Previously, I outlined what I find to be a compelling ontological argument from Alexander Pruss. In the post, we dispelled the idea that there is a single ontological argument and distinguished between a number of families on such arguments. The one we focussed on is a so-called Gödelian ontological argument, named after the famous mathematician Kurt Gödel. Gödelian ontological arguments construct an axiomatic system and use this to prove, as a theorem in the system, that something like God exists; and this is exactly what we did. If you’ll recall, we proved the following theorem:

Theorem 2 There exists a necessary, essentially omniscient, omnipotent, omnibenevolent being.

In fact, we noted that using the same methods we used to prove theorem 2 we could prove that there exists a necessary being who possesses all strongly positive properties. Now, in the proof of this theorem, we made use of a system of modal logic called S5. The goal of this post is to discuss this system, and consider some alternative (and weaker) systems that could be used to arrive at the same result.

Modal claims

“I could have been born in 1920”, “I could have done better in my test” and “contradictions cannot happen” are three examples of what we might call (metaphysical) modal claims. These are claims about what could or could not have been the case. We say that something is (metaphysically) possible if it could have been true; and we say that something it is (metaphysically) necessary if its negation is impossible, that is, it cannot not be true. Modal logic is a symbolic way for us to be rigorous about the claims and arguments we run using modal claims about how reality could have been[1]. We use two symbols: the diamond ◊ symbolises possibility and the square □ symbolises necessity. So ◊A means “possibly A” or “A could have been true”, and □A means “necessarily A”.

We can also talk about this using the language of possible worlds. Recall that a possible world is a maximal description of how reality could have been. By “maximal” we mean that for any proposition, this description either contains that proposition or its negation. In possible world semantics, we understand “possibly A” to say that A is true in some possible world, and “necessarily A” to say that A is true in every possible world.

Now when it comes to modal claims, we have a nice relation between possibility and necessity: it is possible that A if and only if it is not necessary that not A. We can write this as

◊A↔~□~A.

This is equivalent to saying that A is necessary if and only if it is not possible that not A. In possible world semantics, something is true in every possible world just in case there is no world in which it is not true.

From the context of possible worlds, we can understand modal claims as accessibility claims: when we say possibly A, we mean that there is a possible world that we can “see” from the actual world in which A is true. When we say possibly possibly A, we mean that there is a possible world, from which we can “see” another possible world in which A is true.

S5

Besides the one mentioned above, there are other axioms that we would agree apply to such modal claims, but I won’t go into them here. Now S5 is a system of modal logic. What I mean by that is that it is a collection of axioms about modal claims. Many people agree that S5 (or something stronger) correctly describes the metaphysical modality that we’re concerned with (that is, it correctly describes the accessibility relationships between possible worlds). The crucial axiom in S5 is the following (from now on we’ll use S5 to refer to the axiom, and not the system as a whole):

S5. ◊A→□◊A.

What this says, in everyday language, is that if two propositions, A and B, are possibly true, then if B were true, then A would still be possibly true. Put another way, what is actually the case doesn’t affect what could have been the case. Put another way, no matter how things could have gone, if things could have gone a specific way, they always could have gone that way. What it says, in possible world semantics, is that there is one collection of possible worlds. Put another way, that every possible world is accessible from the others. Put another way, “before” there was an actual world, there were possible worlds.

Now it isn’t surprising that many people think that S5 is true. After all, surely whether I go to the shop today doesn’t affect whether or not I could have done something else instead. Or, more generally, it doesn’t seem that what could have happened in reality depends on what actually does happen, if anything it’s the other way around!

Nevertheless we can give two sort of arguments for the truth of S5, both of which come from Alexander Pruss[2,3]. First we start with the intuition that no matter how things could have gone, the way they did go would always have been possible. This is called the Brouwer axiom, and it is strictly weaker than S5 (that is, S5 entails Brouwer, but not the other way around). Because we’re going to be using Brouwer later, we’ll state in symbols,

Brouwer. A→□◊A.

Next, we note that when we’re dealing with metaphysical possibilities, we’re dealing with a sort of ultimate modality. That is, when it comes to talking about the ways reality might have gone, there’s no way we could speak more ultimately about how reality might have gone. If we can think of a more ultimate way, then that would be what we mean by metaphysical possibility. We can capture this as follows: if the diamond symbol captures metaphysical possibility, then it could never be the case that ◊A is false, while ◊◊A is true. After all, in that case we could treat “◊◊” as a modal operator, and it would be more ultimate than simply “◊”[4]. So, when talking about metaphysical possibilities, because we’re talking about ultimates, we have the following,

S4. ◊◊A→◊A.

