Analogical Proportions and Square of Oppositions

Abstract

The paper discusses analogical proportions in relation with the square of oppositions, a classical structure in Ancient logic which is related to the different forms of statements that may be involved in deductive syllogisms. The paper starts with a short reminder on the logical modeling of analogical proportions, viewed here as Boolean expressions expressing similarities and possibly differences between four items, as in the statement “a is to b as c is to d”. The square of oppositions and its hexagon-based extension is then restated in a knowledge representation perspective. It is observed that the four vertices of a square of oppositions form a constrained type of analogical proportion that emphasizes differences. In fact, the different patterns making an analogical proportion true can be covered by a square of oppositions or by a “square of agreement”, leading to disjunctive expressions of the analogical proportion. Besides, an “analogical octagon” is shown to capture the general construction of an analogical proportion from two sets of properties. Since the square of oppositions offers a common setting relevant for syllogisms and analogical proportions, it also provides a basis for the discussion of the possible interplay between deductive arguments and analogical arguments.