A Primer on Proofs and Types

Abstract

This article is intended to be an int roduction (for real novices) to the area of proof systems and type systems. We provide very basic notions and their fundamental properties together with some inforrnal ideas behind them.

We start by describing hov 'logical reasoning' can be formulated into a proof system, and how structures of proofs can be studied based on this system. We then proceed to a concise exposition of the notion of computable functions, which emerged from proof theory in the early 1930's$ (during the development of the incompleteness theorems by Göodel). Next, after a short account of pure (type-free) $\lambda$-calculus in connection with computable functions, we discuss three typed $\lambda$-calculi (the simple type system, the second-order type system, and the calculus of constructions) as representatives of type svstems, with emphasis on their relation to the corresponding intuitionistic proof systems (tlle implicational fragment, the second-order propositional logic, and the higher-order predicate logic, respectively).