Traditional oppositions are at least two-dimensional in the sense that they
are built based on a famous bidimensional object called square of oppositions
and on one of its extensions such as Blanch\'e's hexagon. Instead of
two-dimensional objects, this article proposes a construction to deal with
oppositions in a one-dimensional line segment.Comment: This is the version accepted for publication (South American Journal
  of Logic, 2018

Costa-Leite, Alexandre

English

arXiv

Traditional oppositions are at least two-dimensional in the sense that they are built based on a famous bidimensional object called square of oppositions and on one of its extensions such as Blanché's hexagon. Instead of two-dimensional objects, this article proposes a construction to

deal with oppositions in a one-dimensional line segment

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South American Journal of LogicVol. 4, n. 1, pp. 185–193, 2018ISSN: 2446-6719Oppositions in a Line SegmentAlexandre Costa-LeiteAbstractTraditional oppositions are at least two-dimensional in the sense thatthey are built based on a famous bidimensional object called square ofoppositions and on one of its extensions such as Blanche´’s hexagon. In-stead of two-dimensional objects, this article proposes a construction todeal with oppositions in a one-dimensional line segment.Keywords: theory of opposition, square of opposition, Blanche´’s hexagon,reduction, one dimension, line segment of opposition.IntroductionThe basic theory of oppositions has been developed considering a bidimen-sional structure, i.e., the well known square of oppositions. Later, it has beengeneralized to an hexagon of oppositions by Blanche´ in [4]. Moretti argues in[5] that there is a ‘geometry’ of oppositions, generalizing squares and hexagons,basically, to three-dimensional structures such as cubes and tetradecahedrons.This gives rise to the domain of n-opposition theory. Since works proposed byBeziau in [1] and Moretti in [5], there are now many researches in the field. Thereader should check [7] for an introduction to the square of oppositions and [6]to its main recent developments. Beziau and Read stated in [3] that the theoryof oppositions cannot be identified with the diagram representing this theory(p.315). This means that there are much more on oppositions than what isrepresented in the relations between corners of the square or the hexagon. Thisarticle1, in some sense, can be viewed as an attempt to justify the claim thatthe theory of oppositions is not only the study of these n-dimensional diagrams(n ≥ 2).1A previous version of this paper appeared as a preprint in arXiv (April, 2016). Thanksto Rodrigo Freire, Edelcio de Souza and Fabien Schang for remarks on the constructionsproposed here.186 A. Costa-LeiteConsider a question: is there a way to represent oppositions without two-dimensional objects such as squares or objects of higher dimensions? Theanswer is yes. A construction to formulate this reduction is proposed showingthat there is no need for two-dimensional objects to establish the basic theoryof oppositions. Indeed, one dimension is enough. This means that oppositionscan be defined in a line segment, a piece of one-dimensional space. Moreover,oppositions require precisely at least one dimension to be defined, and thetraditional case of the bidimensional square can be converted step-by-step toit. However, applying the same strategy to Blanche´’s hexagon does not work.But the situation changes with some constraints added at the level of the basicconstruction. So, it is also possible to convert the standard bidimensionalBlanche´’s hexagon to a line segment.This paper shows that line segments are sufficient to define four basic stan-dard oppositions in such a way that the square can be derived from this prim-itive structure. This line segment of oppositions is called here basic construc-tion. Then, this very same basic construction does not work when applied tothe standard hexagon (i.e. Blanche´’s hexagon), but there is a way to generatea similar strategy mutatis mutandis to it. In what follows these constructionsare explained presenting their range and limits.1 Defining oppositions in a line segment1.1 The squareAssume classical logic. Given the framework of first-order logic, the square ofoppositions uses four kinds of categorical propositions (where ϕ is formula):(A) universal affirmative of the form ∀xϕ, (E) universal negative of the form∀x¬ϕ, (I) existential affirmative ∃xϕ and (O) existential negative ∃x¬ϕ. Tak-ing into account that ∀ and ∃ are interdefinable in the presence of negation, it isa matter of taste to decide which one to use to represent these four propositions.They can appear as above, mixing both quantifiers, or with only one kind ofquantifier. Thus, for universal quantifier and negation there is the following:(A) ∀xϕ, (E) ∀x¬ϕ, (I) ¬∀x¬ϕ and (O) ¬∀xϕ. For existential quantificationand negation: (A) ¬∃x¬ϕ, (E) ¬∃xϕ, (I) ∃xϕ and (O) ∃x¬ϕ. In general, in theliterature, there are four traditional oppositions holding between these proposi-tions: contradiction (d), contrariety (c), subcontrariety (sc) and subalternation(s), which are defined in the standard way (see [7]).Further, there are, indeed, other families of concepts satisfying opposi-tional structures: they could be metaphysical statements such as necessity andpossibility (and their derivatives), or deontic propositions containing notionsof obligation and permission, or even statements containing temporal aspectsOppositions in a Line Segment 187such as always, sometimes and never. So, a pure oppositional structure isnot necessarily decorated with categorical statements. Indeed, there are manypossible decorations of the square (see [4, 2, 5]). These concepts fit pretty wellsatisfying the structure of the square of oppositions (or its extensions). They- and similar concepts - are here called categorical-like concepts: these arenotions satisfying the four oppositions and, for this reason, can be arrangedinside the traditional square. A set of categorical-like statements is denotedby C. Oppositions are usually (and historically) represented bidimensionallyusing a two-dimensional object with lenght and width (i.e. a square):AcsdEsdIscOI argue that the above two-dimensional square can be reduced to a simpleone-dimensional structure, that is, a line segment of integers.2 To show howto build this reduction is the first construction suggested here.Assume the set Z of integers and, then, the sets Z+ (positive integers) andZ− (negative integers). If the set of integers does not have zero, it is denoted byZ∗, as usual. Take a line segment such that for each j ∈ Z, j 6= 0, there exists−j, the symmetric of j. In particular, consider a set Z′ = {−r,−q, q, r} ⊆ Z.Then the procedure is: a categorical-like proposition is assigned to each elementj ∈ Z′ in the following way:• j ∈ Z∗+ if, and only if, j is associated to a universal statement;• j ∈ Z∗− if, and only if, j is associated to an existential statement.There is a division of propositions in the sets of universal and existentialstatements. Let C be a set of categorical (or categorical-like) propositions.The function i which connects elements of C to elements of Z′ is a bijectionsuch that: i(A) = q; i(E) = r; i(I) = −r and i(O) = −q.We use α, β for arbitrary propositions ranging on C. Traditional opposi-tions now have to be reformulated inside this new framework. So, clauses foroppositions are defined as follows:1. Propositions α, β are contradictories if, and only if, their assigned num-bers have sum equals to 0, i.e, i(α) + i(β) = 0;2. Propositions α, β are contraries if, and only if, i(α), i(β) ∈ Z∗+;2Simple in the sense that it requires only one dimension.188 A. Costa-Leite3. Propositions α, β are subcontraries if, and only if, i(α), i(β) ∈ Z∗−;4. β is subaltern of α if, and only if, i(β) 6= −i(α) and i(β) ∈ Z∗−;This completes the construction. Let’s show with an example how thesquare can be defined in this way: consider, for instance, that Z′ = {−2,−1,1, 2} ⊆ Z. Thus, given that 1, 2 ∈ Z∗+, it follows that these numbers are as-signed to universal propositions (in which way this association is done is notimportant). In the same manner, from the fact that −2,−1 ∈ Z∗−, these num-bers are connected to existential statements. Without loss generality, considerthat 1 is the number connected to (A), i.e, i(A) = 1 and i(E) = 2. Therefore,the contradictory of (A) is the proposition (O) because i(O) = −1 satisfying,therefore, the condition to be a contradiction, i.e., 1 + (−1) = 0. The same forthe relations between (E) and (I). By construction, (A) and (E) are contrarieswhile (I) and (O) are subcontraries, and these are subalterns of (A) and (E),respectively. (I) is subaltern of (A) given that (I) has associated to it the inte-ger −2 which is, in its turn, different of −1 (note that −i(A) = i(O)). As far asthere are only four oppositions, this procedure can always be done, no matterthe family of concepts considered. This is an example provided to show howa two-dimensional square of oppositions can be reduced to a one-dimensionalline segment of oppositions.Despite the beauty of oppositions represented in a square, these oppositionscan be converted into a one-dimensional line segment, if the number of oppo-sitions remain four. This is the main contribution of this paper. Nevertheless,not all two-dimensional objects can be reduced using this same strategy. Italready fails in the case of the hexagon. So, the basic construction has to beimproved in order to work also for the hexagon. This is of secondary impor-tance here, given that the hexagon does not have the same historical relevanceof the square.1.2 The hexagonBlanche´ proposed an extension of the square, the hexagon, and this new tool(still two-dimensional) can be used to model many situations (see [4], but also[2]). Following Blanche´’s construction, in the (incomplete) diagram below, (U)is defined as the disjunction A ∨ E and (Y) is the conjunction I ∧O.Oppositions in a Line Segment 189UAs??sdEs``cdsI OscYs__s>>How can the reduction strategy, i.e., the basic construction be also ex-tended to Blanche´’s hexagon? The fact that there are propositions of the form(U) and (Y) require some adaptations in the basic construction. The secondconstruction consists in executing these adaptations in order to transform alsothe hexagon into a one-dimensional structure.Take a line segment as above and a set Z′′ = {−s,−r,−q, q, r, s} ⊆ Z.Now, as there are also a disjunction and a conjunction, some changes have tobe made in the way integers are assigned to propositions. To each elementj ∈ Z′′, a proposition is connected as follows:• j ∈ Z∗+ if, and only if, j is associated to a universal statement or adisjunction;• j ∈ Z∗− if, and only if, j is associated to an existential statement or aconjunction;.The division of the class of propositions now contains a set of universal anddisjunctive statements, from one side, and a set of existential and conjunctivestatements, from other side. Let the set C′ be an expansion of the set Cdefined by categorical-like propositions plus a disjunction and a conjunction,and let the function i which connects elements of C′ to elements of Z′′ be abijection such that: i(U) = s, i(A) = q, i(E) = r, i(I) = −r, i(O) = −qand i(Y ) = −s. The number assigned to the conjunction of (I) and (O),that is, (Y), as well to the disjunction of (A) or (E), that is (U), is the sumof both conjuncts (in the first case) and the sum of both disjuncts (in thesecond case). So, define that the integers associated to statements (U) and(Y) are obtained by the sums of their components: i(U) = i(A) + i(E) andi(Y ) = i(I) + i(O). These numbers obtained by sums play an important roleand, therefore, they are called distinct objects: the first one is the positivedistinct object and the second is the negative. Consider a similar strategy as190 A. Costa-Leitethe basic construction and let’s try to reduce the hexagon to a line segment.Suppose, for instance, that Z′′ = {−3,−2,−1, 1, 2, 3} ⊆ Z is given and numbersare associated to propositions: i(A) = 1, i(E) = 2, i(O) = −1 and i(I) = −2.Thus, distinct objects have the following association: i(U) = 3 and i(Y ) = −3,if we consider this particular line segment [−3, 3]. Hence, these statements areobviously in the opposition of contradiction. In this sense, clause (1) above,i.e., contradiction, holds for (U) and (Y):3UYdBut not all clauses from 1-4 are valid. The reason for this is that Blanche´’shexagon contains some unexpected connections between propositions (Y)-(A)-(E) (they are contraries and then clause 2 fails):AccEcYand propositions (U)-(I)-(O) (they are subcontraries and then clause 3fails):UscscIscOIn addition, subalternation also fails. Thus, it is not straighforward tosettle Blanche´’s hexagon in a line segment of integers.4 So, some adaptationsand repairs in the basic construction should have to be done.Assume that α, β, γ are letters for arbitrary propositions. The constructionbelow shows how to reduce Blanche´’s hexagon to a line segment. Let i(γ) be3Moretti (see [6]) remarked that the opposition of contradiction can be characterized, nomatter which dimension is considered, as a certain kind of symmetry. It is a conjecture of thiswork that the whole of n-opposition theory can be reduced to variations of the constructionspresented in this paper.4Note that these last three diagrams are displayed in Blanche´’s hexagon (see [4]). Whenthese relations are added to the diagram, we have the complete hexagon of oppositions.Oppositions in a Line Segment 191a distinct object. For contradictories, condition 1 remains the same as above,but for other oppositions, clauses are the following (note that they have to beapplied in this order: contradiction, contraries or subcontraries, and, by theend, subalternation. Then gaps will be gradually filled, as the oppositions areexcludent, i.e, two propositions cannot be related by two different oppositions):1*. Propositions α, β, γ are contraries if, and only if, (i(α) + i(β)) + i(γ) = 0and i(γ) ∈ Z∗−;2*. Propositions α, β, γ are subcontraries if, and only if, (i(α)+i(β))+i(γ) =0 and i(γ) ∈ Z∗+;3*. β is subaltern of α if, and only if, i(β) 6= −i(α) and i(β) ∈ Z∗−, ori(β) 6= −i(α) and a) i(β) > i(α) and i(α), i(β) ∈ Z∗+ or b) i(β) > i(α)and i(α), i(β) ∈ Z∗− (i(β), in the last condition, is a distinct object);It is not difficult to provide an example of integers to show that the hexagoncan be reduced to it, as done in the case of the square. Assume first - withoutloss of generality - that i(A) = 1. So, i(E) = 2 and, thus, the distinct objecti(U) = i(A)+i(E) = 3. Second, consider that i(O) = −1. It follows that i(I) =−2 and that the other distinct object i(Y ) = i(O) + i(I) = −3. Consequently,(U) and (Y) are contradictories. Note that (i(O) + i(I)) + i(U) = 0, so theseare subcontraries, as