If all truths are knowable, then all truths are known

The title of this post is what’s called Fitch’s paradox of knowability.

It’s a weird result that arises from a few very intuitive assumptions about the notion of knowability. I’ll prove it here.

First, let’s list five assumptions. The first of these will be the only strong one – the others should all seem very obviously correct.

Assumptions

  1. All truths are knowable.
  2. If P & Q is known, then both P and Q are known.
  3. Knowledge entails truth.
  4. If P is possible and Q can be derived from P, then Q is possible.
  5. Contradictions are necessarily false.

Let’s put these assumptions in more formal language by using the following symbolization:

P means that P is possible
KP means that P is known by somebody at some time

Assumptions

  1. From P, derive KP
  2. From K(P & Q), derive KP & KQ
  3. From KP, derive P
  4. From ◇P & (P → Q), derive ◇Q
  5. ◇[P & -P]

Now, the proof. First in English…

Proof

  1. Suppose that P is true and unknown.
  2. Then it is knowable that P is true and unknown.
  3. Thus it is possible that P is known and that it is known that P is unknown.
  4. So it is possible that P is both known and not known.
  5. Since 4 is a contradiction, it is not the case that P is true and unknown.
  6. In other words, if P is true, then it is known.

Follow all of that? Essentially, we assume that there is some statement P that is both true and unknown. But if this last sentence is true, and if all truths are knowable, then it should be a knowable truth. I.e. it is knowable that P is both true and unknown. But of course this can’t be knowable, since to know that P is both true and unknown is to both know it and not know it. And thus it must be the case that if all truths are knowable, then all truths are known.

I’ll write out the proof more formally now.

Proof

  1. P & –KP                Provisional assumption
  2. K(P & –KP)        Assumption 1
  3. ◇(KP & KKP)     Assumption 2
  4. ◇(KP & –KP)        Assumption 3
  5. -(P & –KP)            Reductio ad absurdum of 1
  6. P → KP                 Standard tautology

I love finding little examples like these where attempts to formalize our intuitions about basic concepts we use all the time lead us into disaster. You can’t simultaneously accept all of the following:

  • Not all truths are known.
  • All truths are knowable.
  • If P & Q is known, then both P and Q are known.
  • Knowledge entails truth.
  • If P is possible and P implies Q, then Q is possible.
  • Contradictions are necessarily false.

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