I don't understand what your comments have to do with anything. — Michael

You don't understand the question, what is knowledge?

A the cat is on the mat
B no one knows the cat is on the mat

A is an assertion of knowledge
B contradicts A

A is an assertion of knowledge — Harry Hindu

In practice it may be that asserting a proposition implies that one believes one's assertion (see Moore's paradox), but in formal logic there is a distinction between asserting that a proposition is true and asserting that a proposition is known to be true.

Regardless, your comments have nothing to do with Fitch's paradox. The non-omniscience principle states that ∃p(p ∧ ¬Kp). However, ¬◇K(p ∧ ¬Kp). Therefore, ¬(p → ◇Kp). The knowability principle is false.

In practice it may be that asserting a proposition implies that one believes one's assertion (see Moore's paradox), but in formal logic there is a distinction between asserting that a proposition is true and asserting that a proposition is known to be true. — Michael

But A does not say either way. B tries to clarify the distinction but fails when

C no one know that no knows the cat is on the mat

C takes your principle of non-omniscience to its full conclusion

In practice, meaning it can be useful in the world with formal not necessarily so. I'm more interested in the more useful interpretation.

The non-omniscience principle states — Michael

It seems to state that knowledge and truth are not related.

You're avoiding the questions requesting the definition of the terms you're using but fail to provide any.

What does it mean to be omniscient vs non-omniscient? Don't you have to define knowledge to make sense of that distinction?

Does being non-omniscient mean that we know nothing or that we don't know everything? If the latter then how do we know that what we do know is true? If the former then knowledge is meaningless.

It's knowable because we can look for the cat and see it to be on the mat. In doing so, what was once an unknown truth (1) is now a known truth and what was once a known truth (2) is now a known falsehood. And what was once an unknown truth (3) is now a known falsehood.

3 can never be a known truth. — Michael

Is (2) both true and false? Is (3)?

Is (2) both true and false? — Luke

No. It was true before we knew 1 and false after.

Is (3)? — Luke

No. It was true before we knew 1 and false after.

You're avoiding the questions requesting the definition of the terms you're using but fail to provide any.

What does it mean to be omniscient vs non-omniscient? Don't you have to define knowledge to make sense of that distinction?

Does being non-omniscient mean that we know nothing or that we don't know everything? If the latter then how do we know that what we do know is true? If the former then knowledge is meaningless. — Harry Hindu

We address the problem in formal logic. We start with the two premises that the anti-realist accepts:

Knowability principle
∀p(p → ◊Kp)

Non-omniscience principle
∃p(p ∧ ¬Kp)

We then apply the accepted rules of inference to derive the conclusion:

All truths are known
∀p(p → Kp)

So the anti-realist must reject either the knowability principle or the non-omniscience principle.

Fitch isn't interested in a drawn out debate on what the anti-realist means by knowledge; he's only interested in the internal consistency of their position. So whatever it is they mean by knowledge he shows that their position entails that all truths are known. The anti-realist then has to either accept that or abandon their knowability principle.

Since we don't have access to the registry of things that are, how is one to ascertain that "P is known", as opposed to "persons A, B and C believe that P is true, while person D may disagree"? — Olivier5

The normative standard for making knowledge claims isn't Cartesian certainty, it's evidential. The truth condition for knowledge is part of ordinary usage (which means that contradictory knowledge is impossible).

So I might say, "I thought I knew where my keys were but it turns out I didn't." Similarly, we don't say that people used to know that the Sun orbited the Earth. We say that people used to believe that the Sun orbited the Earth, but they were mistaken (since we now know that the Earth orbits the Sun).

Is (2) both true and false?
— Luke

No. It was true before we knew 1 and false after.

Is (3)?
— Luke

No. It was true before we knew 1 and false after. — Michael

I can see now that I was wrong about this, and I now accept that some truths are unknowable.

Thanks to you and to @Andrew M for your patience and for correcting the errors of my thinking about this.

we don't say that people used to know that the Sun orbited the Earth. We say that people used to believe that the Sun orbited the Earth, but they were mistaken (since we now know that the Earth orbits the Sun). — Andrew M

But back then, they wouldn't say "we believe that the sun orbits the earth". They would rather have said: "we know that the sun orbits the earth". And there was plenty of evidence for it, mind you, though we now understand that this evidence was interpreted incorrectly.

Knowledge is far more complex a process than the letter K, even more complex than the letters Kp....

Nice. Like the fridge argument against Pierce.

Very clear.

But back then, they wouldn't say "we believe that the sun orbits the earth". They would rather have said: "we know that the sun orbits the earth". — Olivier5

And were they right?