i(U) ∈ Z∗+, i.e, it is the positive distinct object. Moreover,(i(A) + i(E)) + i(Y ) = 0, so these are contraries, as i(Y ) ∈ Z∗−, i.e, it is thenegative distinct object. Concerning subalternation, the square is containedin the hexagon, so (I) is subaltern of (A) and (O) is subaltern of (E), as inthe case of the square. For other relations of subalternation, note that (U)is subaltern of (A) and (E), because i(U) > i(A) and i(U) > i(E) whilei(U), i(A), i(E) ∈ Z∗+. Differently, (I) and (O) are subalterns of (Y), given thati(Y ) < i(I) and i(Y ) < i(O), and i(Y ), i(I), i(O) ∈ Z∗−.This is one way to reduce the hexagon to a one-dimensional object, but it isnot argued that there is no other way. Note also that only Blanche´’s hexagon istaken into consideration, because it is the traditional hexagon, although thereare some other available in the literature.2 ConclusionThe standard square of opposition is a two-dimensional object used to organizeand manage categorical-like concepts. There are many notions which fit in thesquare and its regular extensions such as the two-dimensional hexagon, and itsthree-dimensional correlates.192 A. Costa-LeiteThis paper proposed a construction which shows that there is no needto develop two-dimensional squares to represent categorical-like statements orconcepts, instead a one-dimensional line segment based on integers is enough.Although the success in the case of the square, we have showed that the sametechnique does not immediately work for Blanche´’s hexagon and, therefore, itdoes require some adaptations and expansions to transform also the hexagoninto a one-dimensional object. While both squares and hexagons have formu-lations in line segments of integers, these formulations are not straightforwardand intuitive, and thus it seems these results are not so effective and simple asthe pictorial effect of two-dimensional diagrams. However, from the theoreticalviewpoint, it is important to know that there are simple one-dimensional ob-jects able to accommodate theory of oppositions, although the fact that maybe,at this level, they are not so manageable as squares and hexagons. This is aclue that oppositions do not match with the research on diagrams, and in thissense, what authors supported in [3] seems to be rather right:“The diagram has been very important in promoting the theorybut the theory does not reduce to the diagram. The theory startedmany centuries before the basic diagram was drawn and developedbeyond this diagram...The strength of this theory is that it is at thesame time fairly simple but quite rich; it can be applied to manydifferent kinds of proposition, and also to objects and concepts. Itcan also be generalized in various manners, in particular, by con-structing many different geometrical objects.” (pgs. 315-316, in [3])Oppositions are relations between propositions and cannot be identified,therefore, with the study of diagrams, though these are useful tools to explainwhat oppositions are and, in this way, can be largely applied especially forlearning purposes.In general, authors working on the square of opposition generally start withtwo-dimensional constructions like squares and hexagons, and then they jumpto three-dimensional solids and so on. It seems a novelty the construction pro-posed in this paper because it shows that one does not need to go n-dimensional(for n ≥ 2) to develop the theory of oppositions.There are, notwithstanding, some problems which remain open: the ques-tion to determine whether the same procedure can also be applied to solidsand higher dimensions, as well as to more than four oppositions, are very com-plicated and still have to investigated in detail in the domain of line segmentsof oppositions.Oppositions in a Line Segment 193References[1] J-Y. Beziau. New light on the square of oppositions and its namelesscorner. Logical Investigations, 10:218-232, 2003.[2] J-Y. Beziau. The power of the hexagon. Logica Universalis, 6(1):1-43,2012.[3] J-Y. Beziau and S. Read. Square of Opposition: A Diagram and a Theoryin Historical Perspective. History and Philosophy of Logic, 35(4):315-316,2014.[4] R. Blanche´. Structures intellectuelles. Essai sur l’organisationsyste´matique des concepts. Paris: Vrin, 1966.[5] A. Moretti. Geometry of Modalities? Yes: Through n-Opposition Theory.In Aspects of Universal Logic, J-Y. Beziau, A. Costa-Leite, A. Facchini(editors), Travaux de Logique, 17:102-145, 2004.[6] A. Moretti. From the “Logical Square” to the “Logical Poly-Simplexes”.In The Square of Opposition: a general framework for cognition, J-Y.Beziau and G. Payette (editors), 119-156. Bern: Peter Lang, 2012.[7] T. Parsons. The Traditional Square of Opposition. In The Stanford En-cyclopedia of Philosophy, Summer 2015 edition.Alexandre Costa-LeiteDepartment of PhilosophyUniversity of BrasiliaICC Norte, Campus Universita´rio Darcy Ribeiro, Brasilia-DF, BrazilE-mail: costaleite@unb.br

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