Raymond M. Smullyan

A Unifying Principle

At this point we wish to discuss a principle which we introduced in [2] and which simultaneously yields several of the major results in Quantification Theory. The mathematical content of this principle is not really very different from that of our last theorem (Theorem 7, Chapter V), but it is in a form which makes no reference to the particular formal system of tableaux (or to any other specific formal system, for that matter). We believe that this is a good point to discuss this principle while the tableau method is still fresh in the reader’s mind. We shall apply the principle several times in the further course of this study.

Some principles related to Löb's theorem

Publisher Summary Godels second incompleteness theorem can be obtained by purely proof-theoretic methods. This chapter describes Godel and Lobian principles in an abstract setting, which allows simultaneous description of first-order theories, modal logics, and the self-referential systems. It introduces the notion of a provability function—a mapping from sentences to sentences satisfying the analogue of the Hilbert–Bernays conditions. The chapter also considers Kripkes argument and its modifications and describes the modal system K4. The discovery of several facts about K4 is greatly facilitated by use of a basic principle—the translational theorem. The proof of translation theorem for K4 and also for the modal system G is presented. This essentially allows getting new principles of K4 from known ones, which when translated into metamathematical terms, yieldnew Lobian-type constructions from known Godelian-type constructions. The chapter describes metamathematical applications of the results about K4.

First-Order Analytic Tableaux

We use “α”, “β” exactly as we did for propositional logic (only now construing “formula” to mean “closed formula of quantification theory”). We now add two more categories γ and δ as follows.

Axiom Systems for Quantification Theory

We now wish to show how we can use our earlier completeness results to establish the completeness of the more usual formalizations of First Order Logic. We first consider an axiom system Q1 which (except for rather minor details) is a standard axiom system. Its completeness is easily obtained as a consequence of our Unifying Principle. We prefer, however, to emphasize a completeness proof along the following lines. We consider a succession of axiom systems, starting with Q1 and ending with a system Q* 2 each of which seems ostensibly weaker than the preceding. The completeness of Q* 2 easily implies the completeness of Q 1 but it is not immediately obvious that everything provable in Q 1 is provable in Q*2. However, the completeness of Q*2 is an almost immediate consequence of the (strong form) of the Fundamental Theorem. The arguments of this chapter are wholly constructive. We give an effective procedure whereby given any associate of {~X}, a proof of X can be found in Q*2 (and hence also in Q1). We also know from the preceding chapter how given a closed tableau for {~ X} we can find an associate of {~X}. Combining these two constructions, we see how given any proof of X by the tableau method, we can find a proof of X in each of the axiom systems of this chapter.

Analytic versus Synthetic Consistency Properties

We have remarked earlier that magic sets emerged from the completeness proofs of Henkin and Hasenjaeger. In this chapter we wish to discuss the Henkin and Hasenjaeger completeness proofs and their relationship to the completeness proofs of earlier chapters. We conclude this chapter with a new proof of the Unifying Principle—which circumvents the necessity of appealing to systematic tableaux—and we discuss the essential differences and similarities of what are basically two types of completeness proofs; the one along the lines of Lindenbaum-Henkin, the other along the lines of Godel-Herbrand-Gentzen.

The Fundamental Theorem of Quantification Theory

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, Godel, 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.

More on Gentzen Systems

In § 1 of this chapter, we discuss Gentzen’s Extended Hauptsatz. In § 2 we establish anew form of this extension which does not appeal to prenex normal form. In § 3 we consider some variants of Gentzen systems which will play a key role in all 3 subsequent chapters.

Symmetric Completeness Theorems

In this chapter we establish some new and stronger versions of completeness theorems considered earlier. These are closely related to Craig’s Interpolation lemma, and they will play a key role in our final chapter on linear reasoning.

Systems of Linear Reasoning

In Craig’s paper “Linear Reasoning” [1] he considered a system of Quantification Theory in which there are no axioms, but only inference rules, and all rules are 1-premise rules. In each of these rules, the premise validly implies the conclusion. By a derivation of Y from X in the system is meant a finite sequence of lines