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

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