Fitch's "paradox" of knowability
For a basic sketch of the paradox, the Wikipedia article states:
Quoting Fitch's paradox of knowability
I believe the issue lies, not in the truth of the sentence being unknown but, in the sentence itself being unknown. It is impossible to know an unknown sentence, or to know an unknown (anything). It's simply a contradiction in terms.
The Wiki paragraph above can be rendered more simply with p as unknown, instead of an unknown truth. For example:
"...as soon as we know that "p is unknown", then we know the sentence p (what p says), rendering p no longer unknown, so the statement "p is unknown" becomes a falsity."
There is no paradox. It is a truism that "the statement "p is an unknown (truth)" cannot be (both) known (and true at the same time)." Otherwise, it would not be unknown.
It is the sentence p which must be known, not its truth value.
Perhaps all truths (all true sentences) must be known. However, what remains unknown includes all unknown sentences, as well as the unknown truth values of known sentences.
Quoting Fitch's paradox of knowability
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.
I believe the issue lies, not in the truth of the sentence being unknown but, in the sentence itself being unknown. It is impossible to know an unknown sentence, or to know an unknown (anything). It's simply a contradiction in terms.
The Wiki paragraph above can be rendered more simply with p as unknown, instead of an unknown truth. For example:
"...as soon as we know that "p is unknown", then we know the sentence p (what p says), rendering p no longer unknown, so the statement "p is unknown" becomes a falsity."
There is no paradox. It is a truism that "the statement "p is an unknown (truth)" cannot be (both) known (and true at the same time)." Otherwise, it would not be unknown.
It is the sentence p which must be known, not its truth value.
Perhaps all truths (all true sentences) must be known. However, what remains unknown includes all unknown sentences, as well as the unknown truth values of known sentences.
Comments (505)
Either I know what I'm inquiring about or I don't know what I'm inquiring about.
If I know what I'm inquiring about then inquiry is unnecessary.
If I don't know what I'm inquiring about then inquiry is impossible.
Ergo,
Either inquiry is unnecessary Or inquiry is impossible.
Quoting Agent Smith
I'm not arguing along these lines, but I would be interested in an argument for it.
Quoting Agent Smith
I'm not sure I would agree. As the WIki article notes, p is a sentence or a proposition. Such sentences are typically truth apt. I don't consider a field of study, such as calculus, to fit the bill of a truth-apt proposition.
Assuming the law of non-contradiction and the law of excluded middle, either "the box is empty" is true or "the box is not empty" is true. According to the knowability principle, a statement is true if it can be known to be true, and so either we can know that "the box is empty" is true or we can know that "the box is not empty" is true. Now assume that we don't know which of the two is true. From this, either "the box is empty" is true and we don't know that it's true or "the box is not empty" is true and we don't know that it's true.
The problem is that according to the knowability principle, if "the box is empty" is true and we don't know that it's true then it's possible to know that "the box is empty" is true and that we don't know that it's true, which is a contradiction, and that if "the box is not empty" is true and we don't know that it's true then it's possible to know that the "the box is not empty" is true and that we don't know that it's true, which is a contradiction.
Given this contradiction we must either reject the knowability principle or accept that we know which of "the box is empty" and "the box is not empty" is true. And we must do this for every statement and its negation. Therefore if we insist on the knowability principle then we must accept that every true statement is known to be true.
How is this any different than the liar's sentence: "This sentence is false?" It's a grammatically correct sentence that no one would ever speak in real life. Or can you think of a reason for anyone but a philosopher, a13-year-old boy, or a 13-year-old philosopher to say or write it. We've discussed that many times here on the forum. My conclusion - self-referential "paradoxes" are just word games with no intellectual or philosophical significance.
I would say that we (now) know both of these statements, particularly since you have stated them. However, Fitch's argument speaks only of our knowledge - or lack thereof - of true statements.
The argument says that if it is possible to know a true p, then we must know that p is true. I argue that this is not due to our knowledge of p's truth, but due to our knowledge of p: If it is possible to know a true p, then we must know what p states. That is, my point is that the argument equivocates on knowledge of p (i.e. knowing what p states) and knowledge of p's truth value. Btw, I don't disagree with the argument's conclusion. I just don't see it as implying that we must know the truth of any proposition. The argument refers only to those propositions that are true in the first place.
Quoting Michael
I'm curious: do you reject the knowability principle or do you believe that the argument's conclusion endows us with knowing the truth value of every proposition? I don't reject the knowability principle. On what grounds would you?
But we don't know which of the statements is true, which means that we must reject the knowability principle.
Quoting Luke
The argument is that if it is possible to know that p is true then we must know that p is true.
On the grounds that we can't know both that p is true and that we don't know that p is true. That's a contradiction.
That's not Fitch's argument, which assumes the truth of p. It's not that we don't know which statement is true (and which is false); it's that we don't know the statement that is true. So, which of those statements (about the box) is true?
No it isn't. The non-omniscience premise of the argument is that there is some statement p that is not known to be true. We might very well know of the statement, and what it means, just not its truth value. "The box is empty" is one such example. I know of it, I know what it means, but I don't know if it's true. However, the knowability principle entails that if it is true then I know that it is true, which contradicts the fact that I don't know if it's true.
My view is that if it is a true statement, then it cannot be unknown that it is a true statement (see the Wiki quote again). And that's because in order for it to be possible to know that it is true, we must first know the statement and what it means. If it is possible to know that p is true, then we must know that p (is true), And the truth of the statement is presupposed.
Yes, and as the knowability principle is the principle that p is true if it is possible to know that p is true it then follows from what you say here that every true statement is known to be true. That's Fitch's paradox.
I think the argument implies that every known true statement is known to be true:
Quoting Fitch's paradox of knowability
As I said in the OP, this excludes all unknown statements and statements with unknown truth values.
Quoting Fitch's paradox of knowability
This is the heart of darkness - suppose we know something that we suppose we do not know. "the 79 squillionth decimal iteration of pi is a '2'." Well do we know or don't we? Make up your mind, Fitch. The digit is knowable, but 'that it it 2' is knowable only if it happens to be 2, which we don't know. p0, p1... p9 - one of them is an unknown truth, and the others are unknown falsehoods.
Suppose what you cannot even in principle know... arrive at a paradox... everyone gasps at your cleverness.
As one of the ol muppet dudes on the balcony says to the other.....BRILLIANT!!!
Or.....how to take the pristine condition of human reason, and turn it against itself.
I agree with you completely on this.
Quoting unenlightened
That we do and/or do not know something is not about the same sort of temporal possibility/knowability that you describe above. To "suppose we know something that we suppose we do not know" just seems like a pure contradiction.
No, it shows that every true statement is known to be true. I explained this here. I'll try to be even clearer now:
1. if p is true then it is possible to know that p is true
2. the truth value of p is unknown
3. if p is true and the truth value of p is unknown then it is possible to know that p is true and that the truth value of p is unknown (from 1)
3 is a contradiction. I can't know that p is true and know that the truth value of p is unknown. It must be one or the other. Therefore we must reject either 1 or 2.
It's not the truth value of p which is unknown, because we know that p is true. It is the true statement, p, which is unknown.
We don't know that p is true in this case.
That's what I'm disputing about the argument. This is the equivocation I'm talking about.
Then I will offer a specific example of p:
1. if the Riemann hypothesis is true then it is possible to know that the Riemann hypothesis is true
2. we don't know that the Riemann hypothesis is true
3. if the Riemann hypothesis is true and we don't know that the Riemann hypothesis is true then it is possible to know that the Riemann hypothesis is true and that we don't know that the Riemann hypothesis is true
The conclusion is a contradiction, and so we must reject either 1 or 2.
Proposition 1 only says that it is possible to know that the Riemann hypothesis is true. It doesn't state that it is always possible.
Therefore 2 could be one of the cases where it is not possible to know that the Riemann hypothesis is true despite it being true.
You'd need 1 to be "so long as the Riemann hypothesis is true then it is always possible to know that the Riemann hypothesis is true". Otherwise you're left without 3 necessarily following.
What are your views on K(P) [math]\to[/math] KP?
The knowability principle is the principle that a statement is true if and only if it is possible to know that the statement is true. If it is not possible to know that the Riemann hypothesis is true despite it being true then the knowability principle is refuted.
But I'm saying it is possible to know that the RH is true (just not at the same time as knowing that we don't know it's true). In other words, it is generally possible to know that the RH is true (your 1), but not in all circumstances (ie not whilst your 2 is the case). The fact that there exists a circumstance under which something is impossible, doesn't mean that that something is impossible in general.
I think calling something an "unknown truth" is a fallacy or just wrong, since truth is that which is in accordance with fact or reality. So, either we know that something is true or false or we cannot say anything about its truthness or falseness.
Then, "if all truths are knowable" is meaningless because truth is something known by definition!
Besides that, it is also an arbitrary assumption or hypothesis that looks like being used to serve supporting the above mentioned fallacy or wrong statement.
Therefore, I consider the whole construct as unfounded.
1. p??Kp
2. ¬Kp
3. p?¬Kp??K(p?¬Kp)
The logic is straightforward and results in a contradiction.
There might be a teapot in orbit around Jupiter.
You know the sentence "there might be a teapot in orbit around Jupiter"
You do not know if there is a teapot in orbit around Jupiter.
Hence you know an unknown sentence.
IF you don't like the teapot example, substitute any other unknown assertion.
There are paradoxes that are not self-referential.
Further, paradoxes show problems with the grammar of our expressions. If the grammar is inconsistent, we might be able to improve on it. In this case, logic has developed in multiple directions as a result of puzzling over the paradox - the SEP article lists them in some detail.
In 4.3 of the SEP article there is an a account of an attempt to take the timeliness of knowledge into account. The discussion is ongoing.
I don't see that we do. In the proof, p is only ever presented as part of a conditional.
So you are going with the rejection of classical logic - you are happy to introduce statements which are neither true nor false?
Are you accepting intuitionist logic or are you moving to paraconsistent logic?
What if the Riemann hypothesis is false? Then we do not reject 1. It is not enough that we don't know whether p is true; it must also be true. "p" means/entails "p is true". This is where the equivocation lies.
¬Kp could mean that we don't know the content/meaning of p and/or that we don't know the truth of p; that we don't know the Riemann hypothesis and/or that we don't know that it is true.
If I know the sentence, then how is it an unknown sentence?
Quoting Banno
No such example can be given. As the Wiki article tells us:
Quoting Fitch's paradox of knowability
I don't disagree with the conclusion of Fitch's argument, but I don't interpret it to mean that knowability implies the superhuman knowledge of all (known and unknown) true statements, either.
Right, we are supposing, stipulating that the sentence p is an unknown truth, not knowing it, obviously, so what's the problem, where's the paradox? If we come to know that the sentence p is true. then it would no longer be an unknown truth. We would then know that the sentence p was an unknown truth, but is no longer. It seems that changes through time have not been accounted for in this purported paradox of knowability.
The premiss is
?p (p ? Kp)
That's not "it's possible to know an unknown sentence".
Edit: Ah - I see; it's the Wiki rendering that sets the assumption out like that. Have a look at the version in SEP, which avoids this problem.
ditto,
Read the OP and see the Wikipedia proof given there (or see Janus' partial quote above). I am following its use of an unknown p.
So if the thread is about the argument in the Wiki article, it is not about Fitch's paradox.
SEP's proof is much clearer, and does not use the problematic assumption.
SO the proof works with ?p( p & ~Kp) in place of the problematic assumption.
Yes, I had misunderstood the way the sentence was being used, because I was looking at the SEP proof. I was mistakenly trying to make sense of it as a confusion of use and mention. My bad.
You said that it wasn't part of Fitch's paradox. Anyway, I agree that it is impossible to know an unknown sentence. You appeared to be arguing that it was possible only a few posts back. I'm not disputing the argument or its conclusion. I am only disputing the assumption regarding its conclusion: that knowability implies knowledge of all (known and unknown) true statements.
Is it the sentence or it's truth value that is unknown?
SO you accept the assumption ?p (p ? ?Kp) but not the conclusion ?p (p ? Kp)?
I think it would be best to stick to the SEP proof.
Wikipedia is written by people who typically do not fully understand what they're confidently pronouncing on.
Quoting Fitch's paradox of knowability
It is just asserted above that all truths are knowable.
Well, no they're not. Demonstrably. For instance, take the proposition 'X is the case and nobody believes X'. Well, that can be true. But it can't be known to be true.
Or take the view that there are no justifications. It's possibly true. But it could never be known to be, for to know something is to have a 'justified' true belief.
So it would appear demonstrable that not all truths are knowable.
Where's the problem? Is the idea that all truths are knowable supposed to be self-evident or something? It isn't.
Where does it use the wrong assumption?
Quoting Banno
I accept the conclusion, but there is an equivocation whether Kp means knowledge of the sentence or knowledge that the sentence is true.
It can be true, but is it true? The argument speaks only of possible knowledge (of true statements), not of possible truth.
...not in the SEP version...
it seems to me to use Kp as knowing p, not knowing of p...
That's the assumption that I'm challenging. Simply asserting that assumption is not an argument.
Doesn't "p" entail that p is true?
There's a number of blades of grass in the world. So make that X. Now I am fairly certain that nobody currently believes that there are that number of blades of grass in the world. So this proposition: "there are x number of blades of grass in the world and nobody believes it" is true. Yet it can't be known to be true.
And I gave you another example. It is entirely possible that no justifications exist. Well, if that's true - if, right now, there are no justifications for any beliefs at all - then it is true that there are no justifications, yet nobody can know it as knowledge requires justification.
So again, what's the problem?
Then how do you know that "there are x number of blades of grass in the world and nobody believes it" is true?
I don't. No one can. That's the point.
'True' does not mean 'known'.
So there's no problem with there being true propositions that are unknown. I mean, there is a number of blades of grass in the world. And no one knows it. So we already know that there are truths that no one knows.
And it would seem that there are some true propositions that, by their nature, cannot be known.
What's the problem?
Like I say, there's no contradiction involved for 'knowledge' involves truth, but truth does not involve knowledge.
So I don't see any puzzle.
Yep. The interpetation directly after K principle:
It's "know that P", not "know of p".
But you said that it was true? I'm asking how you know that in the first place before you tell me that it can't be known to be true.
I don't see that it matters.
Quoting Luke
Yes. So? That a proposition is true does not entail that it is known to be.
Look, this is very simple: this proposition "X is true and no one believes it" can be true. But when or if it is true, it could not be known. Why? Because to know a proposition is to believe it. And if you believe that in that proposition's truth, then it is false. So, when it is true, no one believes it. And thus it is never known to be true.
You're just confusing truth and knowledge, it seems to me. There's no puzzle here.
But you said that it was true. You both know and don't know that it's true?
You've lost me.
No. Here are tonight's lottery numbers: 1,2, 3,4,5,6.
Imagine they are. Do I know that those are tonight's lottery numbers? No, for my belief was wholly unjustifed.
So, there's a case of a true proposition that I believe to be true and that is not known.
Anyway, as I keep saying, there are countless examples of propositions that are true and not known, and propositions that do not seem capable of being known.
