On the Rules of Proof in the Pure Functional Calculus of the First Order

Abstract

This chapter discusses the pure functional calculus of first order, F 1 p , as formulated by Church. Church gives the definition of the validity of a formula in a given set I of individuals and shows that a formula is provable in F 1 p if and only if it is valid in every non-empty set I . The definition of validity is preceded by the definition of a value of a formula; the notion of a value is the basic semantical notion in terms of which all other semantical notions are definable. The notion of a value of a formula retains its meaning also in the case when the set I is empty. If Z is empty, then an m -ary propositional function (i.e. a function from the m -th Cartesian power I m to the set {f, t}) is the empty set. In such a case, the value of each well-formed formula with free individual variables is the empty set.