A Normative Model of Classical Reasoning in Higher Order Languages

Abstract

The present paper is concerned with a ramified type theory (cf. (Lorenzen 1955), (Russell), (Schütte), (Weyl), e.g.,) in a cumulative version.

§0 deals with reasoning in first order languages. $\mathbb{N}$ is introduced as a first order set.

In §1 and §2 we introduce an extension of a union of higher order languages by means of variables x, y, . . . for constants of arbitrary order, and variables for tuples (c₁, . . ., c _j ) of arbitrary order and arbitrary length $j \in \mathbb{N}^{+}$. So we may simply identify types with orders. In that language ‘type-free’ equations x = y are definable. We extend the assertion games introduced in §0, called the ‘primary game’ and the ‘classical game’, to that language, show that their sentences are non-circular, and that their formulas are invariant under (=). So we may argue classically even with higher order sentences in the classical game.

In §3 we introduce singular description terms.

In §4 we deal with expansions of higher order languages which also contain formulas with indicators and objectual variables, and enable a quantification which combines both substitutional and objectual quantification. One may commute consecutive existential quantifiers that occur in formulas of those expanded languages.