Re: SUO: Re: 3D/4D question
Bill Andersen wrote:
> But, if you want modality as well, then Counterpart Theory is strictly more
> expressive than modal logics with Kripke semantics. It's possible to say in
> Counterpart Theory, for example, that "it is possible that there could have
> been a completely different set of individuals than there in fact are". My
> memory is a little fuzzy - gotta consult my notes, but I think this is an
> example of something which can't be expressed in normal modal logics.
Well, yes and no, Bill. Here's the yes. First of all, let's talk about
modal languages, not logics. Then what you say is true with regard to
the modal language L you get simply by adding (at least one of) the
modal operators [] and <> (expressing necessity and possibility,
respectively, of course) to predicate logic. There is no way of
expressing what "in fact" does in your sentence above in this language.
Here's the no. You *can* express in with a simple extension to L,
namely, in essence, the language you get by adding "in fact" as an
operator! Loosely put, the semantic function of this operator -- more
commonly known as an "actuality" operator -- is to schlep you back to
the actual world W* from the possible world fixed by the modal operator
"could have been" and let you evaluate the sentence in the scope of the
actuality operator in W*. Syntactically speaking, then, such an
operator really has some purchase when it is used in the scope of a
modal operator. Thus, where "A" is our actuality operator, your
sentence above could be expressed as
* <>(x)A(y)(~x=y).
That is, it is possible that everything that exists is distinct from
everything that *actually* exists. Or perhaps more idiomatically, it
could have been the case that everything that would have existed would
have been distinct from everything that actually exists. And in
possible worlds lingo: There is a possible world W such that everything
that exists in W is distinct from everything that exists in the actual
world.
Sound and complete logics have been given for this language with respect
to more-or-less standard Kripke semantics by Harold Hodes and, I think,
Allen Hazen (and probably others). Hodes' work is highly recommended --
not that I don't recommend Hazen's, I just haven't studied it. (I don't
have references at hand, unfortunately, but I recall that the relevant
papers by Hodes came out in the Journal of Philosophical Logic in the
mid-80s.)
Sidebar: Careful with the term "normal modal logic", which you use
informally above. This expression actually has a precise technical
meaning, viz., any extension of the propositional modal logic K, i.e.,
any modal logic extending classical propositional logic that includes
the axiom schema [](p -> q) -> ([]p -> []q) and the rule of
necessitation.
> I think even Pat should be happy with CT - it's extensional and expressible
> in plain-old FOPC.
But, unfortunately, obviously does not capture what we mean when we make
de re modal claims like Brando's "I coulda been a contendah."
-chris
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