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

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.

Tue May 14 00:04:52 PDT 1996