On Jul 26, 2011, at 3:15 AM, sowa@xxxxxxxxxxx wrote:
> Chris,
>
> I believe that the following point is the crux of the misunderstanding: (01)
Indeed, it seems to be. The question is, who is misunderstanding what? ;-) (02)
> CM
> > Dunn's semantics is only a semantics for propositional
> > modal logic. It is far from obvious how it generalizes
> > to quantified modal logic, which would be essential if
> > the framework is going to applicable to the issues that
> > Lewis addresses.
>
> On the contrary, Dunn's method defines the modal operators as an extension to
>whatever logic is used to state the laws and facts. I thought that point was
>so obvious that I didn't mention it. (03)
Not only is it not obvious, it is false. Dunn's is a semantics for
propositional modal logic. Look in particular at the semantic clauses on pp.
89-90 of Dunn's paper: one each for atomic sentences, negations, conjunctions,
and modals. There is no quantificational clause. (04)
> But look at the formal definitions. For any world w, a proposition p is
>possible iff it is consistent with the laws of w, and p is necessary iff it is
>provable from the laws of w.
>
> That definition depends only on your logic and proof theory. (05)
No, it doesn't. The explicit context of the definition is a semantic theory
for propositional modal languages. (06)
> If your logic for stating laws and facts is FOL, you get quantified modal
>logic. (07)
Not so. It is in fact entirely non-trivial how you "get" QML from Dunn's
semantics. The foundational notions of laws and facts are based on an initial
assignment of truth values to all the sentences of a language. To extend this
idea to a language with predicates and quantifiers would mean that one would
assign truth values to atomic and quantified sentences directly instead of
deriving their truth values from an antecedent assignment of denotations to
names/variables and extensions to predicates, as in classical semantics. Dunn
in fact notes in the last paragraph of his paper that such an extension of his
framework to QML would require some sort of substitutional interpretation of
the quantifiers — on the face of it, anyway, a rather radical departure from
the objectual quantification of classical Kripke semantics for QML. I can in
fact imagine ways of doing this that would yield something equivalent to Kripke
semantics for a quantified modal language L — notably, for a given
interpretation I of L, by expressing the quantifier clause substitutionally
with regard to an extension L' of L in which every one of the (perhaps
uncountably many) objects in the domain of I has a name in L'. But, again, that
is a long way from obvious and it is misleading to suggest otherwise. (08)
> Another point: I am not denying the value of any insight or proposal that
>Lewis stated in terms of possible worlds. I'm just saying that replacing
>each w with a pair (L,F) preserves every technical contribution that Lewis
>made. (09)
But I haven't been talking exclusively, or even primarily, about Lewis's
technical contributions. I've been talking *applications* of his world theory
to the solution of semantical and philosophical problems. My point has been
that those applications do not map in any obvious way to Dunn's framework of
laws and facts. (010)
But look, this is a bit of a tempest in a teapot. Both frameworks are rigorous
and both have their own intuitive appeal; either might prove to be a useful
framework for representing information in given context. We should just follow
Carnap's advice here and "let a thousand flowers bloom." But we should also be
clear about what each flower has to offer. :-) (011)
-chris (012)
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