An ontological argument from the possible nondefectiveness of modality

1. Necessarily, if it is necessary that there is no God, then modal reality is bad. (Making the existence of God impossible is terrible!)

2. Necessarily, if something is bad, it is possible for it not to be bad. (The bad is a flaw in something that ought to be better than it is, and what ought to be can be.)

3. So, if modal reality is necessarily bad, then it is possible for modal reality not to be bad. (by 2)

4. So, if modal reality is necessarily bad, then modal reality is not necessarily bad. (by 3)

5. So, modal reality is not necessarily bad. (by 4)

6. So, possibly, modal reality is not bad. (by 5)

7. So, possibly, it is not necessary that there is no God. (by 1 and 6)

8. So, possibly, it is possible that God exists. (by 7)

9. So, it is possible that God exists. (by 8 and S4)

10. Necessarily, if God exists, it is necessary that God exists. (God is a necessary being and essentially divine.)

11. So, it is possible that it is necessary that God exists. (by 9 and 10)

12. So, God exists. (by 11 and Brouwer)

Anti-S5

Suppose narrowly logical necessity L_L is provability from some recursive consistent set of axioms and narrowly logical possibility M_L is consistency with that set of axioms. Then Goedel’s Second Incompleteness Theorem implies the following weird anti-S5 axiom:

• ∼L_LM_Lp for every statement p.

In particular, the S5 axiom M_Lp → L_LM_Lp holds only in the trivial case where M_Lp is false.

For suppose we have L_LM_Lp. Then M_Lp has a proof. But M_Lp is equivalent to ∼L_Lp. However, we can show that ∼L_Lp implies the consistency of the axioms: for if the axioms are not consistent, then by explosion they prove p and hence L_Lp holds. Thus, if L_L∼L_Lp, then ∼L_Lp can be proved, and hence consistency can be proved, contrary to Second Incompleteness.

The anti-S5 axiom is equivalent to the axiom:

• M_LL_Lp.

In particular, every absurdity—even 0≠0—could be necessary.

I wonder if there is any other modality satisfying anti-S5.

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.

A Cosmological Argument based on the Empire State Building

Assume:

1. Necessarily, every exact duplicate of the Empire State Building has a cause.
2. Necessarily, if an exact duplicate of the Empire State Building never changes, then neither it nor any of its parts cause its existence.
3. Possibly, the only contingent beings ever are an unchanging duplicate of the Empire State Building and parts thereof.

Let w be a world where the only contingent beings ever are the unchanging duplicate of the Empire State Building and its parts. By (1), it has a cause. By (2), this cause cannot be the Empire State Building or a part thereof. Since those are all the contingent beings, the cause must be a necessary being, or include a necessary being as a part. So, possibly, there is a necessary being. By S5:

4. There is a necessary being.

Premise (1) is a version of the Causal Principle specialized to the sorts of entities that we are most confident of there being causes of. One might wonder about why one needs the "never changes" in (2). But there is reason for it. Some objects can perhaps be caused by their parts. Imagine a bunch of trees that grow together to form a tower. We could likewise imagine a bunch of moving stony and metallic beings that come together to form an exact duplicate of the Empire State Building. This is ruled out by the "never changes".

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.

S5

In response to a question from a student, I explained S5 as the claim that modal truths don't vary between worlds—a modal truth, we might say, is a proposition of the form "Possibly p" or "Necessarily p" that is true. But actually, that's incorrect as an account of S5. S5 is compatible with there being modal truths in some worlds that do not obtain in others.

To see this, we need to make Robert Adams' distinction betwen truth at a world and truth in a world. On Adams' view, a proposition making de re reference to a particular only exists in a world where that particular exists. Consequently, propositions that reference contingent particulars de re are not necessary beings. Since there are modal truths involving contingent particulars, such as that possibly I will yell "Hurrah!" in five minutes, it follows that some modal truths exist in some worlds but not in others.

Adams' distinction, then, is basically this. We take as the primitive notion being true at a world. We might gloss this as follows: a proposition p is true at w provided that the state of affairs it reports obtains in w. We can then define being true in a world counterfactually: p is true in w iff it is the case that were w actual, p would be true. Since a proposition can only be true if it exists, it follows that p is true in w iff p is true at w and p exists in w. And if we like, we can say that p exists in w iff it is true at w that p exists.

On Adams' view, the proposition that Socrates either does not exist or is human is true at every world. But it is only true in those worlds in which Socrates exists.

Let us then follow Adams in allowing that some propositions exist in only some worlds. Then, in some worlds there will be modal truths that are not true in the actual world simply because they do not exist in the actual world. We now have two ways of defining modal operators. I'll just define possibility, M, since necessity is dual to it (Lp iff ~M~p). We say that M₁p iff there is a possible world w such that p is true at w. We say that M₂p iff there is a possible world w such that p is true in w. These give different results. For instance, if Adams is right about propositions about particulars, then M₁(Socrates never existed) is true but M₂(Socrates never existed) is false.

The modal logic we get with M₂ is no good. According to that modal logic, necessarily Socrates exists, but possibly there are no humans. So the modal logic we want is the one defined by M₁. But this modal logic is just fine for S5: it is basically just a restriction of a Plantingan modal logic with no scruples about non-actual particulars to those propositions that do not reference non-actual particulars de re.

Now we have a picture of modal logic that satisfies S5, even though there are modal truths in other worlds that are not modal truths in our world (because they do not exist as propositions in our world).