Logics for the Relational Syllogistic

TL;DR

This paper explores extending classical syllogistic logic to relational fragments, analyzing their proof systems and computational complexity to better handle relational facts beyond traditional Aristotelian logic.

Contribution

It introduces and studies various relational syllogistic fragments, examining their proof systems and complexity, and assesses the necessity of reductio ad absurdum in these logics.

Findings

Certain relational fragments admit syllogistic proof systems.

Some fragments require reductio ad absurdum for completeness.

The computational complexity varies across different fragments.

Abstract

The Aristotelian syllogistic cannot account for the validity of many inferences involving relational facts. In this paper, we investigate the prospects for providing a relational syllogistic. We identify several fragments based on (a) whether negation is permitted on all nouns, including those in the subject of a sentence; and (b) whether the subject noun phrase may contain a relative clause. The logics we present are extensions of the classical syllogistic, and we pay special attention to the question of whether reductio ad absurdum is needed. Thus our main goal is to derive results on the existence (or non-existence) of syllogistic proof systems for relational fragments. We also determine the computational complexity of all our fragments.