On Jan 14, 2008, at 12:01 PM, Pat Hayes wrote: At 11:11 AM -0600 1/14/08, Chris Menzel wrote: On Sun, 13 Jan 2008, Pat Hayes wrote:
At 3:28 PM -0600 1/13/08, Christopher Menzel wrote:
On Jan 13, 2008, at 1:15 PM, Pat Hayes wrote:
... McCarthy claims that and-distribution applies to both:
(ist c (p & q)) iff ( (ist c p) & (ist c q) )
but there are certainly some cases of time-context where this fails, eg there was one day last year when I was (at various times) in five states, but I have never been in five states all at once. So apparently
(ist thatDay (Pat in Mississippi)) & (ist thatDay (Pat in Kentucky))
but not
(ist thatDay ((Pat in Mississippi) & (Pat in Kentucky)))
Not so clear to me that this is a counterexample to and-distribution. Seems to me that one _could_ do the semantics of ist vis-a-vis temporal contexts so that something that is true with respect to a given interval t has to be true with respect to all subintervals of t.
Yes, you can. And McCarthy uses that semantics, and so concludes (and takes it as axiomatic) that ist distributes over conjunction. But examples from natural language seem to often obviously have the dual interpretation, as here. So for example there is a famous Italian movie, "Last year in Marienbad", all about something that happened in Marienbad the previous year. But it didn't happen everywhere in Marienbad, for the entire year. Examples of this kind of 'contextualization' are rampant in natural language: so what supports the claim that the dual form must be correct, so correct indeed that it is a logical truth?
Well (not that you of all people don't know this), "logical truth" is largely a term of art. What counts as a logical truth is often only decided after we fix a particular semantics. Natural language shows that there are at least two senses of "this happened at that time" and, as you point out, we could take either as primitive (though see below), so it doesn't seem to me that there is necessarily a pre-analytic fact of the matter about the logical truth of and-distribution.
Exactly: that was my entire point.
And so I figured, but as you yourself have noted, we're conducting this discussion in public, so I just wanted to make the record clear. :-)
And yet, as I think you know, advocates of context logic (in particular, of McCarthy's) have insisted that any coherent or rational notion of truth in a context mustsatisfy this axiom, which of course is why it is adopted as an axiom.
Right.
(ist thatDay (Pat in Mississippi)) & (ist thatDay (Pat in Kentucky))
would be false. Granted, in ordinary language, if you travelled from Mississippi to Kentucky on, say, January 5, one can say both that Pat was in Mississippi on Jan 5 and that Pat was in Kentucky on Jan 5. But one could capture this ordinary usage -- and preserve your intuitive data above -- by _defining_ a related notion ist* such that (ist* t P) just in case (ist t' P) for _some_ subinterval t' of t
Or more directly by dualization, (ist* t P) == (not (ist t (not P))).
It's a bit off the point, but I'm dubious about this analysis in general, since (I think) it presupposes that excluded middle holds "internally", that is, that for every context t and proposition P, either (ist t P) or (istt (not P)).
Slightly weaker: (not (ist t (P & (not P)))). I think any notion which fails this has no coherent logic. But we may well disagree here:
I don't think that is a viable principle for at least some notions of contexts
I presume you are thinking of states of belief and other 'psychological' context-ish things.
Actually, no, I was thinking more of Barwise-style situations as limited pieces of space-time, only where the evaluative standpoint is *within* the situation rather than outside it. (I take the latter approach in the paper you allude to below.) On the former approach (typical in situation semantics), if an object is not present in a context, there is no information about it, not even the information, e.g., that it is not present. Hence, internal excluded middle fails right and left.
(though it's probably ok if we're simply identifying contexts with time intervals with no spatial
restrictions).
Its certainly OK, as your own paper on the topic elegantly expounds.
Thanks for the plug. :-) (Paper available at http://cmenzel.org/Papers/occ.pdf.) Actually, internal excluded middle breaks down on my approach as well, but only for potentially paradoxical, "self-referential" propositions whose truth values in a context may not stabilize (more or less in the sense of Gupta's revision theory of truth). You'll have to forgive me if it appeared otherwise, but I was quite certain I wasn't saying anything you were unaware of, Pat. However, it struck me that your claim that "there are certainly some cases of time-context where [and-distribution] fails" without any further qualification was misleading insofar as it suggested (even if you didn't so intend) that one *couldn't* find a reasonable semantics for "ist" on which and-distribution was valid and in terms of which the apparent counterexamples could be explained.
No, my point was that there exists a perfectly reasonable semantics for 'ist' which made it invalid.
Well, we couldn't be in more violent agreement. :-) But now, which of these is the correct view of a time-interval as a 'context'? They can't both be.
Well, it should be pretty clear at this point that I'm leery of the ideathat there is such a thing as "the correct view" here. What we need to do is account for the intuitive data. There might be more than one way of doing that.
So there might be more than one context logic? Isnt it better at this point to admit that there is no actual logic of contexts?
I guess I don't see much difference between between saying there are different theories of context and different logics of context. "Logic" is really just an honorific title we bestow on a formalization of a cluster of concepts whose applicability is particularly broad and hence whose meanings we've attempt to fix, semantically or proof theoretically. That said, in the context of knowledge sharing and the semantic web where it would be good to stick to a single (though flexible) classical logical framework, theories seem like the way to go from a practical point of view.
In fact, its clear that time-intervals-as-contexts are not even one kind of context, but (at least) two. There are (at least) two notions of 'true in an time-interval'. And of course there are more than two: there are notions such as "true almost everywhere" or "true during subintervals which are normal for that kind of proposition" (as in "he was furious for a whole week"), all of which have a valid claim to be a notion of 'true during an interval'. Which was my main point: there is no axiom which is true for ALL of these notions.
Agreed, certainly, though it doesn't follow that there isn't a single notion in terms of which one can give data-preserving *analyses* of the other notions -- which, of course, is not to say there *is* such a notion either. ;-)
Quite. The burden of proof seems to lie elsewhere, and if John McCarthy can't come up with it after over a decade, I think its time to forget the idea.
So let a hundred flowers bloom -- be they logics or theories. :-)
-chris
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