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C00002 00002	shoham[f86,jmc]		Comments on Shoham thesis draft
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shoham[f86,jmc]		Comments on Shoham thesis draft

1986 Oct 20
	Most comments are written in the draft itself.

1986 Nov 9
shoham@yale
hurried comments

	There isn't really a question of syntax vs. semantics.  Often
one's ideas are originally semantic.  However, besides the semantic
formulation, it is possible in every case that I know about to give a
syntactic formulation of the semantic idea at the cost of using higher
order logic.  I will give a second order syntactic formulation of your
chronological minimization.  (Definition 4.1 on p. 113 of the Oct 6 draft).
  I believe chronological ignorance can also be
translated into syntax, but I'm still not familiar enough with it to make
the attempt.  I should remark that I previously found a syntactic version
of Siegel and Bossu.  Also I like chronological ignorance, although I
think it will be part of some more general concept.

	In order to do chronological minimization, we need only give
the ordering with respect to which the minimization will be done.
It's convenient to give the "less or equal" form of the ordering.
We change your notation in order to simplify matters and use
a straightforward second order formulation.  We'll use  p(j,t)
where you write  TRUE(t,t',p).  We're making the predicates
be true at times rather than over intervals, and  j  picks out
a member of your set  S.  Let  p1(j,t)  and  p2(j,t)  be the
two predicates we wish to compare.  We write

(p1 lesseq p2) iff (forall t j)(p2(j,t) & not p1(j,t) implies (exists t' j')
( t' < t & p1(j',t') & not p2(j',t'))).

This is a lot shorter than your definition 4.1, especially when written
with a decent set of logical symbols.  [When you come to Stanford you will
be able to use a computer with a decent set of logical symbols].

The main reason for wanting a syntactic form of the principle
is to take advantage of the notion of proof for reasoning.  All
your results on complexity apply to using the syntactic formulation
as well.

I don't know the relation of this formulation to VAL's pointwise
circumscription.  I'll ask him.

Remarks:

	There are a lot, but I don't have more time at the moment.

	1. I don't agree with your applications to free will,
since I prefer the formulation in the first part of McC and Hayes.

	2. I think you have a "theory of time" rather than a
"logic of time", and it's better that way.

	3. I should send you may formulation in first order logic
of the modality of knowledge using the Kripke accessibility relation
directly.  The point is to introduce the possible worlds as first
order objects.  Your use of S5 with which I sympathize should make
it even easier.