Computability and Logic

19.4. ELIMINATING FUNCTION SYMBOLS AND IDENTITY 257
Now note that every interpretation M of L has a unique expansion to an interpretation
N of L + that is a model of D. The one and only way to obtain such an N is
to take as the denotation R N of the new predicate the relation that holds of a 1 , ...,
a n , b if and only if b = f M (a 1 , ..., a n ). Also, every interpretation P of L ± that is a
model of C has a unique expansion to an interpretation N of L + that is a model of D.
The one and only way to obtain such an N is to take as the denotation f N of the new
function symbol the function that given a 1 , ..., a n as arguments yields as value the
unique b such that R P (a 1 , ..., a n , b) holds. (The truth of C in P is need to guarantee
that there will exist such a b and that it will be unique.)
If S has a model M, by our observations in the preceding paragraph it has an
expansion to a model N of S & D. Then since D implies S ↔ S ± and C, N is a
model of S ± & C. Conversely, if S ± & C has a model P, then by our observations
in the preceding paragraph, it has an expansion to a model N of S ± & D. Then since
D implies S ↔ S ± , N is a model of S.
We now turn to the matter of eliminating the identity symbol, supposing that
function symbols have already been eliminated. Thus we begin with a language L
whose only nonlogical symbols are predicates. We add a further two-place relationsymbol
≡ and consider the following sentence E, which we have already encountered
in chapter 12, and will call the equivalence axiom:
∀x x≡ x &
∀x∀y(x ≡ y → y ≡ x)&
∀x∀y∀z((x ≡ y & y ≡ z) → x ≡ z).
In addition, for each predicate P of L we consider the following sentence C P , which
we will call the congruence axiom for P:
∀x 1 ...∀x n ∀y 1 ...∀y n ((x 1 ≡ y 1 & ... & x n ≡ y n ) →
(P(x 1 , ...,x n ) ↔ P(y 1 , ...,y n ))).
Note that the result of replacing the new sign ≡ by the identity sign = in E or any C P
is a logically valid sentence. For any sentence S, we let S* be the result of replacing
the identity sign = throughout by this new sign ≡, and C S the conjunction of the
C P for all predicates P occurring in S. The precise sense in which the symbol = is
‘dispensable’ is indicated by the following proposition (and its proof).
19.13 Proposition. S is satisfiable if and only if S*&E & C S is satisfiable.
Proof: One direction is easy. Given a model of S, we get a model of S* &E &
C S by taking the identity relation as the denotation of the new sign.
For the other direction, suppose we have a model A of S* &E & C S . We want to
show there is a model B of S. Since E is true in A, the denotation ≡ A of the new sign
in A is an equivalence relation on the domain |A|. We now specify an interpretation
B whose domain |B| will be the set of all equivalence classes of elements of |A|.We
need to specify what the denotation P B of each predicate P of the original language
is to be. For any equivalence classes b 1 , ..., b n in |B|, let P B hold of them if and only