You've yet to explain to me what the problem is supposed to be. You just keep conflating truth and knowledge.
I thought you were making a point about (not) all truths being knowable?
But you cannot justify that they are true. Neither can you justify that "there are no justifications" is true.
Oh! I must be mistaken then. It's been a long time since I read about the paradox. This must be early-onset Alzheimer's. :fear:
There are no justifications.
There. That proposition might be true. Assume it is. Now, we don't know it to be true, do we? We can't. Because if it is true - and assume it is - then no belief is justified.
The knowability thesis is that all truths (i.e. all true statements) are, in principle, knowable.
In order to disprove this, you want me to assume something that might or might not be true? The knowability thesis is about true statements only. If you want to disprove it then use a true statement. You can't just assert that some true statements are unknowable.
That thesis is demonstrably false. I am demonstrating its falsity by providing you with examples of truths that, if true - and it's metaphysically possible that they are - could not be known.
Note, the existence of such truths is not controversial. They've got a name! They're known as 'blindspot propositions'.
Now, again, what is the problem you're trying to raise?
I don't think so. Logic deals in sentences, not meanings of sentences, whatever they are.
Think I'll leave you to it.
My point is that you don't know whether those statements are true or not; they are only possibly true statements. Therefore, they cannot be used to disprove the claim that all true statements are, in principle, knowable. The knowability thesis is not about possibly true statements. You are claiming that if those statements are true, then not all true statements are knowable. That's a big IF. Unless you can show that they are true, then you have not disproven the knowability thesis.
I know! That's the point!
The thesis that every truth is in principle knowable is not, note, the thesis that every truth is actually known. It is that every truth can - in principle - be known.
And it's demonstrably false. There are all manner of propositions that, if true, could not be known. I keep giving you examples. There are LOADS. "No one knows anything" for example.
Now don't reply 'how do you know it's true" - that's the point!! I don't and can't - no one can (save God, of course).
So what problem are you trying to raise? Do you think the knowability thesis has some prima facie plausiblity? It doesn't. It has nothing to be said for it. It's just a false thesis. It may not be obviously false, but it's false upon a bit of reflection.
So what's the problem? Why on earth would one ever think that all truths could be known? It's like thinking all flour is in cakes. No it isn't. There's flour in cakes. But there's no reason to think all flour is cake bound.
The meaning of a sentence is irrelevant to its truth value?
See the OP and the rest of the discussion.
Quoting Bartricks
I can see no reason why any true statement might be unknowable. Let's agree to disagree.
I have asked you umpteen times now to raise a problem. You haven't.
Here are some more problems for us to discuss: the cat/shape problem. My cat has a shape. But some shapes aren't cats. Puzzling.
The hair head problem. My head has hair. But there is some hair that is not on my head. Puzzling.
The language/speak problem. I speak a language. But no language speaks me. Puzzling.
The addition problem. Adding 2 to 2 makes 4. But adding 2 to 3 makes 5. Puzzling.
Cheers.
When can we say we don't know p, a truth? When we're not aware of the justification for p and/or we don't believe p (re JTB theory of knowledge).
p is an unknown truth = p & ~Kp = there are good reasons that p is true but either we're in the dark about those reasons and/or we don't believe p or both.
K(p & ~Kp): Since p is an unknown truth is itself a proposition and we know that, K(p & ~Kp).
The rule that's now applied in Fitch's argument is K(r & s) [math]\to[/math] Kr & Ks. That's to say K(p & ~Kp) [math]\to[/math] Kp & K~Kp.
Kp = We know p i.e. we believe p, p is justified, and p is true.
K~Kp = We know that we don't know p (Socrates).
K~Kp [math]\to[/math] ~Kp (the rule here is Kp [math]\to[/math] p)
~Kp means that we don't believe p and/or we're not aware of the justifications for p [s]and/or p is false[/s].
Kp & ~Kp isn't a contradiction, appearances can be deceptive (not all the conditions of Kp are negated by ~Kp. p is true in both. Coming to justifications it's not that p doesn't have good ones, we just don't know 'em; etc.)
The crux of my argument is that "Kp" conflates the knowledge that:
(a) p (where "p" represents a meaningful proposition); and
(b) p is true
These are both entailed by "Kp".
Note that this is the same distinction that you emphasised earlier between knowing a sentence (e.g. "There is a teapot in orbit around Jupiter") and knowing the truth of that sentence.
Hence, "¬Kp" could mean either that:
(a) p (the meaningful proposition) is unknown; or
(b) p is true is unknown.
Upon further reflection, and thanks in large part to the responses from @Michael, I believe that I am disputing the non-omniscience supposition of the argument:
Quoting SEP article on Fitch's paradox
However, I do not claim omniscience. Instead, I would argue that truth implies knowledge. This is the conclusion of the argument, after all: for all p, if p is true, then it is known that p is true. The reason that the (NonO) statement is false is because p is true implies p is known, so there cannot be any p for which p is true and p is unknown. The reason that p is true implies p is known is because p cannot be true without knowing the meaningful proposition represented by p. Again, this results from the equivocation over the meaning of p and the truth of p.
This is just criticism, @Banno. You only present characterizations (names and adjectives). No argumentation. If you want to disprove my statement-position, you must do it with plausible arguments and/or examples. Can you?
Here:
Quoting Alkis Piskas
You suggest three truth-values - "true", "false" or "cannot say". My bolding. All I was wondering is what variation you might choose. I'm aware of two choices. Intuitionist logic, such that statements are not true until proven, and paraconsistent logic, rejecting ex contradictione quodlibet.
It doesn't make any difference expressed in notation. 3 does not follow from 1 and 2.
1 says that p is possible to know (ie there exists a circumstance in which p is known)
2 says that it is the case that p is not known
3. then claims that it is possible to know p and not know p, but it doesn't follow since it could still be possible to know p, just not in the particular circumstance where one knows that one does not know p.
Saying that it is possible to know p doesn't rule out circumstances where it becomes impossible to know p. It says nothing of the contingency of knowing p, only that there exists a set of circumstances where it could be the case.
*Well, it follows if one holds to classic logic. As @Alkis Piskas pointed out, and as is explained in the SEP article, there are alternatives.
Thanks.
Here's where I'm having trouble (gone to the SEP)
Since p is a proposition (a factual claim about the way the world is) the problem seems trivially solved by saying that some proposition exists for which it is not possible to know the truth. Namely p?¬Kp.
Maybe if I make it clearer you can see:
1. p??Kp (knowability principle)
2. q ? p?¬Kp (define q as something that is true but not known to be true)
3. q??Kq (apply the knowability principle to q)
4. p?¬Kp??K(p?¬Kp) (substitute in the definition of q)
Then that's a denial of the knowability principle. The problem is that if you insist on the knowability principle then the only other way to avoid a contradiction is to deny the non-omniscience principle (i.e. to accept that every true proposition is known to be true).
I see, OK, but I'm not familiar with either intuitionist or paraconsistent logic. I never use and never need to use such terms. 1) They render a discussion to a literary one, 2) They require special knowledge from all the persons involved in the discussion, which might not be available, 3) They might be confusing and/or irrelevant to the subject that is discussed and, most importantly, 4) They do not really add anything that is of essence or importance.
A clear statement/argument talks and can stand by itself, however you call or categorize it.
I think you're the only one guilty of equivocation here. In the context of the argument, Kp means "it is known that the statement p is true". It does not mean "the statement p is known of" or "the meaning of statement p is known".
False, a given observational dataset is compatible with multiple hypotheses. There's no way of knowing which one is the true hypothesis even when one hasta be true (re the scientific method).
The argument doesn't asserting it, but uses it hypothetically to show that consequence,
Not much point in complaining about he use of specialised language in a thread on logic.
Anyway, have you further thoughts, given the consequence of your proposal? Are you happy to throw out classical logic? I think it a very promising line.
Two categories of propositions then:
1. p's that are provable, belief-apt, and true e.g. Biden is POTUS
2. p's that are unprovable though belief-apt and true e.g. scientific hypotheses.
Thanks. That is clearer.
Quoting Michael and
I think it's trivially true that the knowabilty principle cannot apply to propositions about our own knowledge where knowledge is treated as JTB, since the truth of the proposition is contained within the definition of knowing the proposition. To say that "I know I know" is to say (ignoring the justifications for now) "It is true that it is true" which is nonsense. We cannot know things about our knowledge under JTB (one of the reasons I don't like it) because it is senseless to make a truth claim about another truth claim. Whatever uncertainty we had about the first truth claim is automatically propagated to the second.
So, in the terms of the argument, to say that for all p it is possible to know p is to say that for all p it is possible to have p true and be justified believing p.
If q (our substitution) is "p is true" plus some proposition about my knowledge of p, then "p is true" is already a claim (contained within the knowledge claim). "p is true and I don't know p" can be rendered as "p is true and I lack justification", or "p is true and p is not true", or "p is true yet I do not believe p". Or some combination of the three. All of which are clearly contradictory.
Why? Some propositions are provable (e.g. Fermat's last theorem) and others not (e.g. the theory of relativity).
SO, to give an example of who the argument might address, suppose that someone argues that only things that have been proved true are true.
Then for them, if some statement is true, then it has been proven - that's their definition of "true". And if it is proven, it is justified. So we know it. Hence, if a statement is true in this system, it is known.
And it follows from the argument that everything that is true, is known.
So is our conclusion to be that those who thinks that only things that have been proved true are true is muddled, or that Fitch's paradox is faulty?
Well. In my opinion, the whole field of 'truth', and 'knowledge' is made into a quagmire by the use of a JTB definition of knowledge. To my mind, the 'truth' of a proposition is the extent to which it is actually the case, something we ourselves assess by testing the hypothesis that it is actually the case (note, I'm only saying that this is how we test it's truth, not what it's truth actually means). So our knowledge can only ever be a state of the results from those tests. The truth of "p" doesn't enter into it, the results from our latest tests of assuming p is all we ever have. I can't see a place for a mental state (knowledge) which relies on an external state (the truth of something) to be defined. The mental state (knowing that p) doesn't change dependant on p since p might be completely disconnected from our mental state (teapot orbiting Jupiter).
What is often arrived at by way of compromise is a sense that a claim "I know p" and a claim "John knows p " are two different types of claim, with only the latter assessable by JTB. I don't like that solution (though I grant it's coherent), but then we cannot make the claim made in the knowability proposition that 'We' know anything (by JTB) since 'We' necessarily includes 'I'.
Yes, I was considering the same sort of thing. I think this kind of self-referential knowledge is victim to the same problems as other self-referential knowledge/truth claims like the Liar Paradox. Technically speaking, if all meaningful propositions have a truth value and if "this statement is false" is a meaningful proposition then we have a contradiction. But is it really a problem to say that all meaningful propositions except propositions like "this statement is false" have a truth value? Or is that special pleading?
Perhaps we can say (as me and @Banno discussed in the other thread) that empirical truths are subject to the knowability principle, but that the truth of self-referential knowledge claims, counterfactuals, predictions, mathematics, etc. work differently?
Personally, I think statements like this are fine, and I think so on the following ground...
We can only make two kinds of propositions - those about the way the world is, and those about the way the world ought to be. Ignoring the normative for now, and assuming realism, then the world is some way and we determine it to be so by testing the assumption that the world is that way and assessing the result. As such, any idea that 'special pleading' is a fallacy has in it the assumption that the way the world is is simple and contains no special cases. I can see a reason for testing that assumption first, but I can't see a reason why if, on testing that assumption, we find it inadequate, that we shouldn't assume, as our next best assumption, that this is some 'special case'. After all, we've no fundamental reason to assume the world is simple and contains no special cases of otherwise general rules.
I think the same assumption holds even for an idealist. There's no default reason to assume our notions of how we're going to see the world ought contain no special cases of otherwise general rules.
It's not as if the issue hasn't been pretty exhaustively examined. If a special case seems a good solution then, at this late stage, it seems more than a little self-defeatingly stubborn to refuse one.
Quoting Michael
Yes, I think one could almost say that's definitionally true since the dividing out of empirical claims is by finding those to which sense data might apply and to make that delineation one needs to imagine, at least, a way in which one might obtain that sense data (and so 'know' the the proposition).
Claims of the second sort seem to rely more on rule-following and as such encounter the problems Wittgenstein shows about assessing whether a rule is followed, private rules, etc.
I wasn't complaining. I just gave you FOUR reasons why I, personally don't use a specialized language. And also because you asked me what kind of logic I'm using, most probably assuming that I would or should know ...
Re "have you further thoughts, given the consequence of your proposal?": What consequence?
Re "Are you happy to throw out classical logic?": I don't know if I have thrown out any kind of logic, classical or other. See, you are still bound to philosophical "lliterature" and generalities.
I asked you to just disprove my statement-position using plausible arguments and/or examples. You still haven't. So I have to assume that you cannot. I'm not surprised ...
I agree, on Mondays, Wednesdays and Fridays.
While it is interesting to consider the argument in the light of JTB, I don't think that the argument depends on JTB. It has a more general applicability.
Ha! I feel that way about many theories.
Quoting Banno
Your use against idealism seemed apt, certainly. Idealism having it's very own peculiar relationship with the verb 'to know'.
I had thought from his post that was thinking along those lines.
So it would be something like saying that we do know every true proposition, but more propositions keep becoming true as the domain of true propositions expands...
Knowability: p [math]\to[/math] Kp
Non-O: [math]\exists[/math]p(p [math]\land[/math] ~Kp)
Rules:
A. Kp [math]\to[/math] p
B. ?~p [math]\to[/math] ~?p
C. [math]\vdash[/math]p [math]\to \vdash[/math] ?p
D. K(p & q) [math]\to[/math] Kp & Kq
p & ~Kp (instantiation of Non-O)
(p & ~Kp) [math]\to[/math] K(p & ~Kp) (substitute p & ~Kp in Knowability]
1. K(p & ~Kp) (assume for reductio)
2. Kp [math]\land[/math] K~Kp (1, rule D)
3. Kp (Simp 2)
4. K~Kp (simp 2)
5. ~Kp (4, rule A)
6. Kp [math]\land[/math] ~Kp (3, 5 Conj)
7. ~K(p [math]\land[/math] ~Kp) (1 - 6 reductio)
8. ?~K(p [math]\land[/math] ~Kp) (7, rule C)
9. ~?K(p [math]\land[/math] ~Kp) (8, rule B)
10. ~[math]\exists[/math]p(p [math]\land[/math] ~Kp) (from 9)
11. [math]\forall[/math]p(p [math]\to[/math] Kp) (from 12)
:chin:
No problem, @Banno. I know you mean well.
I have never studied Philosopy. I only had a course in College, based on the philosophy of Epictetus. On the other hand I have read really a lot of philosophical books (i.e. books with a philosophical content), but not on Philosophy itself, as a discipline or field of knowledge. Yet, I know about a few common terms, but I use them scarcely, only as a "garnish" or a common reference (e.g. materialism, dualism. etc.). But I can do very well without them! :smile:
I stick to simple logic/critical thinking/reasoning. Sometimes I use the (fuzzy) term "common logic", which is not a logic "common" to all, but the priviledge of only a few! By "common", I mean "simple". My mottos: Simple is beautiful. Simple is efficient.
