First a quick review for those of you who haven't been following along: Fitch's paradox is a formal proof that starts with some mostly innocuous-seeming assumptions and concludes that all truths are known. Since this conclusion is plainly false, the game (and it really is just a game) is to find the flaw in the reasoning.
There is some low-lying fruit: one of the assumptions that goes into the proof is that it is possible to know any truth. That is plainly false because Godel, finite universe, yada yada yada. You can try to do an end-run around this by restricting the domain of the logic to "tractable truths". The problem with this is that tractability inherently involves time, but Fitch's logic does not model time. So in some sense Fitch's conclusion in this case is actually true: if something is not known in the static situation described by the logic, then it cannot be known in that static situation. Hence, all "tractable" truths (to the extent that it is possible to give that word a coherent meaning in a world without time) are in fact known.
An advocate of the tractability approach might try to rescue it by reconstructing Fitch's proof in a logic that did model time. I suspect that this is not possible, and I even suspect that it's possible to prove that it is not possible, but I don't care anywhere near enough to actually try to prove it myself.
What I do want to point out here is that there is actually a much deeper problem: Fitch assumes that it is possible to assign coherent meanings to the words "possible" and "know". In fact, not only does he assume it's possible, he assumes it's *trivial* because he doesn't even *try* to actually define these words. He just tacitly assumes that they have meanings, that these meanings are common knowledge, and that they coincide with the semantics of his modal logic.
In fact, both "know" and "possible" are highly problematic. What does it mean to know something? Siri can tell you the temperature in Buffalo. Does that mean that she "knows" the temperature in Buffalo? Planets move according to Newton's laws, does that mean that they "know" how to solve differential equations?
Even among humans it is far from clear what it means to know something. The subjective sensation of being absolutely convinced of a false proposition is generally indistinguishable from being convinced of a true one. So can it be said that flat-earthers "know" that the earth is flat? Did Ptolemy "know" that the sun revolves around the earth? This matters because Fitch's proof depends on the assumption that anything that is known must be true (KP->P).
But even being true is not necessarily enough. In 1653, Christian Huygens calculated the distance from the earth to the sun and got very nearly the correct answer, but he got it right by pure luck. In fact his calculation was completely bogus, relying on numerology and mysticism to guess that Venus was the same size as the earth. That just happens to be true, but can it be said that Huygens knew it?
The "state of the art" in defining knowledge is to add the condition that a belief must not only be true but properly justified. But that just begs the question: what does it mean to be "properly justified"?
"Possible" is no less fraught. Consider this simple situation: you are about to flip a coin. Can it be said that "it is possible that the coin will land heads-up, and it is possible that the coin will land tails-up"? Most people would say yes. But now consider the situation after you have flipped the coin but before it lands, or after it lands but before you have looked at it. Are both heads and tails still possible? What about after you look? Is it possible that you see "heads" but in fact the coin is "tails" and you are suffering from a hallucination? Is it possible that the coin is neither heads nor tails, but has disappeared or turned into two coins? Even before you flip, are both outcomes really possible, or is your perception that they are both possible merely a product of your inability to predict the outcome?
If you believe that both outcomes are possible before the flip but not after, at what point did the situation change? At the instant the coin landed? Why not a microsecond before, or when it left your hand, or when your brain sent the nerve impulse to your hand to start it spinning?
Possibility can only ever be assessed relative to either ignorance or willingness to suspend disbelief and consider counterfactuals. As I write this, I am wearing a black T-shirt. So relative to my knowledge state, it is not possible that I am wearing a red T-shirt, but relative to your knowledge state it is possible (because I could be lying about wearing a black T-shirt). We can also imagine (counterfactually) some alternate reality that is identical to actual reality except for the color of my T-shirt. These are very different senses of possibility. On the possibility-from-ignorance view, it is not possible that Hillary Clinton won the 2016 election or to solve the halting problem, but on the possibility-from-counterfactuals view, both are possible.
There is an interesting interplay between knowledge and both kinds of possibility. The relationship with possibility-from-ignorance is obvious. If you know P, then relative to your knowledge state, ~P is not possible. On the other hand, your willingness to entertain counterfactuals can also be constrained by your knowledge. Is it possible that the earth is flat? That the square root of two is a rational number? That Santa is real?
Since this whole mess started with a formal proof, let me offer up one of my own. A while back I opined that free will cannot exist in a universe where there is an omniscient, infallible deity. It turns out that you can render the argument as a formal proof. Let KP mean God knows P, and let LP mean P is possible. Let P be an arbitrary universally quantified proposition, and S be the particular proposition "I will choose to sin." Then:
1. KP ∨ K~P (For every proposition, God knows whether or not it is true, definition of omniscience)
2. KP -> ~L~P (If God knows P, then it is not possible that P is false, definition in infallibility)
3. LS ∧ L~S (It is possible that I will choose to sin, and it is possible that I will not choose to sin, definition of free will)
From these three premises we can conclude:
4. KS ∨ K~S (from premise 1)
5. Assume KS (set up for conditional proof)
6. ~L~S (2, 5 modus ponens)
7. KS -> ~L~S (conditional proof, discharge assumption 5)
8. K~S -> ~LS (by analogous conditional proof starting with the assumption K~S, and the logical tautology ~~P -> P)
9. ~L~S ∨ ~LS (From 4, 7 and 8 by the Constructive Dilemma)
10. L~S (from 3, by conjunction simplification)
11. ~LS (from 9 and 10 by disjunctive syllogism)
12. LS (from 3, by conjunction elimination)
13. LS ∧ ~LS (from 11 and 12) -- contradiction
Therefore, premises 1-3 cannot all be simultaneously true, QED.