The objects called cubes of opposition have been presented in the literature in discordant ways. The aim of the paper is to offer a survey of such various kinds of cubes and evaluate their relation with an object, here called “Aristotelian cube”, which consists of two Aristotelian squares and four squares which are semiaristotelian, i.e. are such that their vertices are linked by some so-called Aristotelian relation. Two paradigm cases of Aristotelian squares are provided by propositions written in the language (...) of the logic of consequential implication, whose distinctive feature is the validity of two formulas, A $$\rightarrow $$ → B $$\supset \lnot $$ ⊃ ¬ (A $$\rightarrow $$ → $$\lnot $$ ¬ B) and A $$\rightarrow $$ → B $$\supset \lnot $$ ⊃ ¬ ($$\lnot $$ ¬ A $$\rightarrow $$ → B), expressing two different forms of contrariety. Part of section 1 is devoted to define the notions of rotation and of r-Aristotelian square, i.e. a square resulting from some rotation of an Aristotelian square. In section 2 this notion is extended to the one of a r-Aristotelian cube, i.e. of a cube resulting from some rotation of some square of an Aristotelian cube. This notion is used in the sequel to analyze various cubes of oppositions which can be found in the literature: (1) the one used by W. Lenzen to reconstruct Caramuel’s Octagon; (2) the one used by D. Luzeaux to represent the implicative relation among S5-modalities; (3) the one introduced by D. Dubois to represent the relations between quantified propositions containing positive predicates and their negations; (4) the one called Moretti cube. None of such cubes is strictly speaking Aristotelian but each of them may be proved to be r-Aristotelian. Section 5 discusses the assertion that Dubois cube was anticipated in a paper published by Reichenbach in 1952. Actually Dubois’ construction was anticipated by the so-called Johnson–Keynes cube, while the Reichenbach cube, unlike Dubois cube, is an instance of an Aristotelian cube in the sense defined in this paper. The dominance of such notion is confirmed by J.F. Nilsson’s cube, representing relations between propositions with nested quantifiers, and also by a cube introduced by S. Read to treat quantifiers with existential import. A cube similar to Read’s cube, introduced by Chatti and Schang, is shown to be r-Aristotelian. In section 6 the author remarks that the logic of the formulas occurring in the cubes of Chatti–Schang and Read have the drawback of not satsfying the law of Identity. He then proposes a definition of non-standard quantifiers which satisfies Identity, are independent of existential assumptions and such that their interrelations are represented by an Aristotelian cube. (shrink)

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)

Peters and Westerståhl (Quantifiers in Language and Logic, 2006), and Westerståhl (New Perspectives on the Square of Opposition, 2011) draw a crucial distinction between the “classical” Aristotelian squares of opposition and the “modern” Duality squares of opposition. The classical square involves four opposition relations, whereas the modern one only involves three of them: the two horizontal connections are fundamentally distinct in the Aristotelian case (contrariety, CR vs. subcontrariety, SCR) but express the same Duality relation of internal negation (SNEG). Furthermore, the (...) vertical relations in the classical square are unidirectional, whereas in the modern square they are bidirectional. The present paper argues that these differences become even bigger when two more operators are added, namely the U ( ${{\equiv} {\rm A}\,{\vee} \,{\rm E} }$ , all or no) and Y ( ${\equiv{\rm I} \,{\wedge} \,{\rm O}}$ , some but not all) of Blanché (Structures Intellectuelles, 1969). In the resulting Aristotelian hexagon the two extra nodes are perfectly integrated, yielding two interlocking triangles of CR and SCR. In the duality hexagon by contrast, they do not enter into any relation with the original square, but constitute a independent pair of their own, since they are their own SNEGs. Hence, they not only stand in a relation of external NEG, but also in one of duality. This reflexive nature of the SNEG will be shown to result in defective monotonicity configurations for the pair, namely the absence of right-monotonicity (on the predicate argument). In the second half of the paper, we present an overview of those hexagonal structures which are both Aristotelian and Duality configurations, and those which are only Aristotelian. (shrink)

In 1995 Slater argued both against Priest’s paraconsistent system LP (1979) and against paraconsistency in general, invoking the fundamental opposition relations ruling the classical logical square. Around 2002 Béziau constructed a double defence of paraconsistency (logical and philosophical), relying, in its philosophical part, on Sesmat’s (1951) and Blanche’s (1953) “logical hexagon”, a geometrical, conservative extension of the logical square, and proposing a new (tridimensional) “solid of opposition”, meant to shed new light on the point raised by Slater. By using n-opposition (...) theory (NOT) we analyse Beziau’s anti-Slater move and show both its right intuitions and its technical limits. Moreover, we suggest that Slater’s criticism is much akin to a well-known one by Suszko (1975) against the conceivability of many-valued logics. This last criticism has been addressed by Malinowski (1990) and Shramko and Wansing (2005), who developed a family of tenable logical counter-examples to it: trans-Suszkian systems are radically many-valued. This family of new logics has some strange logical features, essentially: each system has more than one consequence operator. We show that a new, deeper part of the aforementioned geometry of logical oppositions (NOT), the “logical poly-simplexes of dimension m”, generates new logical-geometrical structures, essentially many-valued, which could be a very natural (and intuitive) geometrical counterpart to the “strange”, new, non-Suszkian logics of Malinowski, Shramko and Wansing. By a similar move, the geometry of opposition therefore sheds light both on the foundations of paraconsistent logics and on those of many-valued logics. (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)

