Syntactical and semantical properties of simple type theory

Abstract

In my paper [10] I introduced the syntactical concepts “positive part” and “negative part” of logical formulas in first-order predicate calculus. These concepts make it possible to establish logical systems on inference rules similar to Gentzen's inference rules but without using the concept “sequent” and without needing Gentzen's structural inference rules. Proof-theoretical investigations of several formal systems based on positive and negative parts are published in [11]. In this paper I consider a similar formal system of simple type theory. A syntactical concept of “strict derivability” results from the formal system in [10] by generalization of the axioms and inference rules from first to higher-order predicate calculus and by addition of inference rules for set abstraction by means of a λ-symbol which allows us to form set expressions of arbitrary types from well-formed formulas.