Is there a way to prove that not everything is knowable?

Well, “prove” is a strong word, but Fitch’s paradox makes a pretty strong argument if you ask me. The argument proceeds as follows.

(You don’t need to know formal logic for this answer. I’m just including the formal stuff ’cause I wanna show off and be all fancy. I cribbed the formalizations from the SEP article on Fitch’s Paradox of Knowability.)

We want to argue that not everything is knowable. Take the idea that everything is knowable. Call that idea the K-principle, or KP. KP states that, for any fact P, it is possible for P to be known. Notice that this is not the statement that everything is known, only that everything is knowable. Formally,

(KP) ∀p(p → ◊Kp)

This looks awfully hard and mathy, but it’s not really as scary as it looks. All it’s saying is, “For any statement, P, if P is true, then it is possible for P to be known.” This is the thing we want to disprove.

“Statement” is a little fast and loose. p here is better understood as a proposition, which is kind of like a statement. A proposition is the thing a statement expresses. So if I say the English sentence, “The snow is white”, and then say the German sentence “Der Schnee ist weiss” or the Japanese 雪は白い, then in all three instances, then I’m expressing the same proposition even though the sentences are different.

Now, even if everything is knowable, then we certainly don’t want to say that everything is known. We’re not omniscient, after all! We express this formally as,

(NonO) ∃p(p ∧ ¬Kp).

Which is a fancy mathy way to say “There’s some statement that is true but nobody knows it’s true”.

If I say that there are white houses in the world, then there is at least one house that is white. If I say that cats exist and cats have four legs, then there must be a four-legged cat somewhere. This is dead-simple, so simple it’s almost hard to see. In formal logic, they call this existential instantiation. We can infer specific statements from general ones. So if NonO up there is true, then so is this:

(1) p ∧ ¬Kp

This means, “Somewhere out there, there’s a proposition that is true, but nobody knows it.” Which is just another way of saying that we’re not omniscient. Logically equivalent.

So far, we’ve said that:

If it’s true, you can know it (KP)

We’re not omniscient (NonO)

There’s some fact, p, that we don’t know. (from 2)

If we combine 1 and 3, we get this:

(p ∧ ¬Kp) → ◊K(p ∧ ¬Kp)

This means, “If there’s an unknown fact out there, then we can know that that fact is true, and that it is unknown.”

The problem is…. that’s impossible.

What if I told you, “There is a hedgehog inside of this box, but nobody knows that, including me.” You’d think I was a fucking loon. If A is true and nobody knows A, then I can’t assert that I know it.

But wait! We derived this contradiction, ultimately, from these two premises:

There are unknown truths

All truths are knowable

That’s a reductio ad absurdum. Those two sentences lead to a logical contradiction. They can’t both be true!

So one of the following must be true:

All truths are known (we’re omniscient)

Some truths are unknowable (negation of KP)

Now, there’s some more logical nitty-gritty here. You can probably skip this unless you’re deeply interested in formal logic and/or a masochist (those two might be a case of logical equivalence). I think the idea is intuitively obvious at this stage, but if you really wanna dig into the crap about modality and distributivity of modal operators and what-not, read on.

First of all, if you know that the combination of two statements are true, then you know that each statement is true. This is one of those logical things that is so trivial that it’s almost hard to grasp at first. Basically, if I know the statement, “I have two AND I have two feet”, then I know that I have two hands, and I also know that I have two feet. Dead simple. If you know “A and B”, then you know A and you know B.

Second of all, in order to know something, it has to be true. You can’t know a falsehood. I can’t know the statement, “I have three hands”, because knowledge, by definition, is justified true belief. You can’t know a falsehood. You can know “It is false that I have three hands”, of course, but you cannot know “I do, in fact, have three hands”. Formally, we write these two:

K(p ∧ q) ⊢ Kp ∧ Kq

Kp ⊢ p

We also assume two things about the nature possibility. Two very modest things.

First, any theorem that is true is necessarily true. If it is the case that 1+1=2, then 1+1=2 is a necessary truth. If it is the case that the Collatz Conjecture is true, then the Collatz Conjecture is necessarily true.

Second, if something is necessarily false, then that thing is impossible. It is necessarily false (under Peano arithmetic) that 1+1=3, which means it’s impossible for 1+1=3 to be true.

If ⊢ p, then ⊢□p.

□¬p ⊢ ¬◊p.

So now we’ve said

If you know (A and B), then you know A and you know B.

You can’t know a falsehood.

Theorems are necessarily true.

If it’s necessarily false, it’s impossible. And now we can say….

We know that (p is true and p is unknown) [premise]

We know p and we know that p is unknown (from 8, 12)

We know p, and p is unknown (from 9, 13)

By reductio, we know that 12 is false. That is, we do not know that (p is true and unknown)

15 is necessarily true. (from 10, 15)

Therefore, it is impossible to know (p is true and p is also unknown)

So if you want to defend KP, you must reject NonO. And if you want to keep NonO, you have to reject KP.