Computability and Logic

274 MONADIC AND DYADIC LOGIC
Our induction hypothesis is that if a 1 ,...,a s , a s+1 and b 1 ,...,b s , b s+1 match,
then a 1 ,...,a s , a s+1 satisfy Q in M if and only if b 1 ,...,b s , b s+1 satisfy Q in N .
What we want to show is that if b 1 ,...,b s , b s+1 satisfy Q in N for all b s+1 in N,
then a 1 ,...,a s , a s+1 satisfy Q in M for all a s+1 in M. Therefore it will be enough
to show that if a 1 ,...,a s and b 1 ,...,b s match, where s < r, then for any a s+1 in M
there is a b s+1 in N such that a 1 ,...,a s , a s+1 and b 1 ,...,b s , b s+1 match.
In the degenerate case where a s+1 is identical with one of the previous a i , we may
simply take b s+1 to be identical with the corresponding b i . In the non-degenerate case,
a s+1 belongs to some equivalence class C and is distinct from any and all previous a i
that belong to C. Let the number of such a i be t (where possibly t = 0), so that there
are at least t + 1 elements in C, counting a s+1 . To ensure matching, it will suffice
to choose b s+1 to be some element of C that is distinct from any and all previous b i
that belong to C. Since a 1 ,...,a s and b 1 ,...,b s match, the number of such b i will
also be t. Since t ≤ s < r, and there are at least t + 1 ≤ r elements in C, there will
be at least that many elements of C in N, and so we can find an appropriate b s+1 ,to
complete the proof.
21.10 Corollary. If a sentence involving no nonlogical symbols (but only identity) is
satisfiable, then it has a model of size no greater than r, where r is the number of variables
in the sentence.
21.11 Corollary. If a sentence of monadic logic involving only one variable is satisfiable,
then it has a model of size no greater than 2 k , where k is the number of monadic
predicates in the sentence.
Proofs: These are simply the cases k = 0 and r = 1 of Lemma 21.9.
Proof of Lemma 21.7: This is immediate from Corollary 21.11 and the following, which
is a kind of normal form theorem.
21.12 Lemma. Any sentence of monadic logic without identity is logically equivalent
to one with the same predicates and only one variable.
Proof: Call a formula clear if in any subformula ∀xB(x) or∃xB(x) that begins
with a quantifier, no variable other than the variable x attached to the quantifier
appears in F. Thus ∀x∃y(Fx & Gy) is not clear, but ∀xFx& ∃ yGy is clear. To prove
the lemma, we show how one can inductively associate to any formula A of monadic
logic without identity an equivalent formula A © with the same predicates that is clear
(as in our example the first formula is equivalent to the second). We then note that any
clear sentence is equivalent to the result of rewriting all its variables to be the same (as
in our example the second sentence is equivalent to ∀zFz& ∃zGz). The presence of
identity would make such clearing impossible. (There is no clear sentence equivalent
to ∀x∃yx≠ y, for instance.)
To an atomic formula we associate itself. To a truth-functional compound of formulas
to which clear equivalents have been associated, we associate the same truthfunctional
compound of those equivalents. Thus (B ∨ C) © is B © ∨ C © , for instance,
and analogously for &. The only problem is how to define the associate (∃xB(x)) ©