The fuzzy term "common logic" but not the term "fuzzy logic"?
Sorry, couldn't help myself. :wink:
Within the boundaries of human experience, a proposition is some statement that someone proposes, at some point in time. A proposition is a proposal made by a proposer (?). Before it was proposed, the proposition simply did not exist.
Or if you prefer, it could only exist in the mind of God. Or maybe some superpowerful alien... Not in a human mind.
Likewise, a statement does not exist before it is stated by some author or another. A phrase does not exist before being phrased.
So, within human experience, it makes no sense to say that a proposition no one knows about is true. The proposition needs to exist first. Once it is proposed, then and only then can the question of its truth be asked, and thus be put into existence, and only then, can the question be answered (or not).
Now, in some sense "truth is out there", the world is what it is and not otherwise. But this "truth out there" is not yet phrased in the form of propositions.
No, you did well. You proved my point that one of the bad things about naming or categorizing "logic" is that it may lead to confusion! :grin:
As I said to Luke, this isn't what Fitch's paradox is (necessarily) saying. It's saying that there is some proposition that is not known to be true. That's not the same thing. For example, the Riemann hypothesis is not known to be true. The paradox can be applied to this single proposition (see my next comment).
p is "the Riemann hypothesis is true". q is "the Riemann hypothesis is false". Either p or q is true and neither p nor q is known to be true. Therefore, either p?¬Kp or q?¬Kq. Then applying the knowability principle, either ?K(p?¬Kp) or ?K(q?¬Kq). Both are contradictions.
So either every true proposition is known to be true (abandon non-omniscience) or for some true propositions it is not possible to know that they are true (abandon knowability principle).
Quoting Luke
In the context of Fitch's paradox it means that we don't know the truth of p.
Known by whom, and when? To know is an action done by people, and not a passive state of affairs. Some people know, some people don't. To be known is NOT a quality intrinsic to things.
If I state: "Back in antiquity, people didn't know that the earth orbited around the sun" it means something like: "it was true back then that the earth orbited the sun, and folks weren't aware of it at the time, but now we modern folks are aware of it." So it would be like a truth unknown to antiquity folks, but known to us modern folks.
And indeed it is perfectly possible to know that "In antiquity, people didn't know that the earth orbited around the sun". But of course, the folks back in antiquity didn't know that they didn't know that.
Us, now.
And someone else at another time would have a different knowledge. So there's no such thing as 'a truth known', or 'a truth unknown', in the absolute. It all depends on who does the knowing and when.
This has no bearing on Fitch's paradox.
What I'm trying to say is that we can abandon the principle of non-omniscience (as given) without implying that all (known and unknown) truths must be known. I believe that all the argument implies is that only known truths must be known; or, more to the point, that no unknown truths can be known.
The principle of non-omniscience implies that there are unknown truths which are or can be known. This is simply a contradiction in terms. If a truth is known then it cannot be unknown, and if a truth is unknown then it cannot be known (per modal principle D in the SEP article). If a truth becomes known then it is no longer unknown. To repeat: no unknown truths can be known and only known truths must be known.
According to logic, known truths and unknown truths forever stay that way. Otherwise, we could allow for an unknown truth to become known, but then it would no longer be an unknown truth.
In other words, @unenlightened was right.
The implication for the argument remains what I said earlier: no unknown truths can be known and only known truths must be known. That still doesn't seem very omniscient to me, given what we know.
And yet we don't know which of "the Riemann hypothesis is correct" and "the Riemann hypothesis is not correct" is true, but one of them must be. Therefore not all truths are known.
Well, I'm saying that the argument implies only that known truths are known, which excludes knowing unknown truths. The independent argument given in the SEP article shows that it is impossible to know unknown truths.
Which is a false interpretation. I've explained the logic several times.
I don't believe that you have.
Is "either the Riemann hypothesis is correct or the Riemann hypothesis is not correct" a known truth or an unknown truth? You've said that that's a known truth, but you've also used this to argue that not all truths are known.
On the other hand, it is unknown which one is true, so you cannot claim that one of them is a known truth.
A known truth.
Quoting Luke
Yes, either "the Riemann hypothesis is correct" is an unknown truth or "the Riemann hypothesis is not correct" is an unknown truth.
Quoting Luke
Which is precisely the point. Fitch's paradox entails that we do know which one is true. Given that we don't know which one is true me must reject the knowability principle.
I think it does. To be known is NOT a quality intrinsic to things, therefore 'an unknown truth' or a 'known truth' have no clear meaning. They are not concepts, just noises made with mouths. One would need to state precisely to whom the truth is known or unknown for these phrases to have a meaning.
You disagreed with my claim that the argument implies only that known truths are known. However, in order to show otherwise, you would need to demonstrate that some unknown truth can be known. Since you do not know which one of the above statements is true, then you have not demonstrated knowledge of an unknown truth.
By "known truth" I mean "a proposition that someone knows to be true" and by "unknown truth" I mean "a proposition that no-one knows to be true."
The argument shows that if we assume p ? ?Kp then p ? Kp follows.
Kp ? Kp is a truism that doesn't need Fitch's paradox to prove.
Quoting Luke
No, I need to show that there are no unknown truths, which is what Fitch's paradox does; see above.
If one treats knowledge as a tick-the-box thing, as a feature or commodity, as a mathematical variable that is either present or absent or equal to 12, and existing independently from any particular human knower, then one may indeed end up in a mental glitch.
Just because one can write down K(p) doesn't imply that this scribbling means anything precise.
In order to disprove my claim, which is that the argument demonstrates that only known truths are known, then you would need to show that there are no unknown truths? Doesn't that just support my claim? If there are no unknown truths then only known truths are known.
You seem to want to draw from Fitch's conclusion that not only are known truths known, but also that unknown truths are known, such as that (e.g.) "the Riemann hypothesis is correct". I don't draw this absurd conclusion from the argument.
OK. This has nothing to do with Fitch's paradox.
If there are no unknown truths then all truths are known.
Also, if there are no unknown truths, then only known truths are known.
OK. But it's still the case that the argument shows that, given the knowability principle, all truths are known.
However, it's a fact that some truths aren't known. Either "the Riemann hypothesis is correct" or "the Riemann hypothesis is not correct" is one such truth that isn't known.
Therefore, the knowability principle fails.
So there are unknown truths? Are they knowable?
This is what I am denying, since if an unknown truth becomes known, then it is not an unknown truth.
Quoting Michael
Not according to Fitch's argument.
In reality, yes. However, Fitch's paradox shows that the knowability principle entails that there are no unknown truths. That's why Fitch's paradox shows that the knowability principle is false.
Quoting Luke
Technically speaking Fitch's argument shows that the knowability principle entails that all truths are known. This conclusion is then a reductio ad absurdum to disprove the knowability principle, given that there are unknown truths.
Known by everyone always, or known only by someone at some time? I take it all truths are known implies that no truths are knowable (because they are known)? But if they are known only by someone at some time, would that imply they can be knowable by others, in order to save KP?
BTW, I just realized that my above statement was wrong. And you had the opportunity to easily refute it, if you had paid attention to a detail instead of wondering about what is the type of logic that this statement belongs to. The detail is the word "something". Because one might simply ask: "An apple is 'something'. Can we say that an apple is true or false?" Of course not. It makes no sense. Only a statement or an assertion or a report and that sort of things can be true or false. So my statement was clearly wrong.
Well, another mistake I did was stating that "I stick to simple logic". One might well ask "What is simple logic?", "Simple in comparison with/to what?"[/i], "Simple in what way?", "Why, is there a complicated logic?" and so on. You shouldn't miss that either. I like to have strong "opponents"! :smile:
In philosophical discussions we must pay attention to these things. I'm careless sometimes, too.
The truth is in the detail! :smile:
It does have a bearing, but you are not interested, which is fine.
By someone at some time.
Quoting Luke
In fact the opposite: Kp ? ?Kp.
Quoting Luke
No, because if you address the formal logic of the argument you will see that it entails a contradiction:
a. p ? ?Kp (knowability principle)
b. p ? ¬Kp (some proposition that is true but not known to be true)
c. b ? ?Kb (apply the knowability principle to b)
d. p ? ¬Kp ? ?K(p ? ¬Kp) (substitute in the terms of b)
However, K(p ? ¬Kp) is a contradiction, and so isn't possible, as shown below:
e. K(p ? ¬Kp) (assumption)
f. K(p ? q) ? Kp ? Kq (knowing a conjunction entails knowing each of the conjuncts)
g. Kp ? K¬Kp (from e and f)
h. Kp ? p (knowledge entails truth)
i. Kp ? ¬Kp (from g and i)
i is a contradiction. We cannot know that p is true and not know that p is true. Therefore d is false. Therefore either a (the knowability principle) or b (there is some unknown truth) is false.
This is true, but Fitch's paradox is self-referential. Actually, after looking at it more, including SEP, I'm not sure it is. It seems more like a tautology, or at least a trivial statement, a language game. Calling a particular statement a truth means the same thing as saying it is true. If I know something is true, it isn't unknown.
Where's the language game here?
1. p ? ?Kp (knowability principle)
2. q ? the Riemann hypothesis is correct
3. r ? the Riemann hypothesis is not correct
4. q ? r (law of excluded middle)
5. ¬Kq ? ¬Kr (whether or not the Riemann hypothesis is correct is not known)
6. (q ? ¬Kq) ? (r ? ¬Kr) (from 4 and 5)
7. q ? ¬Kq ? ?K(q ? ¬Kq) (from 1)
8. r ? ¬Kr ? ?K(r ? ¬Kr) (from 1)
9. ?K(q ? ¬Kq) ? ?K(r ? ¬Kr) (from 6, 7, and 8)
10. K(s? ¬Ks) (assumption)
11. K(s ? t) ? Ks ? Kt (knowing a conjunction entails knowing each of the conjuncts)
12. Ks ? K¬Ks (from 10 and 11)
13. Kt ? t (knowledge entails truth)
14. Ks ? ¬Ks (from 12 and 13)
14 is a contradiction, therefore 10 isn't possible, therefore 9 is false, therefore either 1 or 5 is false.
Sorry. Not good with logical symbology.
See here for an explanation in ordinary language.
This is where I think the mathematical formalism is missing something important: the time variable. Knowledge is not static. You are talking of a process of discovery, of the possibility of solving the Riemann conjecture one day. Note that if and when this happens, our knowledge about it will evolve. What we knew not at time t1 will become known at time t2. Which you could write: Kt1(r)<>Kt2(r)
So by adding the time variable, there is no reflexivity anymore. You don't end up knowing what you know not, but knowing now what you knew not back then.
It's called learning.
There exists a being x and a time t such that x knows at t that proposition p is true: ?x?t(Kxtp)
1. p ? ??x?t(Kxtp)
2. p ? ¬?x?t(Kxtp)
3. ??x?t(Kxt(p ? ¬?x?t(Kxtp)))
4. ?x?t(Kxtp ? Kxt(¬?x?t(Kxtp)))
There exists a being x and a time t such that x knows at t that proposition p is true and knows at t that there doesn't exist a being x and a time t such that x knows at t that proposition p is true. This is a contradiction. Therefore 3 is false. Therefore either 1 or 2 is false.
Admittedly this doesn't entail that every true statement is currently known to be true, only that every true statement is known to be true at some point, but that might also be an undesirable conclusion. It's possible that the Riemann hypothesis is never proved nor disproved.
Thanks.
3. ??x?t2(Kxt2(p ? ¬?x?t1(Kxt1p)))
The criterion is 'knowable' not 'known'.
Fitch's paradox shows that if all truths are knowable then all truths are known. Some truths aren't known, therefore some truths aren't knowable.
That's right.
What's difficult to see is if @Olivier5 has a point or has just not understood the logic of the argument. If he has a point, it remains obscure.
The move to "who is it that dos the knowing" is pretty common in phenomenological discourse, but without setting out explicitly how it is relevant to the argument. Notice that the Kvanvig rendering of the argument does take the knower and the time into account. The argument would then proceed into considering the rigidity of the designation, and all that would involve. If @Oliver5 were to proceed in that direction the conversation might become interesting.
SO, Oliver5, are you proposing that the argument suffers a modal fallacy? Can you set it out explicitly?
Self reference in itself is not a problem. This sentence is six word long. This sentence contains thirteen words. No worries. SO saying the paradox involves self reference is neither here nor there.
Otherwise you seem to be making the sam ubiquitous error as others hereabouts.
Can you lay out the argument clearly in plain English?
I just assumed your were adopting the convention of restricting that "something" to propositions. And I understood your "simple logic" to be classical logic.
The principle of charity at work.
@Michael, Don't - it's a trap!
:wink:
If formal logic is merely a self-enclosed game with its own rules and practices differing from the rules and practices of plain language, that's fine, but then it cannot be plausibly claimed that it has any entailments outside of its own boundaries.
You can't have it both ways.
But the conclusion of Fitch's argument can be "translated back" into plain english - and has been, multiple times, in both articles and in this thread. :roll:
Formal logic sets out the grammatical structure of the argument clearly. It is clearer and easier to follow the detail than in an informal argument; that's why we use it.
:roll: So what if the conclusion, but apparently not the argument itself can be translated back into plain English? So what if it is "clearer and easier to follow the detail" in the formal language; the detail should nonetheless be able to be translated into plain language and seen to be valid. If it can't be then it's useless.
This is one.
But far fewer when the formalism is modal.
Bully for you. So rather than set the task for poor @Michael, have a go at it yourself. You have the background, and doing so will give you a much better understanding of how it works.
They were conceived in sin.
All unknown true statements are knowable. (2, from 1)
.......................................................................... (3)
All unknown true statements are known. (4)
This is a good example of how informality introduces problems. I don't see the "knowable/known" distinction in the formal version. There's just "K" and ?K, which makes it clear that the move from "knowable" to "Known" is modal.
IS this better...
(1) All true statements might be known
(2) All unknown true statements might be known (1, sub)
.........................(3)
All unknown true statements are known (4)
??
(KP) All true statements might be known
(NonO) There are unknown truths
(1) There is a truth that is not known (instantiation from NonO)
(2) If there is a truth that is not known, then it might be known that there is a truth that is not known
....(sub (1) into KP)
(3) It might be known that there is a truth that is not known
I'll stop there. that should be enough.
Now, is this a reasonable English representation... and if not, can you do better?
And, for @Janus, do you really think this scholasticism clearer than the more formal presentation?
(edit: fixed 3. )
Quoting Banno
Yes but modality is obscure. Give us the Venn diagram.
Quoting Banno
Or maybe
(1) Some judgements of true statements are knowledge.
??
Hocus pocus.
But, according to the independent argument, starting with the assumption K(p ? ¬Kp) leads to the conclusion ¬?K(p ? ¬Kp). That is, if the conjunction is known, then the conjunction is not knowable.
Just a thought.
At the beginning of the discussion @Agent Smith made reference to Meno's paradox, and I think there could be an interesting parallel to Fitch's. An ambiguity is noted wrt Meno's paradox:
Quoting Meno's paradox ambiguity
I wonder whether the same/similar type of ambiguity applies to your Riemann hypothesis examples. You can say (about unknown truths) which set of statements are truth apt, but not which statements are true. In other words, you can know of unknown truths. but you cannot know them (or which of them) to be true.
