19

Normal Forms

A normal form theorem of the most basic type tells us that for every formula A there is a formula A* of some special syntactic form such that A and A* are logically equivalent. A normal form theorem for satisfiability tells us that for every set of sentences there is a set * of sentences of some special syntactic form such that and * are equivalent for satisfiability, meaning that one will be satisfiable if and only if the other is. In section 19.1 we establish the prenex normal form theorem, according to which every formula is logically equivalent to one with all quantifiers at the beginning, along with some related results. In section 19.2 we establish the Skolem normal form theorem, according to which every set of sentences is equivalent for satisfiability to a set of sentences with all quantifiers at the beginning and all quantifiers universal. We then use this result to give an alternative proof of the Lowenheim¨–Skolem theorem, which we follow with some remarks on implications of the theorem that have sometimes been thought ‘paradoxical’. In the optional section 19.3 we go on to sketch alternative proofs of the compactness and Godel¨ completeness theorems, using the Skolem normal form theorem and an auxiliary result known as Herbrand’s theorem. In section 19.4 we establish that every set of sentences is equivalent for satisfiability to a set of sentences not containing identity, constants, or function symbols. Section 19.1 presupposes only Chapters 9 and 10, while the rest of the chapter presupposes also Chapter 12. Section 19.2 (with its pendant 19.3) on the one hand, and section 19.4 on the other hand, are independent of each other. The results of section 19.4 will be used in the next two chapters.

19.1 Disjunctive and Prenex Normal Forms

This chapter picks up where the problems at the end of Chapter 10 left off. There we asked the reader to show that that every formula is logically equivalent to a formula having no subformulas in which the same variable occurs both free and bound. This result is a simple example of a normal form theorem, a result asserting that every sentence is logically equivalent to one fulfilling some special syntactic requirement. Our first result here is an almost equally simple example. We say a formula is negationnormal if it is built up from atomic and negated atomic formulas using , & , , and

alone, without further use of .

19.1Proposition (Negation-normal form). Every formula is logically equivalent to one that is negation-normal.

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Proof: Given any formula, first replace it by a negation-normal equivalent. Then, using the distributive laws, that is, the equivalence of (B & (C D)) to ((B & C) (B & D)) and of ((B C) & D) to ((B D) & (C D)), ‘push conjunction inside’ and ‘pull disjunction outside’ until a disjunctive normal equivalent is obtained. (It would be a tedious but routine task to rewrite this ‘top down’ description of the process of finding a disjunctive normal equivalent as a ‘bottom up’ proof the existence of such an equivalent.)

If in a formula that is in disjunctive normal form each disjunction contains each Ai exactly once, plain or negated, then the compound is said to be in full disjunctive normal form. (A notion of full conjunctive normal form can be defined exactly analogously.) In connection with such forms it is often useful to introduce, in addition to the two-place connectives and & , and the one-place connective , the zero-place connectives or constant truth and constant falsehood , counting respectively as true in every interpretation and false in every interpretation. The disjunction of zero disjuncts may by convention be understood to be , and the conjunction of zero conjuncts to be (rather as, in mathematics, the sum of zero summands is understood to be 0, and the product of zero factors to be 1).

In seeking a full disjunctive normal equivalent of a given disjunctive normal formula, first note that conjunctions (and analogously, disjunctions) can be reordered and regrouped at will using the commutative and associative laws, that is, the equivalence of (B & C) to (C & B), and of (B & C & D), which officially is supposed to be an abbreviation of (B & (C & D)), with grouping to the right, to ((B & C) & D), with grouping to the left. Thus for instance (P & (Q & P)) is equivalent to (P & (P & Q)) and to ((P & P) & Q). Using the idempotent law, that is, the equivalence of B & B to B, this last is equivalent to P & Q. This illustrates how repetitions of the same Ai (or Ai ) within a conjunction can be eliminated. To eliminate the occurrence of the same Ai twice, once plain and once negated, we can use the equivalence of B & B toand of & C to , and of D to D, so that, for instance, (B & B & C) D is equivalent simply to D: contradictory disjuncts can be dropped. These reductions will convert a given formula to one that, like our earlier example ( P & Q) ( P & R), is a disjunction of conjunctions in which each basic formula occurs at most once, plain or negated, in each conjunct.

To ensure that each occurs at least once in each conjunction, we use the equivalence of B to (B & C) (B & C). Thus our example is equivalent to

( P & Q & R) ( P & Q & R) ( P & R)

and to

( P & Q & R) ( P & Q & R) ( P&Q & R) ( P & Q & R)

and, eliminating repetition, to

( P & Q & R) ( P & Q & R) ( P & Q & R)

which is in full disjunctive normal form. The foregoing informal description can be converted into a formal proof of the following result.