On Aug 8, 2011, at 5:25 PM, John F. Sowa wrote:
> Chris,
>
> Before commenting on today's notes, I'd like to return to Dunn's point
> that his approach could be extended to Quantified Modal Logic (QML)
> by a substitutional interpretation of quantifiers. I reread his paper,
> and he did not rule out other methods. Any method of extending Kripke
> semantics to QML could just as easily be adapted to Dunn's semantics. (01)
Well, I already suggested in my earlier post how one could probably provide a
classical quantification theory for Dunn's semantics by adapting a
substitutional approach. What I strongly disagree with is that it could be done
"just as easily". I have not worked out the details, but it appeared clear to
me it's not for nothing that Dunn suggested extending his semantics with a
substitutional quantification theory; you can't just tack on a classical QT
onto his semantics and go on your merry way the way you can with Kripke
semantics. Since substitutional QT is not in general classical, one would have
to make nontrivial adjustments to get a classical theory. (02)
> CM
>> I'm getting pretty tired of you making Kripke out to be some sort of
>> villain and possible world semantics as vaguely (or explicitly) sinister.
>
> My complaint is not about what Kripke did or said, (03)
But your complaint was exactly that: "if [Kripke] had used the term 'model'
instead of 'possible world', he would have saved a lot of dead trees." (04)
> but about professors
> who teach modal semantics with featureless points and ignore the fact
> that people who implement possible worlds use sets of sentences. (05)
I'm not sure I see the problem with that. It is easy to move back and forth
between the two. The model set corresponding to a "featureless" world is just
the set of sentences true there. (06)
> CM
>> To anyone who is capable of reading Kripke's work on the model theory
>> of modal logic it is *perfectly* clear that his "possible worlds" are
>> nothing more than indices on, essentially, models for classical
>> propositional or (as the case may be) firstorder logic.
>
> I said that in my papers, and I taught that to students before I came
> across Dunn's paper. But I was teaching students who were primarily
> interested in applications to AI, comp. sci., and linguistics. In
> those fields, they represent possible worlds by the *sentences* that
> are true in those worlds. That's D's method, not K's.
>
> It's fine to teach them K's method for historical reasons. But it's
> even more important to show them that D's method is what they actually
> use in their applications. (07)
I am not at all sure that that is true. Be that as it may, Kripke's methods
are of far more than historical interest. They are used ubiquitously  and
compellingly  in theoretical comp sci, AI, and linguistics, as you well know,
so I really don't see how you can say this. (08)
>> formally speaking, once again, the set of "possible worlds" in a model
>> structure is (as far as the models tell us) a set of featureless points...
>
> I grew up as a mathematician. I studied algebraic topology and
> categories of abstract mappings. You don't have to preach to me
> about mappings to and from featureless points. (09)
No one is preaching to you, John. I am well aware of your background and you
know I respect your overall knowledge of logic and math. But we are holding a
public conversation so I was just keeping the background context clear. (010)
> JFS
>>> 1. Logical possibility. A proposition p is possible iff it is not
>>> provably false. Impossible means inconsistent or provably false.
>
> CM
>> This won't do. Provably false in what theory? Consider the continuum
>> hypothesis (CH) ...
>
> I was trying to give a very brief summary of Peirce's many pages of
> writings about multiple modalities. C. I. Lewis and Arthur Prior
> both drew a great deal of inspiration from CSP's writings in their
> pioneering work on modal logics.
>
> CSP was familiar with Cantor's work, but he had obviously not read
> publications that had not been written. I just wanted to say that
> his writings still have a lot to offer for the 21st century. (011)
Fair enough, but you didn't say that explicitly. Readers might have been left
with the impression that CSP's list was correct and exhaustive. (012)
chris (013)
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