Click on image to see Kitty’s reaction upon discovering the Principle of Explosion…

I said a bit about how the idea that God–an omnipotent being–can do the logically impossible is probably not a good idea to accept, whether you are an Atheist attempting to push the Logical Problem of Evil (“Why, your God can’t do that? Not all powerful izzit?”), or a Theist trying to defuse the Logical Problem of Evil. This post expands on that idea and introduces you to a point about logic called the Principle of Explosion.

So let’s begin again with the issue of contention I mentioned–whether an omnipotent being can do the logically impossible, for instance, create a stone so heavy even an omnipotent being cannot move it, microwave a burrito until it’s so hot even an omnipotent being cannot eat it, etc.

Traditionally, most theologians and philosophers of religion thought that the answer is “no”: Omnipotence does not involve the power to do logically impossible things–without thereby implying that it’s not Omnipotence. There are actually quite a number of ways to cash this out but here’s an intuitive one that I like:

Look, we are talking about an omnipotent being, right? So it’s, like, so awesome that it has the power to do anything, right? (See also this.) Let’s say that it can exert an irresistible force. Ok, now turn to the other side–the stone that cannot be moved. So it’s, like, so totally badass that even an omnipotent being cannot move it, right? Let’s say that it’s an unmovable object. So you are asking if a being that is capable of an irresistible force can create an unmovable object. Ok. But if you think a being that is capable of an irresistible force exists, does it make sense for you to also think that an unmovable object exists? And vice versa? Maybe you need to make up your mind, which one do you want to posit: irresistible force or unmovable object? We haven’t really started talking about what an omnipotent being can or cannot do–we aren’t even making any sense to ourselves. Either God–an omnipotent being–exists or not. If a properly unmovable object exists, then God doesn’t exist. If God exists, then there aren’t such things as unmovable objects for God to create.

In any case, as I have already hinted at in the Webinar, Harry Frankfurt–and before him, René Descartes–wouldn’t be impressed by my reasoning. For them, there’s nothing wrong with positing an omnipotent being that can move unmovable objects. By extension, it can be both that God is omnipotent and wholly good such that evil doesn’t exist, and evil exists–if God’s omnipotence is not bound by mere logic, then why should we worry about the Logical Problem of Evil? But as I also mentioned in the Webinar, there’s a heavy price to pay for this “nuclear option”. This has to do with the principle, long recognized since the Medieval thinkers, that ex contradictione (sequitur) quodlibet–“from a contradiction, anything follows”.

As in, anything. Literally.

Modern logicians calls this the “Principle of Explosion“. The below is a standard set of workings to show how you can prove literally anything from a contradiction by means of some simple rules of deductive logic.

(1) P and not-P

(2) P (from (1), by Conjunction Elimination)

(3) P or Q (from (2), by Disjunction Introduction)

(4) not-P (from (1), by Conjunction Elimination)

(5) Q (from (3), (4), by Disjunctive Syllogism)

Don’t be mislead by the specific example. When we say “literally anything”, we meant it–that Q could be anything. Likewise that P. (The move from (2) to (3) may be a bit counterintuitive to most. But here’s the simple way to see what it involves. Now, for any “p or q” statement, as long as one of the two sides–one of the two “disjuncts” is true, the whole statement is true. Let’s say that you knew p to be true already. But if that’s so, then “p or q” will also certainly be true. (Yes, if you already knew that p is true, you probably wouldn’t bother to say “therefore, p or q“. But that’s not the same as saying that the inference is wrong.) For instance:

(1) Iceland is in the tropics and Iceland is not in the tropics.

(2) Iceland is in the tropics. (from (1))

(3) Either Iceland is in the tropics or Singapore is in the Arctic Circle. (from (2))

(4) Iceland is not in the tropics. (from (1))

(5) Singapore is in the Arctic Circle. (from (3), (4), by Disjunctive Syllogism)

And you don’t need to start from an explicit contradiction either. A mere inconsistency is bad enough–after all, to say that there is an inconsistency in a set of propositions just is to say that you can derive a contradiction from it plus some logically necessary premises. And by the way, if you really meant to affirm a premise like (1), what’s happening is that you are affirming both the “p” and also the “not-p“–that’s what it means to affirm a contradiction. They don’t “cancel out each other”, as a few students from a previous semester thought. You are committed to both!

If your conception of omnipotence includes the idea that an omnipotent being is not bound by logic, then it will not be difficult to derive some contradictions. Just consider: suppose you say that God exists and He can create an unmovable object. So from “God, an omnipotent being, exists”, we can derive “Nothing can exist that God cannot move”, which implies that “nothing can exist that cannot be moved”. But from “An unmovable stone can exist”, we can get “something can exist that cannot be moved”–“it’s not true that nothing can exist that cannot be moved”… oops. And from that, explosion begin to do its baleful magic.

By the way, the additional point that God can go on to move this stone doesn’t help one bit for our purposes–it just introduces yet another contradiction! From “God can move the stone”, we will get “the stone is movable”. But just now, we have already said “the stone is unmovable”… you get the idea. And once you allow an inconsistency–and thus the derivation of contradictions–in your system, you can go on to derive pretty much anything.

And that’s not a position that most of us want to be in…

* * * * *

As a historical aside, one of the oldest near-formulations of the principle of non-contradiction in ancient China worked from a line of thinking that is interestingly reminiscent of the paradox of omnipotence:

Once there was a man of Chu selling shields and spears. In praising his shields he said, “The sturdiness of my shields is such that nothing can penetrate them (吾楯之堅，莫能陷也). Again, in praising his spears he said, “The sharpness of my spears is such that there is nothing they don’t (i.e., can’t) penetrate (吾矛之利，於物無不陷也).” In response to his words somebody asked, “How about using your spears to pierce through your shields?” To this the man could not give any reply. Indeed, shields that cannot be penetrated (不可陷之楯) and spears for which there is nothing they don’t (i.e., can’t) penetrate (無不陷之矛) cannot stand together in the same age (不可同世而立)…. (Hanfeizi, ch. 36)

If you insist on the working, here goes:

(1) Nothing can penetrate the shield.

(2) The spear can penetrate everything.

(3) The spear can penetrate the shield (from (2)).

(4) Something can penetrate the shield (from (3)).

(4) contradicts (1).

One very common Chinese term for a contradiction is “spear-and-shield” (矛楯 máodùn), and if you contradict yourself, then you are doing a “spear-and-shield-on-yourself” (自相矛楯 zìxiāngmáodùn).