Computability and Logic

13.3. THE SECOND STAGE OF THE PROOF 159
involving nonlogical predicates. From that point, the rest of the proof will be exactly
the same as where identity was not present. Towards framing the definition of R M ,
note that (E4 ′ ) allows us to give the following definition:
Thus
R M (C 1 *,...,C n *) if and only if R(c 1 ,...,c n )isinƔ*
for some or equivalently any
c i with C i = [c i ].
R M ([c 1 ],...,[c n ]) if and only if R(c 1 ,...,c n )isinƔ*.
Together with (3), this gives (2) of the preceding section. Since as already indicated
the proof is the same from this point on, we are done with the case with identity but
without function symbols.
For the case with function symbols, we pick a distinct object T * for each equivalence
class of closed terms, and let the domain of the interpretation consist of these
objects. Note that (3) above still holds for constants. We must now specify for each
function symbol f what function f M on this domain is to serve as its denotation,
and in such a way that (3) will hold for all closed terms. From that point, the rest of
the proof will be exactly the same as in the preceding case where function symbols
were not present.
(E5 ′ ) allows us to give the following definition:
Thus
f M (T 1 *,...,T n *) = T * where T = [ f (t 1 ,...,t n )]
for some or equivalently any
t i with T i = [t i ].
(4)
f M ([t 1 ]*,...,[t n ]*) = [ f (t 1 ,...,t n )]*.
We can now prove by induction on complexity that (3) above, which holds by
definition for constants, in fact holds for any closed term t. For suppose (3) holds for
t 1 , ..., t n , and consider f (t 1 , ..., t n ). By the general definition of the denotation of
a term we have
( f (t 1 ,...,t n )) M = f M( t M 1 ,...,t M n
)
.
By our induction hypothesis about the t i we have
ti M = [t i ]*.
Putting these together, we get
( f (t 1 ,...,t n )) M = f M ([t 1 ]*,...,[t n ]*).
And this together with the definition (4) above gives
( f (t 1 ,...,t n )) M = [ f (t 1 ,...,t n )]*.