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)

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 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 central aim of this paper is to present a Boolean algebraic approach to the classical Aristotelian Relations of Opposition, namely Contradiction and (Sub)contrariety, and to provide a 3D visualisation of those relations based on the geometrical properties of Platonic and Archimedean solids. In the first part we start from the standard Generalized Quantifier analysis of expressions for comparative quantification to build the Comparative Quantifier Algebra CQA. The underlying scalar structure allows us to define the Aristotelian relations in Boolean terms (...) and to propose a 3D visualisation by transforming a cube into an octahedron. In part two, the architecture of the CQA is shown to carry over, both to the classical quantifiers of Predicate Calculus and to the modal operators—which are given a Generalized Quantifier style re-interpretation. In this way we provide an algebraic foundation for Blanché’s Aristotelian hexagon as well as a 3D alternative to his 2D star-like visualisation. In a final part, a richer scalar structure is argued to underly the realm of Modality, thus generalizing the 3D algebra with eight (2 3 ) operators to a 4D algebra with sixteen (2 4 ) operators. The visual representation of the latter structure involves a transformation of the hypercube to a rhombic dodecahedron. The resulting 3D visualisation allows a straightforward embedding, not only of the classical Blanché star of Aristotelian relations or the paracomplete and paraconsistent stars of Béziau (Log Investig 10, 218–232, 2003) but also of three additional isomorphic Aristotelian constellations. (shrink)

Whereas geometrical oppositions (logical squares and hexagons) have been so far investigated in many fields of modal logic (both abstract and applied), the oppositional geometrical side of “deontic logic” (the logic of “obligatory”, “forbidden”, “permitted”, . . .) has rather been neglected. Besides the classical “deontic square” (the deontic counterpart of Aristotle’s “logical square”), some interesting attempts have nevertheless been made to deepen the geometrical investigation of the deontic oppositions: Kalinowski (La logique des normes, PUF, Paris, 1972) has proposed a (...) “deontic hexagon” as being the geometrical representation of standard deontic logic, whereas Joerden (jointly with Hruschka, in Archiv für Rechtsund Sozialphilosophie 73:1, 1987), McNamara (Mind 105:419, 1996) and Wessels (Die gute Samariterin. Zur Struktur der Supererogation, Walter de Gruyter, Berlin, 2002) have proposed some new “deontic polygons” for dealing with conservative extensions of standard deontic logic internalising the concept of “supererogation”. Since 2004 a new formal science of the geometrical oppositions inside logic has appeared, that is “ n -opposition theory”, or “NOT”, which relies on the notion of “logical bi-simplex of dimension m ” ( m = n − 1). This theory has received a complete mathematical foundation in 2008, and since then several extensions. In this paper, by using it, we show that in standard deontic logic there are in fact many more oppositional deontic figures than Kalinowski’s unique “hexagon of norms” (more ones, and more complex ones, geometrically speaking: “deontic squares”, “deontic hexagons”, “deontic cubes”, . . ., “deontic tetraicosahedra”, . . .): the real geometry of the oppositions between deontic modalities is composed by the aforementioned structures (squares, hexagons, cubes, . . ., tetraicosahedra and hyper-tetraicosahedra), whose complete mathematical closure happens in fact to be a “deontic 5-dimensional hyper-tetraicosahedron” (an oppositional very regular solid). (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)

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)

In this paper, with reference to relationships of the traditional square of opposition, we establish all the relations of the square of opposition between complex sentences built from the 16 binary and four unary propositional connectives of the classical propositional calculus. We illustrate them by means of many squares of opposition and, corresponding to them—octagons, hexagons or other geometrical objects.

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)

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)

In recent years, a number of authors have started studying Aristotelian diagrams containing metalogical notions, such as tautology, contradiction, satisfiability, contingency, strong and weak interpretations of contrariety, etc. The present paper is a contribution to this line of research, and its main aims are both to extend and to deepen our understanding of metalogical diagrams. As for extensions, we not only study several metalogical decorations of larger and less widely known Aristotelian diagrams, but also consider metalogical decorations of another type (...) of logical diagrams, viz. duality diagrams. At a more fundamental level, we present a unifying perspective which sheds new light on the connections between new and existing metalogical diagrams, as well as between object- and metalogical diagrams. Overall, the paper studies two types of logical diagrams and four kinds of metalogical decorations. (shrink)

Logical geometry systematically studies Aristotelian diagrams, such as the classical square of oppositions and its extensions. These investigations rely heavily on the use of bitstrings, which are compact combinatorial representations of formulas that allow us to quickly determine their Aristotelian relations. However, because of their general nature, bitstrings can be applied to a wide variety of topics in philosophical logic beyond those of logical geometry. Hence, the main aim of this paper is to present a systematic technique for assigning bitstrings (...) to arbitrary finite fragments of formulas in arbitrary logical systems, and to study the logical and combinatorial properties of this technique. It is based on the partition of logical space that is induced by a given fragment, and sheds new light on a number of interesting issues, such as the logic-dependence of the Aristotelian relations and the subtle interplay between the Aristotelian and Boolean structure of logical fragments. Finally, the bitstring technique also allows us to systematically analyze fragments from contemporary logical systems, such as public announcement logic, which could not be done before. (shrink)