The definitions of the theories

The relations of reducibility and convertibility were defined in Chapters 1 and 2 via contractions of redexes. The present chapter gives alternative definitions, via formal theories with axioms and rules of inference.

These theories will be used later in describing the correspondence between λ and CL precisely, and will help to make the distinction between syntax and semantics clearer in the chapters on models to come. They will also give a more direct meaning to such phrases as ‘add the equation M = N as a new axiom to the definition of = β … ’ (Corollary 3.11.1).

In books on logic, formal theories come in two kinds (at least): Hilbert-style and Gentzen-style. The theories in this chapter will be the former.

Notation 6.1 (Hilbert-style formal theories) A (Hilbert-style) formal theory T consists of three sets: formulas, axioms and rules (of inference). Each rule has one or more premises and one conclusion, and we shall write its premises above a horizontal line and its conclusion under this line; for examples, see the rules in Definition 6.2 below.

If г is a set of formulas, a deduction of a formula B from г is a tree of formulas, with those at the tops of branches being axioms or members of г, the others being deduced from those immediately above them by a rule, and the bottom one being B.