Hashem and Logic

Can G-d make a square-circle, or a thing which is both red and not-red, or a rock so heavy even He can't lift it? In other words, must G-d obey the laws of logic?

This question is more serious than it seems. In Principia Mathematica, Bertrand Russell derives all of mathematics from the roots of symbolic logic. This means that if Hashem can not defy logic, he also can not make pi=3.5. Even worse, if physicists ever get a theory of everything, or if such a theory exists and is never found, than the laws of nature are forced by the laws of math which in turn are all derivable from the laws of logic. If we answer that paradoxes about Hashem aren't true, we would need to explain, then, how miracles are possible.

The nice thing about logic, however, is that a wide variety of things can be proven as long as you pick the right set of postulates. While all of math including geometry are derivable from boolean logic, there is no indication that reality has to map to Euclid's postulates. (In fact, it doesn't.) Math gives us many models, reality only conforms to one/some of them. Proofs are simply systems for taking a set of postulates and finding their conclusions. The postulates themselves, come before the application of logic.

Both extreme positions are supported. The Ramchal (Pischei Chachmah 30) insists that G-d's omnipotence is absolute, even with regard to things we would regard as impossible. The Rambam, on the other hand, (Moreh 3:15) states:

That which is impossible has a permanent and constant property, which is not the result of some agent, and can not in any way change, and consequently we do not ascribe to G-d the power of doing what is impossible. No thinking man denies the truth of this maxim; none ignore it, but such as have no idea of Logic.... Likewise it is impossible that G-d should produce a being like Himself... to produce a square whose diagonal is equal to one of its sides....
We have shown that according to each of these theories there are things that are impossible, whose existence cannot be admitted, and whose creation is excluded from the power of G-d, and the assumption that G-d does not change their nature does not imply weakness in G-d, or a limit to his power.

R. Aryeh Kaplan, in "Jewish Life - Summer '74" discusses the question of paradox. He raises a number of classical paradoxes:
How can G-d know what I will decide tomorrow, and yet I have free will in that decision?
G-d is unchanging. However, He is now "the One Who created the universe" whereas He wasn't before creation. How?
Can G-d create a stone so heavy even He couldn't lift it?
(I addressed the first two in terms of the inappropriateness of using time-based language when discussing G-d in an earlier entry.)

R' Kaplan explains:

A very good analogy would be trick glasses in which the right lens is red and the left is green. Therefore, if a person wearing such glasses looks at a white paper, he sees it as red with his right eye, and as green with his left. If he looks at it through both eyes he sees some psychedelic mixture of red and green, but under no conditions can he perceive the color white.

With respect to the stone:

The attributes of action would say that He can create such a stone, "G-d is omnipotent and can do all things." The negative attributes would indicate that such a stone could not exist.

So, the authorities are split: no (Ramchal), yes (Rambam), and all of the above (Rabbi Kaplan). That should give me some rein in which to speculate.

When we looked at Divine Attributes, we defined G-d's omnipotence as a negative statement. A declaration about what He isn't. G-d gets results without invoking the notion of "power". Thus, it is meaningless to invoke the notion of "a rock too heavy for Him to lift" as it is to talk about "a song too red." G-d cannot just lift a stone of infinite weight, omnipotence means that weight is a non-issue to what He can lift, just as color is.

The other question is can G-d defy paradox in general. I'd have to agree with the Rambam at least to the extent that some system of logic must apply. Didn't Hashem intend us to use logic to come to understand what we can of Him. If He is above logic, what use is it? How can one say "Since Hashem created logic, therefore ..." as the Ramchal does to start his very argument to conclude that theological answers needn't be logical? How can we the proceed with the rest of this discussion if we didn't already assume that logic works?

Contemporary logic seems to bear out a position very close to the Ramchal's. Human reason seems to be closer modeled by Bayesian probability or Fuzzy Logic than the old Aristotelian-Boolean kind. In English: we are equipped to deal with things other than a black-and-white true vs. false. We can reason about things we can only know are probably true. And while happiness and sadness are opposites, ambivalence, where a person feels both because of different perspective on the same thing, is common. As are dialectics: People can believe "The world was created for me" and "I am dust and ashes" at the same time. Quantum level events conform to a Quantum Logic, which is also non-Boolean and non-Aristotelian. An electron can be in a superposition state, where it's both in one state and another, even though the two contradict; at least until observed. (Don't try to understand that -- I didn't claim it makes sense, just that it's how subatomic particles work.)

Aristotle's Law of Contradiction applies to neither our minds nor the constituents of our atoms. Why need it apply to G-d?

Related to this is my essays on logic and eilu va'eilu (plurality in halakhah) in Mesukim MiDevash for Naso,
and earlier in this blog.

Or, to put it another way -- even if logic is a part of Truth, and therefore of Hashem's essence, which of the many possible systems of logic does that mean? Presumably one of Infinite richness, not the Aristotelian that both the Rambam or the Ramchal were discussing.