Fitch's Knowability Paradox

[Cornelius]

Hi guys!

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. 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 equivalent to the rule of necessitation: if q=¬p can be proven true without assumptions (a tautology), then q is necessarily true, therefore p is impossible).

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 about the material conditional (negated conditionals)

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.

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.

 

[Komodo]

Hi guys!

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:



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.

I'm trying to “translate” modal logic into more everyday language (something I've had mixed success in, at best), using a real-life example for p, and picking up with Axiom C, then with stage 8 of the proof.

C) Suppose all truths are knowable.
8. Now suppose there is, in fact, life on Mars, but it is not known that there is life on Mars (i.e., we don't have any justification for saying that there is life on Mars).
9. Then, since all truths are knowable, it would be possible to know the truth that “there is life on Mars, but it is not known that there is life on Mars.”
10. But this is equivalent to saying, “it is possible to know both that there is life on Mars, and that it is not known that there is life on Mars,” which is self-contradictory. So, we've just proved that it can never be the case that “there is life on Mars, but it is not known that there is life on Mars.”
11. That being the case, if there is life on Mars, it must be known that there is life on Mars. And the same holds for any other particular truth one can name.

So it seems basically like a reductio ad absurdum of the claim that all truths are knowable. Your response, if I read you correctly, is, first, that 10) is not self-contradictory, because it is in fact possible both to know and not to know that there is life on Mars, if you time it just right. That is,

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.

Now, as you observe just below that, modal logic is “timeless” which means that your translation would be disallowed. So much the worse for modal logic, you might say, and my own tendency would be to agree, but maybe that's just my lack of familiarity with modal logic speaking.

I think your second objection actually is another way of protesting the “timelessness” of modal logic. That is, if we say “it could be the case that there is life on Mars without our knowing it,” we are obviously speaking of our present state of knowledge. To say that this may be expressed as p & ¬Kp seems to turn this into an eternal truth, “it is, and always will be the case that there is life on Mars but we shall never know that there is life on Mars,” which of course would not be what anybody would mean by saying “it could be the case that there is life on Mars without our knowing it.”

 

[Temujin]

Hi guys!

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:



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.

This is very interesting, but is your real name Donald Rumsfeld, and why isn't this paradox named after him?

 

[Komodo]

Hi guys!

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 . . .

Maybe the problem is that p & ¬Kp, if translated as "p is true, but is not known to be true," sounds inherently self-contradictory: if "p is true" is part of your statement, aren't you implicitly claiming that you have knowledge of its truth? What if instead we translated it as "there is at least one possible world in which p is true, but is not known to be true." Since the speaker of this sentence would not necessarily be residing in this possible world, he isn't implicitly claiming to know himself that p is true, thus no paradox.

 

[Komodo]

Hi guys!

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 . . .

Also: if Kp means "it is known that proposition p is true," while Lp means "it is possible that proposition p is true," then p → LKp does not mean "if any proposition p is true, then it is possible to know that proposition p is true"; rather, it means "if any proposition p is true, then it is possible that it is known that proposition p is true." In other words, the claim he is reducing to absurdity is not "it is possible for any true proposition to be/become known to be true," it's "it is possible that any (and thus all) true propositions are known to be true," which nobody holds to. (At least not for human beings; if it's meant as a reductio of God's omniscience, it obviously fails, because the proposition "all truths are known" would be the opposite of absurd; given the existence of an omniscient being, it would be necessarily true.)

 

[Komodo]

Hi guys!

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 . . .

Still tinkering here...

In the claim, K(p & ¬Kp), let's take p to mean "Bruce Wayne is Batman." The statement then is asserting "it is known that (Bruce Wayne is Batman, and it is not known that Bruce Wayne is Batman)," which ultimately implies (using the given axioms) the contradiction that "it is known that Bruce Wayne is Batman, and it is not known that Bruce Wayne is Batman."

The self-contradiction of this statement would be removed, however, if we translated the modal language here in a more real-worldish fashion. If we asked people what K(p & ¬Kp) means, where p meant "Bruce Wayne is Batman," I think almost everybody would say it meant something like "it is known to comic-book readers in our actual world, that in the DC world, Bruce Wayne is Batman, but it is not known that Bruce Wayne is Batman." So if this "translation" is valid, then Kp & ¬Kp is not a contradiction, because "K" means two different things (known in our world, known in the DC world) in the two propositions.

And possibly a number of paradoxes and contradictions about knowledge arise from the fact that "it is known that" is so inherently ambiguous; not just (known in our world/known in some possible world) but (it is known now/it will be known at some future time) and (known by all/known by some sub-set). Unless these are disambiguated, maybe the act of "multiplying by K," as in K(p & ¬Kp), should be disallowed in modal logic, like dividing by zero is disallowed in mathematics.

 

[Cornelius]

I'm trying to “translate” modal logic into more everyday language (something I've had mixed success in, at best), using a real-life example for p, and picking up with Axiom C, then with stage 8 of the proof.

