Can God make a rock so heavy, not even He can lift? If yes, then there's something God can't lift. If not, then there's something God can't make. This is such a classic paradox about the logical existence of God's omnipotence that even the late Stephen Hawking mentions it in his A Brief History of Time.
How can omnipotence be logical? After all, this is a "Yes or No" question; it can't be both and it can't be anything else. If a restaurant serves a certain food only at night, but one of the ingredients in it can be prepared only in the morning, it has to be served either in the morning or at night depending on the criteria taken into account, and ends up needing to be served during both and neither at the same time! Clearly it is therefore never served, and might as well not even exist. It's like a restaurant that's open on February 31st and prepares all its paperwork every year for a day that never comes.
But how valid is this objection? If something is logically impossible even for God, such as square circles and married bachelors, does that refute the claim of omnipotence?
Logic and Predicates
These questions basically pit God's power against logic. The temptation is to say that God created logic, but the problem is that God is Himself supposed to operate under it. But once it is realized that logic isn't a "thing" the way a chair is, the issue evaporates, because it is immediately understood that logic is human language's convenient construction for the generalized objects that actually exist: viz., God and his power, and the individual objects (squares, circles, bachelors, and married folks).
The issue then in the end boils down to God's power against Himself. What the question basically asks is, "Is God more powerful than Himself?" - "Can God destroy Himself?" We can already see this as an absurd postulate for omnipotence or any kind of power. True, something non-omnipotent can willingly destroy itself: suicide for example. The 1st century Roman naturalist, Pliny the Elder, even regarded this as the one "advantage" man possessed over God!:
God cannot give himself death even if he wishes, but man can do so at any time he chooses.
But this isn't a strength. Obviously one isn't omnipotent by escaping from life instead of resolving the problems from it.
The whole issue can easily be untangled by understanding something very simple. First of all, Immanuel Kant points out that things like power, logic, or existence aren't predicates. That is, they're not like a chair that has a physical description and illustration in and of itself. These are relationships to one or more such objects. The same goes for love, hatred, information, etc. A good example is infinity, which is not a number, but is generally spoken of as one, and often for convenience's sake treated that way. We give these concepts a word for convenience's sake, which we can easily misuse by manipulating as if these ideas are objects: namely, the fallacy of language. This answers the question of whether God created "existence" and if so, how did He Himself exist? Was it simultaneous? If so, existence is as ancient as God, and has an origin other than Him (i.e. God didn't make "everything," because He couldn't, and so again, isn't omnipotent).
"What's in a name?" "A rose by any other name would still be a rose." Quite right! This basic concept can be seen in many places, whether philosophical or theological. Paul masterfully dismembers this kind of abuse regarding the issue of eating food "polluted" by idols; since the pagan deities don't exist, it can't be a sin once someone has understood this (1 Cor. 8, 10; Romans 14).
So when we say that God "can't" do something, it's because this something isn't a "thing" to be done in the sense we're talking about at all.
This can be shown by the fact that the same exact problem exists with logic itself. Gödel's Incompleteness Theorems show that math will always have axioms. The same for language with Tarski's Undefinability Theorem, and computer science by the Halting Problem. Does this make logic illogical or incomplete in its power somehow? Whether logic is itself a thing or only reflects the power of things is irrelevant here, because either way it's talking about something real (either itself, or the objects). Since the inconsistency exists for finite systems, which have power and are rational, this shows that the kinds of questions like "Can God create a rock too heavy for Him to lift?" are illogically constructed by language. And an infinite number of bigger systems would "solve" the Incompleteness issue for omnipotence, perhaps in a similar way to Zeno's Paradoxes solved by Set Theory.
Can God create Logic?
