Syllogisms in Rudimentary Linear Logic, Diagrammatically

TL;DR

This paper interprets traditional syllogisms within a fragment of propositional linear logic, demonstrating that their provability aligns with diagrammatic reasoning, including extensions with complemented terms.

Contribution

It introduces a diagrammatic calculus for syllogisms in linear logic and extends it to De Morgan-style syllogistics, establishing equivalence between logical and diagrammatic provability.

Findings

Syllogisms are provable iff diagrammatically provable in the fragment.

Extension to De Morgan syllogistics is successful.

Diagrammatic calculus accurately captures syllogistic reasoning in linear logic.

Abstract

We present a reading of the traditional syllogistics in a fragment of the propositional intuitionistic multiplicative linear logic and prove that with respect to a diagrammatic logical calculus that we introduced in a previous paper, a syllogism is provable in such a fragment if and only if it is diagrammatically provable. We extend this result to syllogistics with complemented terms \`a la De Morgan, with respect to a suitable extension of the diagrammatic reasoning system for the traditional case and a corresponding reading of such De Morgan style syllogistics in the previously referred to fragment of linear logic.