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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 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 must satisfy this axiom, which of course is why it is
adopted as an axiom.
The
important thing is that our semantics is able to provide analyses
that
can account for the intuitive data. Whether or not, e.g.,
and-distribution turns out to be a logical truth seems to be
incidental.
Well, I agree; but the point of this thread was my claim that no
axiom, not even this, can be taken to be simply true of all intuitive
uses of 'true in a context'. And as this is the only logical truth
that has been proposed to hold in all cases of truth in a
context, the only conclusion seems to be that there are *no*
axioms which hold for all such cases: so that the 'general upper
ontology' of contextual truth is, in fact, empty. So let us return to
the McCarthy/Sowa claim that 'contexts are whatever satisfy the axioms
of our theory of contexts', and ask what the general theory of
contexts *is*, and we find that it isn't anything. IMO, this
reduces the idea of a 'mathematical' theory of contexts to
triviality.
> In the case of the quantifiers and the modalities, natural
language
> itself has explicit markers for the two dual notions: every vs.
some,
> necessarily vs. possibly. But for the dual interpretations of
temporal
> relativity, there seems to be nothing to lead to one being
preferred
> over the other as the fundamental notion. If anything, I'd say
that
> the 'Marienbad' interpretation is far more common. If we take
that as
> the basic idea, so that (ist t P) means that there is a
subinterval of
> t with P true in it, then ist does
not satisfy McCarthy's axiom.
>
>> On such a semantics
>>
>> (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 (ist t (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. IMO, philosophers have been far
too glib in assuming that states of belief can be logically
inconsistent. There is a lot of evidence that certain kinds of
inconsistency, at any rate, are actively corrected by unconscious
mental processes. (Now, this is a topic worthy of extended debate
:-)
(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.
> But my point is not that his cannot be done - obviously it can -
but
> to question why it is considered axiomatic that ist should be
the
> &-transparent case, rather than ist*? There is a complete
logical
> symmetry between these two ideas.
>
>> (which seems to be the semantics you are assigning to ist
directly
>> above). and-distribution then rightly fails for ist*:
it is
>> unproblematically true that
>>
>> (ist* thatDay (Pat in Mississippi)) &
(ist* thatDay (Pat in
>> Kentucky))
>>
>> and just as clearly false that
>>
>> (ist* thatDay ((Pat in Mississippi) &
(Pat in Kentucky))),
>>
>> Moreover, the reason for the failure would be analyzable in
simple
>> first-order terms as an instance of the general failure
of
>>
>> (exists (x) (P x) & (exists (x) (Q
x))
>>
>> to imply
>>
>> (exists (x) ((P x) & (Q x))).
>
> Er, yes. Of course. That was the whole point (and the notation)
of my
> paper on context mereology.
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.
> 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
idea
that 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?
> 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.
Pat
-chris
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