Miscellaneous

Abstract

In Chap. II, § 5 we defined the relation of substitution Subst αxtβ. For each α, x, t there is at most one β such that Subst αxtβ. However, as we have seen, not every α, x, t possesses a β such that Subst αxtβ. In the following, we shall deal with such exceptions. We want to associate with each α, x, t a unique expression $\alpha \frac{t}{x}$ which, we shall say, is obtained from α by extended substitution of t for x. The expression “extended substitution” is justified by the fact that $\alpha \frac{t}{x} \equiv \beta$ whenever Subst αxtβ.