C) Suppose all truths are knowable.
8. Now suppose there is, in fact, life on Mars, but it is not known that there is life on Mars (i.e., we don't have any justification for saying that there is life on Mars).
9. Then, since all truths are knowable, it would be possible to know the truth that “there is life on Mars, but it is not known that there is life on Mars.”
10. But this is equivalent to saying, “it is possible to know both that there is life on Mars, and that it is not known that there is life on Mars,” which is self-contradictory. So, we've just proved that it can never be the case that “there is life on Mars, but it is not known that there is life on Mars.”
11. That being the case, if there is life on Mars, it must be known that there is life on Mars. And the same holds for any other particular truth one can name.

So it seems basically like a reductio ad absurdum of the claim that all truths are knowable. Your response, if I read you correctly, is, first, that 10) is not self-contradictory, because it is in fact possible both to know and not to know that there is life on Mars, if you time it just right. That is,

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.

Now, as you observe just below that, modal logic is “timeless” which means that your translation would be disallowed. So much the worse for modal logic, you might say, and my own tendency would be to agree, but maybe that's just my lack of familiarity with modal logic speaking.

I think your second objection actually is another way of protesting the “timelessness” of modal logic. That is, if we say “it could be the case that there is life on Mars without our knowing it,” we are obviously speaking of our present state of knowledge. To say that this may be expressed as p & ¬Kp seems to turn this into an eternal truth, “it is, and always will be the case that there is life on Mars but we shall never know that there is life on Mars,” which of course would not be what anybody would mean by saying “it could be the case that there is life on Mars without our knowing it.”

I agree it's a contradiction, but that's only because line 8 assumes that it is not known. If we are more specific and say who doesn't know there's life on Mars, then this becomes clearer. And the timelessness in the modal logic is exactly why these issues arise because that's the problem with Fitch's Paradox. You cannot assume something to be non-timeless and call a universal truth universal. (Sorry for this late reply, I got stuck on some unrelated theological problem and I got distracted by a million things)

 

[Cornelius]

Maybe the problem is that p & ¬Kp, if translated as "p is true, but is not known to be true," sounds inherently self-contradictory: if "p is true" is part of your statement, aren't you implicitly claiming that you have knowledge of its truth? What if instead we translated it as "there is at least one possible world in which p is true, but is not known to be true." Since the speaker of this sentence would not necessarily be residing in this possible world, he isn't implicitly claiming to know himself that p is true, thus no paradox.

No, "p is true" means it is a truth. Knowledge or non-knowledge of it is not assumed. I actually wondered that too. But the statement K~p (knowing a non-truth) would be illogical, because you cannot know a falsehood as if it were a truth: it could be a truth that you know a falsehood is false, but that would simply be Kp, and p is "x statement is a falsehood"

 

[Cornelius]

Still tinkering here...

In the claim, K(p & ¬Kp), let's take p to mean "Bruce Wayne is Batman." The statement then is asserting "it is known that (Bruce Wayne is Batman, and it is not known that Bruce Wayne is Batman)," which ultimately implies (using the given axioms) the contradiction that "it is known that Bruce Wayne is Batman, and it is not known that Bruce Wayne is Batman."

The self-contradiction of this statement would be removed, however, if we translated the modal language here in a more real-worldish fashion. If we asked people what K(p & ¬Kp) means, where p meant "Bruce Wayne is Batman," I think almost everybody would say it meant something like "it is known to comic-book readers in our actual world, that in the DC world, Bruce Wayne is Batman, but it is not known that Bruce Wayne is Batman." So if this "translation" is valid, then Kp & ¬Kp is not a contradiction, because "K" means two different things (known in our world, known in the DC world) in the two propositions.

And possibly a number of paradoxes and contradictions about knowledge arise from the fact that "it is known that" is so inherently ambiguous; not just (known in our world/known in some possible world) but (it is known now/it will be known at some future time) and (known by all/known by some sub-set). Unless these are disambiguated, maybe the act of "multiplying by K," as in K(p & ¬Kp), should be disallowed in modal logic, like dividing by zero is disallowed in mathematics.

Well this doesn't work because p has to be the same in both Kp and ¬Kp. That's why the Paradox is such a little gem. The problem is that modal logic does not acknowledge convergence which is exactly how anything changes due to Space-Time being continuous and not discrete (i.e. you can infinitely divide it - though the issue remains even if it were discrete). How do you learn a fact. If at Moment A you don't know a fact, and at Moment B you know it, and there are no moments in-between, there is no change possible inbetween, and knowledge, learning, etc happened literally out of nowhere. So logically, as the modal logic recognizes but misinterprets due to limitations as in Fitch's Paradox, there has to be a moment where you both know and don't know something. The Paradox captures and exploits that perfectly, which is one of the reasons modal logic is limited and they know it and are trying to fix it. This is essentially Zeno's Paradox but in reverse: you assume the midpoints aren't infinite (in time in this case) and arrive at a contradiction. Convergence solves these issues, but that's not allowed in the Paradox because you impose countability (the logical lines) on something uncountable (continuous space-time moments).

The same works with space. Imagine you kick a ball. If this ball is unkicked until the moment you kick it. The moment that you make contact has to logically precede the moment it is still unkicked. But if you distinguish between two moments like that, then you assume motion stops (at their boundary) and somehow resumes between the two, out of nowhere literally, and you cannot explain that except to So this moment has to by definition be one where it is both kicked and unkicked since you are