Thanks to you and to Andrew M for your patience and for correcting the errors of my thinking about this. — Luke

:up: Thanks for saying so, and for working it through.

Andrew M Nice. Like the fridge argument against Pierce. — Banno

:up:

But back then, they wouldn't say "we believe that the sun orbits the earth". They would rather have said: "we know that the sun orbits the earth". And there was plenty of evidence for it, mind you, though we now understand that this evidence was interpreted incorrectly. — Olivier5

Indeed, and that's the point. When we discover that a former knowledge claim was mistaken, we retroactively downgrade its status from knowledge to belief. We say that they didn't know it after all, since we no longer believe that it was true then.

Another way to think of this is in terms of Ryle's achievement verbs. We can believe or claim that it is raining and be mistaken but we can't know that it is raining and be mistaken, since to know that it is raining is to be correct and for good reason (e.g., we looked out the window).

The distinction between task verbs and achievement verbs or ‘try’ verbs and ‘got it’ verbs frees us from another theoretical nuisance. It has long been realised that verbs like ‘know’, ‘discover’, ‘solve’, ‘prove’, ‘perceive’, ‘see’ and ‘observe’ (at least in certain standard uses of ‘observe’) are in an important way incapable of being qualified by adverbs like ‘erroneously’ and ‘incorrectly’. ... — The Concept of Mind, p134 - Gilbert Ryle

That's the basis for the epistemic principle (B) in Fitch's proof, "Kp ⊢ p".

I don't think so but then, what do I know?

Another way to think of this is in terms of Ryle's achievement verbs. We can believe or claim that it is raining and be mistaken but we can't know that it is raining and be mistaken, since to know that it is raining is to be correct and for good reason (e.g., we looked out the window). — Andrew M

Similarly, it can be shown that, contrary to popular belief, not all chicken can be eaten. Take a live, not yet eaten chicken. Can one eat it one day? Yes but then it would immediately cease to be an uneaten chicken. So an uneaten chicken cannot be eaten.

B(p & ~Bp) - someone at some time has the belief that 'p is true and nobody believes that p is true'. Is this Moore's paradox? — Luke

I was hoping someone would have responded to this point. Did anyone else note this connection between the two paradoxes? Does anyone agree or disagree that these are similar or the same type of paradox?

Similarly, it can be shown that, contrary to popular belief, not all chicken can be eaten. Take a live, not yet eaten chicken. Can one eat it one say? Yes but then it would immediately cease to be an uneaten chicken. So an uneaten chicken cannot be eaten. — Olivier5

Logic says that we're all vegetarians now...

B(p & ~Bp) - someone at some time has the belief that 'p is true and nobody believes that p is true'. Is this Moore's paradox?
— Luke

I was hoping someone would have responded to this point. Did anyone else note this connection between the two paradoxes? Does anyone agree or disagree that these are similar or the same type of paradox? — Luke

Yes, very similar. Interestingly, from SEP:

Frederic Fitch (1963) reports that in 1945 he first learned of this proof of unknowable truths from a referee report on a manuscript he never published. Thanks to Joe Salerno’s (2009) archival research, we now know that referee was Alonzo Church.

...

Church’s referee report was composed in 1945. The timing and structure of his argument for unknowables suggests that Church may have been inspired by G. E. Moore’s (1942, 543) sentence:

(M) I went to the pictures last Tuesday, but I don’t believe that I did. — Epistemic Paradoxes - SEP

Yes, very similar. Interestingly, from SEP: — Andrew M

Oh cool, thanks.

Logic says that we're all vegetarians now... — Andrew M

Specifically, it says that an uneaten chicken cannot be eaten without ceasing to be an uneaten chicken, so we cannot logically speaking eat an uneaten chicken.

Note that we also cannot eat a chicken that has already been eaten. And since a chicken is either eaten or not eaten, it follows that logically speaking, we cannot eat any chicken.

Specifically, it says that an uneaten chicken cannot be eaten without ceasing to be an uneaten chicken, so we cannot logically speaking eat an uneaten chicken.

Note that we also cannot eat a chicken that has already been eaten. And since a chicken is either eaten or not eaten, it follows that logically speaking, we cannot eat any chicken. — Olivier5

You're equivocating. It is possible for us to later eat something that is currently uneaten, or for something that we have eaten to have before that time been uneaten. It isn't possible for us to eat something and for it to remain uneaten.

Interesting observation.

- In Fitch's case, the epistemic operator K is usually assumed to be factive and used in the future-tense in standing for "Eventually it will be known that ...", where K's arguments are general propositions p that can refer to any point in time. So Fitch's paradox is a paradox concerning the eventual knowledge of propositions.

