On Sep 14, 2008, at 2:22 AM, John F. Sowa wrote:
> Pat,
>
> One of the important points that Dunn emphasized is that the
> language in which the laws and facts are stated is a pure FOL
> without any modal operators. He assumed a plain vanilla GOFOL, but
> an extended version of FOL such as Common Logic would be equally
> appropriate. I explicitly used the term "model set", which suggested
> that I was assuming the same kind of FOL that Hintikka assumed for
> his model sets. But the method is the same for any version or subset
> of FOL. (01)
John, I'm not sure, but you seem to be mixing up object language and
metalanguage here. What you call the "language in which laws and
facts are stated" is, I believe, just the usual (non-modal) language
of first-order logic (specifically, the language of first-order set
theory) that it is used in virtually all semantic theories for modal
logic, Kripke's and Hintikka's included. But in each case the
semantic theories in question are provided for *modal* languages. In
Hintikka's case in particular, the model *systems* he uses in the
interpretation of a modal language L are classes of model *sets*,
which in turn are sets of formulas of L that satisfy certain closure
conditions. But the *metalanguage* in which these constructions are
defined is simply that of first-order set theory. (02)
> In Section 4 of the worlds.pdf paper (which you said you read), I
> was very explicit about what I meant by the word 'proposition'. I
> defined a proposition as an equivalence class of sentences under a
> "meaning-preserving translation" (MPT), which I also defined
> explicitly. (03)
Yes, and that is a very nice paper. (04)
> In any case, for the purpose of mapping Kripke's semantics to
> Dunn's, the choice of MPT is irrelevant, since you get the same
> collection of theorems and proofs with any choice discussed in that
> paper. For example, you get the same theorems whether you call p&q
> the "same" proposition as q&p or a "different" proposition. (05)
I'm not sure I see the relevance here, since your notion of
proposition is simply not part of either Dunn's or Kripke's semantics
(though they can no doubt be extended to accommodate it). (06)
> JFS>> Given a Kripke model (K,R,Phi) and for each world w in K,
>>> let M (a Hintikka-style model set) be the set of propositions
>>> true in w.
>
> PH> Whoa. That set is not yet fully defined. What do you mean by
>> 'proposition'? If you mean 'sentence', you have to say what formal
>> language your sentences are written in, because this is not
>> specified by a Kripke (or any other model-theoretic) structure. If
>> you mean something other than 'sentence', I am all ears to hear
>> what it is that you do mean.
>
> Take your pick. Just take the modal language you choose for the
> Kripke
> semantics (some version of FOL with the addition of modal operators),
> then use the base language without the modal operators to state the
> laws and facts for Dunn's semantics. (07)
Again, I think object language and metalanguage are being conflated
here. Dunn's is a semantics for propositional languages. "Law" and
"Fact" are *metalinguistic* notions for Dunn; assertions like
"Sentence A is a law at w" is an assertion in the (nonmodal)
metalanguage (and all it means is that A is mapped to 1 by the "Law"
mapping from sentences to truth values). So to talk about using "the
base language without the modal operators to state the laws and facts"
seems like a category mistake. There is simply the propositional
modal language for which Dunn is defining the notion of an
interpretation (the object language) and the first-order language of
set theory in which he is doing the defining (the metalanguage).
AFAICS, there is nothing corresponding to what you are calling the
"base language" here. (08)
-chris (09)
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