An ontological argument is an attempt to prove the existence of God in a purely conceptual way and they are misunderstood in their strength by both theists and non-theists alike. They are overestimated as being a decisive proof reaching the level of proof seen in abstract mathematics and are underestimated as being a mere play on words or a parlor trick. I say an argument as opposed to the ontological argument because there have been many ontological arguments given by philosophers. The most famous one was given by St. Anselm of Canterbury (1033-1109) in his Proslogion. St. Anselm did not have the power of modal logic or even symbolic logic at his disposal so the argument was purely verbal (which isn’t necessarily a bad thing).

I think the argument can be boiled down to a two premise argument with a conclusion if one wants to tersely use modal logic. What I’ll actually do is extract two assumptions from St. Anselm and rewrite them in modal form and simply use another axiom from modal logic. The axiom is itself controversial but I’m not sure if it’s controversial for any reason other than that it is used in this argument. To my mind, the axiom is intuitive and I would have accepted it if you had not told me it was used in the ontological argument.

Let me give a brief overview of the machinery I’ll need from modal logic. First of all modal logic allows us to systematically talk about necessity and possibility. Certainly some things are true, such as the fact that there is coffee in my mug and that triangles have three sides. However, those two propositions are not true in the same mode. The former, the one about there being coffee in my mug, could have failed to be true in a number of different ways – suppose I had never chosen to fill it with coffee, suppose that I had never brewed any coffee, suppose whoever made the mug never crafted it, etc. What I’m getting at is it’s definitely true but it’s not necessarily true. This is contrasted with the second proposition, that all triangles have three sides. That cannot fail to be true by any stretch of the imagination, that’s necessarily true. Alternatively, you can think of something being necessarily true as being true in all possible worlds – that all triangles have three sides is true in all possible worlds. On the other hand, there being coffee in my mug is certainly true in our world but it is false in some other possible world if the conditions had been different, so it does not reach the level of necessary truth. Symbolically, if A is some proposition, I say that A is true necessarily (or necessarily A) by writing .

On the other hand, propositions can be false but they don’t have to be necessarily false. For example, the sidewalk is currently dry but if the meteorological conditions had been different, it might have rained and it would have been false that the sidewalk is dry. Therefore, I say that the proposition that the sidewalk is wet is actually false but could possibly be true. That is, in some possible world, the sidewalk is wet but not so in the actual world. Furthermore it is not necessarily false that the sidewalk is wet as it would be necessarily false that triangles have four sides – there’s nothing in the intrinsic nature of sidewalks or weather conditions on this particular day and location that make that the case whereas there’s something intrinsic to triangles that forbids them having four sides. Symbolically, if A is some proposition, I say that A is possibly true (or possibly A) by writing .

You may notice there is a connection between the two modal operators, necessarily and possibly, and , respectively. For example, when I say that something is not-possible I can convert this to a statement about the necessity of the falsehood of whatever proposition is in question. When I say that it’s not possible that a triangle have four sides I could just as well say that it’s necessarily false that triangles have four sides. If A is the proposition this triangle has four sides (or some proposition in general) I write , where means not, to say that it’s not possible that this triangle in question have four sides (or not possibly this triangle has four sides) or I may equivalently write to say that it’s necessarily the case that this triangle does not have four sides (or necessarily this triangle does not have four sides). This is a feature of the two modal operators, that we can always switch not-possibly with necessarily-not, or:

More simply, I could just drop the proposition A from that equation and just make the statement about the modal operators:

Which is useful when we need to do symbolic manipulations. One more way to write the above equation is to add another to the right side and then cancel the double negative:

Which makes perfect sense when saying in words – saying something is necessary is to say that it could not have possibly been false and vice versa. This also means that one could actually define the necessary modal operator in terms of not and possibly. We can do something similar with defining the possibly modal operator in terms of not and necessary by returning to the equation and by adding a not to the left side, again cancelling the double negative :

Which also makes sense when saying aloud – to say that something is possible is also to say it’s not necessarily the case that it’s false and vice versa. These formulas are very nice both conceptually are for working in modal logic proofs.

Another thing I will mention is that when we have postulates or theorems, call it A, if they are provable then is also true – the so-called necessitation rule. Also, when we have a conditional and there is a necessary operator out front, we are allowed to distribute it to antecedent and consequent – the distribution axiom:

To my knowledge, everything said above in noncontroversial in modal logic. The next idea is controversial (but I think it’s intuitive). If something is possibly true, then it’s necessarily the case that it’s possibly true:

Or that if something has modal status, that modal status is itself a necessary truth, called axiom 5. Instead of using A let’s use not-A and apply this axiom:

Now take the contrapositive:

The consequent can be simplified using that observation we made above about not-possibly-not being the same as necessarily, , and the antecedent can be simplified using that rule above, :

Expunging the double negative gives:

Now if something is necessarily true – true in all possible worlds – then certainly it must actually be the case in our world, so from we infer A:

Or carrying out the obvious syllogism:

Which is a very interesting result – if something is possibly necessary then it’s actually the case. To my mind, this is not a counter-intuitive result from the vantage of possible worlds. This last antecedent supposes that in some possible world that and if this necessary truth is true in that possible world, because it’s a necessary truth, it must be true in all possible worlds and therefore actually true in ours. So I have no problem with it but perhaps others will see a difficulty I cannot.

