Aristotle's square of opposition in the light of Hilbert's epsilon and tau quantifiers

TL;DR

This paper reinterprets Aristotle's square of opposition using Hilbert's epsilon and tau quantifiers, proposing a logical framework that better models natural language quantifiers and their classical logical relations.

Contribution

It introduces a novel interpretation of Aristotle's quantified sentences through epsilon and tau calculus, bridging natural language and modern logical formalism.

Findings

Two potential squares of opposition are identified.

Under certain conditions, one square satisfies classical opposition relations.

The approach offers a more faithful logical representation of natural language quantifiers.

Abstract

Aristotle considered particular quantified sentences in his study of syllogisms and in his famous square of opposition. Of course, the logical formulas in Aristotle work were not modern formulas of mathematical logic, but ordinary sentences of natural language. Nowadays natural language sentences are turned into formulas of predicate logic as defined by Frege, but, it is not clear that those Fregean sentences are faithful representations of natural language sentences. Indeed, the usual modelling of natural language quantifiers does not fully correspond to natural language syntax, as we shall see. This is the reason why Hilbert's epsilon and tau quantifiers (that go beyond usual quantifiers) have been used to model natural language quantifiers. Here we interpret Aristotle quantified sentences as formulas of Hilbert's epsilon and tau calculus. This yields to two potential squares of opposition and provided a natural condition holds, one of these two squares is actually a square of opposition i.e. satisfies the relations of contrary, contradictory, and subalternation.