Computability and Logic

280 SECOND-ORDER LOGIC
just sets, one-place relation variables may be called set variables.) We suppose that
no symbol of any sort is also a symbol of any other sort. We extend the definition of
formula by allowing relation or function variables to occur in those positions in formulas
where previously only relation symbols (a.k.a. predicates) or function symbols
(respectively!) could occur, and also by allowing the new kinds of variable to occur
after ∀ and ∃ in quantifications. Free and bound occurrences are defined for
the new kinds of variable exactly as they were for defined for individual variables.
Sentences, as always, are formulas in which no variables (individual, relation, or
function) occur free. A second-order formula, then, is a formula that contains at least
one occurrence of a relation or function variable, and a second-order sentence is a
second-order formula that is a sentence. A formula or sentence of a language, whether
first- or second-order, is, as before, one whose nonlogical symbols all belong to the
language.
22.1 Example (Second-order sentences). (In the following examples we use u as a oneplace
function variable, and X as a one-place relation variable.)
In first-order logic we could identify a particular function as the identity function:
∀x f(x) = x. But in second-order logic we can assert the existence of the identity function:
∃u ∀x u(x) = x.
Similarly, where in first-order logic we could assert that two particular indviduals share
a property (Pc & Pd), in second-order logic we can assert that every two individuals share
some property or other: ∀x∀y∃X(Xx & Xy).
Finally, in first-order logic we can assert that if two particular individuals are identical,
then they must either both have or both lack a particular property: c = d → (Pc↔ Pd).
But in second-order logic we can define identity through Leibniz’s law of the identity of
indiscernibles: c = d ↔∀X (Xc↔ Xd).
Each of the three second-order sentences above is valid: true in each of its interpretations.
When is a second-order sentence S true in an interpretation M? We answer this
question by adding four more clauses (for universal and existential quantifications
involving relation and function variables) to the definition of truth in an interpretation
given in section 9.3. For a universal quantification ∀XF(X) involving a relation
variable, the clause reads as follows. First we define what it is for a relation R
(of the appropriate number of places) on the domain of M to satisfy F(X): R does
so if, on expanding the language by adding a new relation symbol P (of the appropriate
number of places) to the language, and expanding the interpretation M to an
interpretation MR P of the expanded language by taking R as the denotation of P, the
sentence F(P) becomes true. Then we define ∀XF(X) to be true in M if and only if
every relation R (of the appropriate number of places) on the domain of M satisfies
F(X). The clauses for existential quantifications and for function symbols are similar.
The definitions of validity, satisfiability, and implication are also unchanged for
second-order sentences. Any sentence, first- or second-order, is valid if and only if
true in all its interpretations, and satisfiable if and only if true in at least one of them.
A set Ɣ of sentences implies a sentence D if and only if there is no interpretation in
which all the sentences in Ɣ are true but D false.