The Aristotelian square of oppositions is a well-known diagram in logic and linguistics. In recent years, several extensions of the square have been discovered. However, these extensions have failed to become as widely known as the square. In this paper we argue that there is indeed a fundamental difference between the square and its extensions, viz., a difference in informativity. To do this, we distinguish between concrete Aristotelian diagrams and, on a more abstract level, the Aristotelian geometry. We then introduce (...) two new logical geometries, and develop a formal, well-motivated account of their informativity. This enables us to show that the square is strictly more informative than many of the more complex diagrams. (shrink)

Given a 4-tuple of Boolean variables (a, b, c, d), logical proportions are modeled by a pair of equivalences relating similarity indicators ( \({a \wedge b}\) and \({\overline{a} \wedge \overline{b}}\) ), or dissimilarity indicators ( \({a \wedge \overline{b}}\) and \({\overline{a} \wedge b}\) ) pertaining to the pair (a, b), to the ones associated with the pair (c, d). There are 120 semantically distinct logical proportions. One of them models the analogical proportion which corresponds to a statement of the form “a (...) is to b as c is to d”. The paper inventories the whole set of logical proportions by dividing it into five subfamilies according to what they express, and then identifies the proportions that satisfy noticeable properties such as full identity (the pair of equivalences defining the proportion hold as true for the 4-tuple (a, a, a, a)), symmetry (if the proportion holds for (a, b, c, d), it also holds for (c, d, a, b)), or code independency (if the proportion holds for (a, b, c, d), it also holds for their negations \({{(\overline{a},\overline{b}, \overline{c}, \overline{d})}}\) ). It appears that only four proportions (including analogical proportion) are homogeneous in the sense that they use only one type of indicator (either similarity or dissimilarity) in their definition. Due to their specific patterns, they have a particular cognitive appeal, and as such are studied in greater details. Finally, the paper provides a discussion of the other existing works on analogical proportions. (shrink)

Some Carrollian posthumous manuscripts reveal, in addition to his famous ‘logical diagrams’, two mysterious ‘logical charts’. The first chart, a strange network making out of fourteen logical sentences a large 2D ‘triangle’ containing three smaller ones, has been shown equivalent—modulo the rediscovery of a fourth smaller triangle implicit in Carroll's global picture—to a 3D tetrahedron, the four triangular faces of which are the 3+1 Carrollian complex triangles. As it happens, such an until now very mysterious 3D logical shape—slightly deformed—has been (...) rediscovered, independently from Carroll and much later, by a logician , a mathematician and a linguist studying the geometry of the ‘opposition relations’, that is, the mathematical generalisations of the ‘logical square’. We show that inside what is called equivalently ‘n-opposition theory’, ‘oppositional geometry’ or ‘logical geometry’, Carroll's first chart corresponds exactly, duly reshap.. (shrink)

Based on an analysis of the category of “infinite judgments” in Kant, we will introduce the logical hexagon of predicate negation. This hexagon allows us to visualize in a single diagram the general structure of both Kant’s solution of the antinomies of pure reason and his argument in favor of Transcendental Idealism.

The paper submits surprising results of systematical investigating a formal-ethical aspect of conjoining Wittgenstein’s, Moore’s, Parmenides’, Gödel’s, and Łukasiewicz’s ideas. A critique of Wittgenstein’s critique of the natural language of ethics and of metaphysics results in submitting and elaborating a new paradigm of metaphysics as formal axiology. In result, the classical metaphysics and ethics of moral rigor are represented as two-valued algebraic systems of metaphysics and formal ethics respectively. By means of this algebraic model, all the well-known scandal-making metaphysical tenets (...) of Parmenides are produced as translations of corresponding algebraic equations from the symbolic language to the natural one. At the level of submitted discrete mathematical model of formal axiology, Parmenides’ metaphysical concepts “consistency” and “inconsistency,” “completeness” and “incompleteness” are compared with Gödel’s logic ones. Formal-axiological meanings of the words “consistency,” “incompleteness,” “being,” “nonbeing,” “movement,” “knowledge,” “belief,” etc., are considered as moral-evaluation-functions determined by one moral-evaluation-variable. Binary moral-evaluation-functions are studied as well. The functions are precisely defined by tables. Precise definitions of “formal-axiological-equivalence,” “formal-axiological-law,” and “formal-axiological contradiction” are submitted. Thus, one can either generate or examine formal-axiological equations of algebra of metaphysics by “computing” relevant compositions of moral-value-functions. Using this “moral-value-table-computation-technique,” one can arrive to a surprising conclusion that both the notorious sentence of Moore and the incompleteness sentence of Gödel are formally-axiologically inconsistent ones: Hence, they are formally-axiologically equivalent. For overcoming the negative psychological effect of such a surprising result, the author has used graphic models explicating the famous Łukasiewicz’s statement “Logic is morality of thought and speech.”. (shrink)

