A standard example in English is the sentence:

“Is not output by the agent when preceded by its quotation” is not output by the agent when preceded by its quotation.

There are examples all over mathematics. For example, there are functions f and g defined as compositions of +, *, cos, exp, such that f(x) = g(x) if and only if the agent never outputs “f(x) = g(x)”.

I describe the sentence as G = ” … G … ” only for ease of expression.

I appreciate you clearly building up the logical paradox and then working through the implied “bullet biting” someone would need to do to make things consistent again. Discarding modus ponens feels quite radical but I’m open to considering it and looking forward to your next post(s) on the subject.

One thought: Can you think of a less contrived statements in the form of G where the statement isn’t self-referential but the reasoning about it still leads to the same dilemma? The current construction feels like it’s relying on a bug in levels of description. In other words, this feels like something that could be solved by the same kind of symbol renaming that most compilers use to avoid collisions in different scope levels — even when they aren’t declared scope levels — so the compiler has to infer them. But this could be bigger problem. How would re-scoping fail as a general tactic to deal with this?

Sorry to offer such a naive solution. It feels like I’m possibly missing an essential part of your argument because it’s unlikely that you wouldn’t have considered a re-scoping scheme already. Still, I’m curious what your thoughts are on this line of reasoning anyway.