(2) Some future affirmations of any true statement not previously affirmed justifiably are justified.
(3) ...................................................
(4) Some previous affirmations of any true statement not previously affirmed justifiably are justified.
Modalities excised (or easily so). Missing line exposed.
What about
(1) There are truths that are not known (instantiation from NonO)
(2) If there are truths that are not known, then it might be known that there
are truths that are not known
....(sub (2) into KP)
(3) It might be known that there are truths that are not known
It seems obvious that there are truths that are not known; for example someone cited the example that the Earth is (roughly) spherical, which at one time was not known. There must be many truths about other planets or yet to be discovered flora and fauna which are not known.
I'm not seeing how the (apparent) fact that there are unknown truths proves either that there are or are not unknowable truths. And I'm also not seeing how there being knowable (in the sense of becoming, obviously not presently, known) unknown truths proves that all truths are known. There must be some (formal) sleight of hand going on, it seems to me.
1. p & ~Kp (assume for reductio)
2. p [math]\to[/math] Kp (premise)
3. p (1 Simp)
4. Kp (2, 3 MP)
5. ~Kp (1 Simp)
6. Kp & ~Kp (4, 5 Conj)
7. ~(p & ~Kp) (1 - 6 reductio)
8. ~p v ~~Kp (7 DeM)
9. ~p v Kp (8 DN)
Either p is false Or we know p (is true).
I'll have a go. It might not be correct (or helpful) but maybe others can chime in to correct and clarify.
Suppose both of these principles:
All truths are knowable (the knowability principle)
We are non-omniscient; there is an unknown truth (the non-omniscience principle)
Combine these principles:
If one of all of the knowable truths (KP) is that we are non-omniscient or that there is an unknown truth (NonO) - in other words, if it is possible to know that there is an unknown truth - then it follows that an unknown truth is knowable.
However, it can be independently shown that an unknown truth is unknowable.
Given the contradiction that an unknown truth is both knowable and unknowable, one of the starting principles (KP or NonO) must be rejected.
However, if we reject the non-omniscience principle (which says that there is an unknown truth) such that there are no unknown truths, then it follows that not only are all truths knowable, but all truths are in fact known.
On the other hand, if we reject the knowability principle (which says that all truths are knowable) such that not all truths are knowable, then it follows that not only is there an unknown truth, but there is an unknowable truth.
Or, as the archived SEP article puts it:
Quoting Archived SEP article
An unknown truth: p is true but we don't know p is true = p & ~K(p v ~p).
We know that p is an unknown truth = K(p & ~K(p v ~p)).
No K(p v ~Kp), no paradox.
Quoting Luke
Good call!
I don't see how it follows from the fact that we know (if we do know) there are unknown truths that an unknown truth is knowable; the fact that there are unknown truths (if there are) is not itself an unknown truth (if it is known).
It isn't that we do know there are unknown truths, it is that it is possible to know there is an unknown truth. If it is possible to know, then it is knowable. These terms are simply synonymous.
A reminder here that this comes from combining the two starting principles, KP and NonO.
Quoting Janus
No, but why do you think it should be?
That's not an instantiation.
https://en.wikipedia.org/wiki/Existential_instantiation
is the rule being used.
But also
https://en.wikipedia.org/wiki/Universal_instantiation
Yes, something like that. Poor logical formalism. The paradox is about the capacity to learn, to know not some truth at time t and then to know it at time t'. This implies that 'the knowledge of x' changes, that it depends on the knower and the time of the knowing, especially if we are talking of a learning process, as Fitch objectively is.
The logical contradiction stems therefore from postulating a change in knowledge in the problem statement, but then ignoring such change in the formalism.
If in the formalism of Fitch you introduce the idea that knowledge changes over time, you may arrive at something that in English means: he now knows what he knew not before. That is an unproblematic statement about learning something new. But erase time from Fitch (or from that bold sentence), and you get: he knows what he knows not, ie a contradiction.
Kp
1. p is true
AND
2. Someone believes p
AND
3. p is justified
~Kp
1. p is false
AND/OR
2. No one believes p
AND/OR
3. p isn't justified
Is Kp & ~Kp a contradiction? No! :snicker:
OK, that seems fine: so it is possible to know there is an unknown truth; that does not mean it is possible to know an unknown truth (which would be a contradiction) but that it is possible to know that there is an unknown truth (which is not a contradiction).
Quoting Luke
I don't think it should be.
Quoting Banno
Right, not an instantiation, but many instantiations? Why should non-omniscience not entail that there be more than one unknown truth?
I'm sure you did. And I'm sure you also aware that assumptions can be big traps. :smile:
Quoting Banno
Thank you for your kindness, Banno. :smile:
That there's an unknown truth can be known = K(p & ~Kp)
Because K(r & s) [math]\to[/math] Kr & Ks, we can say that Kp & K~Kp
Aside: K~Kp = I know that I don't know p (is true). Socratic.
Kp [math]\to[/math] p. Ergo, K~Kp [math]\to[/math] ~Kp.
Kp & ~Kp (contradiction).
Hence, ~K(p & ~Kp). Meno! It isn't possible to know that there's an unknown truth. Inquiry is ~?.
It needs to be singular to substitute in to (2), so as to get (3) right.
If the singular substitutes into (2) as you laid it out, why doesn't the plural substitute into (2) as I laid it out?
I've done so a couple of times: here and here.
I'm not sure if I'm following you, but I'm seeing a problem with this:
Quoting Banno
It seems to me that 'an unknown truth' cannot legitimately be formalised as p but only as (p or ~p) Is that right? does it make sense?
That is to say that I know that there is an umpteenth digit of pi, and can say so, but I cannot say that any particular one of the statements p0 -p9 is true, but only one of all of them. that is what it means for the truth to be unknown.
The formal definition is ?p(p ? ¬Kp): there exists some proposition p that is true and not known to be true.
For example, either "the Riemann hypothesis is correct" is true and not known to be true or "the Riemann hypothesis is not correct" is true and not known to be true, and so either "the Riemann hypothesis is correct" is an unknown truth or "the Riemann hypothesis is not correct" is an unknown truth.
Sounds interesting. Does the proof do this?
Seems to me it uses (~Kp) for "p is not known".
(p v ~p) is p is true or false, not known or unknown.
Fixed.
Yes. I am questioning the legitimacy of that. It seems to be stating a contradiction by asserting p and claiming it to be unknown. If I substitute (p0 or p1 ... or p9) then it is not unknown, but on the contrary that is what is known.
But that's the non-omniscience principle? Without it we must accept that every true proposition is known to be true which is what Fitch's paradox shows follows from the knowability principle.
It's not a contradiction to say "there is intelligent alien life but I don't know that there is." Such a statement is possibly true.
I don't think so. I think the principle needs to be formalised differently, as I indicated.
Quoting Michael
I think it is a contradiction, because it asserts something and denies that it is known. "Either there is intelligent alien life or there isn't, but I don't know which." -- that makes sense.
That doesn't work. p ? ¬p just means "p is true or p is false" and says nothing about what we know.
For example: either my name is Michael or my name is not Michael. This statement is true, but doesn't say that I don't know my name.
Quoting unenlightened
We are able to assert things we don't know. We can make arguments with premises we either don't know or believe to be false, e.g.:
1. My name is Andrew
2. If my name is Andrew then my name is not Michael
3. My name is not Michael
The argument is valid.
1. There is intelligent alien life
2. If there is intelligent alien life then humans are not the only intelligent life
3. Humans are not the only intelligent life
The argument is valid.
So my suggestion is that the non-omnicience principle should go something like:
(p or ~p) and ~Kp and ~K~p.
Can you work with that a little and see how it goes? (My formal logic is fifty years faded)
1. (p ? ¬p) ? ¬Kp ? ¬K¬p
2. (p ? ¬Kp ? ¬K¬p) ? (¬p ? ¬Kp ? ¬K¬p)
3. (p ? ¬Kp) ? (¬p ? ¬K¬p)
4. q ? ?Kq (knowability principle)
5. p ? ¬Kp ? ?K(p ? ¬Kp) (contradiction)
6. ¬p ? ¬K¬p ? ?K(¬p ? ¬K¬p) (contradicton)
In fact from this I'm pretty sure it follows that ?q(q ? ¬Kq), so we're back to the initial formalism.
Well if I am forced to say that because we are not omniscient, there are things we cannot know, I might be able to live with that, at a pinch.
Wait, it doesn't say that, though, it says there is something we don't know, Sorry, brain overheating and I am confused between unknown and unknowable. Need to lie down in a darkened room.
The basic idea is that if there's a truth that isn't known then that implies a related truth that isn't knowable.
Suppose there is some statement t that is true AND no-one knows that t is true (say, Goldbach's conjecture or its negation). That conjunctive statement is itself true but unknowable. Why? Let's assume that someone comes to know that the conjunctive statement is true. That implies that they know that t is true. But that then renders the second conjunct false. The conjunctive statement is therefore false and so not known to be true, which contradicts our initial assumption. So it's not possible to know that the conjunctive statement is true. It's an unknowable truth.
The only way to avoid such unknowable truths is for there to be no unknown truths (i.e., for all truths to be known). That is, for all for truths to be knowable implies that all truths be known.
Why not just accept unknowable truths?
Well, one might prefer to think they are omniscient. ;-)
More seriously, presumably people who have considered Fitch's paradox do accept that. But from Wikipedia:
Quoting Fitch's paradox of knowability - Wikipedia
I see. If I'm a verificationist, then I can be accused of saying that the human race knows all (not that any individual does.) But since I haven't ruled out the expansion of human knowledge, I should show up as reasonable.
I think Wikipedia is talking about truth anti-realism in the second case (not idealism). Deflationary accounts of truth are apt to be anti-realist, redundancy and so forth.
Not according to many here.
A typical deflationist will say that truth only serves a social function. Is someone disagreeing with that?
This same sort of issue shows up in Aristotle with talk about the true/false values of past truths that no longer maintain. For example, "the Colossus of Rhodes is standing," is a proposition that had a truthmaker at one point, but that truthmaker disappeared when the statue fell over. Likewise, the proposition that "p is an unknown truth," is a negative claim about knowledge, and so it has a corresponding falsemaker that ceases to maintain when the truth of p is discovered. By allowing truth values to change over time you solve the problem.
People were unhappy with this storm of changing truth values, but it has, perhaps, been rectified with the idea of possible worlds. We start off with a set of all possible worlds, all those that aren't logically contradicted. As time progresses, the number of worlds consistent with actual events is winnowed down, and so changing truth values is really just the winnowing of possible worlds. Although, I'm not sure you even need possible worlds, you could also just have a set of all truths that has a time stamp on when a proposition was uttered.
It seems to me that the problem only holds up under a narrow set of assumptions. Let's look at how it fares under a few possible viewpoints:
1. Presentism holds that the past and future lack existence. In this case, the unknown truth could be in the set of all truths but it would cease to be as soon as someone knows about p. But really though, the past doesn't actually exist, so the set of all truths never has the contradictory overlap, fixing our problem.
2. This is no issue for eternalism as the future is as existent as the past or present (e.g., block time universes), so if p is ever known, it is not an unknown truth, since the future already exists.
3. For many forms of actualism (i.e. actual occurrences exist, modal truths do not) it seems like this is just the regular occurrence of actual events narrowing the horizon of all possible worlds consistent with actual true propositions. So the "p being an unknown truth" worlds just get shifted from the possible to impossible side of our possible worlds ledger.
There is also the information theoretic approach in which the primary ontological entity is information, that is, propositions. But many of these propositions are "derived" propositions. The only fundemental propositions are about fundemental particles/field excitations as related to each other in space and time. In this view, seeming contradictions are just the result of error and data compression. Broad, high level, derived propositions are multiply realizable because they are compressing information and dropping a lot of it. But in reality, this isn't causing contradictions, the problem is simply that multiple informational microstates are consistent with the truth value of a single macro-proposition.
Thankfully, information has this protean character where it can take multiple forms, and reencoding of information (with relative amounts of compression and error) in forms of self-similarity at different scales (fractal recurrence) allows us to make these derived propositions with some degree of accuracy, but we shouldn't take derived propositions as having ultimate truth values in terms of contradiction as they are multiply realizable.
But in these systems, you're also still talking about truth values given a certain slice of time (generally, most I've seen tend to be actualist or eternalism).
I would think that truth has an important psychological dimension; we (minds) cannot exist without a concept of truth. Even folks who think they have 'deflated truth' do in actual fact believe that it is true that they have deflated truth. They don't usually believe that they have come to some social agreement to pretend that truth was deflated.
Right. They just don't believe that "is true" adds anything except emphasis.
Of course. I think you're misunderstanding. A deflationist does not have a problem with using the word "true" in the normal way. She just resists piling unwarranted projects on top of that normal usage.
She would say we shouldn't be bewitched by
language.
A most bewitching Wittgensteinian proposition... both recognising the power that a language holds upon individuals speaking it, warning them against it in fact, but then offering no practical method to deliver them from the spell of their spell.
I will agree that one needs to use words with care, that they are 'treacherous' in some sense. But there are solutions, such as the pragmatic, instrumentalist approach: any given problem will require a certain degree of precision in language for its resolution, much beyond which it is useless to go. That's similar to the standard practice in math and physical sciences.
Its downside is of course that its very practicality misses on the creative, poetic and polysemic virtues of language. The bewitching can be a feature, in an explorative way. But then, even Witty's defiance towards language is perhaps missing on that, on the accidental creativity of the bewitching.
Anyway, this is an aside.
Quoting Tate
Would you have examples of these unwarranted projects?
If truth is a property of statements, talk of "unknown truths" might give us unstated statements. Not good.
Excellent. I made a similar remark on a related thread sometime back, about yet unproposed propositions.
Although I think @Michael avoids this specific objection to Fitch by chosing as an example a mathematical hypothesis already stated (Riemann's) but whose truth value is yet unknown.
I wasn't objecting to Fitch there. Just giving an example of bewitchment of language leading to metaphysical conclusions.
We can escape Fitch by just saying we don't know if the status of Riemann's hypothesis is knowable.
The anti-realist Wikipedia mentioned would just say we're using figures of speech when we say that.
Right. Thanks for the clarification.
Yes, that's perfectly fine.
My solution is to add the time variable to the formalism. Knowledge evolves over time.
So I would write:
Suppose p is a sentence that is an unknown truth; that is, the sentence p has been proposed, it 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 today; and, if all truths are knowable, it should be possible one day to [s]know[/s] learn that "p was an unknown truth" up untill that day.
Yes.
"The problem is that according to the knowability principle, if "the box is empty" is true and we don't know that it's true then it's possible to know that "the box is empty" is true and that we don't know that it's true, which is a contradiction, and that if "the box is not empty" is true and we don't know that it's true then it's possible to know that the "the box is not empty" is true and that we don't know that it's true, which is a contradiction."