- In Moore's case, the epistemic operator B is assumed to be non-factive and referring only to the present state of the world, in standing for "It is presently believed that", where B's argument is the present state of the world s that changes over time. So Moore's paradox is a temporal paradox referring to the indistinguishability of the concepts of belief and truth in the mind of a single observer with respect to his understanding of the present state of the world, in spite of the fact the observer distinguishes these concepts when referring to the past and future state of the world.

- Only in the case of K is there the general rule K p --> p , since knowledge is assumed to be true, unlike beliefs that aren't generally regarded as truthful , except in the case of the present tense if Moore's sentences are rejected for all s, in which case it is accepted that for all s, ~ (s & ~B s). This premise is equivalent to saying that for all s, ( s --> B s).

-The argument for Fitch's knowability conclusion (p --> K p) starts from a weaker knowability premise that (p --> possibly K p). On the other hand, Moore's sentences, if rejected, are rejected a priori as being grammatically inadmissible, meaning that (s --> B s) is accepted immediately and doesn't require derivation.

Chicken-edibility principle
∀c(c → ◊Ec)
(if a chicken exists, it can be eaten)

Non-omnigallinavorous principle
∃c(c ∧ ¬Ec)
(there exist chicken that are not eaten)

We then apply the accepted rules of inference to derive the conclusion:

∀c(Ec → Ec → ¬◊Ec) ∧ ∀c(¬Ec → ¬◊Ec)
(all eaten chicken have already been eaten and can't be eaten anymore, and all uneaten chicken cannot be eaten either, otherwise they wouldn't be uneaten chicken)

We then apply the accepted rules of inference to derive the conclusion:

∀c(Ec → Ec) ∧ ∀c(¬Ec → ¬◊Ec)
(all eaten chicken have already been eaten and all uneaten chicken cannot be eaten, otherwise they wouldn't be uneaten chicken anymore) — Olivier5

What rules of inference get you there?

Chicken-edibility principle
∀c(c → ◊Ec)
(if a chicken exists, it can be eaten)

Non-omnivorous principle
∃c(c ∧ ¬Ec)
(there exist chicken that are not eaten) — Olivier5

Also the symbols here make no sense. I think you need something like:

∀x(Cx → ◊Ex)
For all things, if that thing is a chicken then it is possible to eat that thing.

∃x(Cx → ¬Ex)
There is at least one thing that is a chicken and hasn't been eaten.

I am using the accepted rules of inference, not the unaccepted ones.

Which ones? Write them out.

You realize I'm pulling your leg, right?

Indeed, and that's the point. When we discover that a former knowledge claim was mistaken, we retroactively downgrade its status from knowledge to belief. We say that they didn't know it after all, since we no longer believe that it was true then. — Andrew M

But this misses the point that what we used to call knowledge wasn't knowledge in light of new observations, but observations is what allowed us to assert knowledge that we didn't have in the first place. So how do we know that we've made every possible observation to assert we possess knowledge? Seems to me that either knowledge is not related to truth as Michael's non-omniscient principle seems to state:

some truths are unknowable — Luke

or "knowledge" is a useless term and we can only ever believe our assertions.

Or, we re-define knowledge to be a set of rules that we have adopted for interpreting some observation, like the sun moving across the sky, and the rules (knowledge) can change with new observations.

In Fitch's case, the epistemic operator K is usually assumed to be factive and used in the future-tense in standing for "Eventually it is known that ...", where K's arguments are general propositions p that can refer to any point in time. — sime

I’ll try and come back to the rest of your post, but if the above is correct, then this would seem to contradict @Michael’s claim that a proposition can be known to be true at one time and then known to be false at a later time. If K refers only to what is eventually known, then a proposition which is ultimately known to be false cannot earlier be known to be true.

this would seem to contradict Michael’s claim that a proposition can be known to be true at one time and then known to be false at a later time — Luke

The proposition "Joe Biden is President of the United States" was known to be false in 2016 and is known to be true now.

Another way to think of this is in terms of Ryle's achievement verbs. We can believe or claim that it is raining and be mistaken but we can't know that it is raining and be mistaken, since to know that it is raining is to be correct and for good reason (e.g., we looked out the window). — Andrew M

This is circular.

You can look out the window at the moment your trickster brother sprays the window with a hose.

Is it possible to believe a truth? How would that be different than to know a truth? How do we ever know that we have all the evidence necessary to assert knowledge over belief?

It seems to me that knowledge can only ever be a present or past state, never a future state. We can know what we know and know what we knew but never know what we will know.

EDIT:
Now that I think about it, it seems that knowledge is only a present state, kind of like the current fashion trend.