An interesting but elementary argument against the logical consistency of God’s character would be the classically known Stone Paradox: “The old question of whether God can create a stone too heavy for Him to lift” . The argument can be schematized as such:
- (1) God is an omnipotent being.
- (2) An omnipotent being is one who is all powerful.
- – (a) God can create a rock that is too heavy for Him to lift.
- – (b) God cannot create a rock that is too heavy for Him to lift.
- (3) If God can create a rock that is too heavy for Him to lift, then He is not omnipotent – from (1), (2), and (a)
- (4) If God cannot create a rock that is too heavy for Him to lift, then He is not omnipotent – from (1), (2), (a), and (b)
- (5) Therefore, God is not omnipotent – from (1), (a), (b), (3), (4) and (5)
However, in J.L. Cowan’s The Paradox of Omnipotence Revisited (1974), he runs through a more simpler version that can otherwise be stated in a mere 4 step syllogism :
- (1) Either God can create a stone which He cannot lift, or He cannot create a stone which He cannot lift.
- (2) If God can create a stone which He cannot lift, then He is not omnipotent (since He cannot lift the stone in question).
- (3) If God cannot create a stone which He cannot lift, then He is not omnipotent (since He cannot create the stone in question).
- (4) Therefore, God is not omnipotent.
An interesting response to this argument, so as it goes, is that in order to avoid the conclusion we can either (a) deny God’s omnipotence or (b) deny God’s existence. Since, (b) simply suggests that omnipotence is a consequence of God’s perfect character, and the lack of omnipotence follows to an imperfection – God therefore would not exist (or, perhaps could not exist). However, the often so common response to this paradox is that even God cannot do the logically impossible. As usual, a few examples are often given in regards to demonstrating this to be so: God cannot draw a square circle, make the universe exist and not exist at the same time, etc.
Referring back to his 1965 essay The Paradox of Omnipotence, we see his argument that “[t]here are predicates… which are not self-contradictory and which are thus not logically vacuous as are those so far been considering, predicates which, indeed, are not vacuous at all, which not only can be but are truly applied, and which yet raise difficulties for the concept of omnipotence .
Earlier in Cowan’s essay, he was suggesting that “expressions, statements, phrases, descriptions, or predicates” are the only things that can be considered self-contradictory (330). These sort of grammatical/logical sentiments (predicates, statements, etc.) if they are in fact self-contradictory cannot be truly applied to anything (i.e., they cannot exist in reality). “Thus, when we have noted that God cannot draw a square circle, we still have not noted any thing which God cannot draw” (330). “Therefore,” says Cowan, “we have still not provided any valid objection to the doctrine that God can do anything.”
In consequence, Cowan suggests that in fact “There are predicates… which are not self-contradictory and which are thus not logically vacuous” (as quoted before; p. 330). He then writes our paradox with an interesting analysis:
Consider the old question of whether God can create a stone too heavy for Him to lift. There does not seem to be any inconsistency here. The expression “making something too heavy for the maker to lift” is not, like “drawing a square circle,” one which we must refuse to apply to any activity whatsoever, but is, rather, one which we readily and correctly apply to many quite simple, homely, everyday activities. I myself have preformed such and know many others who have. Can God preform such activities? Can He do what even I have done? Can He make something too heavy for the maker to lift? (331)
However, supposing I think Cowan’s mistake is that of definitions, consider this argument where x can represent any being [soon to be defined]:
- (2) Either x can create a stone which x cannot lift, or x cannot create a stone which x cannot lift.
- (2) If x can create a stone which x cannot lift, then, necessarily, there is at least one task which x cannot preform (namely, lift the stone in question).
- (3) If x cannot create a stone which x cannot lift, then, necessarily, there is at least one task which x cannot preform (namely, create the stone in question).
- (4) Hence, there is at least one task which x cannot preform.
- (5) If x is an omnipotent being, then x can preform any task.
- (6) Therefore, x is not omnipotent.
This was formulated by C. Wade Savage (1967) in response to Cowan, “in a manner not requiring reference to God.” Substituting some being x with “God” according to the deductive scheme above can be (at length) defined as:
There exists necessarily a person without a body who necessarily is eternal, perfectly free, omnipotent, omniscient, perfectly good, and the creator of all things’. I understand by God’s being eternal that he always has existed and always will exist. By God’s being perfectly free I understand that no object or event or state in any way causally influences him to do the actions that he does – his own choice at the moments of action alone determines what he does. By God’s being omnipotent I understand that he is able to do whatever it is logically possible that he can do. By God’s being omniscient I understand that he knows whatever it is logically possible that he know. 
According to this definition, it would be surely absurd to suggest that God can create a stone heavier than Him to lift. For, God would have to create a stone heavier than infinity – a logical impossibility. By omnipotence, the theist only maintains that God in the magnitude of His power is only able to do that which is logically possible. In other words, He cannot create a stone that is “heavier” than Him because such a stone would be an absurdity.
To further the previous point: this argument assumes that weight is an intrinsic property of the stone, whereas we know that weight is merely a gravitational force which can be cancelled out so as to be equal to zero. The intrinsic property is the stone’s inertial mass, and thus, the only legitimate question is whether or not God can accelerate the stone by applying a sufficiently large force. A stone of infinite mass requires an infinite force to accelerate it. If God can create a stone of infinite mass he can also create a force of infinite magnitude and thus move the stone. The same must be true of any non-infinite mass and force. But by definition no stone can have a mass greater than infinity so there is no possible stone that God cannot move.
-  J.L. Cowan, “The Paradox of Omnipotence” in The Impossibility of God, ed. Michael Martin and Ricki Monnier (Promotheus: 2003) p. 330
-  J.L. Cowan, “The Paradox of Omnipotence Revisited” (1974) Ibid., p. 337
-  Ibid. (1965), p. 330
-  Richard Swinburne, The Existence of God, 2nd edn. (Oxford University Press: 2004) p. 7