This treatment is similar to Davis's (1980) treatment of domain circumscription. Pat Hayes (1979) pointed out that the same ideas would work.
The intuitive idea of circumscription is saying that a tuple
satisfies the predicate P only if it has to.
It has to satisfy P if this follows from
the sentence A. The model-theoretic counterpart of circumscription
is minimal entailment. A sentence q is minimally entailed
by A, if q is true in all minimal models of A, where a model
is minimal if as few as possible tuples satisfy the
predicate P. More formally, this works out as follows.
Definition. Let M(A) and N(A) be models of the sentence A.
We say that M is a submodel of N in P,
writing 
, if M and N have the same domain,
all other predicate symbols in A besides P have the
same extensions in M and N, but the extension of P in M is
included in its extension in N.
Definition. A model M of A is called minimal in
P if 
only if 
. As discussed by
Davis (1980), minimal models don't always exist.
Definition. We say that A minimally entails q with respect
to P, written 
provided q is true in all
models of A that are minimal in P.
Theorem. Any instance of the circumscription of P in A is true in all models of A minimal in P, i.e. is minimally entailed by A in P.
Proof.
Let M be a model of A minimal in P. Let 
be a
predicate satisfying the left side of (1) when substituted
for 
. By the second conjunct of the left side P is an
extension of . If the right side of (1) were not
satisfied, P would be a proper extension of . In that case,
we could get a proper submodel 
of M by letting agree
with M on all predicates except P and agree with on P. This
would contradict the assumed minimality of M.
Corollary. If 
, then 
.
While we have discussed minimal entailment in a single
predicate P, the relation 
, models minimal in P and Q,
and 
have corresponding properties and a corresponding
relation to the syntactic notion 
mentioned
earlier.