Chris, (01)
I admit that we were arguing about different issues. (02)
The immediate purpose of Dunn's semantics is not to replace
a modal logic with a nonmodal logic. The first step is to
replace a world w with a pair (M,L). The result of that
step does not in any way change the modal logic that Kripke's
method or Dunn's method is being used to model. (03)
CM> I think object language and metalanguage are being
> conflated here. (04)
I admit that I should have emphasized the distinction: Dunn's
method enables much of modallike reasoning to be done at the
metalevel of talking about the sets (M,L). But that kind of
reasoning is distinct from the proof procedures of the original
modal logic for which either Kripke's or Dunn's method could be
used to define the semantics. (05)
CM> What you call the "language in which laws and facts are stated"
> is, I believe, just the usual (nonmodal) language of firstorder
> logic (specifically, the language of firstorder 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. (06)
I completely agree with that point. (07)
CM> I'm not sure I see the relevance [of discussing the way the
> term 'proposition is defined] 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). (08)
I only mentioned that because Pat explicitly asked how I defined
the term 'proposition'. We can reach the same conclusions with
just the word 'sentence', and I probably should have used that
word in order to avoid an extraneous issue. (09)
> So to talk about using "the base language without the modal operators
> to state the laws and facts" seems like a category mistake. (010)
In my original paper, I didn't use the word 'base language', and
I shouldn't have mentioned it here. The original modal logic is
quite distinct from the language used to state M and L. Dunn
mentioned that it might be necessary to use some modal operators
to state M and L if the original modal logic allowed iterated
modalities. But that is a separate issue. (011)
To be clear, I should have emphasized the following distinctions: (012)
1. The original modal logic, whose semantics can be defined in
equivalent ways by either Kripke's method or Dunn's method. (013)
2. The details of how Kripke's models are mapped to Dunn's models. (014)
3. The new opportunities for modallike reasoning that become
possible by metalevel reasoning about the sentences in M and L.
These reasoning methods are distinct from the proof procedures
used with the original language in #1. (015)
As you mentioned, the methods of #1 and #3 are probably not logically
equivalent. And that's actually a good argument for using Dunn's
method for certain kinds of problems. But as I claimed, the methods
of either K or D can support exactly the same proof procedures used
for the original language #1. (016)
John (017)
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