Are any truths unknowable? When I first saw this, my impulse certainly was to say No — at least not in principle! We poor finite beings have our limitations, limited intellectual storage capacity and low processing speed, admittedly. But those limitations are contingent, and malleable. So, leaving those aside, surely if anything is true it could be known?
In 1963, just as I was entering graduate school, Frederic Fitch published his ‘Knowability Paradox’ (though he did not call it that!), his proof that
[Fitch] if every truth is knowable then every truth is known.
The argument is simple. Suppose A is true and no one knows that A is true. Then the following statement is true:
(B) A and it is not known that A
Now suppose, per absurdum, that all truths are knowable. Then it is possible that someone knows that B. In that eventuality, a fortiori two things are also known:
(C) It is known that A
(D) It is known that it is not known that A
But if something is known then it is true. So we have to infer from (D) that it is not known that A. This is in contradiction with (C).
That’s it, our supposition that there was a truth, A, which was not known (and assuming that indeed all truths are knowable) has been reduced to absurdity .
There is a reason why magicians never explain how their tricks are done. The reason is that, if they do, you will feel very disappointed — the magic is gone, it was all too simple, there was really nothing to it.
When we see how simple Fitch’s argument really was, we have to feel disappointed, in view of the initial air of deep, profound paradox. But it is also clear from the history of this paradox and the many reactions philosophers had (it’s all in a book published in 2008, see below) that the disappointment is not definitive. We still feel uneasy: is it really impossible to know that B is true?
PLAN: first, here, I will delve into Fitch’s argument to see what it requires by way of logic. Then, in the next post, I will explain the resolution that leaves me feeling at peace with the paradox.
We begin, following Hintikka and many writers since, by regarding “It is known that” as the sort of propositional attitude assimilated to modal logic. That include classical sentential logic, which we will take for granted. Let “It is known that” be abbreviated as K, and let us take it to be governed by two postulates.
First of all, the verb ‘know’ is factive. That means:
[Factivity] For every statement X, if it is known that X then X is true.
If X is not true then you may think or believe that X, but you cannot know that X, falsehoods are not the kind of thing that can be known. And secondly we postulate that to know that two things are true is to know that each of them is true.
- KA ⊃ A (‘facticity’)
- K(A & B) ⊃ [KA & KB] (‘&-distribution’)
Now we add two suppositions. The first is that there is some true proposition that is not known, and then, to start a Reductio ad Absurdum, that someone knows that.
3. A & ~KA …. supposition, assumed factually true
4. K(A & ~ KA) …… supposition, to start Reductio argument
Then we argue:
5. KA & K ~ KA from 4. by &-distribution
6. ~KA from 5, second conjunct, by facticity
7. ~K(A & ~KA) from supposition 4. and the contradiction between 5. and 6.
Is there anything dubitable in the logic employed? Certainly, and I’ll note all of it here, but it is not going to do away with the force of the argument, as you’ll see.
The first is that 4. is too strong a supposition for this purpose. We are only allowed to suppose that 3. is knowable, not that it is known! But the retort is that if we can get a contradiction from supposing something, X, then it is also the case that a contradiction follows from the very possibility that X, as well. (This point we will revisit below, when we look at the argument through a ‘possible world semantics’ lens.)
Secondly, Reductio is disputed as a logical principle by the Intuitionists, and it is not even valid classically once supervaluations are admitted. But the form of Reductio here employed is just the half of the principle that the Intuitionists do admit! And there is no sign anywhere that supervaluations would have any relevant point of entry.
Thirdly, a bit more telling, perhaps, is that postulate 2., &-distributivity, is not as innocent as it looks. If A implies B then A is logically equivalent to (A & B). So from 2. we get the corollary that anything and everything logically implied by what is known is also known. But we are not perfect logicians, and often do not see the logical implications of what we know.
There is a familiar retort, since this objection has often come up in discussions of knowledge and its logic. I already signaled it at the outset: we say “in principle” or “implicitly”, bracketing the shortcomings due to our mortal finitude.
Now what? There is one more thing we can do: we can look at the semantic analysis of modal logic, in the possible world form, to see how the suppositions are to be imagined as realized. And we should do this, to clear up the objection that supposition 4. was too strong.
A model structure <W, R> consists of a non-empty set W (the ‘worlds’) and a binary relation R (‘knowledge-accessibility’, ‘relative knowledge possibility’). Intuitively, if x and y are worlds then xRy if and only if all that is known in world x is true in world y. Put another way, what is known in x is precisely and only all what is true in all the worlds to which x bears relation R.
We can put this more precisely, by entering some notation: let R(x) be the set {y: xRy}, and let |A| be the set of worlds in which A is true. Then the foregoing amounts to:
KA is true in world x if and only if R(x) is part of |A|
Relation R is reflexive, so that if KA is true in world x then (since xRx) A is true in x. So our postulate 1. is true in all worlds. And &-distributivity holds just because, if R(x) is part of |A & B| then it is part of each of |A| and |B|, by the usual truth-table rules for &. So postulate 2. is also true in all worlds.
We should add that “It is possible that A” is true in a world exactly if A is true in any world at all. (We could add notation for this, but it would not make the argument clearer.)
Now we focus on a particular world, call it α, ‘our world’, ‘the actual world’. We suppose that a certain statement A is true in α (that is, α is a member of |A|), but it is not known to be true in α (that is, A is false in some world in R(α)). It follows that (A & ~KA) is true in α.
Enter now the principle that every truth is knowable. That implies that it is possible that (A & ~KA) is known — which, in this analysis, means that there is some world z such that:
K(A & ~KA) is true in z.
Accordingly also:
KA is true in z
K(~KA) is true in z,
and therefore also, by facticity,
~KA is true in z
and here we have our contradiction.
(A quick note aside: Notice how this world z came into play. It has nothing to do with our world α, nor with the set R(α), it just came in because the knowability principle brought in simple possibility. There is a noticeable disconnect here that I hope to exploit.)
So either we give up the knowability principle, or we give up the idea that some statement A could be true in our world even though we do not know it to be true. To put is more grandiosely, either the very possibility of knowledge is limited or we are omniscient.
This is the end of the scrutiny of the argument. You could be content at this point. Maybe you were taken in by the opening question, but now you know better. How could any but a silly, un-self-critical philosopher ever have believed that there aren’t any unknowable truths?
But I’m not content at this point. I’m going explore what looks to me like a loophole: that there may be a difference between something being possibly known and being possibly actually known.
SOURCES
“A Logical Analysis of Some Value Concepts”. Frederic B. Fitch. The Journal of Symbolic Logic 28 (2) (Jun., 1963), pp. 135-142.
Joe Salerno, “Knowability Noir: 1945-1963” in J. Salerno, New Essays on the Knowability Paradox, Oxford 2008.
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