Aristotle's square of opposition in the light of Hilbert's epsilon and   tau quantifiers
  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. 
Fabio Pasquali and Christian Retor\'e
math.LO