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What is called a truth-functional formula in (alternational) normal form is built up of sentence letters 'p', 'q', etc., or their negations 'i', 'a', etc., or both, by using only the notations of conjunction and alternation, and in such a way as to subject alternations never to conjunction but only vice versa: thus 'pq v firs v f'. This paper is concerned with the problem of reducing an arbitrary truth-functional formula to a shortest equivalent in normal form. Part of the content of my last previous paper on the subject* will be presented anew in an improved way, and a further theorem will be established. I shall not assume familiarity with my previous papers. Let us sharpen our terminology. Sentence letters and their negations are called literals. Literals and conjunctions of them are called fundamentalformulas, provided that none contains the same letter twice. Fundamental formulas and alternations of them are called normal, and are said to have those fundamental formulas as their clauses. On these definitions, a formula is convertible to nornal form only if it is not self-contradictory; but there is no serious loss in setting aside the self-contradictory cases. A prime implicant of a formula 4) is a fundamental formula that logically implies 4 but ceases to when deprived of any one literal. A normal formula will be called uniliterally redundant if it is equivalent to what remains of itself on dropping some one occurrence of a literal. Obviously then a normal formula is uniliterally redundant if and only if not all its clauses are prime implicants of it. Consequently, in particular,
W. V. Quine (Sun,) studied this question.