This site is supported by Nobility Studios.

Idea: Fitch's paradox of knowability

16 posts in this topic

Posted

Some philosophers claim that it is possible to argue from simple premises that anti-realism positions which adhere to the knowability principle are mistaken. The knowability principle holds that "p (p →

Share this post


Link to post
Share on other sites

Posted

Thanks Rexia

I am interested, but somewhat circular to me

please do assume ignorance of prior K on the matter

I need help in understanding what you are saying/trying to say

Some philosophers claim that it is possible to argue from simple premises that anti-realism positions which adhere to the knowability principle are mistaken.

could you expound a bit more? which philosophers?

( I presume why they are mistaken is explained below?)

The knowability principle holds that "p (p →

Share this post


Link to post
Share on other sites

Posted (edited)

$p (p &

Edited by TheBeast
1 person likes this

Share this post


Link to post
Share on other sites

Posted

$p (p &

Share this post


Link to post
Share on other sites

Posted

Is Fitch's paradox a threat to anti-realism?

Fitch's paradox is an epistemic paradox and not a logical paradox.

(1) Anti-realism asserts the knowability principle (or the doctrine of verificationism) that all truths are knowable.

(2) If all truths are knowable, then by Fitch's paradox, all truths are known.

(3) It is false that all truths are known.

(4) Anti-realism is mistaken.

See: http://plato.stanford.edu/entries/fitch-paradox/

http://consequently.org/papers/notevery.pdf

http://sites.google.com/site/knowability/home

Share this post


Link to post
Share on other sites

Posted

I'm having trouble, though, moving from the inference that

(1) p is known and it is known that p is unknown

to

(2) p is not knowable and an unknown p is not knowable.

This inference is needed to reveal a contradiction in assuming that all truths are knowable and that there is an unknown truth.

Not sure if this makes any sense, pg?

Share this post


Link to post
Share on other sites

Posted

Hi Rexia, I've only familiarised myself with Fitch's paradox in the last half hour or so, and I'm far from an expert on logic, but here are my thoughts. I found this section of the article you linked to helpful in understanding the paradox:

[fieldset=Not Every Truth Can Be Known (at least, not all at once), Restall, G.]

Share this post


Link to post
Share on other sites

Posted (edited)

I'm having trouble, though, moving from the inference that

(1) p is known and it is known that p is unknown

to

(2) p is not knowable and an unknown p is not knowable.

This inference is needed to reveal a contradiction in assuming that all truths are knowable and that there is an unknown truth.

Not sure if this makes any sense, pg?

Rexia, thanks for raising the paradox, it is indeed very interesting

i am not sure I am following the argment completely (yet)

I am probably not sophisticated enough, and I admit I

cling to simplicity and naivety of views

but

(1) p is known and it is known that p is unknown

to

(2) p is not knowable and an unknown p is not knowable.

I dont think i can accept either statement above, they both seem

paradoxical in themselves - remind me, where do you get 1) and 2) from?

i also need to understand if

not knowable and

unknown are the same, or equivalent, or different in any way

(please address)

additionally, I think there may be something missing from such a formulation, for example, the individual knower point of view

all truths are knowable (but not at the same time, by the same person due to cognitive limitations, right?)

so there is a temporal issue - and time is a bit controversial ....

Also I reminded myself of some ancient postulates

whereby knowing is the product/function of the intellect, so basically

the unknown is what the intellect can't see due to its limitations

http://www.themystica.com/mystica/articles/b/buddhi.html

(btw mysticism is a recategorization of prearistotelic scientific knowledge)

and thanks for the precious links.... cool stuff....

I will not be offended if my remarks are missing the point, and if you have no particular comment to add, but any explanation of where you come from

would be appreciated

pg

Edited by Pgalaxy

Share this post


Link to post
Share on other sites

Posted

Not sure if this helps either, Rexia, but the Stanford article states (in "The Paradox of Knowability" section):

Also presupposed are two modest modal principles: first, all theorems are necessary. Second, necessarily

Share this post


Link to post
Share on other sites

Posted

Thanks, pg and parsec.

I am studying the formulation of Fitch

Share this post


Link to post
Share on other sites

Posted

Rexia

I am studying the formulation of Fitch’s paradox as stated in the article on Fitch’s paradox in the Stanford Encyclopedia of Philosophy. I decided to re-write it in words, instead of in logical notation, as below. But I am not sure that I’ve understood the proof. Can check if this makes any sense, pg and parsec?

I can tell you what makes or does not make sense *to me* :-)

A contradiction is reached from assuming the following:

(1) All truths are knowable. (Assumption)

(2) There is an unknown truth. (Assumption)

If (2) is true, then it is a truth that there is an unknown truth. Hence,

(3) It is knowable that there is an unknown truth.

Now, assume for reductio that

(4) There is a known truth and it is known that there is an unknown truth.