We revisit a typological puzzle due to Horn (Doctoral Dissertation, UCLA, 1972) regarding the lexicalization of logical operators: in instantiations of the traditional square of opposition across categories and languages, the O corner, corresponding to ‘nand’ (= not and), ‘nevery’ (= not every), etc., is never lexicalized. We discuss Horn’s proposal, which involves the interaction of two economy conditions, one that relies on scalar implicatures and one that relies on markedness. We observe that in order to express markedness and to (...) account for a bigger typological puzzle, namely the absence of lexicalizations of ‘XOR’ (= exclusive or), ‘all-or-none’, and many other imaginable logical operators, one must restrict the basic lexicalizable elements to a small set of primitives. We suggest that an ordering based perspective, following Keenan and Faltz (Boolean semantics for natural language, 1985), makes the stipulated primitives that we arrive at more natural. We also propose a modification to Horn’s proposal, based on recent work on implicatures, in which only the implicature condition is operative and in which markedness is part of the definition of the alternatives for scalar implicatures rather than an independent condition. (shrink)

This discussion explores the possibility of distinguishing a tighter notion of contrariety evident in the Square of Opposition, especially in its modal incarnations, than as that binary relation holding statements that cannot both be true, with or without the added rider ‘though can both be false’. More than one theorist has voiced the intuition that the paradigmatic contraries of the traditional Square are related in some such tighter way—involving the specific role played by negation in contrasting them—that distinguishes them from (...) other pairs of incompatible statements constructed from the same conceptual materials. Prominent among examples, these other nonstandard pairs are the ‘new contraries’ presented by Robert Blanché’s hexagon(s) of opposition. With special, though not exclusive, attention to these cases, we investigate whether contrariety in the distinguished sense can be captured by adding to the incompatibility condition the further demand that the pair of statements concerned can be represented as the results of applying some sentence operator to the content in its scope, for one of the pair, and, for the other, the application of that same operator to the negation of that content. For one of the two cases, a Blanché case, of nonstandard contrariety singled out for attention, the question of whether such a representation is available is settled at the end of Section 4, and then in a more satisfying way in Section 5, though for the other case, noticed by Peter Simons, the question remains open, after some tentative discussion in one subsection, 6.2, of an Appendix (Section 6). (shrink)

The hexagon of opposition is an improvement of the square of opposition due to Robert Blanché. After a short presentation of the square and its various interpretations, we discuss two important problems related with the square: the problem of the I-corner and the problem of the O-corner. The meaning of the notion described by the I-corner does not correspond to the name used for it. In the case of the O-corner, the problem is not a wrong-name problem but a no-name (...) problem and it is not clear what is the intuitive notion corresponding to it. We explain then that the triangle of contrariety proposed by different people such as Vasiliev and Jespersen solves these problems, but that we don’t need to reject the square. It can be reconstructed from this triangle of contrariety, by considering a dual triangle of subcontrariety. This is the main idea of Blanché’s hexagon. We then give different examples of hexagons to show how this framework can be useful to conceptual analysis in many different fields such as economy, music, semiotics, identity theory, philosophy, metalogic and the metatheory of the hexagon itself. We finish by discussing the abstract structure of the hexagon and by showing how we can swing from sense to non-sense thinking with the hexagon. (shrink)

Duality in Logic and Language [draft--do not cite this article] Duality phenomena occur in nearly all mathematically formalized disciplines, such as algebra, geometry, logic and natural language semantics. However, many of these disciplines use the term ‘duality’ in vastly different senses, and while some of these senses are intimately connected to each other, others seem to be entirely … Continue reading Duality in Logic and Language →.

The aim of this paper is to elucidate the relationship between Aristotelian conceptual oppositions, commutative diagrams of relational structures, and Galois connections.This is done by investigating in detail some examples of Aristotelian conceptual oppositions arising from topological spaces and similarity structures. The main technical device for this endeavor is the notion of Galois connections of order structures.