Anselm and Brouwer

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.

1. E!(God) → □E!(God).

2. ∀x[if Conceivable(x) and  ∼ E!(x), then: E!(x) → ⋄ ∼ E!(x)].

3. So, not: (Conceivable(God) and  ∼ E!(God)).

4. 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:

5. 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

6. If q, then □⋄q,

where L is necessity, and the following principle about subjunctive conditionals:

7. 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:

8. 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):

9. 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.

Another intuition concerning the Brouwer axiom

Suppose the Brouwer axiom is false. Thus, there is some possible world w₁ such that at w₁ our world w₀ is impossible. Here’s another world that’s impossible at w₁: the world w_⊥ at which every proposition is both true and false. Thus at w₁, possibility does not distinguish between our lovely world and w_⊥. But that seems to me to be a sign that the possibility in question isn’t making the kinds of distinctions we want metaphysical modality to make. So, if we are dealing with metaphysical modality, the Brouwer axiom is true.

You can't run this argument if you run this one.

A heuristic argument for the Brouwer axiom

Suppose that:

1. We cannot make sense of impossible worlds, but only of possible ones, so the only worlds there are are possible ones.

2. Necessarily, a possible worlds semantics for alethic modality is correct.

3. Worlds are necessary beings, and it is essential to them that they are worlds.

Now, suppose the Brouwer axiom, that if something is true then it’s necessarily possible, is not right. Then the following proposition is true at the actual world but not at all worlds:

4. Every world is possible.

(For if Brouwer is false at w₁, then there is a world w₂ such that w₂ is possible at w₁ but w₁ is not possible at w₂. Since w₁ is still a world at w₂, at w₂ it is the case that there is an impossible world.)

Say that the “extent of possibility” at a world w is the collection of all the worlds that are possible at w. Thus, given 1-3, if Brouwer fails, the actual world is a world that maximizes the extent of possibility, by making all the worlds be possible at it. But it seems intuitively unlikely that if worlds differ in the extent of possibility, the actual world should be so lucky as to be among the maximizers of the extent of possibility.

So, given 1-3, we have some reason to accept Brouwer.

An asymmetry between the theistic and atheistic modal ontological arguments

A simple version of the modal ontological argument goes as follows:

1. Necessarily: If there is a God, then necessarily there is a God. (Premise)

2. Possibly, there is a God. (Premise)

3. So, possibly necessarily there is a God. (By 1 and 2)

4. So, there is a God. (By 3)

It is well-known that there is a very similar argument for the opposite conclusion. Just replace (2) by the premise that possibly there is no God, and you can change the conclusion to read that there is no God. So it seems we have a symmetric stalement. Though, perhaps, as Plantinga has noted, we get to conclude from the arguments that the probability that God exists is 1/2, which when combined with other arguments for theism (or maybe with a particularly plausible version of Pascal’s Wager?) it could be useful.

However, interestingly, the symmetry is imperfect in a way that I haven’t seen mentioned in the literature. Consider this atheistic ontological argument:

5. If there is a God, then necessarily there is a God. (Premise)

6. Possible, there is no God. (Premise)

7. So, it’s not necessary that there is a God. (By 6)

8. So, there is no God. (By 5 and 7)

This argument differs from the theistic one in two ways. First, the atheistic argument can get away with premise (5) which is formally weaker (given Axiom T) than (1). This is not a big difference, since (5) is no more plausible than (1).

But there is a bigger difference. In the theistic argument, to derive (4) from (3) requires the somewhat controversial Brouwer Axiom of modal logic (which follows from S5). But the atheistic argument does not need any axioms of modal logic, besides the uncontroversial modal De Morgan equivalences behind the inference of (7) from (6).

My first thought on noticing this asymmetry was that the atheistic argument is somewhat superior to the theistic, at least when the audience isn’t sure of S5 (or Brouwer).

My second thought was that the atheistic argument is more subject to the objection that its possibility premise begs the question. For the conclusion follows more directly from the possibility premise, and that makes a question-beggingness objection a little bit more plausible.

I don’t know exactly what to think now.

Anyway, nothing earthshaking here. For those of us who think S5 is true, the differences are pretty small. But it’s worth remembering that the symmetry is imperfect.

Adams' ontological argument

Robert Adams' modal ontological argument in his piece on Anselm in The Virtue of Faith seems not to get much attention. Adams' modal ontological argument doesn't need S5: it only needs the Brouwer axiom p→LMp, namely that if p is true, it not only is possible, but it is a necessary truth that p is possible. Here is a version of Adams' argument. Let G be the proposition that God exists. Then as God is by definition a necessarily existent and essentially divine being, that God exists entails that God necessarily exists:

1. L(G→LG).

Add that possibly God exists:

2. MG.

The proof is simple:

3. MLG. (By 1 and 2 and K)
4. ~G→LM~G. (Brouwer)
5. MLG→G. (Contraposition on 4)
6. G. (Modus ponens on 3 and 5)

And by an application of 1, 6, axiom M (the necessary is true) and modus ponens we can even conclude LG, that necessarily God exists.

This doesn't use S4. So worries about the transitivity of possibility are irrelevant here.

Griffin attributes the Brouwer-based argument to Leibniz.