Formalizations Of Substitution Of Equals For Equals

Abstract

Inference rule "substitution of equals for equals" has been formalized in terms of simple substitution (which performs a replacement even though a free occurrence of a variable is captured), contextual substitution (which prevents such capture), and function application. We show that in connection with pure first-order predicate calculus, the function-application and no-capture versions of the inference rule are the same and are weaker than the capture version. We discuss the deductive apparatus needed for the no-capture version to be as powerful as the capture version.