Pat, (01)
Duality may make two different kinds of operations look symmetrical,
but that symmetry can hide enormous differences in practice. (02)
> The relationship between ist and ist* is symmetrical. They are
> duals. We could adopt either as the real meaning of 'true in
> a temporal context' and then the other would be defineable:
> and only one of these strategies gives McCarthy's axiom (the
> other gives the dual axiom, ie transparency under DISjunction.)
> So what justifies the choice of one rather than the other as
> fundamental? Is there any linguistic evidence for this? (03)
There is an overwhelming amount of evidence of many different kinds: (04)
1. The only two logical operators that can be directly observed
are 'and' and 'there exists'. You can see A, and B, and C,
and D... That implies that things that cause the percepts
(or hallucinations) of things like A, and B, and C, and D...
must exist. (05)
2. A negation can never be observed. A failure to observe something
might mean that you haven't looked hard enough, you were looking
in the wrong direction, it was smaller than you expected, etc.
Russell once joked that he could not get Wittgenstein to admit
that the statement "There is no hippopotamus in this room" was
absolutely certain. But W. was right. There is no way to prove
that a very tiny hippopotamus -- perhaps an embryo of a hippo or
one that had been miniaturized by a reducing ray -- wasn't lurking
under the rug or that they weren't at the zoo and merely imagining
that they were in an apartment at Cambridge U. A negation can only
be inferred from observation plus many assumptions about what would
be the case if the opposite were true. (06)
3. Any operator defined in terms of negation is just as problematic,
if not more so. You might be uncertain whether you saw Sam or Joe,
but you didn't see their disjunction. You saw a man whom you
couldn't identify for certain. But that does not mean you
saw a disjunction. (07)
4. Although 'every' is dual to 'some', you only need one observation
to see *some cow* but it's impossible to see *every cow*. (08)
5. Implication is impossible to observe. See Hume, for example. (09)
6. In theorem proving, conjunction and the existential quantifier
are the easiest to deal with. But disjunctions in the consequent
cause the highly efficient Horn-clause subset of logic to become
NP complete. FOL without universal quantifiers is as easy to
deal with as propositional calculus. But universals require a
much more complex approach. (010)
7. Finally, every observation sentence can be stated in NL (or in any
version of logic) with only conjunctions and existentials. All
languages express those two operators in essentially equivalent
ways. But all other operators are problematical. Negations raise
many different kinds of puzzles (negation as failure, as absence,
as denial, as prohibition, etc.). Some languages have inclusive
disjunction, some have exclusive disjunction, some have both, and
some are not clear about which is intended. If-then isn't always
clearly expressed in all languages, and when something like it is
available, it is more likely to be some kind of "strict" implication
rather than material implication. And various languages express
different variations of universal quantifiers in different ways,
none of which, of course, represent an observation. (011)
> The point however is that the context approach to temporal reference
> does not allow tenses: it sets out to replace them by context-
> reference.... The actual inner sentences are always in a 'present'
> tense (actually, are tenseless.) (012)
I agree. That is what I usually do: attach a relation for point-
in-time (PTim) to the outside of a context box and put tenseless
statements inside. The problems and paradoxes aren't caused by
what's inside the context box, but by problematical cases on the
outside. This would require more discussion, but I believe that
problems arise because of inadequate axioms and representations
on the outside, not the inside of the context box. (013)
> It is no good quoting dictionaries in discussions like this. There
> are FAR more notions of 'context' than this. McCarthy gives the
> following examples... (014)
But all of those examples involve relations and axioms on the
*outside* of the context box. That was the point I was making:
it is only necessary to have one very simple syntactic mechanism,
such as a box. Then the open-ended number of special cases for
the semantics can be handled with axioms outside the box. (015)
> That is not the point. The point is that others use it [the
> word 'context'] to justify bad scholarship, claim originality
> for old ideas, and repeat old mistakes. (016)
That's their problem, not mine. I'm just using the word for one
very simple syntactic mechanism with one simple claim: all the
hard stuff goes on outside the box. (017)
> There is no such definition, and McCarthy (IMO, correctly) says
> so explicitly. The only definition is the context logic itself. (018)
I quoted a very clear definition from a typical dictionary:
a context is a piece of text. Context logic is not inside the
box, but in whatever axioms anyone wants to write on the outside.
As I said before, there are infinitely many axioms that people
might state about contexts, and they should be allowed to do so. (019)
> What have CGs have to do with a discourse? The NL notions of
> context all seem to arise in some kind of conversational or
> discourse setting: but logics aren't used in such settings.
> They are used to make assertions, not to have conversations with. (020)
Discourse and conversations are just text. I might use multiple
boxes to enclose different turns of the conversation, and put
relations on the outside of the boxes to relate them. A speech
act can be represented by a statement in a context and an
intensional verb outside that context. See McCarthy's Elephant
article, for example. He certainly intended to use logic
for such things -- I'll admit it's still a wide-open research
area, but that does not mean logic is irrelevant to it. (021)
> You havn't said what "true in a context" means. (022)
Of course not. That is the purpose of the open-ended range of
axioms that can be stated outside the context box. (023)
> And what are some of those axioms? Or even ONE of them? (024)
For starters, you can take any axioms that anybody has ever
proposed for contexts and assert them outside the context box.
I don't believe that any finite number can cover all the uses. (025)
John (026)
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