Gentzen's three consistency proofs for elementary number theory have a common aim that originates from Hilbert's Program, namely, the aim to justify the application of classical reasoning to quantified propositions in elementary number theory. In addition to this common aim, Gentzen gave a “finitist” interpretation to every number-theoretic proposition with his 1935 and 1936 consistency proofs. In the present paper, we investigate the relationship of this interpretation with intuitionism in terms of the debate between the Hilbert School and the Brouwer School over the significance of consistency proofs. First, we argue that the interpretation had the role of responding to a Brouwer-style objection against the significance of consistency proofs. Second, we propose a way of understanding Gentzen's response to this objection from an intuitionist perspective.

Bertrand Russell presented the very first theory of types in the appendices of The Principles of Mathematics. I will argue that he was led to the theory due to Frege's argument against any philosophical account of classes that assimilates a singleton with its sole member. By so doing I will attempt to show that the original theory of types was not meant to be a mere technical solution to the set-theoretic paradox but a philosophical account of what classes are in themselves.

In this paper we propose a semantics in which the truth value of a formula is a pair of elements in a complete Boolean algebra. Through the semantics we can unify largely two proofs of cut-eliminability (Hauptsatz) in classical second order logic calculus, one is due to Takahashi-Prawitz and the other by Maehara.