If "the box is empty is true" and we don't know that it is true it does not follow that it's possible to know that "the box is empty is true" and that we don't know that it's true, at the same time. We don't know that it's true, but we may come to know that it's true, and if we come to know that it's true, it will no longer be the case that we don't know that it's true; and hence there is no contradiction. Am I missing something? I'm finding it impossible to see why anyone would think there is a paradox here. If I am missing something it should be explainable, no?
Quoting Andrew M
We don't know if that statement is true, though; someone might know but isn't telling, so it's truth is merely being stipulated. It is unknown whether anyone knows the truth of Golbach's conjecture, but not unknowable, because someone may demonstrate that they know that it is true or false.
It does according to the knowability principle: if a proposition is true then it is possible to know that the proposition is true (p ? ?Kp).
a) "the box is empty" is true and we don't know that it's true
The above is a proposition which, if true, entails that it is possible to know that it's true (a ? ?Ka).
I'm sorry, but I don't see why "1.", if it is true, entails that it is possible to know that it is true. In other words, we don't know whether the knowability principle is itself true, but we do know that we don't know everything. The stumbling block in the argument, for me, remains the fact that time is apparently being ignored, and it is that ignore-ance that creates the apparent paradox, as far as I can tell. I am very open to being corrected, but no one seems able to explain what it is that I'm purportedly missing.
Because that's what the knowability principle says. If some proposition p is true then it is possible to know that proposition p is true, and in this case:
p. "the box is empty" is true and we don't know that it's true
OK, assuming the knowability principle is itself true, the case doesn't contradict it anyway, because it says that ""the box is empty" is true and we don't know that it's true" not ""the box is empty" is true and we can't know that it's true".
But there are two parts to proposition p:
1) "the box is empty" is true
2) we don't know that "the box is empty" is true
If we know part 1) then we can't know part 2) and vice-versa. Therefore it's impossible to know that proposition p is true. But if it's impossible to know that proposition p is true then, according to the knowability principle, proposition p isn't true.
What about this: is the truth of the proposition that there are unknowable propositions itself unknowable? We might want to say that it is, because if there are unknowable propositions then we could never know there are, just because they are unknowable.
But then it would follow that there is at least one unknowable truth, that it is unknowable as to whether there are unknowable truths; and that is a contradiction, because it would also follow that we know that there is at least one unknowable truth.
We know that one of these must be true, as per the law of excluded middle (and assuming for the sake of argument that we don't know whether or not the box is empty):
a) the box is empty and we don't know that it's empty
b) the box is not empty and we don't know that it's not empty
But we can never know either to be true because that would be a contradiction. We can't know that the box is empty and that we don't know that the box is empty, therefore we can't know a. We can't know that the box is not empty and that we don't know that the box is not empty, therefore we can't know b.
However, either a or b must be true. Therefore, either a or b is an unknowable truth. And if either a or b is an unknowable truth then the knowability principle is false.
Quoting Janus
No. Both a) and b) are known to be unknowable propositions.
Yes, or else that the verificationist holds a contradictory view.
Quoting Fitchs Paradox of Knowability - SEP
Quoting Tate
Yes.
Quoting Tate
OK, though from SEP again:
Quoting Fitchs Paradox of Knowability - SEP
Goldbach's conjecture was just an example. The point is that if there is any unknown truth (i.e., if we are not collectively omniscient), then there is also a related unknowable truth.
Yes, I agree, but we do know that one of them is true, we just can't know which one without chaging the state of the game.
Quoting Michael
Read again; I wasn't referring to a) or b).
I'm still not getting it from that angle but I think this shows that there is at least one unknowable truth:
Is the truth of the proposition that there are unknowable propositions itself unknowable? We might want to say that it is, because if there are unknowable propositions then we could never know there are, just because they are unknowable.
But then it would follow that there is at least one unknowable truth, that it is unknowable as to whether there are unknowable truths; and that is a contradiction, because it would also follow that we know that there is at least one unknowable truth.
Quoting Andrew M
I don't know what kind of anti-realism the SEP is talking about.
Do you?
I think this or a form of it is the obvious solution. If I imagine a database of all possible propositions, with a truth value column, I can just as well imagine duplicates of many propositions with them being differentiated by a timestamp column. This would allow you to have the set of all true propositions without timing becoming a source of contradiction.
But you can also look at the truth of a thing being something progressive through time. True propositions about the thing sprout up and die away with time. So, the truth of a tree is the acorn, the sappling, the tree, and the mature branch that yields another acorn, all together. "The flower does not refute the bus."
Or, with more detail at the risk of being more convoluted:
Whew...
Quoting SEP
OK, though it's not clear to me what you are objecting to.
Quoting Janus
No. False propositions are unknowable in the sense that you can't know what is false. And, in the absence of omniscience, Fitch's paradox shows that there are true propositions that are unknowable. That demonstration is how we know that there are unknowable truths.
Quoting Banno
:up:
Right. I think my failure to note this distinction may have caused some issues earlier in the discussion.
In an attempt to justify the scare quotes in the title of the OP, I will explain why I find the results of Fitchs argument unsurprising.
As noted in my penultimate post, a contradiction arises from the combination of the knowability principle and the non-omniscience principle.
If we reject the non-omniscience principle and retain the knowability principle, it follows from Fitchs argument that all truths are not only knowable but known. This is unsurprising given our omniscience!
If we reject the knowability principle and retain the non-omniscience principle, it follows from Fitchs argument that there is not only an unknown truth but an unknowable truth. This is unsurprising as it prevents our omniscience! It is also unsurprising given that not all truths can be known!
I'm not strictly objecting to anything. I'm just not seeing how it follows from there being unknown truths, that there are unknowable truths.
As I pointed out with my example we know that it is unknowable as to whether there are unknowable truths, because we can never be sure that there are not unknowable truths. But then I've just said that it it is knowable that it is unknowable as to whether there are unknowable truths, from which it seems to follow, paradoxically that we do know there are unknowable truths, or at least one, at any rate.
Yes. Fitch is just poor logical formalism (or poor English) passing for a paradox. @Janus and others made the same point.
I must admit I did not understand the rest of your post... :worry:
The two principles of Fitch's argument are that all truths are knowable (Knowability Principle - KP) and that there is an unknown truth (Non-Omniscience Principle - NonO). If we take the unknown truth (of NonO) to be one of the knowable truths (of KP), then it follows that an unknown truth is knowable. However, it can also be independently proven that an unknown truth is unknowable. This contradiction leads us to reject either KP or NonO. If we reject NonO, then it follows that all truths are known. If we reject KP, then it follows that there is an unknowable truth. Hope this helps.
Perhaps see here (bold mine):
Quoting Kant, the Paradox of Knowability, and the Meaning of Experience - Andrew Stephenson
Because Alice can (speculatively) say of an unknown truth, t, that "t is true and no-one knows that t is true".
Alice's statement will, in turn, be an unknown truth. While someone could come to know that t is true, no-one could come to know that Alice's statement is true. (Though one could potentially come to know that Alice's statement was true in the past, but not now.)
Quoting Janus
But we don't know that, since it is false. We instead know, per Fitch's paradox, that there are unknowable truths.
Yes.
Quoting Luke
Yes. Though note there is nothing in Fitch's argument that precludes humans from coming to know the unknown (but knowable) truths.
Someone could come to know the unknown truth, t, but no-one could come to know Alice's statement about t is true? Couldn't Alice come to know that their statement is true, at least? What do you make of @Michael's earlier claims in this discussion regarding the Riemann hypothesis and its being an unknown truth that it is correct (or else an unknown truth that it is incorrect)? Can't we all come to know the truth of Michael's statement(s)?
I would have thought that it was the unknown truth (of NonO) that becomes unknowable upon the rejection of the knowability principle, rather than a statement regarding the unknown truth. The SEP article appears to show only the rejection of the NonO side of things. Do you know of any literature that speaks to the rejection of the KP side?
Knowability principle (modal logic variant): p [math]\to[/math] ?Kp
Non-O: [math]\exists[/math]p(p & ~Kp)
Instantiation of Non-O: p & ~Kp
Input p & ~Kp in the Knowability principle and we get: (p & ~Kp) [math]\to[/math] ?K(p & ~Kp)
Compare the two bolded + underlined statements (vide infra).
~?K(p & ~Kp) contradicts ?K(p & ~Kp)
In other words, Fitch's argument is rather interesting in that the reductio ad absurdum argument is itself a reductio ad absurdum argument. A Zen moment for me!
I think you misunderstand Fitch's paradox. It is a reductio ad absurdum against the knowability principle. So, Fitch's paradox is literature that speaks to the rejection of the KP side. Fitch is saying "if you accept the knowability principle then this implausible conclusion follows, therefore we must reject the knowability principle."
No.
Quoting Luke
Perhaps we can - who knows? But they are not the unknowable truths that Fitch's paradox expresses.
Quoting Luke
It's the latter. In the SEP proof, line 1 asserts that p is an unknown truth. Line 3 asserts that it is possible to know the conjunction from line 1. Finally, line 3 is shown to be false. The essential point here is that p and "p & ~Kp" are different statements - the former is unknown (but potentially knowable), the latter is unknowable.
Quoting 2. The Paradox of Knowability - SEP
Folks, in outline, the SEP proof works as follows:
Part 1
Assuming KP and NonO, we derive line (3)
Part 2
Assuming A,B,C,& D, we derive Line (9)
Conclusion:
Line (9) contradicts line (3);
hence, one of the assumptions here is wrong.
Or we need an alternative logic.
A,B,C,D are unassailable (I'm sure that won't stop someone here making the attempt...)
Hence there is a contradiction between KP and NonO. They cannot both be true.
So someone who maintains that KP is true must deny NonO - they admit omniscience.
Hence, if all truths are knowable, everything is known.
The SEP article states:
Quoting SEP article
Doesn't "~Kp" therefore mean that "it is not known by someone at some time that'? That is, p is unknown.
I don't see why "p & ~Kp" is unknowable.
Moreover, "p & ~Kp" is the conjunction of the non-omniscience principle, which looks like what the SEP calls an unknown (not an unknowable) truth:
It is only once the knowability principle is rejected that there is an unknowable truth.
The argument may have implications for KP, but what is presented in the SEP article is what follows from rejecting the NonO principle (my emphasis):
Quoting SEP article
As Banno says (despite accusing me of getting it wrong):
Quoting Banno
And besides, I find it logically interesting to consider the rejection of each side. Not to mention that @Janus raised a question about unknowability which follows from rejecting the KP side instead of the NonO side.
How is that any different to what I said here and here?
Quoting Banno
...and yet we do not know everything, and hence must reject KP.
The purpose here is to show that such versions of antirealism as accept KP are committed to an unacceptable conclusion, andhence we must reject KP.
The whole of the literature is the rejection of KP...
Then just reject the knowability principle. I don't understand the problem.
Because knowing it renders it false.
There wasn't a problem.
As per Banno's summary of the argument:
Quoting Banno
The above describes what follows when NonO is denied. But given the contradiction between KP and NonO, KP could also be denied. I am merely interested, for the sake of symmetry or completeness, to see what follows if KP is denied. What follows is that there is an unknowable truth. A further discussion about unknowability also occurred when Janus asked how we get from an unknown to an unknowable truth in the argument.
Yes, my mistake. I mistook @Andrew M to be saying that "p & ~Kp" stands for an unknowable truth.
What follows from the knowability principle being denied has nothing to do with Fitch's paradox.
Assume that John argues that an omniscient God exists and that we have free will. Jane provides an argument to show that if an omniscient God exists then we don't have free will.
You then want to know what follows from an omniscient God not existing, which has nothing to do with Jane's argument.
I find it epistemologically interesting that if we reject NonO then all truths are not only knowable but known, and if we reject KP then there is not only an unknown but an unknowable truth. These both follow from Fitch's argument, so I wouldn't say it has nothing to do with it. Is it wrong to have an interest and be curious about the argument?
Quoting Michael
So? Maybe I'm curious to know whether we have free will.
Then it's a topic for another discussion, not this one.
Quoting Luke
This is where you're misunderstanding Fitch's paradox. It isn't showing that if we reject the non-omniscience principle then all truths are known or that if we reject the knowability principle then some truths are unknowable; it's showing that if we accept the knowability principle then all truths are known.
That a rejection of the non-omniscience principle entails that all truths are known is a truism, and that a rejection of the knowability principle entails that some truths are unknowable is a truism. This has nothing to do with Fitch's paradox.
Is the knowability principle that 'all truths are known'? No.
Neither is NonO that 'there is an unknowable truth'.
I see. That makes sense. If I say that truth only has a social function, then there are no unknowable truths, and I would be comfortable saying all truths are known. Fitch's target is trying to do more with truth. That's interesting, thanks.
My thinking was that p is just a true proposition and "p & ~Kp" represents that it is an unknown truth. You now appear to be saying that it is this unknown truth which follows from the argument as unknowable:
Quoting Andrew M
Whereas, you previously said that it was Alice's statement about the unknown truth which becomes unknowable.
Yes, which is what I said above ("the former is unknown").
Quoting Luke
Because that's what the proof shows. "<>K(p & ~Kp)" (line 3 in the SEP proof) is proved to be false.
The reason is that knowing "p & ~Kp" would entail knowing p and also not knowing p which is impossible.
Quoting Luke
No, p is the unknown truth. The above conjunction asserts that about p (i.e., that p is true and that p is not known).
Quoting Luke
p is the unknown truth and that is expressed by the above conjunction. The conjunction itself is unknowable.
Quoting Luke
They say the same thing. Alice's statement is "p & ~Kp".
:up:
If the unknown truth is expressed by "p & ~Kp", then it is not expressed by "p". The unknown truth expressed by "p & ~Kp" is equivalent to your "t":
Quoting Andrew M
If the unknown truth "t" is equivalent to the expression "p & ~Kp", then what Alice can (speculatively) say of the unknown truth, t, (via substitution) is that ""p & ~Kp" is true and no-one knows that "p & ~Kp" is true." This would make "p & ~Kp" knowable, but you have told me that:
Quoting Andrew M
This is why I said in my initial response that:
Quoting Luke
To clarify, p is the unknown truth and that p has the characteristics of being unknown and true is expressed by the conjunction "p & ~Kp".
So to summarize:
p is an unknown truth. "p & ~Kp" asserts that p is an unknown truth. p is true and knowable. "p & ~Kp" is true but not knowable.
p is equivalent to my earlier t. "p & ~Kp" is equivalent to my earlier "t is true and no-one knows that t is true".
Hope that clears it up.
Fair enough.
Quoting Andrew M
Isn't the unknown truth "p & ~Kp" both knowable and unknowable, according to the argument?
I'm not sure if I get your meaning. Indexicals could certainly preform the function of a timestamp by fixing a proposition's referent as well. Why is that not a solution?
As noted in my earlier post, I think the problem here is deeply rooted to one's ontology and one's conception of time.
For eternalists, this does seem like a problem, but a referent to the time the proposition refers to seems like it would resolve the issue.
For presentists, I'm not sure if a contradiction ever actually exists. Only propositions about the present can be true.
