Matt Hart comments:
. . . most of what we conceive is possible. So if we say that
1) In 80% of the cases, if 'Conceivably, p' then 'Possibly, p'
2) Conceivably, God exists
3) Pr(Possibly, God exists) = 80%
4) If 'Possibly, God exists' then 'necessarily, God exists'
5) Pr(Necessarily, God exists) = 80%,
we seem to get by.
I had made the point that conceivability does not entail possibility. Hart agrees with that, but seems to think that conceivability is nondemonstrative evidence of possibility. Accordingly, our ability to conceive (without contradiction) that p gives us good reason to believe that p is possible.
What is puzzling to me is how a noncontingent proposition can be assigned a probability less than 1. A noncontingent proposition is one that is either necessary or impossible. Now all of the following are noncontingent:
Necessarily, God exists
Possibly, God exists
God does not exist
Necessarily, God does not exist
Possibly, God does not exist.
I am making the Anselmian assumption that God (the ens perfectissimum, that than which no greater can be conceived, etc.) is a noncontingent being. I am also assuming that our modal logic is S5. The characteristic S5 axiom states that Poss p --> Nec Poss p. S5 includes S4, the characteristic axiom of which is Nec p --> Nec Nec p. What these axioms say, taken together, is that what's possible and necessary does not vary from possible world to possible world.
Now Possibly, God exists, if true, is necessarily true, and if false, necessarily false. (By the characteristic S5 axiom.) So what could it mean that the probability of Possibly, God exists is .8? I would have thought that the probability is either 1 or 0. the same goes for Necessarily, God exists. How can this proposition have a probability of .8? Must it not be either 1 or 0?
Now I am a fair and balanced guy, as everyone knows. So I will deploy the same reasoning against the atheist who cites the evils of our world as nondemonstrative evidence of the nonexistence of God. I don't know what it means to say that it is unlikely that God exists given the kinds and quantities of evil in our world. Either God exists necessarily or he is impossible (necessarily nonexistent). How can you raise the probability of a necessary truth? Suppose some hitherto unknown genocide comes to light, thereby adding to the catalog of known evils. Would that strengthen the case against the existence of God? How could it?
To see my point consider the noncontingent propositions of mathematics. They are all of them necessarily true if true. So *7 + 5 = 12* is necessarily true and *7 + 5 = 11* is necessarily false. Empirical evidence is irrelevant here. I cannot raise the probability of the first proposition by adding 7 knives and 5 forks to come up with 12 utensils. I do not come to know the truth of the first proposition by induction from empirical cases of adding. It would also be folly to attempt to disconfirm the second proposition by empirical means.
If I can't know that 7 + 5 = 12 by induction from empirical cases, how can I know that possibly, God exists by induction from empirical cases of conceiving? The problem concerns not only induction, but how one can know by induction a necessary proposition. Similarly, how can I know that God does not exist by induction from empirical cases of evil?
Of course, *God exists* is not a mathematical proposition. But it is a noncontingent proposition, which is all I need for my argument.
Finally, consider this. I can conceive the existence of God but I can also conceive the nonexistence of God. So plug 'God does not exist' into Matt's argument above. The result is that probability of the necessary nonexistence of God is .8!
My conclusion: (a) Conceivability does not entail possibility; (b) in the case of noncontingent propositions, conceivability does not count as nondemonstrative evidence of possibility.