Have any of you encountered Fitch's Knowability Paradox [Wikipedia link]? It's actually a really interesting little idea that (supposedly) there are unknowable truths or else knowing

*ANY*truth means the knower is omniscient. Here's the modal logic from the wiki:

This can be formalised with modal logic.KandLwill stand forknownandpossible, respectively. ThusLKmeanspossibly known, in other words,knowable. The modality rules used are:

The proof proceeds:

(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 pcan be proven false without assumptions, thenpis impossible (which is equivalent to the rule of necessitation: ifq=¬pcan be proven true without assumptions (a tautology), thenqis necessarily true, thereforepis impossible).

The last line states that if

1. Suppose K(p& ¬Kp)2. Kp&K¬Kpfrom line 1 by rule (B) 3. Kpfrom line 2 by conjunction elimination 4. K¬Kpfrom line 2 by conjunction elimination 5. ¬ Kpfrom 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& ¬Kp9. 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→Kpfrom line 10 by a classical tautology about the material conditional (negated conditionals) pis true then it is known. Since nothing else aboutpwas assumed, it means that every truth is known.

It's very simple and makes you really go "Huh?" but I think the usual interpretation against omniscience is flawed for two reasons. First of all, the assumption that you can know and not know something at the same time is false. If you think about it, the moment you learn something, inevitably that moment in time is when you both know and don't know it. This is just a fancy Zeno Paradox of infinite midpoints but with time. Think of a ball you kick. The point at which your shoe ends and the ball begins is inevitably a single point in space, since space is continuous (that is, it can be infinitely "zoomed in" and you can never get to a smallest unit of space - proof of this is the Weyl Tile Argument and the fact that the Pythagorean theorem works in real life and not just on paper: otherwise a right triangle with equal legs measuring one unit would equal the hypotenuse! (like a king's movements in chess, where the diagonal squares are "1" square moves legally)). If this is true for space, then it must also be true for time seeing how the two are connected in Relativity.

Yet Line 1, which is translated as "I know (learn) a truth (p) that I didn't know ~Kp" means no one can ever learn, and exposes this, in my opinion, deficiency.

The other objection is the interpretation of what constitutes "omniscience". The modal logic looks at all known truths from the point of view of a world without contingency. What I mean is that imagine you never find out your middle name. Does that mean you could have never known it? According to the timeless modal logic above, "No," because you never found it out, and if you could've you would've. It makes sense in a way, but the conclusions drawn that truths are

*unknowable*and not

*unknown*is based on a worldview of contingencies - validity; not truth (whether you do or don't). In a sense, the Paradox interprets what you already know to be the totality of knowledge with respect to you, should you never learn anything at your maximum amount of knowledge. This is a tautology: of course you know only what you know; as far as you're concerned that's all there is to know and the rest is unknowable. You can't imagine what you don't know. I remember reading how a girl born blind imagined clouds as huge gray dull walls around the world in the sky until surgery gave her sight and she couldn't believe how totally different and beautiful they were. Basically if the knowledge never reaches you, as far as you're concerned, it's unknowable. That doesn't mean you can't learn anything, because the paradox is talking about the maximum point of knowledge by default, but this is easily missed.