## Saturday, April 9, 2011

### Does Conceivability Entail Possibility?

The following should probably be pretty clear to some people. Just another way of showing that conceivability does not imply possibility. The proof will go somewhat informally with a few hidden premises, and I'm not even sure if it's any good.

Suppose
(1) If I can conceive of a possible world, then such a world exists. (This doesn't mean it's actual of course.)
Further,
(2) I can conceive of a possible world which is empty.
(3) Therefore, there is an empty possible world.
(4) I can conceive of a necessary being.
One can easily deduce from (4) that
(5) Therefore, there is no empty possible world.
(6) Therefore, there is an empty possible world and there is no empty possible world.
(7) Therefore, ~(1). (because 1 implies a contradiction, viz. 6)

By "empty possible world" I mean a world in which no entities exist. Some may object that this is harder to understand than at first sight, and may in fact be inconceivable. For those who agree that it is conceivable we've shown that (1) is false. For the skeptic's sake let's take a less controversial example. Let's say that in every possible world there exists at least one entity. This isn't to say that in each world it is the same entity, i.e. we're not saying there exists something x such that x exists in all possible worlds. To illustrate the point let's use a little bit of basic sets. Take two sets A = {1,2} and  B = {3,4). These two sets are disjoint. If you tried to find a set which has as its elements/members all x such that x is in A and x is in B you'd come up with an empty set. In other words, they have no members in common. In an analogous way we can conceive of two possible worlds being "disjoint", that is, not having any entities in common. So we can say

(8) I can conceive of two "disjoint" possible worlds.
(9) Therefore, there are two disjoint possible worlds.

Remember, this is consistent with our rejection of (2).

(10) If there are two disjoint possible worlds, then there are no entities common to all possible worlds.
(11) Therefore, there are no entities common to all possible worlds
(12) If a necessary being is possible, then there are some entities which are common to all possible worlds.
(13) Therefore, there are some entities common to all possible worlds. (One could easily show that this follows.)

We've of course come to our flat contradiction with (11) and (13) and shown (1) to be false.

Without (1) how can we learn about metaphysical possibility? It seems to me that we'll have to go back to those good ol' essences, thus raising the further question of how we can know both essences and their implications. I wonder though how we can figure out why anything exists at all. We can't turn to essences to learn about the modal properties of being. Being has no "essence", since it's a transcendental. The falsity of (1) also has implications for those who want to create modal arguments for God. We can't just go based on God's conceivability. It's interesting to note that Robert Maydole's proof of the Maximally Great Being's (MBG) metaphysical possibility seems to rely on the very essence of the MGB.

P.S. Now we've got Ss. Augustine, Anselm, and Bonaventure praying for us on the side. :-)