Formulas of first-order logic in distributive normal form

Abstract

It was shown by Jaakko Hintikka that every formula of first-order logic can be written as a disjunction of formulas called constituents. Such a disjunction is called a distributive normal form of the formula. It is a generalization of the disjunctive normal form for propositional logic. However, there are some significant differences between these two normal forms, caused chiefly by the impossibility of defining the constituents in such a way that they are all consistent. Distributive normal forms and some of their properties are studied. For example, the size of distributive normal forms is examined, and although we can't determine exactly how many constituents (of each form) are consistent, it is shown that the vast majority are inconsistent. Hintikka's definition of trivial inconsistency is studied, and a new definition of trivial inconsistency is given in terms of a necessary condition for the consistency of a constituent which is stronger than the condition which Hintikka used in his definition of trivial inconsistency. An error in Hintikka's attempted proof of the completeness theorem of the theory of distributive normal forms is pointed out, and a similar completeness theorem is proved using the new definition of trivial inconsistency.