Fitch's Paradox: Knowability of All Truths

[Kumar8434]

https://en.wikipedia.org/wiki/Fitch's_paradox_of_knowability It begins by assuming that 'All truths are knowable' and then logically proves that that assumption means 'All truths are already known.' The proof is like this:

Suppose p is a sentence that 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.

But isn't this plain wrong just by common sense? Because 'All truths are knowable' certainly doesn't mean 'All truths are already known'. Because consider a universe in which there's a house in which only one person lives. There is nothing else in the universe. There are only two truths about the universe. Both truths are written on two different pieces of paper, one piece is hidden in the kitchen and the other in the balcony. So, both truths are knowable to the person, they just have to check the kitchen and the balcony, but that clearly does not mean 'both truths are already known to the person. So, something MUST be wrong with the proof, right?

 

[mfb]

The truth of "the sentence p is an unknown truth" is not universal - it depends on our knowledge. That is contrary to how mathematical statements are usually phrased.

There are only two truths about the universe. Both truths are written on two different pieces of paper, one piece is hidden in the kitchen and the other in the balcony.

What about the knowledge "there are only two truths about the universe"? Or "the person doesn't know any truth yet"?

 

[Kumar8434]

The truth of "the sentence p is an unknown truth" is not universal - it depends on our knowledge. That is contrary to how mathematical statements are usually phrased.
What about the knowledge "there are only two truths about the universe"? Or "the person doesn't know any truth yet"?

So you're saying that there can't be a finite number of truths. But I don't think that makes much difference. In such a universe, infinite number of truths written on different pieces of paper are floating in the universe. All papers are numbered. The person has a device in which they have to put in the the number and they get co-ordinates of the truth paper. And they have a space shuttle to reach anywhere. So, each of these truths is knowable but is not yet known to the person.

 

[Kumar8434]

The truth of "the sentence p is an unknown truth" is not universal - it depends on our knowledge. That is contrary to how mathematical statements are usually phrased.
What about the knowledge "there are only two truths about the universe"? Or "the person doesn't know any truth yet"?

Let's not make it complicated. The person just has an infinite list of truths. They can read whichever truth they want.

 

[Kumar8434]

I personally don't think saying that something can't be known and true at the same time is the same saying something is an unknowable truth. The sentence 'p is an unknown truth' is just a clever sentence which is no longer true when p is known. This sentence is not really an unknowable truth. Unknowable truth must mean that no one has the capabilities to find that truth out. Like what happens in time intervals smaller than plank time or what's inside black holes. Unknowable truth can't mean some truth which can be known but is no longer true when known.

 

[Stephen Tashi]

https://en.wikipedia.org/wiki/Fitch's_paradox_of_knowability It begins by assuming that 'All truths are knowable' and then logically proves that that assumption means 'All truths are already known.' The proof is like this:

So, both truths are knowable to the person, they just have to check the kitchen and the balcony, but that clearly does not mean 'both truths are already known to the person. So, something MUST be wrong with the proof, right?

No. We'd have to understand the precise rules of modal logic to comment on the correctness of the proof. The conclusion of a correct proof may be wrong for a particular example if the proof is based on assumptions that don't apply to the example.

The proof in the Wikipedia does not employ concepts of possibility or knowledge that have a time stamp. So "knowable" in that article does not specify a time when the knowledge is gained. As remarked in the article, the assumption C:

(C) p → LKp – all truths are knowable.

can be disputed. And, without any notion of a time stamp, an expression such as $Kp$ doesn't provide information such as "Knowable on Dec 28, 2017" or "Knowable on or before Jan 3, 2025".

In fact, the statement that is supposed to be paradoxical "All truths must be known" is only offensive because the word "known" is used - indicating the time stamp "known as of the current time". The literal interpretation of the conclusion of the proof, $p \rightarrow Kp$ is that "All truths are knowable", without any specification about a time when the knowledge is gained.

