Prenex Normal Form

Please note that the content of this book primarily consists of articles available from Wikipedia or other free sources online. A formula of the predicate calculus is in prenex normal form if it is written as a string of quantifiers followed by a quantifier-free part (referred to as the matrix). Every formula is equivalent in classical logic to a formula in prenex normal form. Every first-order formula is logically equivalent (in classical logic) to some formula in prenex normal form. There are several conversion rules that can be recursively applied to convert a formula to prenex normal form. The rules depend on which logical connectives appear in the formula. The equivalences are valid when x does not appear as a free variable of ¿; if x does appear free in ¿, it must be replaced with another free variable. There are four rules for implication: two that remove quantifiers from the antecedent and two that remove quantifiers from the consequent.