Partial Functions in an Impredicative Simple Theory of Types

Abstract

A functional notation is not a necessity for a predicate logic since a function of n arguments can be represented as a predicate of n + 1 arguments. But a functional notation in a predicate logic with identity can greatly simplify some assertions, and for this reason a functional notation is frequently assumed for predicate logics, both first order and higher. But a functional notation that is admitted as primitive in a predicate logic must of necessity be interpreted as a notation for total functions, not partial functions, over the domain of the functions. The traditional way of introducing a notation for partial functions into a predicate logic with an assumed or defined identity is using the notation _(lx) F of Russell’s definite descriptions that is read “the x such that F”. But the traditional manner of introduction requires the treatment of what Quine has called “the waste cases”; that is when there is no x or more than one x such that F. The purpose of this paper is to demonstrate that the tableaux method of formalizing logics permits the introduction of definite descriptions without the need to provide a denotation for waste case definite descriptions. As a result the distortions of meaning that result from Quine’s treatment of the waste cases is avoided. The technique is illustrated by introducing a notation for partial functions into an impredicative version ITT of the simple theory of types. The resulting logic ITTf is shown to be a conservative extension of ITT. The tableaux proof theory of ITT is of independent interest both for its motivation and for the strength of its proof theory. The logic has a nominalist motivation appropriate for a logic intended for applications in computer science. Its extension of the membership of the type of the individuals of the simple theory of types avoids the abuses of use and mention that can result when higher order predication is given a nominalist interpretation. The proof theory does not require an axiom of infinity. As a result, the definition of both well-founded and non-well-founded recursive predicates is much simpler than in the simple theory of types with an axiom of infinity.