(5)

I'd say the sense stops here for me :-)

I think if you take into account what you have added later

Not knowable and unknown- If a proposition is not knowable, it seems to follow that it is unknown. If a proposition is not knowable, it *cannot* be known, or it is impossible to be known. As a consequence, that proposition is also unknown.

then, there is no contradiction:

all truths are knowable,

but not all truths are known

not sure if thats what Stanford says though...

let me know if I am getting something wrong

PG

If knowledge entails truth, then

(6) There is a known truth and there is an unknown truth (from 4).

(7) Contradiction! We therefore deny (4):

(8) It is false that (there is a known truth and it is known that there is an unknown truth).

(9) (8) is a theorem and all theorems are necessary.

(10) It is necessarily false that (there is a known truth and it is known that there is an unknown truth).

(11) A necessarily false proposition is impossible.

(12) It is impossible that there is a known truth and it is known that there is an unknown truth.

(13) It is impossible that it is known that there is an unknown truth.

(14) It is not knowable that there is an unknown truth.

Now compare (3) and (14):

It is knowable that there is an unknown truth.

It is not knowable that there is an unknown truth.

(15) Contradiction! We therefore deny (2):

(16) It is false that there is an unknown truth.

(17) All truths are known.

Hence,

(18) All truths are knowable entails that all truths are known.

---------- Post added at 08:30 ---------- Previous post was at 08:28 ----------

This above is a proof-theoretic approach to the knowability paradox.

Not knowable and unknown- If a proposition is not knowable, it seems to follow that it is unknown. If a proposition is not knowable, it *cannot* be known, or it is impossible to be known. As a consequence, that proposition is also unknown.

But if a proposition is unknown, it doesn’t follow that it is not knowable. For it might be the case that an unknown proposition is knowable. For example, it is unknown to us that a spider lurks in the corner of the room. But it is not unknowable that a spider lurks in the corner of the room.

Knowable and known- Just as “not knowable” and “unknown” mean differently, “knowable” and “known” mean differently. If a proposition is knowable, this by no means implies that it is known. But if a proposition is known, then it seems to mean also that it is knowable. For a proposition has to be knowable before it can be known.

Another thing- The main difference between a proposition that is unknowable and a proposition that is unknown is that one proposition is necessarily so. If a proposition is unknowable, then it is necessarily unknown.

Share this post


Link to post
Share on other sites

Posted

But if a proposition is unknown, it doesn’t follow that it is not knowable. For it might be the case that an unknown proposition is knowable. For example, it is unknown to us that a spider lurks in the corner of the room. But it is not unknowable that a spider lurks in the corner of the room.

I agree with you here, but I think that the logic of Fitch's "paradox" is sound. But perhaps there is a problem with the logic, or perhaps the logic is correct but there is a problem with our current understanding of knowledge and truth which makes it seem paradoxical. I think the problem lies with logic. As I said earlier, if we allow for p, which logic must, then p, being a true proposition, must have the qualities of knowledge attributed to it by Fitch's logic. To reiterate, I'm far from an expert on logic, but the problem could lie with the assumption (as you presented it): (5) "knowledge entails truth". Could it not be, instead, that truth entails knowledge? I'm not sure that the logic would turn out the same if we were to alter this assumption.

I found this paper last night, which may be of help:

[fieldset=On Keeping Blue Swans and Unknowable Facts at Bay. A Case Study on Fitch’s Paradox, Brogaard, B.]What has come to be called the knowability paradox was first published by Frederic Fitch as Theorem 5. It is equivalent to the claim that if every truth is knowable, then every truth will be known:

(T5) ϕ → ◊Kϕ |-- ϕ → Kϕ

where ◊ is possibility, and ‘Kϕ’ is to be read as ϕ is known by someone at some time. Let us call the premise the knowability principle and the conclusion near-omniscience. Here is a way of formulating Fitch’s proof of (T5). Suppose the knowability principle is true. Then the following instance of it is true: (p & ~Kp) → ◊K(p & ~Kp). But the consequent is false, it is not possible to know p & ~Kp. That is because the supposition that it is known is provably inconsistent. The inconsistency requires us to deny the possibility of the supposition, yielding ~◊K(p & ~Kp). This, together with the above instance of the knowability principle, entails ~(p & ~Kp), which is (classically) equivalent to p → Kp. Since p occurs in none of our undischarged assumptions, we may generalize to get near-omniscience, ϕ → Kϕ. QED.

(T5) is today considered by many to be a paradox for a number of related reasons, among others, that it threatens to show that the very thesis that is thought to discriminate a mature semantic anti-realism from a na

Share this post


Link to post
Share on other sites

Posted

Thanks Parsec

I read with interest the paper you point to

at some point it says

It would be interesting to find out what the malady is. Is there a deep metaphysical point to be made about why these different positions give rise to this distinctive kind of paradox?

I think the malady referred to is widespread: we look at the constructs of our mind as if they were the reality we are observing. So the paradox is in the mind of the beholder, so to speak - its the product of a subjective observation/perception.

again

In conclusion: the present case study indicates that Fitch-like paradoxes present a major obstacle, not only to semantic anti-realism, but also potentially to a number of other anti-realisms. The Fitch-like paradoxes give anti-realists reason to restrict. Restriction leads to timid anti-realism. Fitch

Share this post


Link to post
Share on other sites

Posted

Thanks, pg. I tried to explain the problem again, see below.

