Computability and Logic

244 NORMAL FORMS
Proof: The proof is by induction on complexity. The base step is trivial, since an
atomic formula is already negation-normal. Most cases of the induction step are trivial
as well. For instance, if A and B are equivalent respectively to negation-normal formulas
A* and B*, then A & B and A ∨ B are equivalent respectively to A*&B* and
A* ∨ B*, which are also negation-normal. The nontrivial case is to prove that if A is
equivalent to the negation-normal A* then ∼A is equivalent to some negation-normal
A † . This divides into six subcases according to the form of A*. The case where A*is
atomic is trivial, since we may simply let A † be ∼A*. In case A* is of form ∼B, so
that ∼A*is∼∼B, we may let A † be B. In case A* is of form (B ∨ C), so that ∼A*is
∼(B ∨ C), which is logically equivalent to (∼B & ∼C), by the induction hypothesis
the simpler formulas ∼B and ∼C are equivalent to formulas B † and C † of the required
form, so we may let A † be (B † & C † ). The case of conjunction is similar. In case A*
is of form ∃xB, so that ∼A*is∼∃xB, which is logically equivalent to ∀x∼B, by the
induction hypothesis the simpler formula ∼B is equivalent to a formula B † of the required
form, so we may let A † be ∀xB † . The case of universal quantification is similar.
In the foregoing proof we have used such equivalences as that of ∼(B ∨ C) to
∼B & ∼C, to show ‘from the bottom up’ that there exists a negation-normal equivalent
for any formula. What we show at the induction step is that if there exist
negation-normal equivalents for the simpler formulas ∼B and ∼C, then there exists
a negation-normal equivalent for the more complex formula ∼(B ∨ C). If we
actually want to find a negation-normal equivalent for a given formula, we use the
same equivalences, but work ‘from the top down’. We reduce the problem of finding
a negation-normal equivalent for the more complex formula to that of finding such
equivalents for simpler formulas. Thus, for instance, if P, Q, and R are atomic, then
can be successively converted to
∼(P ∨ (∼Q & R))
∼P & ∼(∼Q & R)
∼P &(∼∼Q ∨∼R)
∼P &(Q ∨∼R)
the last of which is negation-normal. In this process use such equivalences as that
of ∼(B ∨ C) to∼B & ∼C to ‘bring junctions out’ or ‘push negations in’ until we
get a formula equivalent to the original in which negation is applies only to atomic
subformulas.
The above result on negation-normal form can be elaborated in two different
directions. Let A 1 , A 2 , ..., A n be any formulas. A formula built up from them using
only ∼, ∨, and &, without quantifiers, is said to be a truth-functional compound of
the given formulas. A truth-functional compound is said to be in disjunctive normal
form if it is a disjunction of conjunctions of formulas from among the A i and their
negations. (A notion of conjunctive normal form can be defined exactly analogously.)
19.2 Proposition (Disjunctive normal form). Every formula is logically equivalent to
one that is in disjunctive normal form.