Now, S4 and Brouwer entail S5: apply Brouwer to ◊A to get □◊◊A. Now if we have A→B, then clearly □A→□B (if B always follows from A, and A is true in all possible worlds, then so is B). Applying this to S4, gives us □◊◊A→□◊A. Thus, ◊A→□◊◊A→□◊A, which is S5.

The second argument I quote straight from Pruss’ book[5]:

…one might say that precisely those propositions are [metaphysically] possible which the fundamental laws of metaphysics allow. But the collection F of all the fundamental laws of metaphysics could not be different from what it is – that is central to its being the collection of the fundamental laws of metaphysics – and the “could not” here is surely metaphysical. Moreover, what F allows cannot have been different. If it were different, that would presumably be because a collection of laws might permit different things in different circumstances. Suppose that C is some collection of fundamental laws that permits different things in different circumstances. But then there would need to be further metaphysical laws as to what the laws in C collectively permit under what circumstances, and barring a vicious regress of more and more basic laws, there would have to be fundamental laws specifying what the laws in C permit. And these laws couldn’t be in C, since then the laws in C would not permit different things in different circumstances. Therefore, if C permits different things in different circumstances, then C does not contain all the fundamental laws, in the way that F does. Thus, what F permits could not be different, and hence modality could not have been different. And that is what axiom S5 says.

He mentions a third “argument” for S5: our best accounts of metaphysical modality entail S5. There’s not enough space or time to go through these. You can check here, and section 2.2.6 here for an overview of them. If you want a fuller treatment I recommend Pruss’ book, Actuality, Possibility, and Worlds.

Just Brouwer

So S5 seems very plausible both on intuitive grounds and because of the arguments mentioned in the previous section. Nonetheless, I like to minimise the strength of the assumptions in a given argument, and I suspect this can be done here. For the rest of this post, we’re going to look at alternatives to S5 as used in proving theorem 2. We mentioned the Brouwer axiom in the previous section. I claim that we only need it to get the theorem to work.

The challenge that arises in proving theorem 2 with just the Brouwer axiom is that we can’t guarantee that G has necessary existence in the actual world. With S5, we have that ◊□A→□A[6], so if A=”N exists”, then if N possibly necessarily exists, it follows by S5 that N necessarily exists. However, with Brouwer, all that follows is N exists. This is a challenge because we needed N to have necessary existence to prove anything else about it.

I think this challenge can be overcome with a little more specificity: pick the original uniqualizing strongly positive property to be one that entails necessary existence. We’ve seen two candidates so far: aseity and being the paradigm of moral goodness. Consider aseity, for example. Then possibly N exists a se. Thus possibly N exists necessarily. Thus, by Brouwer, N exists. But aseity is an essential property of N, so N exists a se. Thus N exists necessarily.

Essentiality of Contingency

Could we weaken S5 in a different way? I believe so. Consider the following principle:

Essentiality of Contingency: contingent existence is an essential property of contingent beings.

This seems very plausible, after all, what could make a contingent being necessary? What does this principle look like when put into modal language? Let C be any being, then the principle says,

EoC. ◊(C fails to exist)→□◊(C fails to exist).

It follows from this, however, that necessary existence is essential too. To see this let N be a some being. Then using the contrapositive of EoC, we have that ◊□(N exists)→◊~◊(N fails to exist)→~□◊(N fails to exist)→~◊(N fails to exist)→□(N exists).

Now one of our axioms from the previous post was that necessary existence is positive. But, now we see that necessary existence is essential, so it’s actually strongly positive. So, we run the argument as normal: pick any uniqualising positive property and necessary existence as two positive properties. Then this being exists in the actual world and is necessary. The argument follows from this the same way it did in proof we gave for theorem 2 in the previous post.

Notes

1. To be clear, modal logic can be applied to more than metaphysical modalities. It has been applied to tense, ethics, in epistemology, and can be easily applied to strictly logical possibilities too. In this post I use the phrase “modal claim” to designate specifically metaphysical modal claims unless otherwise stated.
2. Giving arguments for modal axioms seems very strange. I mean, how might we argue for the first modal axiom I gave?
3. If you’re interested in this stuff, I recommend his book (from which I got these arguments), Actuality, Possibility, and Worlds.
4. For more details on this, I refer you to Pruss’ book.
5. Actuality, Possibility, and Worlds (pg 17) In the book he talks about C_0 and C, I’ve changed C_0 to F in the quote for clarity.
6. We showed this in the previous post. To see this, start with ~A in S5: ◊~A→□◊~A. Then take the contrapositive: ~□◊~A→~◊~A. Then we have that ◊□A→◊~◊~A→~□◊~A→~◊~A→□A.