Computability and Logic

21.2. MONADIC LOGIC 273
the forms of such arguments comes at a price, namely, that of the undecidability
of the contemporary notions of validity and satisfiability. For, as the results listed
above make plain, undecidability sets in precisely when two-place predicates are
allowed.
Let us get straight to work.
21.2 Monadic Logic
Proof of Lemma 21.9: Let S be a sentence of monadic logic with identity involving
k one-place predicates (possibly k = 0) and r variables. Let P 1 ,...,P k be predicates
and v 1 ,...,v r the variables. Suppose M is a model of S.
For each d in the domain M =|M| let the signature σ (d) ofd be the sequence
( j 1 ,..., j k ) whose ith entry j i is 1 or 0 according as Pi
M does or does not hold
of d [if k = 0, then σ (d) is the empty sequence ( )]. There are at most 2 k possible
signatures. Call e and d similar if they have the same signature. Clearly similarity is
an equivalence relation. There are at most 2 k equivalence classes.
Now let N be a subset of M containing all the elements of any equivalence class
that has ≤r elements, and exactly r elements of any equivalence class that has ≥r
elements. Let N be the subinterpretation of M with domain |N |=N. Then N has
size ≤2 k · r. To complete the proof, it will suffice to prove that N is a model of S.
Towards this end we introduce an auxiliary notion. Let a 1 ,...,a s and b 1 ,...,b s
be sequences of elements of M. We say they match if for each i and j between 1
and n, a i and b i are similar, and a i = a j if and only if b i = b j . We claim that if
R(u 1 ,...,u s ) is a subformula of S (which implies that s ≤ r and that each of the us
is one of the vs) and a 1 ,...,a s and b 1 ,...,b s are matching sequences of elements
of M, with the b i all belonging to N, then the a i satisfy R in M if and only if the b i
satisfy R in N . To complete the proof it will suffice to prove this claim, since, applied
with s = 0, it tells us that since S is true in M, S is true in N , as desired.
The proof of the claim is by induction on complexity. If R is atomic, it is either of
the form P j (u i ) or of the form u i = u j . In the former case, the claim is true because
matching requires that a i and b i have the same signature, so that Pj
M holds of the
one if and only if it holds of the other. In the latter case, the claim is true because
matching requires that a i = a j if and only if b i = b j .
If R is of form ∼Q, then the as satisfy R in M if and only if they do not satisfy
Q, and by the induction hypothesis the as fail to satisfy Q in M if and only if the bs
fail to satisfy Q in N , which is the case if and only if the bs satisfy R in N , and we
are done. Similarly for other truth-functional compounds.
It remains to treat the case of universal quantification (and of existential quantification,
but that is similar and is left to the reader). So let R (u 1 ,...,u s ) be of form
∀u s+1 Q(u 1 ,...,u s , u s+1 ), where s + 1 ≤ r and each of the us is one of the vs. We
need to show that a 1 ,...,a s satisfy R in M (which is to say that for any a s+1 in
M, the longer sequence of elements a 1 ,...,a s , a s+1 satisfies Q in M) if and only if
b 1 ,...,b s satisfy R in N (which is to say that for all b s+1 in N, the longer sequence
of elements b 1 ,...,b s , b s+1 satisfies Q in N ). We treat the ‘if’ direction and leave
the ‘only if’ direction to the reader.