February 20, 2006

This is what we did in Kant today

[] is the modal operator 'necessarily' or 'in all possible worlds'

P1. [](x is a substance -> x exists)
P2. Santa Claus is a substance
3. [](x is a substance) -> [](x exists)
4. x is a substance -> [](x exists)
5. Santa claus is a substance -> [](Santa Claus exists)
C6. [](Santa Claus exists)

So Santa Claus exists necessarily, if you believe that being a substance is a sufficient condition for existence.


Drew said...

I find these logic posts quite fascinating. It's been years since I studied formal logic, and even then, not nearly so advanced a level as this. But I'm curious about P2.

The why I look at it, being a substance is not a sufficient condition for existence, but rather existence is a necessary condition for being a substance. It's obvious that Santa Claus, a man, would have to be a substance, but only if he existed. In other words, existence is a sine qua non for being a substance.

I have to confess that I was quite baffled by the recondite proof of the existence of God you posted recently. Couldn't quite wrap my head around that one. In fact, I think there's something of a capricious nature to these puzzles. By making very subtle alterations in the logical rules, it's possible to construct the most absurdly illogical "proofs" that appear to be perfectly reasonable.

Noumena said...

Both "being a substance is a sufficient condition for existence" and "existence is a necessary condition for being a substance" would be formalized as P1. The defenders of P1 in class rejected P2 in the same way you did, but the conclusion of this argument -- that Santa Claus necessarily exists -- is too funny to pass up. Try it with something we'd normally take to be a substance, but that only contingently exists, like yourself (or your Cartesian ego).

Logic puzzles have been some of the most fruitful in the history of philosophy. I've heard that more has been written on the Ontological Argument (a purported, purely logical proof of the existence of God) and its intellectual descendants than any other family of arguments in history. It would be interesting to write a metaphilosophical piece on why philosophers are so fascinated by puzzles like this.

===== my solution =====
Since we're using modal logic, 'exists' is vague. Contemporary modal logic is cashed out in talk of possible worlds, so there are three ways we can narrow down the meaning of 'exists':
1) x exists =df x is in some possible world;
2) x exists =df x is in the actual world (this one, the one we live in); and
3) x exists =df x is in all possible worlds.

If we use meaning 1, then P1 is automatically true -- if x is a substance, then it's already in some possible world or another. But then it's so weak as to be uninteresting, since pretty much everything is in some possible world or another. If we use 3, then P1 is most likely false -- the intuition that makes this argument a problem in the first place is simply that being a substance isn't enough to guarantee that you show up in every possible world.

If we use 2, then P1 is probably false again, since it now asserts that all the substances show up in the actual world. If you think some things would be substances if they existed, ie, that other possible worlds have substances that the actual world doesn't (like Santa Claus, or your goateed and evil counterpart), then you're going to reject P1.

MosBen said...

Actually we're looking for proof of Drew's clean shaven, non-evil counterpart.