On the completeness of the classical sentential logic

E.W. Beth
1958 Indagationes Mathematicae (Proceedings)  
We consider a version of the classical sentential logic in which atoms p, q, r, ... , negation-, and implication--+ are used. It is well known that this subject may be approached in three ways: (A) We define the notion of a logical identity (tautology, or valid formula) by means of truth-tables for the· connectives-and --+. (B) We set up a calculus of sequents. (C) We select an axiom system and rules of deduction, and we consider the formulas deducible from the axioms. The completeness theorem
more » ... xpresses the fact that, essentially, the methods (A), (B) and (C) lead to the same result. In particular, a formula is deducible if and only if it is a logical identity. The equivalence of the approaches under (A) and (B) is a relatively simple matter; it is also rather easy to prove that every formula deducible from the axioms is a logical identity. Therefore, it will not be necessary here to go into these matters. However, the problem of proving that every logical identity is deducible is not quite as simple. In fact, several proofs of this most substantial part of the completeness theorem have been given, but they are either .rather tedious or they rely on metamathematical results of a relatively advanced kind, which in turn would become much more easily accessible if the completeness proof were given in advance. Therefore, the following discussion may still present some interest. In some earlier publications, I have introduced the method of semantic tableaux for testing whether a given formula X is, or is not, a logical identity. For our present purpose, it will be sufficient to mention the following rules for the construction and closure of such a tableau. (i) The formula X to be tested is inserted in the right column as the only initial formula. (ij) If U appears in a left (right) column, then U is inserted in the conjugate right (left) column [i.e., in the right (left) column of the same (sub-)tableau].
doi:10.1016/s1385-7258(58)50060-2 fatcat:xsplulvjjrcffhtpxy7i5x27pq