The logical hexagon (or hexagon of opposition) is a strange, yet beautiful, highly symmetrical mathematical figure, mysteriously intertwining fundamental logical and geometrical features. It was discovered more or less at the same time (i.e. around 1950), independently, by a few scholars. It is the successor of an equally strange (but mathematically less impressive) structure, the “logical square” (or “square of opposition”), of which it is a much more general and powerful “relative”. The discovery of the former did not raise interest, (...) neither among logicians, nor among philosophers of logic, whereas the latter played a very important theoretical role (both for logic and philosophy) for nearly two thousand years, before falling in disgrace in the first half of the twentieth century: it was, so to say, “sentenced to death” by the so-called analytical philosophers and logicians. Contrary to this, since 2004 a new, unexpected promising branch of mathematics (dealing with “oppositions”) has appeared, “oppositional geometry” (also called “n-opposition theory”, “NOT”), inside which the logical hexagon (as well as its predecessor, the logical square) is only one term of an infinite series of “logical bi-simplexes of dimension m”, itself just one term of the more general infinite series (of series) of the “logical poly-simplexes of dimension m”. In this paper we recall the main historical and the main theoretical elements of these neglected recent discoveries. After proposing some new results, among which the notion of “hybrid logical hexagon”, we show which strong reasons, inside oppositional geometry, make understand that the logical hexagon is in fact a very important and profound mathematical structure, destined to many future fruitful developments and probably bearer of a major epistemological paradigm change. (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)

Sceptics vis-à-vis introspection often base their scepticism on ‘phenomenological disputes’, ‘introspective disagreement’, or ‘introspective disputes’ (Kriegel, 2007; Bayne and Spener, 2010; Schwitzgebel, 2011): introspectors massively diverge in their opinions about experiences, and there seems to be no method to resolve these issues. Sceptics take this to show that introspection lacks any epistemic merit. Here, I provide a list of paradigmatic examples, distill necessary and sufficient conditions for IDs, present the sceptical argument encouraged by IDs, and review the two main strategies (...) to reject such a scepticism. However, both types of strategies are unsatisfactory. In order to save introspection from the looming sceptical threat, I advocate a deflationary strategy, based on either an ‘Argument from Perceptual Kinship’ or an ‘Argument from Ownership’. In the end, there cannot be any genuine IDs, for nothing can fulfil the reasonable conditions for IDs. What looks like IDs may instead be indicators of phenomenal variation. Debates that look like IDs may then arise even if introspection were a perfect method to know one’s mind. Thus, scepticism vis-à-vis introspection based on IDs rests on shaky grounds. (shrink)

This paper first investigates logical characterizations of different structures of opposition that extend the square of opposition in a way or in another. Blanché’s hexagon of opposition is based on three disjoint sets. There are at least two meaningful cubes of opposition, proposed respectively by two of the authors and by Moretti, and pioneered by philosophers such as J. N. Keynes, W. E. Johnson, for the former, and H. Reichenbach for the latter. These cubes exhibit four and six squares of (...) opposition respectively. We clarify the differences between these two cubes, and discuss their gradual extensions, as well as the one of the hexagon when vertices are no longer two-valued. The second part of the paper is dedicated to the use of these structures of opposition for discussing the comparison of two items. Comparing two items usually involves a set of relevant attributes whose values are compared, and may be expressed in terms of different modalities such as identity, similarity, difference, opposition, analogy. Recently, J.-Y. Béziau has proposed an “analogical hexagon” that organizes the relations linking these modalities. Elementary comparisons may be a matter of degree, attributes may not have the same importance. The paper studies in which ways the structure of the hexagon may be preserved in such gradual extensions. As another illustration of the graded hexagon, we start with the hexagon of equality and inequality due to R. Blanché and extend it with fuzzy equality and fuzzy inequality. Besides, the cube induced by a tetra-partition can account for the comparison of two items in terms of preference, reversed preference, indifference and non-comparability even if these notions are a matter of degree. The other cube, which organizes the relations between the different weighted qualitative aggregation modes, is more relevant for the attribute-based comparison of items in terms of similarity. (shrink)

The paper first introduces a cube of opposition that associates the traditional square of opposition with the dual square obtained by Piaget’s reciprocation. It is then pointed out that Blanché’s extension of the square-of-opposition structure into an conceptual hexagonal structure always relies on an abstract tripartition. Considering quadripartitions leads to organize the 16 binary connectives into a regular tetrahedron. Lastly, the cube of opposition, once interpreted in modal terms, is shown to account for a recent generalization of formal concept analysis, (...) where noticeable hexagons are also laid bare. This generalization of formal concept analysis is motivated by a parallel with bipolar possibility theory. The latter, albeit graded, is indeed based on four graded set functions that can be organized in a similar structure. (shrink)