Moreover, very bare ontologies would have it that only a very small set of all possible semantic propositions are actually meaningful, and would exclude these examples anyhow in favor of a binary representation of what most people would call the "physical world." It seems like the paradox needs certain assumptions unless I am missing something.
The paradox reminds me a bit of problems in physics around information and entropy (which is really the truth value of propositions about the configuration of particles if you think about it). We have elaborate statistical ways of knowing about systems based on the possible configurations given X,Y, etc.
But, in reality, none of these "possible" configurations are actually possible aside from the one that actually obtains. The entire intellectual apparatus is based on a finite being's ability to know X about Y (the same can arguably be said for epistemology). Now in physics, we can throw out paradoxes that result from infinite information, such as Maxwell's Demon, by simply pointing out that said demon violates the laws of physics by needing to collect potentially infinite information to do his thing, and such infinite information cannot physically exist. But in the world of epistemology we can talk about sets of all true propositions (something also potentially infinite).
Perhaps there is a similar issue here where we are attempting to define truth from an absolute perspective, when really it is about information X can have about Y, as it has to be in physics.
a. "p" is an unknown truth
b. "p" was an unknown truth
These are not the same proposition.
According to the knowability principle, if a proposition is true then it is possible to know that proposition. Therefore, if a is true then it is possible to know a. You've only argued that we can know b. Knowing b is not the same as knowing a.
Your proposition a. can likewise be true untill such a time when it becomes false.
According to the knowability principle, if a proposition is true then it is knowable. Therefore, if a proposition is not knowable then it is not true.
As Fitch's paradox shows, a isn't knowable. Therefore, according to the knowability principle, a isn't true.
It seems to me though, that the knowability principle ought to apply equally to 'unknown truths' and 'unknown falsehoods'. A false proposition is the mirror image of a true one: its negation.
If p --> possibility of Kp, then non p --> possibility of Knonp
If you follow Berkeley on "to be is to be perceived," (at least as far a knowledge is concerned) then I don't think you have an issue. The truth of propositions like "no one knows that Theseus is standing" cannot be perceived, as the perception of said truth entails that the knower does, in fact, know that Theseus is standing (the paradox in a universalist view). But if to be is to be perceived then this imperceivable "truth" isn't true, since truths presumably cannot lack being.
This would indeed entail that "all truths are known," but rather than being a paradox it is simply trivial, a result of the ontology.
For people who don't buy into those sorts of Berkelean arguments about being this might seem facile, but consider that, if being can exist outside perception, that would entail that you're committed to "truths that cannot be perceived." But if you have truths that cannot be perceived then clearly "all truths are knowable," cannot obtain. The difficulty I see for this position is this: what difference can any necessarily unknowable truth ever make to anyone? It seems like a totally extraneous ontological entity that can't do any lifting.
Now, Berkeley would say all truths are known because God knows them (God is always a big help at resolving issues). Hegel would have the paradox driving the engine of the dialectical and progress towards the Absolute. In a Hegelian system the two truths result in a new entity, a world where "no one knew (past tense)" that Thesus is standing, but things have progressed and now someone does know -> being into becoming. The being of both truths creates a contradiction, the becoming of our world has one proposition pass into a present tense. This could be formalized nicely, but instead we're more likely to get a page long run on sentence about how this is the progression of Spirit (or some shit like that, smart guy, not the easiest style). These sorts of contradictions then are what drive the process of becoming that we exist in, as opposed to static "being" which is also a contradiction.
----
On a side note: You can get the same paradox to show up with truthmakers thrown in:
-Theseus is standing. (Truthmaker: Theseus standing)
-No one knows Theseus is standing. (Truthmaker: everyone's lack of knowledge of the fact that Theseus is standing, presumably Theseus as well, perhaps he is asleep)
-Persumably, knowledge can only be of true things. "No one knows the Earth is flat," would not cause this paradox if the world is actually round.
-Thus, the truthmaker for "no one knows Theseus is standing" relies on the very same truthmaker as "Theseus is standing," plus an added truthmaker about the state of knowledge relative to said truthmaker amongst all entities. The paradox emerges from this sharing of a single truthmaker.
Just a different way to view the same problem but I think some may find it more intuitive.
You can see how this isn't an issue with a Berkeley inspired system because the truthmaker for the thing no one knows about doesn't exist (granted, a sleeping man is a bad example here because people arguably still have perceptions while asleep).
No. Line 3 of the SEP proof asserts that "p & ~Kp" is knowable, i.e., "<>K(p & ~Kp)". "<>K(p & ~Kp)" is then subsequently proved to be false. Therefore "p & ~Kp" is not knowable. As the comment after Line 3 says:
Quoting 2. The Paradox of Knowability - SEP
In that case there would be no contradiction, but as the SEP proof asserts:
Quoting 2. The Paradox of Knowability - SEP
The contradiction means that one of the premises is false (KP or NonO). Not that "p & ~Kp" is both knowable and unknowable.
If KP is false, "p & ~Kp" can be true but not knowable. If NonO is false, "p & ~Kp" is never true and so also not knowable.
But there is no contradiction unless p & ~Kp is both knowable and unknowable.
Fitch's paradox shows that a contradiction follows from KP and NonO. Per the law of non-contradiction, contradictions are false. Thus it's false that "p & ~Kp" is both knowable and unknowable. So we need to reject at least one of KP or NonO, not conclude that the contradiction is true.
Quoting Andrew M
That still seems wrong to me. The proposition is an assumption or stipulation: let's assume or stipulate that p and that we don't know p. There doesn't seem to be any problem with that until what seems like the absurd idea of "knowing" (the truth of, presumably) that proposition is introduced.
The alternative I proposed:
Quoting Janus
Does seem to show that we do know that there is at least one unknowable truth; that it is unknowable as to whether there are unknowable truths, although I was wrong above to say that is a contradiction, because we are not knowing an unknowable truth but the knowable truth that there is at least one unknowable truth.
If the contradiction is not that p & ~Kp is both knowable and unknowable, then what is the contradiction?
Quoting 2. The Paradox of Knowability - SEP
It's not merely an assumption or stipulation though, it's the justifiable proposition that there is some particular truth that isn't presently known. That can be anything from Goldbach's conjecture to whether there's any milk left in the fridge (assuming no-one knows that).
Quoting Janus
The consequence, though, is that either the knowability principle or non-omniscience has to be given up. That's a problem for philosophical positions that assume those two principles. From SEP:
Quoting Fitchs Paradox of Knowability - SEP
Quoting Janus
I agree with your conclusion, but not your argument. First, we already know there are unknowable truths via Fitch's proof (and that we're not omniscient). Second, if we didn't have that proof (or others that I may not be aware of), then we wouldn't know whether there were unknowable truths or not.
That's the contradiction. However it's not true that a proposition can be both knowable and unknowable is it?
Right, but neither should the contradiction imply that p & ~Kp is necessarily unknowable. If the contradiction is false, then p & ~Kp is either knowable or unknowable.
If we accept that an unknown truth is knowable, that seems almost trivially true.
It is only if we reject that triviality and accept that an unknown truth is unknowable that the seemingly absurd result follows that all truths are known.
But upon reflection, it doesnt seem so absurd. The reason it would be impossible to come to know an unknown truth is because there are no further unknown truths to know; because all truths are (already) known.
In any case:
Quoting Andrew M
That may be true, but if it is unknowable as to whether there are unknowable truths, which seems easy enough to show, then we know there is an unknowable truth, no?
There's a jump to 'there are no unknown truths'. You've gone from a specific situation where a certain unknowln proposition is (somehow) known to be true. That is the problematic situation. The specific case. The generalization that there are no unknown truths is not supported by the problems of assertiing Assertion X, which we do not know since it is unknown is true. In the general case no one is claiming to know that any specific unknown truth is true. What we do is go by the experience that we find out new true things and there are likely more, but we, by definition, do not know what these are and can make no claims about any specific unknown truth.
Possibly. What's your reasoning?
Quoting Janus
The move from unknown to unknowable is given in the "independent result" in lines 4-9 of the SEP proof. The logic of that reductio argument is beyond my understanding, and I would welcome someone to explain it. However, I don't dispute its conclusion.
Actually, it doesn't follow. All knowable truths could be known with only unknowable truths left. But then surely new truths are arising every moment, so it seems absurd to think that there could be no unknown truths; we (collectively) would have to be constantly up to the minute.
Quoting Luke
Yeah, I don't comprehend it either, as I said, but I also accept the conclusion (although not on account of the "paradox") that there must be unknowable truths.
As I understand it, the conclusion of the independent result is not that there must be unknowable truths. The conclusion of the independent argument is that it is impossible to know an unknown truth. It follows from this in the SEP proof that there does not exist an unknown truth (at line 10) and that all truths are known (at line 11).
It does imply that. If the independent result (from Lines 4 to 9) doesn't convince you, can you come up with a concrete instance where p & ~Kp can be known? See also the example below.
Quoting Janus
Let me try a concrete example. Suppose there is milk in the fridge and no-one knows there is.
It's thus true that there's milk in the fridge and no-one knows there is.
That true statement is unknowable. Why? Because anyone coming to know that there's milk in the fridge (say, by looking) would render the statement false (since the second conjunct would be false). The statement doesn't change from an unknown truth to a known truth. It changes from an unknown truth to a known falsity.
That's it. If one holds that all truths are knowable then Fitch's proof requires that they either change their position (i.e., reject that all truths are knowable) or, else, hold that all truths are known (i.e., reject non-omniscience).
Quoting Janus
We do know that there are unknowable truths, as the above example demonstrates.
Correct me if I'm wrong, but doesn't this only hold if you take as a premise that "all truths are knowable." The issue is the existence of an unknown truth that cannot be known, as its being known entails it no longer being true.
But if you accept that there are unknowable truths then you're not in any difficulty. So, this seems to me like a potentially major problem for verificationalism or versions of epistemology where truth is actually about attitudes and beliefs (but not all of such systems, I think Bayesian systems escape unscathed), yet not much of a problem for other systems.
I am honestly flummoxed by SEP's list of systems that would be imperiled by this. Berkley seems like he can get out of this easily due to the fact that the unperceived truth of p doesn't exist, and anyhow God definitionally knows all truths for him already. Peirces system is also an odd one on the list. The "end of inquiry" would presumably be once we know the truth of p's referent, and so the paradox of one truth passing away as another is recognized is just part of the pragmatic process of gaining knowledge. I'm not even sure logical positivism is hit that hard. After all, it was the basis for the Copenhagen Interpretation of Quantum Mechanics, which very much supposed unknowable truths. The conclusion was simply that statements about the true absolute position/velocity of a particle were meaningless. Copenhagen is generally criticized for being incoherent, but this is because it creates a totally arbitrary boundary between quantum systems and classical ones not because it discounts statements about unmeasurable values.
Is "if" in the wrong place, or does it just need an "only"?
Quoting Andrew M
You seem to be saying that the truth of the statement "It's true that there's milk in the fridge and no-one knows there is" is unknowable, which seems reasonable, since I don't know there's milk in the fridge unless I open it but then if I do that someone knows there is milk in the fridge. But when I open the fridge I know (excluding weirdness like the milk coming to be there only when I looked) that the statement was true before I looked. So, again, there seems to be a time element involved.
If I go down the 'weirdness' rabbit hole and say that when I look and see the milk I still don't know that the milk had been there prior to my looking, then all bets are off.
Aye, there's the rub. If a truth is knowable, then it can come to be known; that is, it can change from being unknown to being known. However, as you note, the statement "p & ~Kp" does not (and cannot) change from being unknown to being known. Therefore, the starting suppositions make it impossible for an unknown truth to become a known truth. The starting suppositions give the impression that all truths are knowable and that we should be able to come to know an unknown truth. But if "p & ~Kp" cannot possibly change from being unknown to being known, then of course it is unknowable: it's a rigged game from the outset. It follows only from this logical impediment that it is impossible to know an unknown truth, that no truths are knowable, and that all truths are known. These conclusions can safely be ignored, however, given that the confidence trick does not allow for an unknown truth to become a known truth.
No it doesn't.
"There are 163 coins in the jar" was an unknown truth before someone counted, and then it became a known truth.
Quoting Luke
I don't know what you mean by it being "rigged". It just shows that the knowability principle is wrong. Some truths are, in fact, unknowable.
To borrow @Andrew M's example:
Suppose there are 163 coins in the jar and no-one knows there is.
It's thus true that there's 163 coins in the jar and no-one knows there is.
That true statement is unknowable. Why? Because anyone coming to know that there's 163 coins in the jar (say, by counting) would render the statement false (since the second conjunct would be false). The statement doesn't change from an unknown truth to a known truth. It changes from an unknown truth to a known falsity.
I mean that the unknown truth "p & ~Kp" of NonO cannot possibly become a known truth. If that is impossible from the outset, then so is knowability.
The result of the argument seems to be that all unknown truths are unknowable, as there is no unknown truth of the form "p & ~Kp" that can change into a known truth or that can become known. That all unknown truths are unknowable is just as absurd as the result that all truths are known.
No it isn't. There are some things which are unknown truths which can become known, e.g. the number of coins in a jar.
These are two different propositions:
1. There are 163 coins in the jar
2. There are 163 coins in the jar and no-one knows there is
It is possible that both propositions are true. It is possible that neither proposition is known to be true. It is possible to know the first proposition. It is not possible to know the second proposition. Therefore, the knowability principle is false.
Presumably, the unknown truth of the number of coins in a jar is not expressed as "p & ~Kp", since this is unknowable. So how would you express the unknown truth about the number of coins in a jar?
1. p
2. p ? ¬Kp
Assume p is true. Both 1 and 2 are true. Neither 1 nor 2 are known to be true. 1 can be known to be true. 2 can't be known to be true.
1. does not express that it is unknown
2. expresses that it is unknown, but it is unknowable.
Therefore, the number of coins in the jar remains unknowable.
It isn't. We can count the coins and then we will know how many coins are in the jar.
Quoting Luke
Which is why it is possible to know it.
Quoting Luke
Which is why the knowability principle is wrong.
Then this should be able to be expressed in the argument. If it cannot be expressed in the argument, then it is not a failure of the knowability principle, but a failure of logic. Otherwise, accept the logic and the number of coins in the jar is unknowable.
Quoting Michael
I asked how you would express (in logical notation) that it was unknown.
2 does that.
Quoting Luke
I don't understand what you're asking for here. The argument simply shows that if you take the knowability principle and the non-omniscience principle as premises then it follows that the non-omniscience principle is false. It is then up to the reader to decide whether to accept that the non-omniscience principle is false or to reject the knowability principle.
So why can't you just accept that the knowability principle is wrong? Some truths are, in fact, unknowable.
2 (when expressed as "p & ~Kp") is unknowable, which means that so is the number of coins in the jar.
Quoting Michael
It is not possible to know any proposition of the form "p & ~Kp", which means that all unknown truths (expressed in this way, at least) are unknowable. An unknown truth cannot become a known truth, and vice versa. The result of the argument is therefore that all (known) truths are known and all unknown truths are unknowable, and never the twain shall meet. The conclusion is not a failure of KP, but a failure of logic.
On the one hand, you want me to accept the argument's implication that there is at least one unknowable truth, and that therefore KP must be rejected.
