Short Proofs of Tautologies Using the Schema of Equivalence

It is shown how the schema of equivalence can be used to obtain short proofs of tautologies A, where the depth of proofs is linear in the number of variables in A. Eq (A, B, C arbitrary formulas) is the propositional pendant of the schema of identity. It can be argued that, apart form the usual propositional tautologies and inference schemas which are given as axiomatizations of propositional logic (e.g., modus po-nens, modus tollens, case distinction, chain rule), the schema of equivalence is also used extensively in mathematical reasoning. However, it seems that Eq has not been used or investigated in the proof theory of propositional logic to any signiicant extent. A related rule, which has been presented by Sch utte 1960] (see Satz 2.9), is the following: C(T) C(F) C(A) S where A and C are formulas and T and F are the logical constants true and false, respectively. Using S, we can derive Eq uniformly for A, B, C in a constant number of steps: (1) T , T) ? C(T) , C(T) (2) F , T) ? C(F) , C(T) (3) A , T) ? C(T) , C(T) from (1), (2) by S