That’s all by way of introduction. Let’s now formulate an ontological argument, letting G be the proposition God exists. I will use that interesting result to propose the following simple argument:

St. Anselm’s (modified modal) Ontological Argument:

P1)

P2)

As you can see, it’s a simple inference using modus ponens and is therefore obviously valid. We now turn to the question of whether it’s sound. The work showing P1 is true is all to be found above so if you have a problem with P1 then you have a problem with modal logic or its application here, not with anything coming from St. Anselm. And one should have good independent reason for rejecting P1 not reject it ad hoc. So, there’s not much more to say about P1. Now, I will show that P2 follows from two assumptions coming from St. Anselm – I’ll call them A1 and A2:

A1)

A2)

Why accept either of the two assumptions? One might accept A1 because of one’s a priori conception of what God is – that if God were to exist, He would not merely exist contingently (in some possible world but not all) but exist necessarily. If God could somehow fail to exist in some possible worlds, that is, only exist contingently, then this seems to be out of line with our conception of what God is, in that God is supposed to be maximally great and part of this, presumably, is not having some prior conditions be met in order to exist, which is the case for existing contingently, but would have to additionally exist necessarily. There’s lots more to say about A1, but that’s the idea – if God were to exist, which doesn’t mean He does (at this stage), it’s merely an antecedent, then God would have to exist necessarily not contingently.

Why accept A2? Either A2 is true or it is false. If it were false then this is saying that is not even possible for God to exist and, to my mind, there are no successful attempts to show that God’s existence is not possible, that is, necessarily false, recalling that . Examples of things being necessarily false include logically self-contradictory propositions like triangles with four sides. So, the lack of compelling reason on the negative side may persuade us to accept A2 or a simple agnosticism about the mere possibility of God’s existence, as opposed to being absolutely impossible, may also encourage us to accept A2. Also notice that A2 is not saying (God exists), so it is not question begging.

Let’s now show how the truth of A1 and A2 will give us P2 and complete the argument. Let’s begin with the contrapositive of A1:

Recall that if this is a postulate, then it’s necessarily true by the necessitation rule:

We now use the distributive axiom:

Let me rewrite the antecedent using the rule that :

Let’s also notice that A2 is logically equivalent to as we observed near the beginning of the post, by the very meaning of the possibility modal operator. Using that writing of A2 and the last line, we infer by modus tollens:

Which is precisely P2. So we see that P2 follows from the truth of A1 and A2, which means if you accept A1 and A2 and have no general problems with P1, then you’re logically committed to the conclusion, G, that God exists.

Now, I must confess that St. Anselm’s ontological argument does not, for me, have the same compelling power that the various cosmological arguments have. But at the same time it’s not nonsense and it suggests that in order for the atheist to get around this sort of argument, he’d have to argue for the negation of A2, which is which is to say he needs to argue that it’s necessarily false that God exist, not merely that it’s false but possibly true – if he says it’s false but still possibly true, he’s locked into A2. This is an amusing point because many unsophisticated atheists will compare God to a unicorn or cosmic superhero. But here’s the thing – there’s nothing logically contradictory about either of these, which is what you need to show absolute impossibility, otherwise you’re committed to A2. There are no unicorns but it is true that there are possibly unicorns. As for A1, I think an atheist should accept this since a failure to exist necessarily as opposed to merely contingently is a fault that would be picked up by the atheist, in that God would fail to exist in some possible worlds, so this seems to be some defect. If an appendix is going to be used against God (argument from bad design) then I think a failure to exist in a possible world if He does exist should also be noted by the atheist.

Another confession – as someone who is highly sympathetic to Thomism (or probably already is a Thomist) I share in the suspicion of a priori arguments for the existence of God in favor of a posteriori arguments, such as the cosmological arguments. So I’m suspicious of ontological arguments in general, not just St. Anselm’s. Besides the fact that this argument is a priori, the notion of using possible worlds and modal operators as things more fundamental to God, as though there are possible worlds in the platonic sense, and God just so happens to be an element of some of them seems to get the roles reversed in terms of which, possible worlds or God, is more metaphysically fundamental. Even if we interpret these possible worlds in the platonic sense as real, existing abstract objects, the question then arises how in the world these possible worlds arise and what makes it possible that there are possible worlds at all.

This isn’t to say I don’t find this argument extremely interesting; I wouldn’t have written a blog post to it if that were not the case. However, like other arguments, these serve to direct the intellect to a particular location, not bend your arm behind your back and force you to believe – arguments do not force you to believe anything. But I hope this post does illustrate that St. Anselm’s ontological argument isn’t just silly nonsense and mere playing with words – there’s something you need to grapple with here.