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 ...∀xk∃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.KeywordsNormal FormOrder LogicConjunctive Normal FormDisjunctive Normal FormHigh Order LogicThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.