The subheadings below are not found in Mavrodes' article. I have added them to try to help break up the argument into more manageable chunks.
[Further examples of self-contradictory things might be married bachelors, hot coldness or two dimensional spheres].
Mavrodes begins by referring back to Aquinas who argued that God can do anything which is not self-contradictory (impossible) but God cannot do the impossible. For example, God cannot make a square circle.
Mavrodes agreed with Aquinas and argued that we do not regard the inability to do the impossible as a lack of power.
Thus far Mavrodes defends Aquinas’ claim. We would (probably) not say that God lacked power because he could not do the impossible.
Next Mavrodes sets out a version of the paradox of the stone.
‘A more involved problem, however, is posed by this type of question: can God create a stone too heavy for Him to lift? This appears to be stronger than the first problem, for it poses a dilemma. If we say that God can create such a stone, then it seems that there might be such a stone. And if there might be a stone too heavy for Him to lift, then He is evidently not omnipotent. But if we deny that God can create such a stone, we seem to have given up his omnipotence already. Both answers lead us to the same conclusion.’
I.e. they both lead us to the conclusion that God is not and cannot be omnipotent.
Initially, it seems that we cannot use Aquinas’ response here. Mavrodes states that ‘x is draw a square circle’ is clearly self-contradictory and thus impossible. However, ‘x is able to create a stone that x cannot lift’ does not appear to be self-contradictory.
Think about the objects made. A square circle is impossible. A very heavy stone is not. This means that whilst not being able to create a square circle does not undermine God’s omnipotence the inability to create a stone that he cannot lift does suggest that there are things that God should be able to do (because they are possible) but can’t.
BUT, Mavrodes argues that actually we can use Aquinas’ defence here because creating a stone too heavy for God to lift would still involve doing a self-contradictory thing. In other words, despite the apparent difference, the examples are actually the same.
To see how and why this is the case we need to look very carefully at how the argument is set out. Mavrodes’ article sets this out in a paragraph, but it may be clearer in bullet points.
This is an DICHOTOMY either A or B must be true. There are no other options.
First we will assume that he is NOT omnipotent.
So, if we assume that God is NOT omnipotent, there is no paradox and no problem. However, the solution is trivial and tells us nothing (we assumed God was not omnipotent, we concluded that there are somethings he can do and some he cannot).
Let's start again and assume this time that God IS omnipotent.
Thus, if God is omnipotent then the scenario is equivalent to creating a square circle, and if God is not omnipotent then we conclude that there are some things God cannot do.
As Mavrodes points out, it is God’s omnipotence that makes the existence of a stone too heavy for God to lift a logical impossibility.
We could take issue with this argument by disputing the claim that ‘a stone too heavy for God to lift’ is a logical impossibility. We could argue it is not the equivalent of a square circle. Mavrodes would not agree with this line of argument, but he points out that if we do argue this way then we would run into other problems. We would be committed to saying that it is a logical possibility and if it is a logical possibility then an omnipotent being would be able to do it. We could set that argument out as follows.
Mavrodes suggests that the objector would become trapped. Either they must say that it is logically impossible (which means that we cannot reasonably expect even an omnipotent being to do it) or it is logically possible (which means that God would be able to do it). Mavrode writes ‘The objector cannot have it both ways’.
Suppose we take a different approach. We accept that the paradox of the stone demonstrates that God either cannot create a stone so big he cannot lift it or he can create something that he cannot lift.
We might decide that the lesser of two evils is to say God can lift everything and thus God cannot create a stone so heavy he cannot lift it. Intuitively (and colloquially), this would suggest that he cannot create really, really heavy things.
Thus, although the philosopher might have begun by compromising on God's omnipotence, it is possible to claim that they have not really 'lost' any aspect of the doctrine at all. Mavrodes' asked:
‘..what he has given up. Is it the unlimited power of God to create stones? No doubt. But what stone is it that God is now precluded from creating? The stone too heavy for Him to lift, of course. But we must remember that nothing in the argument required the theologian to admit any limit on God’s power with regard to the lifting of stones. He still holds that to be unlimited. And if God’s power to lift is infinite, then His power to create may run to infinity also without outsripping the first power. The supposed limitation turns out to be no limitation at all, since it is specified only by reference to another power which is itself infinite.’
Mavrodes’ final point is that he has not proved that God IS omnipotent, only that the specific arguments used to try to prove that he cannot be have failed.