From owner-cse663-fa08-list@LISTSERV.BUFFALO.EDU Sun Sep 21 16:51:03 2008
Date: Sun, 21 Sep 2008 16:50:54 -0400
From: "William J. Rapaport"
Subject: 663: Quantified Modal Logic I: The Barcan Formula
To: CSE663-FA08-LIST@LISTSERV.BUFFALO.EDU
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Subject: Quantified Modal Logic I: The Barcan Formula
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Because I decided not to cover quantified modal logic in lecture, in the
interests of time, here is the first of two supplementary discussions of
it.
Notation:
Ax for "for all x"
L for the necessity operator (the "box")
> for the material conditional (the "horseshoe")
P is an n-place predicate possibly containing an occurrence
of the free variable x
The following formula, named after the logician Ruth Barcan Marcus, is
called the "Barcan formula" (BF):
Ax.LP(x) > LAx.P(x)
Question: What is the domain of interpretation for x?
There are two ontological possibilities:
* "possibilism": Assume that there is a single domain containing
all possible objects
* "actualism": Assume that each possible world contains its own objects
that exist in it.
BF says: If everything is necessarily such that it satisfies P,
then it is necessary that everything satisfies P.
>From this, it follows that all objects that exist in every possible
world exist in the actual world.
I.e., the domain of any possible world is a subset of the domain of
the actual world.
This is a version of possibilism.
But perhaps BF is not the case. Then its antecedent is true but its
consequent is false.
I.e., even if everything fallin gunder the actual range of the
quantifier satisfied P in every possible world, there could be something
*else* in some other possible world that failed to satisfy P in *that*
world.
This is a version of actualism.
Bottom line: Your ontological views about the nature of possible
worlds will determine whether BF is true or false.