The Fundamental Theorem of Quantification Theory

Abstract

There is one very basic result in Quantification Theory which appears to be less widely known and appreciated than it should be. It results from the cumulative efforts of such workers as Herbrand, Gödel, Gentzen, Henkin, Hasenjaeger and Beth. We have referred to it in [2] as a form of Herbrand’s theorem, though this is perhaps unfair to the other workers mentioned above. This theorem is indeed Herbrand-like in that it gives a procedure which associates with every valid formula of Quantification theory a formula of propositional logic which is a tautology. This theorem easily yields the completeness theorem for the more conventional axiomatization of First Order Logic (which we study in the next chapter), but it yields far more. The beauty of this theorem is that it makes absolutely no reference to any particular formal system of logic; it is stated purely in terms of a certain basic relationship between first order satisfiability and truth-functional satisfiability. In view of all those considerations, we feel justified in referring to this theorem as the Fundamental Theorem of Quantification Theory.