On Sep 14, 2008, at 2:22 AM, John F. Sowa wrote: (01)
> 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. (02)
Yes, I do know Dunn's work. (03)
> 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.
>
> 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'. (04)
Sure, but I wasnt certain you were using the same sense in the email. (05)
>
> I defined a proposition as an equivalence class of sentences
> under a "meaningpreserving translation" (MPT), which I also
> defined explicitly. (06)
Yes, I'm familiar with your construction, of course. But now we get
back to my earlier objection. Propositions, in this sense, might as
well be sentences: a set of propositions is just a set of sentences
with an equivalence defined on it. (07)
> (We had talked about that idea a few years
> ago in connection with the IKRIS project.) The simplest example
> of a "meaningpreserving translation" is the identity, which
> implies that each syntactically distinct sentence states a
> distinct proposition, but any other MPT would do as well.
>
> 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. (08)
Indeed, so my point applies in either case. Your mapping as stated
refers to the set of propositions/sentences, but does not specify
which set. A Kripke structure does not itself define a particular
language to interpret against it, so your mapping from Kripke to Dunn
seems to be underdefined.
>
>
> JFS>> Given a Kripke model (K,R,Phi) and for each world w in K,
>>> let M (a Hintikkastyle 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 modeltheoretic) 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. (09)
That isn't a mapping from Kripke to Dunn. Its a mapping from (Kripke
plus a choice of modal language) to Dunn. I'm still waiting to hear
how to map form Kripke to Dunn. (010)
If you can show a metatheorem to the effect that the essential aspects
of the translation are independent of the choice of formal language,
that might be a good first step. But I don't actually think this is
true. (011)
Pat (012)
>
>
> John
>
>
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