I think the issue between us is that I am meaning something other than what you think I am meaning. Perhaps it is my fault for using a term like "understand' without fully defining it.
To me, "understanding" means coming to a conclusion about something, even if subconsciously. Actually, especially subconsciously.
To me, it has the dictionary definition of comprehension.
Formally, my definition for understanding would be:
Given a rational agent A and a statement P, A understands P if and only if A can infer new or existing facts from P (combined with A's existing knowledge base, of course).
I suspect that is different from what you mean when you say "understand."
Probably, but your definition seems to lack something. For example, we have Godel's Incompleteness Theorems, and demonstration of the Halting Problem, deriving in a general sense from such devices. So if you're proposing we don't understand, you have to cull this part out. You seem to want to do this:
Note that in this case, your statements about a self-referential sentence lead to new facts -- but those facts are not inferred from the self-referential statement itself.
...but you introduce other problems. For example, "True=True" is a true statement, and we certainly can't derive any
new facts from it. That squeezes you down to "existing" facts. And, okay, we can actually start with T=T and derive things from it, if
that is what you mean, but then, if that's the case, what about the false claim "T=F"? Not only can you not derive
new facts from this, but I think you'd find it hard to derive
existing facts from it. Does this mean that per your formal definition, we do not understand the statement "T=F"?
I would argue that this starts to veer so far off of the colloquial, dictionary definition of "understanding" that perhaps it'd be better to bind the formal definition to a different word altogether.
And to explore the same thing from another perspective, let's revisit sudoku puzzles, but in a broader sense. Allow me to introduce three technical terms: a "valid sudoku puzzle" refers to the ordinary, official sudoku puzzles... these have exactly one solution. A "bad sudoku puzzle", let's say, has more than one solution (bad in the sense that it's "not a good one"--note that, for applicability to the subject at hand, a sudoku puzzle that has one solution, but is overspecified would be counted toward the good case). And finally, an "invalid" sudoku puzzle is one that has no solution. Those three categories seem like very useful things to analyze to me.
Now, certainly, you realize that sudoku puzzles
are self referential, by their very nature, and are logical puzzles as well. They can (ahem!) easily be reformulated as a series of logical statements (given you have a nice set of 81 variables, sufficient lifetime, and sufficiently insufficient "having-a-life", to deal with the combinatorically huge number of statements). As such, if your claim is taken at face value, it is equivalent at least to saying that the category of "invalid sudoku" puzzles is beyond human understanding (even if we could show how they are invalid). Are you also going to claim that bad sudoku puzzles are beyond our understanding? Are you claiming, as well, that good sudoku puzzles are? I'm not sure I know how to apply your proposed formal definition to this scenario and get a useful result out of it, nor do I know what I'm supposed to infer that's particularly shocking even if said formal definition says that I don't understand one or all three classes of sudoku puzzles. I'm still perfectly capable of solving them, and even classifying them, so it makes no
pragmatic difference in this regard if I'm formally accused of not understanding one or all three categories.
Do you agree that our definitions are not the same?
Sort of. I'm still not quite sure I understand what your definition is... at least formally; that is, I'm not quite sure how to practically apply your definition to the scenario, or to (especially) "antitautological" claims, or what utility I'm to derive from it that is related to the dictionary understanding of "comprehension" once I find that I formally don't understand something.