Abstract

In section 1, I expose in an informal way the rules - and the logical rules - on the proofs of the universal statements and existential statements, and the rules - and the logical rules - on the deductions from these statements. In section 2, I show how Hilbert’s operators τ and ϵ allow a representation of the universal statements and existential statements which is strictly related to the logical rules on the proofs of these statements and to the logical rules on the deductions from these statements, so that we may say that Hilbert in the introduction of the operators τ and ϵ aimed to propose a kind of proof-theoretical representation of the universal statements and existential statements. In section 3, I show the logical naturalness and the logical depth of this representation of universal and existential statements, since τ-axiom and ϵ-axiom - which are the implicit definitions of these operators - arise in a very naturalway froma deep analysis of what happens when we try to prove (in sequent calculus) the sequents ∀xA ⊢ ∀xA and ∃xA ⊢ ∃xA from the identity axiom A ⊢ A.