On the other hand, you do not accept the argument's implication that we cannot come to know mundane unknown truths such as the number of coins in a jar.
That p ? ¬Kp is unknowable isn't that p is unknowable. The number of coins in the jar is p. We can know p.
This is where you have a fundamental misunderstanding that I don't know how to explain to you. Maybe like this?
a) p
b) a is not known to be true
Both a and b are true. Neither a nor b are known to be true. It is possible to know a but not possible to know b.
Quoting Luke
No it doesn't.
Look, smarter people than both of us have addressed Fitch's knowability paradox. None of them have argued that it somehow entails that all truths are unknowable; instead they accept that it shows that either the knowability principle is false or that every truth is known. Their solution to the problem (where they want to keep some form of the knowability principle) is to change the knowability principle. See Tennant's and Dummett's responses as detailed here.
I'm asking you how else "p is unknown" could be expressed in logical notation - other than as "p ? ¬Kp", and other than as your mere assurance outside of logical notation that p is unknown.
Quoting Michael
I understand the conjunction. I don't see how this contradicts what I said.
p ? ¬Kp is how you express it.
The problem is that you seem to go from "p ? ¬Kp" is unknowable to "p" is unknowable. And that just doesn't follow.
Please tell me where I am going wrong here:
The unknown truth that is the number of coins in the jar is expressed as: p ? ¬Kp
It is impossible to know the unknown truth: p ? ¬Kp
Therefore, it is impossible to know the unknown truth that is the number of coins in the jar.
Here are two propositions:
1. the cat is on the mat
2. the cat is on the mat and the mat was bought from Ikea
Both are true, and even though the first proposition doesn't express it, the mat was bought from Ikea (as explained by the second proposition). And it's possible that I (eventually) know that the cat is on the mat but not that the mat was bought from Ikea (so I know the first but not the second).
Similarly:
3. the cat is on the mat
4. the cat is on the mat and nobody knows that the cat is on the mat
Both are true. And even though the third proposition doesn't express it, nobody knows that the cat is on the mat (as explained by the fourth proposition).
The issue is that it is possible to (eventually) know 3 but it isn't possible to (eventually) know 4.
Are you saying that we can change the expression of the unknown truth in Fitchs proof to p instead of p & ~Kp?
:up:
Quoting Janus
Yes. While you're down the rabbit hole, be sure to check out the quantum superposition version: |milk in the fridge> + |no milk in the fridge>. :-)
:up:
Quoting Luke
No, whether a statement is unknowable or not is conditional on the content of the statement. As @Michael points out, unknown truths that don't mention that they're unknown can be known.
Quoting Luke
Of course the statement is intentionally constructed to give that result. But it has real consequences for "any theory committed to the thesis that all truths are knowable" (from SEP).
So is there a way to express an unknown truth in logical notation without mentioning that it is unknown?
Sure, just don't mention it's unknown. So instead of "p & ~Kp", that would be "p". With the milk example, that would be "there's milk in the fridge". It's also true that it's initially unknown but since the statement doesn't mention that, its truth status doesn't change when someone comes to know it.
How does that express that it is unknown?
It doesn't. That information is part of the context. The statement doesn't mention it. It also doesn't mention a host of other things, such as whether it's lite or full cream milk, whether it's in Alice's fridge or Bob's fridge, and so on.
I'm sorry but I just don't know how to fix your confusion. I've tried my best.
Then we can simply express the unknown truth in Fitchs proof as p and the problem goes away: there are no unknowable truths.
EDIT: Does Fitchs proof allow for some unknown truths to be expressed as p and others to be expressed as p & ~Kp?
No, that's just changing the subject. There are unknowable truths regardless of whether there's a proof about them.
Quoting Luke
That a proposition is true is expressed by "p". That a proposition is unknown is expressed by "~Kp". If those two ideas need to be expressed together, then the conjunction symbol is used. "p" by itself implies nothing about whether the proposition is known or unknown, but it is nonetheless one or the other.
You want to disregard Fitch's proof, but I'm the one changing the subject?
Either an unknown truth is expressed as p & ~Kp and it follows that we must reject KP because some/all unknown truths are unknowable, or else an unknown truth is expressed as "p" and it follows that we need not reject KP because all truths are knowable.
The result of Fitch's proof is that some truths are unknowable. However, if one of its suppositions brackets off and excludes those unknown truths that do not mention they are unknown, then that leaves most unknown truths as knowable.
You want to say, in essence, that Fitch's proof affects only those unknown truths that mention they are unknown. Fine. There are some unknown truths which are unknowable, and it is only those unknown truths which mention that they are unknown. But unless it is necessary for an unknown truth to mention that it is unknown, then all truths are knowable.
Is there a reason why an unknown truth must mention that it is unknown, or can any unknown truth be expressed as "p"? If we can simply re-express the unknown truths of Fitch's proof such that they do not mention that they are unknown, then all truths are knowable. If this re-expression is possible, then knowing these truths is possible.
However, my point is that we can safely ignore these unknowable truths since they can be re-written without self-reference; the unknown truths on which they are based can be re-written such that they do not mention they are unknown. If the only unknowable truths are those that mention they are unknown, then there is no loss of information or knowledge which comes from expressing these unknown truths as p instead of p & ~Kp.
Yes.
Quoting Luke
Regardless of the symbols you use to express the proposition, it is impossible to know that the cat is on the mat and that nobody knows that the cat is on the mat.
Whats the issue with just accepting that some truths are unknowable?
But this would only apply to a small subset of unknown truths, those that follow the form "no one knows that X" where X is a true proposition.
And while this set of truths cannot be known, the past tense version "no one knew that X was true," can be known. I don't see this as a huge problem.
Indeed, I'm not even sure if this problem holds for an eternalists view of time in the first place. If all moments in time are real, then the issue is simply that the knowledge of the truth of the "known one knows that X is true" proposition has to occur simultaneously with the discovery of X being true. But the addition of a past tense would really just be an artifact of our languages' inability to transcend the present. . After all, if the past is real then there is a reality where "no one knows that X is true" is still true and someone in the future can have knowledge of this past truth. If this holds, the truthmaker of the the fact that "no one knows that X is true," still exists even after someone knows X.
This doesn't seem too dicey to me. There are plenty of good empirical reasons to accept eternalism (e.g., physics)
Has someone claimed it's all quantum yet?
It'll happen.
It just seems counterintuitive to me that any unknown truths should be unknowable in priniciple. If the only unknowable truths are that 'p is true and no one knows that p is true', then that's merely a quirk of logic that has little effect on substantive knowability. It is still knowable that p is true. The only reason we cannot know 'p is true and no one knows that p is true' is because knowing the first conjunct would falsify the second. I don't see why this should be "of concern for verificationist or anti-realist accounts of truth", as the WIkipedia article states.
Quoting Banno
Not the best comment, this, o pompous one.
Then read up on Tennants and Dummetts responses. Theyre in that SEP article. Tennants is the simplest:
[quote]Tennant (1997) focuses on the property of being Cartesian: A statement p is Cartesian if and only if Kp is not provably inconsistent. Accordingly, he restricts the principle of knowability to Cartesian statements. Call this restricted knowability principle T-knowability or TKP:
(TKP) p??Kp, where p is Cartesian.
Notice that T-knowability is free of the paradoxes that we have discussed. It is free of Fitchs paradox and the related undecidedness paradox.
Ironically, this is relevant given empiricism tells us that knowing a quantum object's velocity makes its position unknowable and vice versa. Another point for unknowable truths. It's all quantum! (...you brought this on yourself).
And the rest of us. :cry:
Succsessful invocations surely merit a ban for witchcraft?
The knowability principle is like the proposition that all swans are white. When someone discovered that some swans were black, then that refuted the original proposition. Regardless, the original proposition was false independent of that discovery.
Similarly "p & ~Kp" was a counterexample to the knowability principle before Fitch ever formulated his proof.
Quoting Luke
For an example of why the counterexample matters, consider Peirces pragmatic theory of truth, i.e., that truth is what we would agree to at the limit of inquiry. Since no-one would ever plausibly agree that "p & ~Kp" is true, does it follow that it is never true? Presumably not, and so the theory either needs to be rejected or else qualified in some way.
If the latter, please note that in practice it is often extremely hard to prove that some proposition is true, beyond any doubt. We almost never 'know X to be positively true'. What we do instead is eliminate theories that are proven false.
So from a pure epistemic view point, the knowability principle is false because contradicted by day-to-day experience, and by our knowledge that we know very little. That'd be why most examples given on this thread are mathematical, as the only domain of knowledge where certainty applies.
Surely it is never true. If a statement is known to be true, then it cannot also be unknown to be true ("by somebody at some time"). Which is what the independent result tells us.
It's a trick of logic. Every "p" remains knowable, but not when put into a conjunction with "~Kp". Therefore, it cannot be known both that p is true and p is unknown to be true. That's just word play (or logic play) which does not affect every (other) "p" being knowable.
The same could be done for other propositional attitudes. For example, desires (D):
D(p & ~Dp) - someone at some time has the desire that 'p is true and nobody desires that p is true'. Is this undesirable?
Or beliefs:
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?
It's like a liar paradox for propositional attitudes. But less paradoxical and more nefarious.
From the little I've read, they seem to be looking to qualify the theory in some way (as Andrew put it). For example:
Quoting SEP article
I accept that the problematic statement (form) "p & ~Kp" is inconsistent. My only qualification is that it's a kind of logical loophole that doesn't really affect knowability. I accept that it's unknowable, but it's also trivial: "If I know something then I can't also know that it's unknown." Okay, so what?
Then the claim that if a proposition is true then it is knowable is wrong. One must instead claim, as Tennant does, that if a Cartesian proposition is true then it is knowable.
I accept that. But it is only wrong in the sense that one cannot both know the proposition and know that it is unknown. Knowing it negates its being unknown. If it's known then you cannot know it to be unknown.
Yes, that's exactly the point. It is true but can't be known. Therefore, the (unrestricted) knowability principle is false.
It means to know that something is true, e.g., that it is raining (say, as a consequence of looking out the window).
Quoting Olivier5
Mathematical certainty isn't required for the ordinary use of "know". However it does require a higher bar then mere opinion or guesswork (i.e., there need to be good reasons, or evidence, or justification for making knowledge claims). But the knowability principle is false not because we don't know some things, but because we can't know some things (i.e., propositions of the form "p & ~Kp").
"p & ~Kp" is sometimes true. There have been plenty of examples in this thread.
Quoting Luke
That's right. But "<>K(p & ~Kp)" (which is never true) is a different proposition to "p & ~Kp" (which can be true).
Quoting Luke
It's not "word play" if one's theory of truth depends on the knowability principle being true. Consider again Peirces pragmatic theory of truth, i.e., that truth is what we would agree to at the limit of inquiry. If there is milk in the fridge and no-one knows there is, is the statement "there is milk in the fridge and no-one knows there is" true? According to Peirce's theory, it isn't true. But that's mistaken.
What if one person knows the proposition as true and another knows it as false? Is it 'known' then?
Quoting Andrew M
Fitch is easily solved by noting that knowledge evolves over time. Lamest paradox ever.
But yes, there are many things we cannot know, such as the things in themselves, as Kant explained, or whether it rained on a given site 36,785,477 years, 278 days and 4 hours ago, or what your wife thinks.
I accept that, according to the logic, "p & ~Kp" is unknowable. However, I don't think this is an issue for knowability, but an issue for logic.
"p & ~Kp" is supposed to represent an unknown truth. The logic of Fitch's proof absurdly implies that an unknown truth cannot become known. The problem, as I have stated in several recent posts, is the conjunct of ~Kp. But that is only a problem in logic, not a problem in reality. In reality, coming to know that p is true means that it has become known and is no longer unknown, not that we impossibly know both that p is true and that p is unknown to be true. Logic holds one set of truths to be eternally known and the other to be eternally unknown, and those sets can never change. But in reality, those known and unknown truths are not eternal and do change; what is unknown can become known.
You're right. I meant to say that it is never known to be true.
Quoting Andrew M
According to logic, if it is true and unknown that there is milk in the fridge, then it can never become known.
It doesn't. I thought we went over this? You seemed to understand it here:
Quoting Luke
We did, but I didn't realise then, and wasn't making the point then, that the issue was with logic and not with knowability.
You claim that we can know "p" even though we can't know "p & ~Kp". But that implies that we can't come to know anything that is unknown to be true. That's surely a problem - not just for knowability but for everyday reason. Isn't it? That's just as absurd as the result of Fitch's proof that 'all truths are known'.
I'm saying that we can retain knowability by acknowledging that logic cannot account for any changes from a truth being unknown to its being known. This failing of logic creates the paradox. The paradox dissolves in everyday reason where we obviously can come to know unknown truths.
No, it doesn't. Imagine these two propositions:
1. "the cat is on the mat" is true
2. "the cat is on the mat" is true and is written in English
To represent these in symbolic logic we would do something like:
1. p
2. p ? E(p)
Even though 1 doesn't say anything about p being written in English, p is in fact written in English. Just look at the previous sentence; it is written in English even though it doesn't say it about itself. A proposition doesn't need to state every fact about itself.
And the same with unknown truths:
3. p
4. p ? ¬Kp
Even though 3 doesn't say anything about p being unknown, p is in fact unknown. We can come to know 3, in which case an unknown truth has become a known truth. But we can never know 4 as that would be a contradiction.
3 only says that p is true, not that it is true and unknown.
I know. But as I said above, a statement doesn't need to state every fact about itself.
Your claim that I am quoting is written in English, even though it doesn't say so about itself. Your claim that I am quoting contains 44 letters, even though it doesn't say so about itself.
So we can do it as two propositions:
a) p
b) p is unknown
p is an unknown truth. When we come to know a we no longer know b (because b is false). And so c can never be known:
c) p and p is unknown
Yes, that's the reason that we can't know both a) and b). Again, I'm not disputing the logic, only its implications.
We cannot know both a) and b) means that we cannot come to know an unknown truth. Which is absurd.
No it doesn't.
As I've said before, I just don't know how to explain this to you any more clearly than I already have.
Quoting Fitch's proof
Isn't that unknowable?
The knowability principle: p [math]\to[/math] Kp.
1. K = Knowable
p is true and p is unknown: p & ~Kp
We know that p is true and p is unknown: K(p & ~Kp)
2. K = Know(n)
Inconsistency in the meaning of symbol K (compare 1 and 2).
No. It's p ? ?Kp.
? is the symbol for "it is possible that".
(p & Kp) (only) [math]\to[/math] ?K(p & Kp).
How do we get to K(p & ~Kp) [math]\to[/math] Kp & ~Kp?
:brow:
It's in the article.
(A) K(p ? q) ? Kp ? Kq
(B) Kp ? p
1. K(p ? ¬Kp) Assumption [for reductio]
2. Kp ? K¬Kp from 4, by (A)
3. Kp ? ¬Kp from 5, applying (B) to the right conjunct
3 is a contradiction so 1 isn't possible, so ?K(p ? ¬Kp) is false.
