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2018 | OriginalPaper | Chapter

Aristotelian and Duality Relations Beyond the Square of Opposition

Authors : Lorenz Demey, Hans Smessaert

Published in: Diagrammatic Representation and Inference

Publisher: Springer International Publishing

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Abstract

Nearly all squares of opposition found in the literature represent both the Aristotelian relations and the duality relations, and exhibit a very close correspondence between both types of logical relations. This paper investigates the interplay between Aristotelian and duality relations in diagrams beyond the square. In particular, we study a Buridan octagon, a Lenzen octagon, a Keynes-Johnson octagon and a Moretti octagon. Each of these octagons is a natural extension of the square, both from an Aristotelian perspective and from a duality perspective. The results of our comparative analysis turn out to be highly nuanced.

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previous chapter Iconic Logic and Ideal Diagrams: The Wittgensteinian Approach

next chapter Syllogistic with Jigsaw Puzzle Diagrams

Similar conclusions were reached in [41], but there, one could still object that the loss of correspondence beyond the square is merely due to the fact that from a duality perspective, the hexagon is not a natural generalization of the square of opposition. Such an objection cannot be raised against the conclusions drawn in this paper.

Note that the definitions of contrariety and subcontrariety can both be seen as weakened versions of that of contradiction. It can also be shown that contradiction is the most informative of the Aristotelian relations [43].

Furthermore, the contradiction relation is usually visualized by means of central symmetry, so that all pairs of contradictory propositions are represented by diagonals that intersect each other in the Aristotelian diagram’s center of symmetry [10, 12, 14].

In this paper we will not distinguish between different geometrical representations of the same set of PCDs. For example: (i) 3 PCDs can be visualized as a hexagon or as an octahedron; (ii) 4 PCDs can be visualized as an octagon or as a cube [12, 14].

Under the group-theoretical isomorphism between the Klein 4-group for the duality functions and \(\mathbb {Z}_2 \times \mathbb {Z}_2\), id corresponds to (0, 0) (apply no negations at all), eneg to (1, 0) (only apply external negation), ineg to (0, 1) (only apply internal negation) and dual to (1, 1) (apply both external and internal negation). If the duality function f corresponds to \((i,j) \in \mathbb {Z}_2 \times \mathbb {Z}_2\), we thus get \(f(O(\alpha _1,\dots ,\alpha _n)) = \lnot ^i O(\lnot ^j\alpha _1,\dots ,\lnot ^j\alpha _n)\) (with the usual definitions \(\lnot ^0 \varphi := \varphi \) and \(\lnot ^1\varphi := \lnot \varphi \)).

Note that the ADM includes EQ (logical equivalence) as the Aristotelian counterpart of id. Strictly speaking, EQ is not one of the Aristotelian relations, but it is closely related to them [43], and it is implicitly present whenever we write multiple, logically equivalent propositions in a single vertex of an Aristotelian diagram. (Each vertex thus has an EQ-loop to itself.) Note, in this context, that the square of opposition is sometimes also referred to as ‘the square of opposition and equipollence’ [33].

This rhyme is incomplete, because as we have seen above (Fig. 2), internal negation (post) should not just be associated with contrariety, but also with subcontrariety [29, Footnote 54]. However, this omission can be explained in terms of the famous non-lexicalization of the O-corner [22]. The fact that no A are B is the internal negation of all A are B (i.e. no \(\equiv \) all \(\lnot \), or in Latin: nullus \(\equiv \) omnis \(\lnot \)) is a contingent, empirical fact about English (resp. Latin), and should thus be captured by the rhyme. By contrast, the fact that some A are not B is the internal negation of some A are B (i.e. some not \(\equiv \) some \(\lnot \), or in Latin: aliquis non \(\equiv \) aliquis \(\lnot \)) is almost analytically true, and thus need not be captured by the mnemotechnic rhyme.

For reasons of space, we only consider the modal hexagon, which extends the modal square in Fig. 1(b). In exactly the same way, one could also extend the other two squares in Fig. 1(a)/(c) to hexagons, and draw the same conclusions about them.

If \(O_2\) is n-ary, the composed operator \(O_1\circ O_2\) will also be n-ary. Furthermore, \(O_1\) will be assumed to be unary, but this assumption is not essential.

To avoid cluttering the diagrams, we will henceforth not explicitly show the CD- and eneg-relations. These occur exactly at the diagram’s diagonals, which intersect each other in the diagram’s center of symmetry (recall Footnote 3).

Compare the ADMs for the modal JSB hexagon and Buridan’s modal octagon in Figs. 4 and 6, and note the absence of \(\varnothing \) in the latter.

In general, for an n-ary operator, we have \(n+1\) independent negation positions, viz. 1 external negation and n internal negations (one for each argument position).

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Title: Aristotelian and Duality Relations Beyond the Square of Opposition
Authors: Lorenz Demey
Hans Smessaert
Publisher: Springer International Publishing
Book: Diagrammatic Representation and Inference
Print ISBN: 978-3-319-91375-9

Electronic ISBN: 978-3-319-91376-6

Copyright Year: 2018
DOI: https://doi.org/10.1007/978-3-319-91376-6_57

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