^{p1}(α_{1} (A/Ā, –c1, q1)) ^{P2}(α_{2} (A/Ā, –c2, q2). Such definition generalizes easily to the dual theses of the n-composed theses.

20 The description of the different steps of the dialectical process thus defined also suggests other types of plans than the one which has been emphasized here. Alternative plans can notably highlight a part related to the step of determination of the truth value of the considered thesis, or well to the dual thesis of this latter.

21 Alternatively, one could also consider the following variation:

1. From an analytic point of view

1.1 From the viewpoint of A^{0}

1.1.1 Praise of A^{+}

1.1.2 Blame of A^{–}

1.2 From the viewpoint of Ā^{0}

1.2.1 Praise of Ā^{+}

1.2.2 Blame of Ā^{–}

2. From a synthetic point of view: the complementarity between A^{+} and Ā^{+} and between A^{–} and Ā^{–}

22 A variation of this type of plan consists evidently in assimilating the part 3 with the conclusion.

23 For this last type of thesis whose structure is ^{+}(A^{+}) ^{–}(Ā^{–}), it is also possible to recur to another type of plan which emphasizes more the dual thesis ^{+}(Ā^{+}) ^{–}(A^{–}). Such a type of plan proves to be close to the classical dialectic plan and stresses on the dual thesis of the considered thesis, e.g. ^{+}(eclecticism^{+}) ^{–}(compartmentalization^{–}). Such a type of plan presents then the following structure:

1. Thesis

1.1 The advantages of expertise^{+}

1.2 The dangers of superficiality^{–}

2. Dual thesis

2.1 The necessity of eclecticism^{+}

2.2 The risk of compartmentalization^{–}

3. The necessary synthesis between eclecticism^{+} and expertise^{+}, and superficiality^{–} and compartmentalization^{–}