In Stanford Encyclopedia of Philosophy, Joe Salerno provides the following introduction to Fitch’s paradox of knowability:
“The paradox of knowability is a logical result suggesting that, necessarily, if all truths are knowable in principle then all truths are in fact known. The contrapositive of the result says, necessarily, if in fact there is an unknown truth, then there is a truth that couldn’t possibly be known. More specifically, if p is a truth that is never known then it is unknowable that p is a truth that is never known. The proof has been used to argue against versions of anti-realism committed to the thesis that all truths are knowable. For clearly there are unknown truths; individually and collectively we are non-omniscient. So, by the main result, it is false that all truths are knowable. The result has also been used to draw more general lessons about the limits of human knowledge. Still others have taken the proof to be fallacious, since it collapses an apparently moderate brand of anti-realism into an obviously implausible and naive idealism.” (Bolding by me)
Note the statement in bold letters: “For clearly there are unknown truths; individually and collectively we are non-omniscient.”
The above statement can only be true in the past and present time. However, the Knowability thesis does not constrain knowledge to the state of the arrow of time. What is an unknown truth today can be a known truth tomorrow for some reason that is beyond the scope of this argument. Let us say by chance, for example. Thus, we are not justified to say that “For clearly there are unknown truths” because the consequent and the antecedent in the statement “if all truths are knowable in principle then all truths are in fact known” refer to infinite time and existence. However, to say that “there are unknown truths” can only refer to present or past time and existence.
Therefore, the kind of omniscience implied by the knowability thesis in Fitch’s proof applies in the limit of infinite time and existence (a religious person – not me specifically – could claim that it applies only to an eternal God). A limit is only an asymptotic state and can never be reached exactly. As such, the kind of omniscience implied by Fitch’s proof is not problematic because it is never achieved given a positive arrow of time. As a matter of fact, the presence of an arrow of time contributes to a constant emergence of new truths that may be unknown at a specific point in time but known in its future.
Since omniscience is never achieved in a world with an arrow of time, Fitch’s proof poses no threat to the Knowability thesis and anti-realist accounts of truth.
Joe Salerno argued the following in response to the above claims:
“The claim is that, clearly, there are some truths that will never be known by any human. To deny this is to claim a number of interesting things. It is to claim that it is not clear that we will never know obscure truths about, say, the precise number of breaths taken by Caesar in his lifetime, etc.”
I argue the above claim should be taken as a postulate rather than a self-evident truth. I can propose a counter argument. Suppose that our reality is manifested by a computer simulation (see “It from Bit” and pancomputantionalism) and the program keeps record of every event that has taken place in spacetime as well as of all spacetime processes (sequences of events in spacetime). Further assume that there is a possibility of getting access to that program in the future. Then, every truth is knowable, including Ceasars’s breath count. But that is not necessarily omniscience since the future output of the program may depend on a stochastic process.
The real issue is whether the type of omniscience implied by Fitch’s proof poses a threat to the anti-realist thesis. I argue it does not, because this type of omniscience is never achieved even if all truths are knowable and this is because of the arrow of time even if, metaphysically speaking, the nature of reality supports the Knowability thesis.
In conclusion, I insist that Fitch’s proof is more of a modern form a scholastic argument and it is mind boggling why it has attracted so much attention 300 or more years after the end of the Scholastic Philosophy era.