In referencing paradoxical logic (as opposed to Aristotelian) I think I said, Parmenides, but I meant Heraclitus…
‘In opposition to Aristotelian logic is what one might call paradoxical logic, which assumes that A and non-A do not exclude each other as predicates of X. Paradoxical logic was predominant in Chinese and Indian thinking, in Heraclitus’ philosophy, and then again under the name of dialectics in the thought of Hegel and Marx. The general principle of paradoxical logic has been clearly described in general terms by Lao-Tse: “Words that are strictly true seem to be paradoxical.” And by Chuang-tzu: “That which is one is one. That which is not-one, is also one.”‘
“Call the set of all sets that are not members of themselves “R.” If R is a member of itself, then by definition it must not be a member of itself. Similarly, if R is not a member of itself, then by definition it must be a member of itself.”
What can this possibly mean, other than we are very capable of creating “impossible definitions?”
Similarly, we can talk of the sign in the barber shop that says, “I shave anyone who does not shave himself” and ask, does the barber shave himself? If he does, then he doesn’t and if he doesn’t then he does.
In other words, “this sentence is false.”
The only possible meaning that we can derive from such statements is that A is not A.
These kinds of paradoxes reveal how easy it is to ask impossible questions and make impossible definitions. These examples inevitably call into question the meaning of meaning. What does it mean to mean something?
Can God create a rock so big that even God can’t move it? If yes, then God is not all-powerful. If no, then God is not all-powerful.
This question does not reveal a problem with God, but the concept of an “all-powerful” being. If we mean to say that “all-powerful” means being able to do what cannot, by definition, be done, then we essentially say, “this sentence is false,” or, what is impossible is possible, or that the barber, who only shaves those who do not shave themselves, shaves himself.
In all of these paradoxes, we are essentially made to assume that A is not-A.
The question, in other words, makes no sense – unless we redefine what it means to “make sense.”
This is the meaning of meaning.
Therefore, we must admit that what is impossible is impossible, even for God. This does not contradict omnipotence, unless we define omnipotence as the ability to do the impossible, at which point we have flushed logic and sense down the toilet completely. This is the “Alice in Wonderland” world where we can accept as true the statement, “this sentence is false.” Once this is accepted, nothing is true, nothing is false.
It is important to note that we arrive at this disagreeable point by first saying that we allow what is not allowed (the impossible is possible).
We could just as well ask if God can create a spherical triangle, or if God knows the answer to the question, how many fingers does a blue have?
It is easy to ask God questions that He cannot answer. This does not put a limitation on God. Rather, it shows our ignorance of what meaning means.