  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.
