Computability and Logic

272 MONADIC AND DYADIC LOGIC
associated with the sentences out of which it is compounded. For instance, (B & C)* is
B*&C*, and analogously for ∨.IfA is ∀xF(x), then A*is(F(1)& ...&F(m))*, and
analogously for ∃. Then A comes out true in the interpretation if and only if A* does.]
Putting our three observations together, we have proved the following.
21.5 Lemma. For each n, the n-satisfiability problem for first-order logic is solvable.
To show that the decision problem for a class Kissolvable, it is sufficent to show
how one can effectively calculate for any sentence S in K a number n such that if
S has a model at all, then it has a model of size ≤n. For if this can be shown, then
for K the satisfiability problem is reduced to the n-satisfiability problem. The most
basic positive result that can be proved in this way concerns monadic logic, where
only one-place predicates are allowed.
21.6 Theorem. The decision problem for monadic logic is solvable.
A stronger result concerns monadic logic with identity, where in addition to oneplace
predicates, the two-place logical predicate of identity is allowed.
21.7 Theorem. The decision problem for monadic logic with identity is solvable.
These results are immediate from the following lemmas, whose proofs will occupy
section 21.2.
21.8 Lemma. If a sentence involving only monadic predicates is satisfiable, then it has
a model of size no greater than 2 k , where k is the number of predicates in the sentence.
21.9 Lemma. If a sentence involving only n monadic predicates and identity is satisfiable,
then it has a model of size no greater than 2 k · r, where k is the number of monadic
predicates and r the number of variables in the sentence.
Before launching into the proofs, some brief historical remarks may be in order.
The first logician, Aristotle, was concerned with arguments such as
All horses are mammals.
All mammals are animals.
Therefore, all horses are animals.
The form of such an argument would in modern notation be represented using oneplace
predicates. Later logicians down through George Boole in the middle 19th
century considered more complicated arguments, but still ones involving only oneplace
predicates. The existence had been noticed of intuitively valid arguments involving
many-place predicates, such as
All horses are animals.
Therefore, all who ride horses ride animals.
But until the later 19th century, and especially the work of Gottlob Frege, logicians
did not treat such arguments systematically. The extension of logic beyond the
monadic to the polyadic is indispensable if the forms of arguments used in mathematical
proofs are to be represented, but the ability of contemporary logic to represent