Computability and Logic

250 NORMAL FORMS
can be the domains of subinterpretations that are closed under the functions f A that
are the denotations of function symbols of the language, or in other words, that contain
the value of any of these functions for given arguments if they contain the arguments
themselves.
When B is a subinterpretation of A, we say that A is an extension of B. Note that
the notions of extension and subinterpretation pertain to enlarging or contracting the
domain, while keeping the language fixed.
Note that it follows from (S1) and (S3) by induction on complexity of terms that
every term has the same denotation in B as in A. It then follows by (S1) and (S2)
that any atomic sentence has the same truth value in B as in A. It then follows
by induction on complexity that every quantifier-free sentence has the same truth
value in B as in A. Essentially the same argument shows that, more generally, any
given elements of B satisfy the same quantifier-free formulas in B as in A. Ifan
∃-sentence ∃x 1 ...∃x n R(x 1 ,...,x n ) is true in B, then there are elements b 1 , ..., b n
of |B| that satisfy the quantifier-free formula R(x 1 , ..., x n )inB and hence, by what
has just been said, in A as well, so that the ∃-sentence ∃x 1 ...∃x n R(x 1 , ..., x n )is
also true in A. Using the logical equivalence of the negation of an ∀-sentence to an
∃-sentence, we have the following result.
19.7 Proposition. Let A be any interpretation and B any subinterpretation thereof.
Then any ∀-sentence true in A is true in B.
19.8 Example (Subinterpretations). Proposition 19.7 is in general as far as one can go. For
consider the language with just the two-place predicate