Proof
Suppose p is a sentence which is an unknown truth; that is, the sentence p is true, but it is not known that p is true. In such a case, the sentence "the sentence p is an unknown truth" is true; and, if all truths are knowable, it should be possible to know that "p is an unknown truth". But this isn't possible, because as soon as we know "p is an unknown truth", we know that p is true, rendering p no longer an unknown truth, so the statement "p is an unknown truth" becomes a falsity. Hence, the statement "p is an unknown truth" cannot be both known and true at the same time. Therefore, if all truths are knowable, the set of "all truths" must not include any of the form "something is an unknown truth"; thus there must be no unknown truths, and thus all truths must be known.
This can be formalised with modal logic. K and L will stand for known and possible, respectively. Thus LK means possibly known, in other words, knowable. The modality rules used are:
| (A) | Kp → p | - knowledge implies truth. |
| (B) | K(p & q) → (Kp & Kq) | - knowing a conjunction implies knowing each conjunct. |
| (C) | p → LKp | - all truths are knowable. |
| (D) | from ¬p, deduce ¬Lp | - if p can be proven false without assumptions, then p is impossible (which is similar to the rule of necessitation: if p can be proven true without assumptions, then p is necessarily true). |
The proof proceeds:
| 1. Suppose K(p & ¬Kp) | |
| 2. Kp & K¬Kp | from line 1 by rule (B) |
| 3. Kp | from line 2 by conjunction elimination |
| 4. K¬Kp | from line 2 by conjunction elimination |
| 5. ¬Kp | from line 4 by rule (A) |
| 6. ¬K(p & ¬Kp) | from lines 3 and 5 by reductio ad absurdum, discharging assumption 1 |
| 7. ¬LK(p & ¬Kp) | from line 6 by rule (D) |
| 8. Suppose p & ¬Kp | |
| 9. LK(p & ¬Kp) | from line 8 by rule (C) |
| 10. ¬(p & ¬Kp) | from lines 7 and 9 by reductio ad absurdum, discharging assumption 8. |
| 11. p → Kp | from line 10 by a classical tautology |
The last line states that if p is true then it is known. Since nothing else about p was assumed, it means that every truth is known.
Read more about this topic: Fitch's Paradox Of Knowability
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