Structures of Opposition and Comparisons: Boolean and Gradual Cases

Abstract

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 (hexagon, cubes) for discussing the comparison of two items. Comparing two items (objects, images) 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.

Approaching the Square of Oppositions in Terms of the f -indexes of Inclusion and Contradiction

“…Further work in this line will focus on trying to relate our work with other approaches. On the one hand, one can find gradual versions of the square (and other structures) of opposition in the literature: in [6] the square is in fact a 4-tuple of truth-degrees which should fulfill certain relations among them in order to properly generalize the different notions of opposition, in [10] graded structures of opposition are used in fuzzy natural logic. On the other hand, in this paper, we identify the segment AO with the index of inclusion, and the segment EI as the index of contradiction, and the obtained results allow extracting information in terms of opposition.…”

“…Due to the features above, the square of opposition has received some criticisms, but it is still considered as a reference for current research on this topic [5,6]. In fact, in those works different opposition structures are studied both from a classical and a fuzzy standpoint.…”

A Formal Analysis of Generalized Peterson’s Syllogisms Related to Graded Peterson’s Cube

“…At this point, we also recall the classic cubes of opposition that were proposed by Moretti and Keyne. Gradual extensions to these two structures have been made and deeply studied by Dubois et al in [36]. In the possibility theory, a graded extension of these cubes of opposition was analyzed by Dubois in [37].…”

Another Side of Categorical Propositions: The Keynes–Johnson Octagon of Oppositions

“…But Aristotle never included this case into his own syllogistic, where subject terms are always affirmed. On the other hand, a number of contemporary works extended the famous square of opposition into a logical cube (Dekker 2015;Demey and Smessaert 2018;Dubois et al 2020), without always mentioning the central contribution to Keynes in this respect by revealing the 'other side' of categorical propositions (whenever the subject term is denied). Our point is not to implement these formal developments but, rather, to present a complete analysis of Keynes' octagon in three steps: first, a historical introduction to Keynes' logical achievements (Section 1); second, a look at the logical structure that was pioneered by Keynes (Section 2); and third, a formal semantics suggested to make sense of the logical relations between the eight propositions of Keynes' octagon.…”