Michael Gabbay

Term Sequent Logic

We consider a term sequent logic for the lambda-calculus. Term sequents are a judgement form similar to the logical judgement form of entailment between sentences, but denoting equality or reducibility between terms. Using term sequents, it is possible to treat lambda-terms almost like logical sentences, and to use proof-theoretic methods to establish their properties. We prove a cut-elimination result for untyped lambda-calculus and describe how this generalises the usual confluence result. We give a notion of uniform proof for lambda-terms, and suggest how this can be viewed as a mixed logic-programming/functional programming framework with the ability to assume arbitrary reductions. Finally, we discuss related and future work.

A Formalist Philosophy of Mathematics Part I: Arithmetic

In this paper I present a formalist philosophy mathematics and apply it directly to Arithmetic. I propose that formalists concentrate on presenting compositional truth theories for mathematical languages that ultimately depend on formal methods. I argue that this proposal occupies a lush middle ground between traditional formalism, fictionalism, logicism and realism.