Assumption and Substitution

Abstract

The forms of assertion concerning canonical and noncanonical sets and elements, introduced in the previous chapters, are all categorical in the sense that the terms of the assertions may not depend on any assumptions. In this chapter, the type-theoretic forms of assertion will be generalized to the hypothetical case, ie, to the case where the terms may depend on assumptions . In the first section, hypothetical assertions are related to the concept of function; the definitions of the hypothetical forms of assertions follow in the next section. In the third section I present a version of the type-theoretic substitution calculus. The fourth, fifth, and sixth sections are concerned with the generalization of the material of previous chapters to the hypothetical case. The elimination rules of intuitionistic type theory are given in the seventh section. The eighth and last section of this chapter deals with the Curry-Howard correspondence.