I was reading Anselm’s replies to Gaunilo, and was struck by this:
Furthermore: if it can be conceived at all, it must exist. For no one who denies or doubts the existence of a being than which a greater is inconceivable, denies or doubts that if it did exist, its non-existence, either in reality or in the understanding, would be impossible. For otherwise it would not be a being than which a greater cannot be conceived. But as to whatever can be conceived, but does not exist – if there were such a being, its non-existence, either in reality or in the understanding, would be possible. Therefore if a being than which a greater is inconceivable can be even conceived, it cannot be nonexistent.
Let → indicate subjunctive conditionals. Let E!(x) say that x exists.
E!(God) → □E!(God).
∀x[if Conceivable(x) and ∼ E!(x), then: E!(x) → ⋄ ∼ E!(x)].
So, not: (Conceivable(God) and ∼ E!(God)).
So, if Conceivable(God), then E!(God).
The ∀x quantifier in (2) is problematic, since it ranges over beings that don’t exist. Perhaps we can read it substitutionally. Let’s suppose we can finesse this issue.
What interests me in (2) is that big conditional in it is most plausibly as seen as a special case of:
- If C(p) and q, then p → ⋄q,
where C(p) says “conceivably p”, which may or may not be the same as “possibly p”.
We can prove (5) from the Brouwer Axiom
- If q, then □⋄q,
where L is necessity, and the following principle about subjunctive conditionals:
- If C(p) and Lr, then p → r,
namely that necessities would still hold no matter what conceivable things happened (to get (5) from (6) and (7), let r be ⋄q). Principle (7) is very plausible if conceivability is possibility: if a possible thing happened, anything necessary would still be true. It’s less plausible if conceivability is not possibility.
And, of course, the Brouwer Axiom is controversial, albeit not quite as much as S5. I initially hoped that the use of the subjunctive conditional in (2) allowed Anselm to get by with something weaker than Brouwer. But not so if the route to (2) goes through (5) and possibility implies conceivability (PIC). For we get Brouwer from (5), PIC and the very plausible principle:
- If p is possible, then we do not have p → r and p → ∼ r.
For suppose that contrary to Brouwer we are at a world where q is true, but ⋄q is false at some accessible world. By PIC, ∼ ⋄q is conceivable. Let p be ∼ ⋄q. Then C(p) and ∼ p. But clearly ∼ ⋄q → ∼ ⋄q. If we had (5), we would have ∼ ⋄q → ⋄q, and contradict (8).
So, if Anselm’s argument for (2) goes through (5), we don’t have an improvement over Brouwer. But we can still get (2), given some very plausible assumptions, and the following special case of (5):
- If C(∼q) and q, then ∼ q → ⋄q.
And I feel that (9) has some plausibility above (6), at least if conceivability is the same as possibility. For suppose q is true and ∼ q is possible. Then it seems somewhat plausible that q is possible in the ∼ q-worlds that are closest to the actual world. Maybe. But maybe there is some way to derive Brouwer from (9) and additional plausible premises.