But let's explore whether God could have created logic as Martin Luther and possibly Descartes believed. A simple demonstration to the opposite would be the following. Let's make a contradiction be true:
Let's modify this by specifying:
- "A = not-A" (or ~A in symbolic logic)
Could God do this? Sure, why not - He split the Red Sea. Well replace "A" with "God" and what do you get? Now, someone could easily say, "Well this applies to everything but God." Fair enough, after all we're not being logical here. And sure, God isn't necessarily a "thing" to contain, one might argue. But it seems to my mind that we're using logic to define "illogic" and that in the end the whole argument is either meaningless, or we're back to the original conclusion that logic simply isn't a "thing". After all, we can always go back to the question of "Can God destroy Himself?" to illustrate that this train of thought is simply a dead end.
- "A exists and A doesn't exist."
This is regardless of dialetheism and its solution to "This statement is false" and related dilemmas. This only serves to show the fallacy of language pointed out above, since logic and truth do not break down because of that dilemma. The following section, however, explores this a bit.
There are two more things that can relate here. There's a little property in modal logic known as a Vacuous Truth (good Wiki intro here). It deals directly with the above issue of "non-logic" and "existence."
A good example the article uses is, "If there are no cell phones in a room, are all cell phones in that room on or off?" The answer is both! You can easily prove this with De Morgan's Laws and some other simple modal/formal logic rules. The answer is strongly related to the (valid!) concept of premises and conclusions in logic. For example, an argument is invalid only if its premises are true, but its conclusions false (T->F). If an argument has false premises and a true conclusion (or a false one) - (F->T and F->F), it is valid! It isn't sound (T->T), but valid. This is because of the difference between the actual and the potential.
Take, for example, a geometric shape "X". Let's say that "X" is a circle. Isn't it true that "If X is a square, then X has four sides?" But shape "X" is not a square, and doesn't have four sides. So the logic has a false premise and conclusion (with respect to reality), but it would be true, if "X" were a square. It's valid but untrue.
The same is true of false premises with true conclusions. Imagine there are two countries at war: Country A and B. A soldier from Country A comes offering information to the other. He sympathizes more with them. Country B then has to decide whether he's telling the truth or is a spy trying to lead them into a trap. They reason: "If what he says is true, we're already prepared as best as we can so he's of no use, we should send him back. If he's lying or is trying to trick us, it's the same result - send him away." So the true conclusion is valid for them to send him away regardless of whether his statements are true or false.
So what does this mean? Just as the vacuous truth's conclusion is both True and False, so also the original question, "Can God make a rock so heavy, not even He can lift?" has both a "Yes," and "No," answer. This means that the question itself is invalid. It's not that you "can't" ask the question; it's that it doesn't make sense due to the error in the situation, ultimately because of the fallacy of language. It's the difference between asking whether a bucket is full or empty, whereas there simply is no bucket. On the other hand, one might be inclined to say, "the answer is both 'yes' and 'no,' and God is above logic, just like the vacuous truth, which 'exists' (?)":
Person A: Can God make a rock so heavy not even He can lift?
Here the disconnect between language and logical reality is a lot more obvious, but in the end, the principle is the same as the objection.
Person B: Yes.
Person A: So there's something God can't lift?
Person B: No
The bottom line is that having a question like this in no way invalidates omnipotence if it is understood properly, any more than logic and truth are invalidated by asking whether the statement, "This statement is false," is true or false. There are many similar problems and resolutions in mathematics as well, such as Russell's Paradox.
Set Theory and Cardinality
The second connection to this. How many natural numbers are there (1, 2, 3...)? Infinite. How many fractions? Infinite. How many rational numbers? Infinite. How many real numbers (rational and irrational)? More than infinite. There's actually a way to show that, although there are an infinite number of rational numbers, the number of irrationals "outnumbers" them! This is Cantor's Diagonal Proof and an excellent explanation can be seen here (Youtube - "The Banach-Tarski Paradox"; from 1:57-6:04, particularly 2:55-6:04).
If something can exist that's "bigger than the biggest," does it mean God could've created numbers? And since math is basically logic, does it mean He could've made logic itself? And how does "bigger than the biggest" relate to omnipotence and our original question, or whether an omnipotent force can destroy itself like a non-omnipotent one can? I personally don't have a clue.