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X¬A(x) MP, lines 1 and 4 6. ∃xA(x) Abbreviation of line 5 We can extract the content of the proof of Theorem K11, and create an Add-∃x Rule that handles even more cases. We need to be careful that the hidden use of Axiom 4 is acceptable. In the following, the clauses requiring that A(t) is the result of replacing every free occurrence of x in A(x) with t and t is free for x in A(x) insure that ∀x¬A(x) → ¬A(t) is a correct instance of Axiom 4. This allows us to modify the proof of K11 to obtain a proof of ∃xA(x) from A(t).

We wanted a good way to show that formulas were logically valid. The following two very important theorems say that this is the proof system that we want. Theorem (Completeness Theorem for Predicate Calculus). (G¨ odel 1930) If A is logically valid, then A. Theorem (Soundness Theorem for Predicate Calculus). (G¨ odel 1930) If A, then A is logically valid. 12. A PROOF SYSTEM FOR PREDICATE CALCULUS 53 Summarizing, we can prove a formula A if and only if it is logically valid. The relationship between K and the logically valid formulas is exactly the same as the relationship between L and the tautologies.

Be careful! Our strategy only works on leading quantifiers whose scope is the whole formula. Other situations require gyrations of a different sort. The next shortcut will make these gyrations easier. 14 Rule T Since Axiom 1, Axiom 2, Axiom 3 and MP are all included in K, every proof in L is also a proof in K. The following rule lets us use all the results we could have proved in L, even if we didn’t actually do the proof. RULE T: Any instance of a tautology may be inserted as a line in a predicate calculus proof.