[FOM] A minor issue in modal logic
rlindauer at gmail.com
Mon Jul 5 16:43:02 EDT 2010
On Sat, Jul 3, 2010 at 10:53 AM, Keith Brian Johnson
<joyfuloctopus at yahoo.com> wrote:
> The fundamental question, I think--I'll give its origin in a moment--is this: For any proposition p, where "Np" means "It is necessarily true that p" and "Np[a]" means "p is necessarily true in the actual world," are Np and Np[a] equivalent--or, if not, does Np[a] at least entail Np?
There is a vagueness between what is necessary in this world and
necessary tout court for possible-world semantics.
We want to say that there are some truths that are necessary tout
court (1+1=2) and some truths that might not be necessary p: (Ex x >
aleph-1) depending on our Discursive logic (viz the logic in which we
carry out our conversation about what is necessary). Someone
"speaking cantorian" assumes (p), whereas a weak finitist rejects (p).
What this does, though, is just point out that possible-world
semantics is not adequate for discussions of the rational
revise-ability of logical laws; For that, I think, only a
dialectical/discursive logic is adequate.
If we take possible-world talk as a discursive mechanical tool of talk
rather than "real", it fits into a dialectical logic nicely whereas
the contrary is not so. Our "ordinary english" dialectical logic can
recognize the limitation of the possible-world talk, and recognize
that the limiting game of possible-worlds is not applicable to the
question e.g. of how to decide on which possible-world logic is "the
only correct one".
A realist about possible worlds and logical laws must commit to the
idea that there is exactly one possible set of logical laws, including
set of possible-world-semantics. Such a realist, I think, must also
be committed to either a relatively weak finitism (relative to ZF,
eg.) or a relatively strong para-consistent logic at this point which
doesn't recognize simple truth or falseness (Hartry Field's lectures
on the rational revise-ability of logic apply here).
Such a realist could continue to make sense of 'Necessarily true in
world-a but not necessarily true in world-b' though with a rather
simple but generally unused modality, eg:
Nec (b -> c)
b is true in W1
b is false in W2
c is "necessary-contingent"
or, roughly in English, c is a consequent of b, but b is not necessary.
The English, in this case, is more sensible than the corresponding
possible-world talk which only tends to confuse the matter.
Applying to the God-case.
Def: God is that thing which must exist.
Lemma: Everything which must exist exists in every possible world.
Conclusion: God must exist in every possible world.
We can, discursively, reject the lemma that everything that must exist
exists in every possible world, since surely there are facts that are
"necessary-contingent" in the sense shown above - that must exist only
conditionally upon some other feature of that world.
It may, nevertheless, be that God is a special case since there may be
some things that must exist in every possible world; however, our
possible-world logic is not sufficiently strong to decide this matter.
But suppose the proposition: "There are some things that must be the
case in every possible world is false." Unfortunately this
proposition is a Russelian paradox - if it is true, it is not true in
every possible world, and there may be a possible world in which it is
false (according to it, there must be). In that world, it will be the
case that in every possible world God exists. If there is any such
world in which God must exist in every possible world, I suspect, then
God must exist in every possible world.
But, as I said, the "way out" is again to recognize that this
possible-world talk is just a game and we could decide to play it
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