|Date:||Sat, 30 Jul 2011 06:17:38 -0400 (EDT)|
> Dunn's semantics is a semantics
Dunn's semantics specifies a way of deriving a pair of laws and facts (L,F) for each world w of a Kripke model. The simplest derivation is to let L be the set of necessary propositions of w, and let F be the set of true propositions of w. The logic used to express L and F is the same as the base (non-modal) logic of the K model.
D's semantics also provides a way of deriving a K model from any specification of possible worlds in terms of any laws and facts chosen for each of the worlds. For any world w, F can be any consistent set of propositions that is closed under deduction in whatever non-modal logic you prefer. The propositions in F are declared to be true of w. L is any subset of F that is closed under deduction in the same logic.
Since L and F are both assumed to be true, all sentences in L and F map to the truth value true; sentences not in F map to false. But the sentences in L and F are stated in any NON-modal language one chooses. I prefer FOL or the Common Logic extension to FOL.
Dunn's construction defines Kripke's accessibility relation in terms of those non-modal sets. The result is to define a modal logic that adds modal operators as prefixes to sentences in the same base logic.
> ... the validity of
you use Dunn's method to derive specifications for the worlds by starting
from the (L,F) pairs, you can choose whatever sets of individuals you
prefer. If you like, you can choose to have exactly the same individuals
in all sets of facts -- that would give you a set of worlds that
all contain the same individuals. As a result, you could use Barcan's
formula for that set of worlds.
Both Matthew and Lewis talk about worlds that they imagine (or generate or conjure up -- choose any verb you prefer). The method that they use corresponds to starting with some kind of specification.
For each world, the facts would consist of all true sentences implied by that specification. The laws would consist of the deductive closure of all general sentences (i.e., any statement that goes beyond the subset that can be expressed using just conjunction and the existential quantifier).
Therefore, I would claim that the procedure that Matthew and Lewis use to derive (generate, imagine, conjure up, or whatever) their worlds corresponds more closely to Dunn's method than to Kripke's.
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