The Lower Predicate Calculus: Introduction
In the Propositional Calculus we have considered solely complete statements or propositions and some of the ways (viz. the truth-functional ways) in which these complete propositions can be combined or operated on to form complex propositions. The only variables we have used have been propositional variables, i.e. variables which take as their values whole propositions, whether simple or complex. A variable such as ‘p’ can be taken to represent any proposition we please, but it cannot stand for anything less than a complete proposition, i.e. it cannot stand for any part of a proposition, unless that part is itself a complete proposition. One result of this is a certain limitation in the applicability of PC methods. A valid inference can have its validity exhibited by the techniques of PC if, but only if, when we replace each distinct whole proposition within it by a distinct variable, the result is an inferential schema which can be shown to be valid by PC methods as described in Part I. We have seen a number of examples of this procedure in earlier chapters.