Computability and Logic

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 Ɣ of sentences
there is a set Ɣ* of sentences of some special syntactic form such that Ɣ and Ɣ* 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 Löwenheim–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 Gödel 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 ∼.
19.1 Proposition (Negation-normal form). Every formula is logically equivalent to one
that is negation-normal.
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