Following a previous article (cf. Costa-Leite, A. (2018). Oppositions in a line segment,  South American Journal of Logic, 4(1), pp.185-193) in which 

logical oppositions are defined in a line segment, this 

article goes one step further and proposes a method defining them

using a zero-dimensional object: a point

Costa-Leite, Alexandre

English

PhilSci Archive

Oppositions in a pointAlexandre Costa-LeiteDepartment of PhilosophyUniversity of Brasilia, BrazilAbstractFollowing a previous article (cf.[3]) in which logical oppositionsare defined in a line segment, this article goes one step further andproposes a method defining them using a zero-dimensional object: apoint.1 IntroductionIn the environment of a two-valued logic such as first-order logic, (logical)oppositions are regularly four: contradiction, contrariety, subcontrariety andsubalternation. By the very beginning, these relations were displayed usingthe square of opposition and, later, Robert Blanche´ in [2] showed how thesquare induces another bidimensional object: the hexagon of opposition.Recently, some authors investigated generalizations of the square to hexagonsand to three-dimensional solids and so on (cf. [1], [4] and [5]).In [3], it is argued that a line segment of integers is enough to define logicaloppositions in such a way that there is no need to use n-dimensional objects(n ≥ 2) to get representations of standard logical oppositions. The procedureproposed works for the square of opposition and, with some adaptations,works as well for the hexagon of opposition. New connections with respectto the line segment have been established by Fabien Schang in [6].Currently, a different direction is pursued but taking into account theresults of [3]. Instead of a one-dimensional object such as a line segment ofintegers, this paper defines oppositions in an abstract dot, a point, a zero-dimensional object. Therefore, there is no need to use any dimension at all1Figure 1: Reducing dimensions of diagramsto define logical oppositions.1Figure 1 reflects the scheme of reductions concerning dimensions of dia-grams used to represent oppositions. The square is initially transformed intoa line segment which, in its turn, is transformed into a mathematical point.2 Oppositions without dimensionsIn the construction suggested in [3], the set Z is taken into considerationand categorical statements are associated to positive and negative integersallowing oppositions to be defined in a line segment. This produces, therefore,a conversion of the square and the hexagon of opposition into a line segmentof opposition (for precision and details cf. [3]).In order to get started, consider classical two-valued first-order logic.21When I have developed the basic construction to convert the square and the hexagoninto a line segment, a one-dimensional structure, Jean-Yves Be´ziau asked to me (in per-sonal communication, 2019) whether it would be possible to define oppositions withoutdimensions. This article is a reply to his question. Thanks to Edelcio de Souza and PeterArndt for comments and discussions.2In an n-valued logic, with n > 2, other oppositions could appear.2The following construction defines all four oppositions by means of a mathe-matical dot, a point, a fragment of zero-dimensional space: it does not work,however, for all points in general. Differently, it works precisely for an uniqueand special point in the real line and the reasons for this fact will be clear indue time.2.1 The square in a pointLet exp(a) denotes the exponent of a (this notation is somewhat essentialbecause the construction uses physical distinctions between exponents withthe same base). Again, as in [3], let C = {A,E, I, O} be a set of categorical-like sentences containing statements of the form A, E, I and O (these areforms of sentences in a square of opposition), and for m 6= n and m,n 6= 0,let Z′ = {−n,−m,m, n} ⊂ Z be a set of integers used to generate exponentsof a given number a. Let a ∈ Z and let Z] be a set of the following type:Z] = {a−n, a−m, am, an}. Now, consider a bijection e : C → Z] such thate(A) = am, e(E) = an, e(I) = a−n and e(O) = a−m. Thus, each member ofC is assigned to a member of Z] in the following way: positive exponents areconnected with universal statements and negative exponents are connectedwith existential statements. Let α, β be variables for elements of C. So,e(α) and e(β) are numbers of the form aj, for j ∈ Z′ while exp(e(α)) andexp(e(β)) refer to the exponents of a.Logical oppositions have to be redefined in this new setting according tothe laws below:1. α and β are contradictories if, and only if exp(e(α)) + exp(e(β)) = 0;2. α and β are contraries if, and only if exp(e(α)) and exp(e(β)) arepositive integers;3. α and β are subcontraries if, and only if exp(e(α)) and exp(e(β)) arenegative integers;4. β is subaltern of α if, and only if, exp(e(β)) 6= −exp(e(α)) and exp(e(β))is a negative integer.It is easy to see that the above construction does not lead to a zero-dimensional object for all a such that a ∈ Z∗.3 Take, for instance, a = 2 and3We always take a 6= 0 because exponentiation is defined with base a.3Figure 2: Oppositions in a zero-dimensional objectlet Z′ = {−2,−1, 1, 2}. In this case, the function e generates: e(A) = 21,e(E) = 22, e(I) = 2−2 