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C00002 00002	partia[e82,jmc]		Partial models and partial relativization
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partia[e82,jmc]		Partial models and partial relativization

	1. We want partial models in order to give a more mathematical
idea that achieves some of the goals people have been trying to achieve
by talking about resource limited deduction.

	Let formulas be in prenex form.  Suppose we have two domains,
D1 ⊂ D2, called the inner and outer domains.  The predicate
symbols are given interpretations on the outer domain.  Function
symbols are defined on the inner domain but may take values in the
outer domain; if they are to be iterated then they must take values
in the same domain as their arguments.

	Universallly quantified vaiables are considered to range over the inner
domain and existentially quantified variables over the outer domain.
The truth of a formula in a partial interpretation is computed in the
usual way except for this.  It should be noted that systems of formulas
that may be equivalent w/r standard semantics may be inequivalent
w/r some partial semantics.

	Our goal is to be able to handle some mathematical counterfactuals.
Consider some axiomatization of arithmetic that includes some definition
of prime number.  Consider the counterfactual conditional sentence

"If 51 were a prime, then 2*51=102 would be a prime".

or

	prime(51) ⊃ prime(2*51).

If we encounter such a sentence in an argument and don't know whether
51 is a prime, we would like to be able to consider the sentence
false or at least invalid.  Our idea is that there will be a reasonable
axiomatization of arithmetic, including a definition of prime number,
which will have some partial models in which 51 is a prime.  However,
in these models 102 will not be prime, so the counterfactual will be
false in these models and therefore invalid with respect to this
class of partial models.  We would like it to turn out that such
classes of partial models correspond to the state of mind of a person
who doesn't know whether 51 is prime but can see that multiplying
it by 2 will surely make it non-prime.

	2. In conversation with Kurt Konolige (1982 July 29), he
suggested that if partial models were a useful notion, they would
have a syntactic counterpart.  This syntactic counterpart might
be the following notion of partial relativization.

In relativization,  ∀x. ...  is replaced by  ∀x.p(x) ⊃ ...  , and
∃x. ...  is replaced by  ∃x.p(x) ∧ ... .  In partial relativization,
we use an inner formula to replace universal quantifiers and an
outer formula to replace existential quantifiers.  Thus we have

∀x.p1(x) ⊃

and

∃x.p2(x) ∧

where we presumably have  ∀x.p1(x) ⊃ p2(x).