 

[TeethWhitener]

It's somewhat unclear what the proof has to do with modal logic. Defining K as "known" is nonstandard. Typically the dual operators in modal logic are necessity and possibility. In which case, the modal axiom (D) from the Wiki article
$$\neg p \vdash \neg Lp$$
seems awfully close in spirit to $\neg p \longrightarrow \neg Lp$. Assuming that duality holds ($K \equiv \neg L \neg$), this proposition is immediately equivalent to $\neg p \longrightarrow K\neg p$, or by substitution, $q \longrightarrow Kq$ (which is the "paradox"). But this assertion is an extraordinarily strong axiom, known in modal logic as TRIV (the trivial axiom). In fact, the Kripke model of this system is degenerate: one reflexive world, and it's a maximal system (meaning that adding any axiom to it that's not a direct theorem makes the system inconsistent).

Edit: it just occurred to me that maybe they're using K as a predicate, and not as a modal operator. In which case, the usage is very non-standard (e.g., the absence of quantifiers). Regardless, the statement $\neg p \longrightarrow \neg Lp$ is still equivalent to TRIV: $q \longrightarrow \square q$, where $\square$ is the necessity operator dual to $L$. To see why this axiom is so strong, consider the contrapositive, which basically says "If p is possible, then p is true."

 

[Stephen Tashi]

A slight digression - there is another amusing paradox described in section "Intuitive problems with denotic logic" in the current Wikipedia article on modal logic, https://en.wikipedia.org/wiki/Modal_logic.

 

[TeethWhitener]

A slight digression - there is another amusing paradox described in section "Intuitive problems with denotic logic" in the current Wikipedia article on modal logic, https://en.wikipedia.org/wiki/Modal_logic.

I find this dubious. Wiki asserts that the phrase "If you have stolen some money, it ought to have been a small amount" is represented by $\square(K\longrightarrow Q)$, but "If you have stolen some money, it ought to have been a large amount is represented by $\square(K \longrightarrow (K \land \neg Q))$. Why isn't the second phrase simply $\square (K\longrightarrow \neg Q)$? (K = "You stole some money" and Q = "The money was a small amount")

The first phrase plus the K axiom and modus ponens gives $\square K \longrightarrow \square Q$.
The modified second phrase plus the K axiom and MP gives $\square K \longrightarrow \square \neg Q$.
Combining these two and using simple negation introduction from propositional logic gives $\{ (\square K \longrightarrow \square Q),(\square K \longrightarrow \square \neg Q)\} \vdash \neg \square K$, or "It's not the case that you ought to steal some money." Logically this is not equivalent to $\square \neg K$ ("You ought not to steal money"), but the two statements are compatible, and you can add $\square \neg K$ as an axiom to the system without risking logical inconsistency (that is, $\square \neg K$ is a stronger statement than $\neg \square K$).

 

[Erland]

The formal derivation in the Wikipedia link is correct

https://en.wikipedia.org/wiki/Fitch's_paradox_of_knowability

but the problem is the formalization of the argument. It is assumed that "p is knowable" can be formalized as LKp, which means "It is possible that p is known". That is not a reasonable formalization. If we say that something is known, it is implicitly assumed that it means that it is known right now. But if we say that something is knowable, it means that is possible that it is or will be known at some time, perhaps in the future.

For example, suppose we are back at October 1st 1959. Let p be the sentence "The are fewer dark regions on the far side of the Moon than on the near side", which happens to be true. It was not known at October 1st 1959 that p is true, and it was not even possible that it was known, because no one had seen the far side of the moon by then and no spacecraft had photographed it. So at that time, LKp was false. But p was certainly knowable by then, in the above sense. In fact, within a month from that date, the Soviet spacecraft Luna 3 flew over the far side of the Moon and sent the first photographs ever of it to the Earth, and these photographs showed that p is true.

Therefore, Fitch's argument is flawed, in my opnion.

On the other hand, Fitch's conclusion is true nevertheless, because it is obvious that not all truths are knowable.
For example, let N be the number of iron atoms in the World Trade Center at 10:00 a.m. June 5th 1980, and let p be the sentence "There was N iron atoms in the World Trade Center at 10:00 June 5th 1980" (where N is replaced by its actual number). p is true, but we can safely assert that it will never be possible to find that out, so p is unknowable.