Our propositional variables p and q will take declarative statements as substituends. Let K be the epistemic operator ‘it is known by someone at some time that.’ Let ◊ be the modal operator ‘it is possible that’.
Suppose the knowability principle (KP)—that all truths are knowable by somebody at some time:
(KP) ∀p(p → ◊Kp).And suppose that collectively we are non-omniscient, that there is an unknown truth:
(NonO) ∃p(p ∧ ¬Kp).If this existential claim is true, then so is an instance of it:
(1) p ∧ ¬Kp.Now consider the instance of KP substituting line 1 for the variable p in KP:
(2) (p ∧ ¬Kp) → ◊K(p ∧ ¬Kp)It follows trivially that it is possible to know the conjunction expressed at line 1:
(3) ◊K(p ∧ ¬Kp)However, it can be shown independently that it is impossible to know this conjunction. Line 3 is false.
The independent result presupposes two very modest epistemic principles: first, knowing a conjunction entails knowing each of the conjuncts. Second, knowledge entails truth. Respectively,
Also presupposed are two modest modal principles: first, all theorems are necessary. Second, necessarily ¬ p entails that p is impossible. Respectively,
(A) K(p ∧ q) ⊢ Kp ∧ Kq (B) Kp ⊢ p
Consider the independent result:
(C) If ⊢ p, then ⊢□p. (D) □¬p ⊢ ¬◊p.
Line 9 contradicts line 3. So a contradiction follows from KP and NonO. The advocate of the view that all truths are knowable must deny that we are non-omniscient:
(4) K(p ∧ ¬Kp) Assumption [for reductio] (5) Kp ∧ K¬Kp from 4, by (A) (6) Kp ∧ ¬Kp from 5, applying (B) to the right conjunct (7) ¬K(p ∧ ¬Kp) from 4–6, by reductio, discharging assumption 4 (8) □¬K(p ∧ ¬Kp) from 7, by (C) (9) ¬◊K(p ∧ ¬Kp) from 8, by (D)
(10) ¬∃p(p ∧ ¬Kp).And it follows from that that all truths are actually known:
(11) ∀p(p → Kp).The ally of the view that all truths are knowable by somebody is forced absurdly to admit that every truth is known by somebody.
In plain English:
Given a few very reasonable assumptions, we can prove that any truth that is knowable is also known. Here are the assumptions:
1. Knowledge implies truth (if some fact is known, then that fact is true).
2. If we know the conjunction of two truths (e.g. it's raining outside and my car is wet) then we know the two truths themselves (e.g. we know that it's raining outside and we know that my car is wet).
3. Contradictions are necessarily false.
4. All truths are at least possibly knowable.
If we assume that there is some truth that is unknown, and that such a truth is knowable, then we can derive a contradiction. It follows with logical certainty that if any truth is unknown, then it is unknowable. This means that if all truths are knowable, then all truths are known.
Strengthening the Argument
There is still one problem with the argument. Assumption 4 may very well be called into dispute. What reason do we have to think that all truths are knowable? Fair enough.
Let's switch assumptions by swapping out:
4. All truths are at least possibly knowable.
and swapping in:
4. Some particular truth is at least possibly knowable.
Let's call that particular truth q.
Now, let's replace the KP with a modified KP
(KP') ∃q(q ∧ ◊Kq) - There is some truth that is true and possibly knowable.
(NonO') (q ∧ ¬Kq) - This truth is also unknown.
(POS) ◊K(q ∧ ¬Kq) - The fact that this truth is unknown is at least possibly knowable.
If this existential claim is true, then so is an instance of it:
(1) q ∧ ¬Kq.
It is possible to know (1):
(2) (q ∧ ¬Kq) ∧ ◊K(q ∧ ¬Kq)
Given (2), we apply conjunction elimination:
(3) ◊K(q ∧ ¬Kq)
However, it can be shown independently that it is impossible to know this conjunction. Line 3 is false. If (3) is true, then there is some possible world where K(p ∧ ¬Kp) holds. If that world can be shown to be contradictory, then (3) is also false.
(4) K(q ∧ ¬Kq) Assumption [for reductio]
(5) Kq ∧ K¬Kq from 4, by (A)
(6) Kq ∧ ¬Kq from 5, applying (B) to the right conjunct
(7) ¬K(q ∧ ¬Kq) from 4–6, by reductio, discharging assumption 4
(8) □¬K(q ∧ ¬Kq) from 7, by (C)
(9) ¬◊K(q ∧ ¬Kq) from 8, by (D)
Line 9 contradicts line 3. So a contradiction follows from KP' and NonO'. Even when we modify the knowability principle, we still discover that if some truth is unknown, then it is unknowable.
Again, in plain English
Since"All truths are at least possibly knowable" has its difficulties, let's assume that some truths are unknowable. Does this rescue us from the paradox? Not at all. With a slight modification, we can use this argument to show that all knowable truths are known. This seems quite absurd. Does nobody truly gain new knowledge? Did we always know that the world is round? Did we always know that George Washington would be elected first President of the United States?
The only other way out is to deny that it is possible know (1), which puts us into a difficult spot. When we say that we can know (1), all we are saying is that we can know that there are unknown but knowable things. For example, we don't know my exact internal body temperature as I write this, but we sure could know it. Knowledge of this information is not impossible.
The philosophical literature for this field is rich, and yet philosophers cannot come to a consensus as to which of these three principles is false. Attempts to revise the logic have generally resulted in implausible solutions, and each of the three assumptions seems quite plausibly true. There is simply no agreed solution as to how this problem should be resolved.
Philosophers generally try to jettison assumption 4 in order to solve the paradox, but that leads to some difficulties with truth and knowledge. On atheism, truth and meaning generally have to be defined in terms of our ability to know them. And regardless, the modified version of the argument does not require assumption 4, anyway.
Now, the question arises: why should we think of this as a paradox at all? A paradox is something that seems logically airtight, but implies an absurd conclusion. Curry's Paradox, for example, allows you to prove any conclusion that you want. That is what makes it a paradox. Fitch's paradox has no such strange implications. It just implies that all knowable truths are known. But what if you believe in an omniscient God?
The only argument against an omniscient God being the solution to this problem is that truth or meaning is somehow determined by linguistic communities. Yet I see know reason why we should believe this is true. The very definition makes truth both subjective and relative - a postmodern position indeed. It's hard to see how such definitions could be true or even meaningful. If meaning or truth is not absolute, then it's hard to see how the statement "meaning/truth is not absolute" is not true in an absolute sense. And therefore we should not look for truth and meaning to be grounded in relevant expressions in a human linguistic community. We should look to God as the ground and anchor of meaning and truth.
Is Fitch's Paradox an actual paradox? If you do not believe in God, then it seems impossible to resolve. If you do believe in an omniscient God, then it is not even a problem. There is someone who knows all knowable truths: God.