On Simplifying the Matrix of a WFF

Abstract

In [3], [4], and [5] Joyce Friedman formulated and investigated certain rules which constitute a semi-decision procedure for wffs of first order predicate calculus in closed prenex normal form with prefixes of the form ∀x₁ ^...∀x_k∃y₁ ^...∃Ym∀z₁ ^...∀z _n . Given such a wff QM,where Q is the prefix and M is the matrix in conjunctive normal form, Friedman’s rules can be used, in effect, to construct a matrix M* which is obtained from M by deleting certain conjuncts of M. Obviously, ⊢QM⊃QM* Using the Herbrand-Gödel theorem for first order predicate calculus, Friedman showed that ⊢QM if and only if ⊢QM*. Clearly if M* is the empty conjunct (i.e., a tautology), ⊢QM* so ⊢QM. Friedman also showed that for certain classes of wffs, such as those in which m ≤ 2 or n = 0 in the prefix above, ⊢QM if and only if M* is the empty conjunct. Hence for such classes of wffs the rules constitute a decision procedure. Computer implementation [4] of the procedure has shown it to be quite efficient by present standards.