At my Shabbos Shuva Drasha this year I incidentally and very briefly touched on this topic. I mentioned an article by R. Aryeh Kaplan, that I read when it first came out about 35 years ago that attempted to deal with the question of "Can Hashem create a rock too heavy for Him to lift?", as well as some other, similar dichotomies. Regarding the rock-too-heavy question, R. Kaplan says that it really is not a paradox; it only seems so because of a dual-vision that is the result of our looking at Hashem in two different ways. But he really misses the mark on this - his logic is terribly faulty. His approach is as follows:
As we know, we are unable to know anything about Hashem's essence. So when we say things about Hashem, we are either referring to (a) Hashem's actions - such as Creator, Splitter of the Yam Suf etc., which does not say anything about what Hashem is, but rather the effects His actions have had on the world, or (b.) we talk about what Hashem is not, rather than what He is. When we say, for example, that Hashem is "strong," we really mean Hashem is not weak; when we say He is all-knowing, we mean He possesses no ignorance; when we say He is one, we mean He is not plural or composite, etc.
So far so good. No Chiddushim here. But then R. Kaplan goes on to explain that the reason we perceive the Rock question as paradoxical is because Hashem's omnipotence - which is the attribute of Hashem that tells us He could create a rock as heavy as He wants - is referring to Hashem's actions (type "a" statement above) whereas Hashem's not being impotent - which is what tells us a rock too heavy for Him to lift cannot exist - is referring to what Hashem is not (type "b" statement above), and looking at Hashem in these two different ways at once is like looking at a white paper through two different colored lenses - one green and one red, for example - which will cause you to be unable to see the whiteness of the paper.
He also compares it to wave particle duality, which means that light sometimes seems as if it is made of waves and at other times it seems as if it is made of particles. One explanation for this seeming discrepancy is the "the widely used Copenhagen interpretation, in which wave–particle duality is one aspect of the concept of complementarity, that a phenomenon can be viewed in one way or in another, but not both simultaneously".
So the reason it seems to us that the Rock question creates a paradox, R. Kaplan says, is because it views Hashem at once through the "lens" of His negative attributes (i.e. what He is not - in this case, His lacking any weakness) and the "lens" of His action attributes (in this case, His being able to do anything).
But that is not correct at all. According to R. Kaplan, the rock paradox would not apply to any omnipotent being - it would only apply to an omnipotent being who has no positive attributes. According to him, a being who really does have infinite strength (a positive attribute) would not be subject to the paradoxical question of "can he create a rock too heavy for him to lift," since we would not be looking at such a being through dual lenses. This of course is not true. The paradox can be leveled equally at a being who possesses infinite strength (positive attribute) as much as a being who is "not weak" (negative attribute).
The rock-too-heavy paradox is neither created nor does it have anything at all to do with the use of a negative attribute and an action attribute together. To illustrate, let's not talk about negative attributes or action-attributes. Let's not even talk about Hashem altogether. Let's we are talking about a human being. I will claim this human being is all-powerful. This human being is not kulo poshut, and so when we say he is strong, we mean literally that he has strength.
Can I still ask, paradoxically about this person: Can he make a rock too heavy for him to lift? Of course I can. Even though we are not using "negative attributes" or "action attributes" when referring to him. Thus, what R. Kaplan claims, that the paradox exists due to the fact that we look at Hashem with "negative attributes" together with "action attributes" is not true in the slightest.
The paradox is not weakened at all if we eliminate "negative attributes" from the equation. Negative attributes are simply a way that we avoid giving Hashem parts, since he is Kulo poshut. but even if He would not be Kulo poshut, the paradox would be just as strong.
The rock paradox is created by the idea of omnipotence in general. Omnipotence itself is self-contradictory. If something can do anything and everything - can it conjure something that it cannot destroy? If I am all-powerful can I make myself weak? If I say I can defeat everyone in this room, can I defeat myself? That's the reason this paradox exists. Not because of different types of Midos of Hashem.
The paradox simply asks: Can something that can do anything limit what it can do?
This is equally true whether "can do anything" is a negative attribute or a positive attribute.
Here's another way to illustrate this: In R. Kaplan's analogies - the red and green lenses and the light wave duality - at the end of the day, the vision that the person looking through the glasses sees of the paper being greenish-red, is a distortion. The impressions that each of the two lenses he is using lenses causes are false. It is the tool with which we look at the paper - or the light - that filters our vision and returns an impression that is ultimately distorted. That is how the "paradox" in the case of the glasses and the light is resolved. The paradox is an illusion because the facts that seem to contradict each other - the paper is green vs. the paper is red - are both false. When one or more of the facts that create the paradox is false, you have no more paradox.
But that does not work in the case of Hashem and the rock. There, neither impression is false. Are we saying that just like the lens creates a false vision of a green paper, so too Hashem's omnipotence is similarly false, merely a wrong impression created by some distorted means of looking at Hashem? Chas v'sholom. Are we saying that the impossibility of a rock too heavy for Hashem to lift is really false, that such a rock can indeed exist, and the fact that we think it cannot is due to some distorted way of looking at things? Chas v'sholom.
R. Kaplan's two-lens theory only works if you are willing to say that at least one side of the paradox is not accurate. Is that what he wants to say about Hashem? What exactly is there in the paradox that is not accurate? Fact: Hashem is all-powerful, and fact: A rock that Hashem cannot lift cannot exist.
So R. Kaplan has answered nothing. The fact that we are using a negative attribute together with an action attribute has as much to do with the paradox as the price of tea in China.
Light is either (a) waves, (b-) particles, (c-)neither, or (d) both - there is no fourth possibility. Any explanation we give of duality must result in one of those options. And when we talk about Hashem's omnipotence, Hashem can either (a) create a rock too heavy for Him to lift or ( b-) he can't. There is no third alternative. Although we can say the paper is NEITHER green nor red, and we can say light is NEITHER waves nor particles, regarding Hashem, we are are still stuck with our paradox because we still remain with the fact that Hashem can do anything He wants and also that nothing can be stronger than Him.
So R. Kaplan's understanding is incorrect. His analogy to the colored glasses or light wave duality is apples and oranges.
The truth is, his attempt to explain the paradox stood no chance to begin with. R. Kaplan's approach could conceivably work in a case where you observe a single thing through two different media, thereby attributing one or both of the paradox-creating facts to media distortion. But "negative attributes" and "action attributes" are not media through which we observe Hashem. They are simply different types of facts. Undeniable, unfiltered, unadulterated facts. It is like saying "light travels at 300MM per second" and also "light is not a frog." Neither of those statements are due to media distortion. They are just facts. So too when we say "Hashem can create anything," and "Hashem can lift any rock." Those, too, are facts. There is no media to which we could attribute distortion of those facts. R. Kaplan's idea that media distortion is the cause of the Rock paradox can never even get off the ground.
As regards to the paradox itself, the truth is there is none. The answer is simply "No." Hashem cannot create a rock too heavy for Him to lift, but that is not a problem, because the reason He cannot do that is not due to any limitations or weaknesses on His part. But rather, a "rock too heavy for Hashem to lift" is like saying "a book that is not a book," or "danger that is not a sakanah," or a HFGHFHGFHF$^%$#. Hashem cannot make any of those either. Not because He is not strong enough, but because the question does not provide Him with anything to make.
See above for details.