Logical Extensions of Aristotle’s Square (original) (raw)

Logical

2015

We start from the geometrical-logical extension of Aristotle's square in [Bla66], [Pel06] and [Mor04], and study them from both syntactic and semantic points of view. Recall that Aristotle's square under its modal form has the following four vertices: A is α, E is ¬α, I is ¬ ¬α and O is ¬ α, where α is a logical formula and is a modality which can be defined axiomatically within a particular logic known as S5 (classical or intuitionistic, depending on whether ¬ is involutive or not) modal logic. [Béz03] has proposed extensions which can be interpreted respectively within paraconsistent and paracomplete logical frameworks. [Pel06] has shown that these extensions are subfigures of a tetraicosahedron whose vertices are actually obtained by closure of {α, α} by the logical operations {¬, ∧, ∨}, under the assumption of classical S5 modal logic. We pursue these researches on the geometrical-logical extensions of Aristotle's square: first we list all modal squares of opposition. We show that if the vertices of that geometrical figure are logical formulae and if the sub-alternation edges are interpreted as logical implication relations, then the underlying logic is none other than classical logic. Then we consider a higher-order extension introduced by [Mor04], and we show that the same tetraicosahedron plays a key role when additional modal operators are introduced. Finally we discuss the relation between the logic underlying these extensions and the resulting geometrical-logical figures.

The Modal Logic of Aristotelian Diagrams

Axioms

In this paper, we introduce and study AD-logic, i.e., a system of (hybrid) modal logic that can be used to reason about Aristotelian diagrams. The language of AD-logic, LAD, is interpreted on a kind of birelational Kripke frames, which we call “AD-frames”. We establish a sound and strongly complete axiomatization for AD-logic, and prove that there exists a bijection between finite Aristotelian diagrams (up to Aristotelian isomorphism) and finite AD-frames (up to modal isomorphism). We then show how AD-logic can express several major insights about Aristotelian diagrams; for example, for every well-known Aristotelian family A, we exhibit a formula χA∈LAD and show that an Aristotelian diagram D belongs to the family A iff χA is validated by D (when the latter is viewed as an AD-frame). Finally, we show that AD-logic itself gives rise to new and interesting Aristotelian diagrams, and we reflect on their profoundly peculiar status.

Logical Squares for Classical Logic Sentences

Logica Universalis, 2016

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 (CPC). We illustrate them by means of many squares of opposition and, corresponding to them-octagons, hexagons or other geometrical objects.

Logic-Sensitivity of Aristotelian Diagrams in Non-Normal Modal Logics

Axioms, 2021

Aristotelian diagrams, such as the square of opposition, are well-known in the context of normal modal logics (i.e., systems of modal logic which can be given a relational semantics in terms of Kripke models). This paper studies Aristotelian diagrams for non-normal systems of modal logic (based on neighborhood semantics, a topologically inspired generalization of relational semantics). In particular, we investigate the phenomenon of logic-sensitivity of Aristotelian diagrams. We distinguish between four different types of logic-sensitivity, viz. with respect to (i) Aristotelian families, (ii) logical equivalence of formulas, (iii) contingency of formulas, and (iv) Boolean subfamilies of a given Aristotelian family. We provide concrete examples of Aristotelian diagrams that illustrate these four types of logic-sensitivity in the realm of normal modal logic. Next, we discuss more subtle examples of Aristotelian diagrams, which are not sensitive with respect to normal modal logics, but...

GENERALIZATIONS AND COMPOSITIONS OF MODAL SQUARES OF OPPOSITION (second draft)

The paper aims at showing that the notion of an Aristotelian square may be seen as a special case of a variety of different more general notions:1) the one of a subAristotelian square, 2) the one of a semiAristotelian square 3)the one of an Aristotelian cube, which is a construction made up of six semiAristotelian squares, two of which are Aristotelian. Furthermore, if an Aristotelian square is seen as a special ordered 4-tuple of formulas, there are four 4-tuples with the same properties describing different rotations of the same square, i.e. of the standard Aristotelian square. The second part of the paper focuses the notion of a composition of squares. After discussing some possible alternative definitions, a privileged notion of composition is identified, thus opening the road to introducing and discussing the wider notion of composition of cubes.

Logical Geometries and Information in the Square of Oppositions

Journal of Logic, Language and Information, 2014

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 (such as the square) and, on a more abstract level, the Aristotelian geometry (a set of logical relations). We then introduce two new logical geometries (and their corresponding diagrams), 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.

A pr 2 01 4 The Square of Opposition in Orthomodular Logic

2020

In Aristotelian logic, categorical propositions are divided in Universal Affirmative, Universal Negative, Particular Affirmative and Particular Negative. Possible relations between two of the mentioned type of propositions are encoded in the square of opposition. The square expresses the essential properties of monadic first order quantification which, in an algebraic approach, may be represented taking into account monadic Boolean algebras. More precisely, quantifiers are considered as modal operators acting on a Boolean algebra and the square of opposition is represented by relations between certain terms of the language in which the algebraic structure is formulated. This representation is sometimes called the modal square of opposition. Several generalizations of the monadic first order logic can be obtained by changing the underlying Boolean structure by another one giving rise to new possible interpretations of the square. Mathematics Subject Classification (2000). 03G12; 06C1...

Béziau’s Contributions to the Logical Geometry of Modalities and Quantifiers

Studies in Universal Logic, 2015

The aim of this paper is to discuss and extend some of Béziau's (published and unpublished) results on the logical geometry of the modal logic S5 and the subjective quantifiers many and few. After reviewing some of the basic notions of logical geometry, we discuss Béziau's work on visualising the Aristotelian relations in S5 by means of two-and three-dimensional diagrams, such as hexagons and a stellar rhombic dodecahedron. We then argue that Béziau's analysis is incomplete, and show that it can be completed by considering another three-dimensional Aristotelian diagram, viz. a rhombic dodecahedron. Next, we discuss Béziau's proposal to transpose his results on the logical geometry of the modal logic S5 to that of the subjective quantifiers many and few. Finally, we propose an alternative analysis of many and few, and compare it with that of Béziau's. While the two analyses seem to fare equally well from a strictly logical perspective, we argue that the new analysis is more in line with certain linguistic desiderata.