and e(O) = 2−1. Despite the fact that all oppositionsare satisfied, considering a = 2, it induces a line segment in the interval[1/4, 4]. Indeed, more generally, everytime there is an a ∈ Z such that a > 1and the greatest exp(a) is k, then a line segment of the form [1/ak, ak] isobtained. If a < 1, a 6= 0 and the greatest exp(a) is k, then the line segmenthas the form [a, ak].So, in order to get oppositions defined in a point, we need a special pointa in Z. This is the number 1, the surprising point : the unique point, inthis method, in which all oppositions can be defined without leading to aline segment. Let’s check why. Consider a = 1 and let Z′ = {−2,−1, 1, 2}.Therefore, the function e associates e(A) = 11, e(E) = 12, e(I) = 1−2 ande(O) = 1−1. So, in all cases, we have the number 1 and oppositions pre-cisely defined in it. A and O (and E and I) are contradictories because theexp(e(A))+ exp(e(O)) = 0 (exp(e(E))+ exp(e(I)) = 0). We also have A andE are contraries because that exp(e(A)) and exp(e(E)) are positive integers.In the same way, I and O are subcontraries as exp(e(I)) and exp(e(O)) arenegative integers; By the end, notice that I is subaltern of A given thatexp(e(I)) 6= −exp(e(A)) and exp(e(I)) is a negative integer. The same holdsfor E and O. Figure 2 represents how oppositions can be defined in the point1 (in the center) with its different manifestations using different exponents.Mutatis mutandis, following the construction proposed in [3] combinedwith the technique above, the two-dimensional hexagon of opposition can4also be converted to a point, and this is let as an exercise to the reader. So,both the square and the hexagon can be transformed step-by-step into a linesegment and into a point.3 ConclusionThe theory of logical oppositions, as it has been developed in the literature,uses n-dimensional diagrams (n ≥ 2) to represent contradiction, contrariety,subcontrariety and subalternation. Nevertheless, it is possible to define op-positions using a line segment (one-dimensional space) and, as showed here,even no dimension at all is required, considering that oppositions can bedefined in the surprising point 1.A criticism to this approach has been raised by Peter Arndt from theUniversity of Du¨sseldorf (in personal communication, 2020). He argues that,despite the fact that only the number 1 is used to define oppositions, theone-dimensional character of the construction appears in the exponents. Thisrestores, therefore, one-dimensionality. It is somewhat reasonable to acceptthat there is, indeed, a parasitary connection between oppositions withoutdimensions defined in a point and those defined in a line segment. The worthand essential remark here is that all representations used of the number 1refer to the same object, and logical oppositions make use of only a singlemathematical dot to formulate all oppositions in a zero-dimensional diagram.The construction developed shows the zero-dimensional point of opposition,though it is dependent of a line segment in the exponents, what decides thequestion is the dimension of the diagram, and in the case of the point ofopposition, it is its base which tell us in which dimension the diagram is, notthe exponents. So, it is legitimate to speak about oppositions without dimen-sions. A next step would be to find a zero-dimensional point of oppositionwithout any kind of dependence. Moreover, it is important to say that theconstructions suggested in [3] and the method here are not unique ways toreduce dimensions of diagrams, maybe some other possible reductions couldalso be achieved.Two-dimensional or three-dimensional diagrams to represent oppositionsare very well-known, and they have been used widely in the literature (espe-cially squares). The work of Alessio Moretti in [5] contains the foundationsof what is called n-opposition theory with some exploitations concerning di-mensions of simplexes, but there is no attempt there to define all oppositions5simultaneously in a given simplex of dimension one or zero. Again, con-cerning the plurality of dimensions, a natural ontological point is to checkwhether Ockham’s razor should be applied in the case of diagrams. Whyshould we proliferate dimensions using complicated diagrams if we can usea line or a point? Is there any special reason for this fact? The existence ofsimple diagrammatic abbreviations such as a line segment of opposition anda point of opposition challenges the limits of the researches on the square andits extensions.References[1] Beziau, J-Y. (2012). The power of the hexagon. Logica Universalis, 6(1),pp.1-43.[2] Blanche´, R. (1966). Structures intellectuelles. Essai sur l’organisationsyste´matique des concepts. Paris: Vrin.[3] Costa-Leite, A. (2018). Oppositions in a line segment. South AmericanJournal of Logic, 4(1), pp.185-193.[4] Moretti, A. (2004). 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, pp. 102-145.[5] Moretti, A. (2012). From the “Logical Square” to the “Logical Poly-Simplexes”. In: The Square of Opposition: a general framework for cog-nition. J-Y. Beziau and G. Payette (editors), pp. 119-156. Bern: PeterLang.[6] Schang, F. (2018). End of the Square? South American Journal of Logic,4(2), pp.485-505.6

Oppositions in a point

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