Computability and Logic

22
Second-Order Logic
Suppose that, in addition to allowing quantifications over the elements of a domain,
as in ordinary first-order logic, we allow also quantification over relations and functions
on the domain. The result is called second-order logic. Almost all the major theorems
we have established for first-order logic fail spectacularly for second-order logic, as
is shown in the present short chapter. This chapter and those to follow generally
presuppose the material in section 17.1. (They are also generally independent of each
other, and the results of the present chapter will not be presupposed by later ones.)
Let us begin by recalling some of the major results we have established for first-order
logic.
The compactness theorem: If every finite subset of a set of sentences has a model,
the whole set has a model.
The (downward) Löwenheim–Skolem theorem: If a set of sentences has a model,
it has an enumerable model.
The upward Löwenheim–Skolem theorem: If a set of sentences has an infinite
model, it has a nonenumerable model.
The (abstract) Gödel completeness theorem: The set of valid sentences is semirecursive.
All of these results fail for second-order logic, which involves an extended notion
of sentence, with a corresponding extension of the notion of truth of a sentence in
an interpretation. In introducing these extended notions, we stress at the outset that
we change neither the definition of language nor the definition of interpretation:
a language is still an enumerable set of nonlogical symbols, and an interpretation
of a language is still a domain together with an assignment of a denotation to each
nonlogical symbol in the language. The only changes will be that we add some new
clauses to the definition of what it is to be a sentence of a language, and correspondingly
some new clauses to the definition of what it is for a sentence of a language to
be true in an interpretation.
What is a second-order sentence? Let us refer to what we have been calling ‘variables’
as individual variables. We now introduce some new kinds of variable: relation
variables and function variables. Just as we have one-, two-, three-, and more-place
predicates or relation symbols and function symbols, we have one-, two-, three-, and
more-place relation variables and function variables. (Since one-place relations are
279