Generalization of the Truth-relevant Semantics to the Predicate Calculus

1.1 Truth-relevance There are Boolean formulae such that their value can be determined by a subset of their variables. Consider for example A = P v ~P v Q. When v(P) is T then v(A) = T regardless of the value of Q. When v(P) = F then v(A) = T also regardless of the value Q. The set of variables occurring in A is {P, Q}. We say that the subset {P} is truth-determining for A; for all the valuations of {P}, i.e. v(P) = T and v(P) = F, we can determine the value of A regardless of the other variables. Definition 1.1.1: A set of propositional variables is truth-determining for a proposition A iff the value of A can be determined as true or false on all assignments of T, F to the set. If P is true then (Q-> P) is true regardless of the value of Q, but then the entire formula (1.1) is true (regardless of the value of Q.) Suppose P is false. Then (1.1) is true regardless of the value of Q. In either case [i.e. v(P) = T or v(P) = F] the value of (1.1) can be determined without any knowledge of the value of Q. Thus {P} is truth-determining for (1.1).