Re: Proposed definition of "possible world"
Chris,
Nicola, among others, has been using the term "possible world"
in his definitions. That is a model-theoretic notion that he
found useful in defining ontological terms. I objected to that
notion because it has never been given any formal definition
of any kind. In its place, I suggested a replacement for
the term "possible world" that does have a formal definition.
>I'll have to study Dunn's semantics in more detail, but I do find it
>very puzzling that a *semantics* for a modal language should intimately
>involve proof theoretic notions, or lack (as far as I can see) any
>notion of interpretation for the elements of the object language.
One of the most common axioms for modality is that all
tautologies (i.e. anything provable from the logical axioms)
are necessarily true. That is certainly proof theoretic.
George Boolos wrote a whole book about proof theory, using
the box symbol of modal logic to mean "is provable":
George Boolos, _The Logic of Provability_, Cambridge UP, 1993
The notion of relating modality to privileged propositions
called laws is as old as Aristole, and in trying to explain
causal modality, Rudolf Carnap said
I propose the following definition: a statement is
_causally true_ or C-true, if it is a logical consequence
of the class of all basic laws.
That is in Ch. 21, "The Logic of Causal Modality", of _An
Introduction to the Philosophy of Science_, Dover, NY, 1966.
And what elements of the object language are you talking about
that lack any "notion of interpretation"? What I was objecting
to is Nicola's use the the term "possible world", which has
no formal definition of any kind. (At least not in Kripke
or Montague's work -- and David L. has lots of talk and no
formal definitions. Then Dov Gabbay suggested the "innovation"
of replacing "possible world" with "theory" -- which I heartily
applaud.
John