Commonsense tells us that if a proposition is unknown, it does not follow that it is unknowable. Common sense also tells us that if all truths are knowable, it does not follow that all truths are known. Just because something is possible to know does not imply that it is already known.

However, according to SEP, Frederic Fitch held that we can logically prove that there is an unknown truth entails that there is an unknowable truth. Its contraposition is the controversial thesis that all truths are knowable entails that all truths are known.

In SEP, the authors assume that all truths are knowable and that there is an unknown truth, and show that it leads to a contradiction. The contradiction implies the need to deny that there is an unknown truth, from which it follows that all truths are known.

I

Share this post


Link to post
Share on other sites

Posted

Thanks Rexia

I appreciate the simplification of the problem - I think I understand

so what you are asking, really, is whether Fitch's Paradox makes sense

(or rather, is it logicallly valid?)

However, according to SEP, Frederic Fitch held that we can logically prove that there is an unknown truth entails that there is an unknowable truth.

So the question is:

are all unknown truths knowable?

or

not all truths are knowable

maybe we need to define the values of

uknown

vs

unknowable

I think its safe to assume different levels of knowledge exists, and maybe even different levels of truths, so to dig deeper we may have to start refining our language for this analysis (need help with that)

perfect knowledge (whereby all truths are knowable and known)

does not belong to the world as we know it, although through different

stages of enlightenment some glimpse of it may be attainable

p

Thanks, pg. I tried to explain the problem again, see below.

Commonsense tells us that if a proposition is unknown, it does not follow that it is unknowable. Common sense also tells us that if all truths are knowable, it does not follow that all truths are known. Just because something is possible to know does not imply that it is already known.

However, according to SEP, Frederic Fitch held that we can logically prove that there is an unknown truth entails that there is an unknowable truth. Its contraposition is the controversial thesis that all truths are knowable entails that all truths are known.

In SEP, the authors assume that all truths are knowable and that there is an unknown truth, and show that it leads to a contradiction. The contradiction implies the need to deny that there is an unknown truth, from which it follows that all truths are known.

I

Share this post


Link to post
Share on other sites

Posted

Thanks Rexia

I appreciate the simplification of the problem - I think I understand

so what you are asking, really, is whether Fitch's Paradox makes sense

(or rather, is it logicallly valid?)

However, according to SEP, Frederic Fitch held that we can logically prove that there is an unknown truth entails that there is an unknowable truth.

So the question is:

are all unknown truths knowable?

or

not all truths are knowable

maybe we need to define the values of

uknown

vs

unknowable

I think its safe to assume different levels of knowledge exists, and maybe even different levels of truths, so to dig deeper we may have to start refining our language for this analysis (need help with that)

perfect knowledge (whereby all truths are knowable and known)

does not belong to the world as we know it, although through different

stages of enlightenment some glimpse of it may be attainable

p

Thanks, pg. I tried to explain the problem again, see below.

Commonsense tells us that if a proposition is unknown, it does not follow that it is unknowable. Common sense also tells us that if all truths are knowable, it does not follow that all truths are known. Just because something is possible to know does not imply that it is already known.

However, according to SEP, Frederic Fitch held that we can logically prove that there is an unknown truth entails that there is an unknowable truth. Its contraposition is the controversial thesis that all truths are knowable entails that all truths are known.

In SEP, the authors assume that all truths are knowable and that there is an unknown truth, and show that it leads to a contradiction. The contradiction implies the need to deny that there is an unknown truth, from which it follows that all truths are known.

I?m not sure, though, what makes it a theorem that there is an unknown truth entails that there is an unknowable truth. If a truth is unknown, it does not seem to follow that it is unknowable. Yet this is what Fitch?s theorem tells us.

A formal definition, pg?

Let K be the knowledge operator. Let  be the possibility operator.

Then,

that there is an unknown truth = p (p & ~Kp)

that there is an unknowable truth = p (p & ~Kp)

For an informal definition, let an unknown truth be a truth that we do not know. For example, that there is water on Mars could be a truth that we do not know. Let an unknowable truth be a truth that we cannot know. For example, that God created the universe could be a truth that we cannot know.

The question does seem to be whether or not all unknown truths are knowable. In SEP, the authors of the entry on Fitch’s paradox claim that we can prove, independently of the paradox, that some unknown truth is not knowable. So, not all unknown truths are knowable.

If we reflect on the history of science, or, even on our personal life, we see that we have come to know truths that we previously did not know. Only if unknown truths were knowable could we have come to know truths that we did not know before.

Some unknown truths are therefore knowable, even if not all unknown truths are knowable. Sorry to seem going around in circles! I think there must be more, but it is difficult to get out of this loop.

Share this post


Link to post
Share on other sites

Create an account or sign in to comment

You need to be a member in order to leave a comment

Create an account

Sign up for a new account in our community. It's easy!


Register a new account

Sign in

Already have an account? Sign in here.


Sign In Now