Relational completeness of a set of syllogistic constants

Abstract

Formerly, we offered the language and set theoretical semantics of positive syllogistic the alphabet of which contains all syllogistic constants. These constants are interpreted as the signs of different relationships between two non-empty sets (the extensions of general terms). Among these relationships we give accent to ‘Eulerian’ relationships: (1) equality, (2) strict inclusion of the first set into the second, (3) strict inclusion of the second set into the first, (4) overlap, (5) exclusion. The other relationships can be represented as various combinations of Eulerian. Every constant $k$ in the ‘universal’ syllogistic language is encoded with a sequence of numbers from 1 to 5 in accordance with the diagrams where the proposition of the form $SkP$ is true. We introduce the notion of the definability of a syllogistic constant in a ‘local’ syllogistic language that contains only some of such constants, and the notion of the relational completeness of a set of initial constants in a ‘local’ language. We set out the completeness criteria for a set of syllogistic constants. We prove that such a set is relationally complete iff it contains a constant including either 2 or 3, a constant including either 1 or 4, a constant including either 1 or 5, a constant including either 4 or 5.