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C00002 00002	valnot[s88,jmc]		Notes on Lifschitz's Circumscriptive Theories
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valnot[s88,jmc]		Notes on Lifschitz's Circumscriptive Theories

In VAL's formalism, suppose there is only one predicate P ranging
over a two element domain.

T(P) ≡ P(1) ∨ P(2)

and we have

VPP(1,1) ∧ VPP(1,2) ∧ VPP(2,1) ∧ VPP(2,2)

Then the circumscriptive theory is inconsistent.  

It seems to me that a system permitting joint minimization might be more
powerful.  This can be illustrated with a single predicate assuming its
domain contains more than one element.  Consider the formula

p ≤ P ≡ ∀xy(¬VPP(x,y) ⊃ (p(y) ≡ P(y))) ∧ ∀x(p(x) ⊃ P(x)).

We then want to minimize w/r this ordering.

It seems to me that with this kind of circumscription using the same
T(P) and VPP axioms, we get

∀x(P(x) ≡ x = 1) ∨ ∀x(P(x) ≡ x = 2).

The real issue is whether my scheme or his is more expressive.