A 32-vertex generalization of the logical square: "Logical Lantern" for propositions in V.I. Markin's Universal language for traditional positive syllogistic theories

Abstract

In this work is constructed a visual representation for logical relations (contradiction, subalternation, contrariety, subcontrariety) among propositions about each of the possible relations between two non-empty sets. Those relations between sets are equity, strict inclusion, reversed strict inclusion, intersection, exclusion and their combinations (disjunctions); also is used the constant of falsity "none".This representation follows the results by V.I. Markin, who created a universal language for traditional positive syllogistic theories. This language includes syllogistic constants for propositions about each of the 32 possible relations between two non-empty sets (5 basic constants; complex constants, constructed from basic ones; constants for truth and falsity). Among these 32 constants are the traditional vertices of the square of opposition: A ("All S are P"), E ("No S are P"), I ("Some S are P"), O ("Some S are not P").The proposed scheme shows contradiction and subalternation and allows to deduce contrariety and subcontrariety relations among the propositions in question. It allows to easily find all the propositions that are in a given relation with some (arbitrary) chosen proposition. It also allows to find the relation between two chosen propositions. The scheme also helps to think about the square of opposition and similar diagrams on a meta level. Among the things it helps to think about are relations that are similar to those in the square of opposition, although not themselves present in the square. They are what we called "exhaustive n-place contrariety" and "exhaustive n-place subcontrariety" (n is a natural number, n is greater than 2) – logical relations that are similar to usual contrariety (subcontrariety), but hold among more than two propositions and have an additional property: all the n propositions in this relation taken together are both truth-incompatible and falsity-incompatible.The proposed scheme can be used as a whole, or in parts (in a reduced way) - as diagrams that would show the logical relations among some but not all the possible propositions about relations of two non-empty sets.Such diagrams can be seen as variations of the Lantern, and the Lantern – as their generalization. One of the variations is the square of opposition, one more – Blanché hexagon. Other possible variations are more or less similar to the ones just mentioned and can have the same or different properties, making the Lantern – together with V.I. Markin's Universal language for traditional positive syllogistic theories, on which the Lantern is based, – a "common denominator", an instrument to compare and study the properties of different diagrams.