Don't leave it ambiguous then. If truths are either known or unknown, then this can be expressed as:
1. p ? Kp; or
2. p ? ¬Kp
1. is knowable. 2 is unknowable. I imagine you will find that the paradox occurs for all unknown truths.
Or we write it as:
a) p
b) ¬Kp
a is knowable, b is not knowable, a ? b is not knowable.
It's really straightforward logic. Fitch et al. know what they're talking about. You haven't found some fundamental flaw with their reasoning.
I thought you said "p" could either be known or unknown?
It can, but Fitch's paradox takes an example of an unknown truth to show what follows.
Okay, but I removed the ambiguity by expressing known and unknown as:
Quoting Luke
To which you said "we write it as":
Quoting Michael
That's either making it ambiguous again (if "p" can be either known or unknown), or refuting what you said earlier (if "p" represents "p is known").
It doesn't make it ambiguous. b is a second (true) proposition that asserts that p is unknown.
To repeat an example I gave earlier:
1. "the cat is on the mat" is true
2. "the cat is on the mat" is written in English
Is it ambiguous whether or not "the cat is on the mat" is written in English? No; it's explicitly stated in 2. So then apply the same understanding to:
3. "the cat is on the mat" is true
4. "the cat is on the mat" is not known to be true
Then you misunderstood that I was expressing both known and unknown truths.
Quoting Luke
This removes the ambiguity of your unknown truth expressed merely as "p".
Then:
Quoting Luke
It's not ambiguous because of the second premise:
a) p
b) p is unknown
:snicker: So it was you all along!
Can't know what isn't so. From Fitch's proof:
Quoting 2. The Paradox of Knowability - SEP
Quoting Olivier5
Noting that knowledge evolves over time doesn't help those theories that depend on the knowability principle.
Quoting Fitchs Paradox of Knowability - SEP
I'm not sure how that answers the question above. The point is that the statement above is a counterexample to various antirealist theories.
Quoting Fitchs Paradox of Knowability - SEP
Our dispute is over your claim that there are knowable unknown truths.
If all truths can be expressed as either:
1. p ? Kp [known]; or
2. p ? ¬Kp [unknown]
Then which of these are knowable?
1 is knowable.
But this doesn't address what I said before. You clearly just don't understand logic.
It is unknowable that p is true and that somebody knows p is true? Why is it unknowable?
You claim that "p" can be unknown and knowable.
But if all truths are expressible as 1. and 2. above, then what other "p" is there? Where is this knowable unknown truth?
a the cat is on the mat
b nobody knows that the cat is on the mat
Both a and b are true. This means that, even though a doesn't say so about itself, a is an unknown truth. Compare with:
c Michael is a man
d Michael is 34 years old
Even though c doesn't say so, it is about a 34 years old. When presented with both c and d it doesn't make sense to say that Michael's age is ambiguous because c doesn't say anything about Michael's age. It doesn't matter what c says about Michael's age because d provides that information.
And by the same token, it doesn't matter what a says about whether or not it is known that the cat is on the mat (it says nothing about knowledge) because b provides that information.
So with that in mind, given the truth of b it then follows that a is an unknown truth even though a doesn't refer to itself as being unknown.
Now, it is possible to know a and it is possible to know b, but as Fitch's paradox shows, it isn't possible to know the conjunction a ? b even though the conjunction a ? b is true, thereby showing that the (unrestricted) knowability principle is false (there is at least one truth that is impossible to know).
Likewise.
Every truth ("p") is either known ("p & Kp") or unknown ("p & ~Kp"). There are no other known or unknown truths.
Your mistake (and mine, too, previously) is in thinking that a truth either mentions that it is unknown or does not. However, the expression "p & ~Kp" does not "mention" that it is unknown. Instead "p & ~Kp" represents that p is true AND unknown; "p" represents only that p is true; and "p & Kp" represents that p is true AND known. This accounts for all known and unknown truths.
If there is some other way to express that p is both true AND unknown, then I welcome you to provide that expression.
p means "the cat is on the mat"
¬Kp means "it is not known that the cat is on the mat"
p ? ¬Kp means "the cat is on the mat and it is not known that the cat is on the mat"
p is an unknown truth but is knowable
¬Kp is a known truth
p ? ¬Kp is an unknown truth and is not knowable
It's that simple.
If p is an unknown truth, then it is represented by "p ? ¬Kp".
It's that simple.
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"?
In other word, the concept of knowledge is mistreated here, cheapened, overly simplified when made an absolute. Knowledge is not something that exists objectively out there. It's something that people do.
You just don't understand symbolic logic, so address the argument in natural language.
1. the cat is on the mat
2. it is not known that the cat is on the mat
3. the cat is on the mat and it is not known that the cat is on the mat
1 is an unknown truth but is knowable
2 is a known truth
3 is an unknown truth and is not knowable
Quoting Michael
What does "p ? ¬Kp" represent if not that the cat is on the mat AND that it is not known that the cat is on the mat?
"p" does not represent that p is true and unknown; only that p is true.
Quoting Michael
Why is 1 an unknown truth? It could equally be a known truth. I have removed this ambiguity in my post above, yet you continue to ignore it.
Its an unknown truth because 2 says so. Do you not understand than an argument can have more than one premise? Your reasoning here is ridiculous.
I didn't realise that they were premises; I thought they were unrelated statements.
Quoting Michael
If knowing 2 makes 1 unknown, then how is 1 knowable?
That is, if 'the cat is on the mat' is true (as a result of 1) AND unknown (as a result of 2), because of the relationship between 1 and 2, then how can 1 be knowable?
This would mean that "p ? ¬Kp" is knowable.
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.
It seems to me that b renders a as a meaningless string if scribbles.
If no one knows the cat is on the mat then from from where does A follow? Why was A stated in the first place? How is it possible to positively assert that which is not known?
We could go on ad infinitium with
c no one knows that know one knows the cat is on the mat
d no one knows that no one knows that no one knows the cat is on the mat
etc.
With each subsequent statement rendering the prior statement as useless.
The question is, what is knowing? How does knowledge relate to truth? Have you ever claimed to know something and later found it was not true?
I might believe it to be so? e.g. intelligent alien life exists, the real part of every nontrivial zero of the Riemann zeta function is 1/2, and it will rain tomorrow.
But to be more formal, it follows from the non-omniscience principle ?p(p ? ¬Kp) that there is some p such that:
1. p is true, and
2. p is not known to be true
We might not know what specific p satisfies this criteria, but that's irrelevant. It is not possible to know p ? ¬Kp and so therefore the unrestricted knowability principle is false.
But one has reasons to believe alien life exists and that it will rain tomorrow. What reasons does one have to know that know one knows alien life exists or that it will rain tomorrow?
And then we can always cancel out the prior statement with a subsequent statement that no knows the prior statement is true. What prevents sliding down the slippery slope? Have you ever claimed to know something and found that it was not true?
How is belief different than knowledge?
I don't understand what your comments have to do with anything. If we accept the non-omniscience principle then there is some p which is true and not known to be true, so we address:
1. p is true, and
2. p is not known to be true,
3. therefore, p is true and p is not known to be true
It's not possible to know 3, therefore the knowability principle is false.
We don't need to know a real example of p for the logic to work.
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
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.
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.
Quoting 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.
Is (2) both true and false? Is (3)?
No. It was true before we knew 1 and false after.
Quoting Luke
No. It was true before we knew 1 and false after.
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.
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).
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.
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....
Very clear.
Quoting Olivier5
And were they right?
:up: Thanks for saying so, and for working it through.
Quoting Banno
:up:
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).
That's the basis for the epistemic principle (B) in Fitch's proof, "Kp ? p".
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.
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?
Logic says that we're all vegetarians now...
Quoting Luke
Yes, very similar. Interestingly, from SEP:
Quoting Epistemic Paradoxes - SEP
Oh cool, thanks.
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.
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)
What rules of inference get you there?
Quoting 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.
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:
Quoting 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.
Ill try and come back to the rest of your post, but if the above is correct, then this would seem to contradict @Michaels 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.
The proposition "Joe Biden is President of the United States" was known to be false in 2016 and is known to be true now.
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?
EDIT:
Now that I think about it, it seems that knowledge is only a present state, kind of like the current fashion trend.
Logically speaking, you can't have your chicken and eat it too.
To be clear, the difference with that to the knowability paradox is that "p & ~p" is a contradiction - it can never be true. Whereas "p & ~Kp" is not a contradiction. It can be true, but never known to be true.
We don't. But "every possible observation" is not the standard for making knowledge claims or forming beliefs. Good evidence is. If good counter-evidence emerges, then we should change our minds and retract the former claim.
Quoting Harry Hindu
In which case you wouldn't know it was raining, you would just think you did.
Quoting Harry Hindu
Yes. To know it also requires good reason, or evidence, or justification.
Quoting Harry Hindu
Your question assumes a standard of infallibility or Cartesian certainty. But you can say that you know it is raining (or not) by simply looking out the window. That's the relevant standard for making knowledge claims.
I'm not saying you're wrong; I'm merely noting that what you have said appears to contradict what @sime has said. Does the Fitch proof use a non-standard meaning of "knowledge", perhaps?
I note that the SEP article defines the epistemic operator "K" as:
Quoting SEP article
This also appears to be different to sime's statement that:
Quoting sime
Yes, but for the exact same reason than you can't eat an uneaten chicken. Fitch says that one cannot know an unknown truth, because as soon as one knows it, it cease to be an unknown truth. Likewise the Olivier5 chicken paradox states that one cannot eat an uneaten chicken, because as soon as one eats it it ceases to be an uneaten chicken.
Like in Fitch, one of two things follows from the Olivier5 chicken paradox: either not all chicken can be eaten, or all chicken have already been eaten (omnigallinavorousism).
I lean toward the former: not all chicken can be eaten.
You haven't explained the logic behind your "chicken paradox". And as I mentioned here your symbols were wrong anyway.
Quoting Olivier5
And as I said here, you're equivocating. There's a difference between saying that we cannot come to know something that wasn't known before and saying that something cannot be both known and known to be unknown. Fitch is saying the latter.
Non-O: There's an unknown truth = p & ~Kp
Substituting (p & ~Kp) in the knowability princple, we get:
(p & ~Kp) [math]\to[/math] ?K(p &~Kp)
Now, foe Fitch's argument to work, the following hasta be true:
?K(p &~Kp) [math]\to[/math] K(p & ~Kp). None of the rules used by Fitch in the SEP article allow this move. Also, intuitively, it looks/feels wrong.
Compare with:
1. If God is omnipotent then it is possible for God to create a rock that he cannot lift
2. If God creates a rock that he cannot lift then ...
Fitch is using the same reasoning:
1. If p is true and not known to be true then it is possible to know that p is true and not known to be true
2. If it is known that p is true and not known to be true then ...
Such an important step and the rule is left unmentioned. Odd!
My "chicken paradox" follows the exact same structure as the "Fitch paradox" and should thus rightly be called the "chicken transposition of the Fitch paradox".
If there is a flaw in my chicken paradox -- as I strongly suspect is the case :razz: --, then the exact same thing is wrong with Fitch.
You pointed yourself to that flaw here, as I and many others have done before you, about the non-chicken version of Fitch.
Let me walk you through this. You pointed out:
Quoting Michael
Transposing your point to Fitch (eat --> know)
It is possible for us to later know something that is currently unknown, or for something that we know to have before that time been unknown. It isn't possible for us to know something and for it to remain unknown.
Note the flagrant similarity with this point of mine, about Fitch:
Quoting Olivier5
Quoting Olivier5
The first flaw in your proposed paradox is what I explained here. Your symbols are wrong. It should be:
?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.
Fitch's paradox, however, uses the correct symbols.
Quoting Olivier5
That's not a flaw with Fitch's paradox. That's me explaining to you how you're misinterpreting/misrepresenting Fitch's paradox by using ambiguous wording that leads to equivocation.
Everything is a goat.
yet,
How can such a straight forward argument lead to such a counterintuitive conclusion?
The problem must be with the second premise. Hence there must be a goat that is uneaten, and yet eats everything.
The alternative, that the Great Goat eats itself, is unpalatable.
The exact same critique can be made about Fitch, but for some reason you fail to see it.
It can't be made about Fitch because his premises work. You just don't seem to understand propositional logic.
?x(Px ? ?Kx)
For all things, if that thing is a proposition then it is possible to know that thing.
?x(Px ? ¬Kx)
There is at least one thing that is a proposition and hasn't been known.
That doesn't address what I was saying about your argument. Formal logic is concerned with the relationship between propositions. In the case of a ? b, both a and b are propositions. In your argument you want a to "stand in" for a chicken, which doesn't make sense. Chickens aren't truth apt and can't be the antecedent of a material implication.
Or perhaps you meant for a to be the proposition "the chicken exists"? In which case the consequent of your material implication, ?Ea, says that it is possible to eat the proposition that the chicken exists, which is of course absurd; you can't eat a proposition.
Fitch's argument, however, correctly utilises formal logic. p ? ?Kp: if the chicken exists then it is possible to know that the chicken exists.
As I suggested to another earlier, if you don't understand formal logic then address the argument in natural language. The reasoning is the same. If you accept the knowability principle and the non-omniscience principle then it follows that all truths are known. Therefore, you must either reject the non-omniscience principle or the knowability principle.
Premise 1: Goats eat everything.
Premise 2: Eating is asymmetric. That is, if A eats B, then B does not eat A.
Therefore:
Conclusion: There is at least one non-goat.
Not sure where you got the "can eat" from?
As Wittgenstein said in On Certainty
"I know" seems to describe a state of affairs which guarantees what is known, guarantees
it as a fact. One always forgets the expression "I thought I knew".
If the epistemic usage of "to know" is considered to be the same as "to be certain", then knowledge changing over time is no big deal for the verificationist and simply means that one's beliefs are changing as the facts are changing. But this doesn't necessitate contradiction.
For instance, if p is "Novak is Wimbledon Champion", then p today, and hence K p (assuming verificationism). Yet on Sunday it might be the case that ~p and hence K ~ p. But any perceived inconsistency here is merely due to the fact that the sign p is being used twice, namely to indicate both Friday 8th July and Sunday 10th July.
If instead p is "Novak is Wimbledon Champion with respect to the years 2011, 2014, 2015, 2018,2019, 2021" and q is "Kygrios is 2022 Wimbledon Champion" then we will still have K p whatever happens, even though the domain of the operator 'K' has enlarged to include q.
Of course, not every observation, such as the contents of a fridge, has an obvious time-stamp that places the observation into an order with every other observation of the fridge, but contradictions can at least be averted by using fresh signs to denote present information. "Never the same fridge twice".
It does, it's the exact same logic. The original version says one cannot know an unknown truth. The chicken version of Fitch says one cannot eat an uneaten chicken. There's no fundamental difference between the two ideas. They are both equally ridiculous.
No it doesn't. It says that if you accept the knowability principle and the non-omniscience principle then it follows that all truths are known.
As I said here, you're equivocating. The phrase "one cannot know an unknown truth" is ambiguous and you're using the wrong interpretation. It can mean one of these:
1. If some p is not known to be true then it is not possible to (ever) know p
2. It is not possible to know that p is true and that p is not known to be true
The first is false, the second is true.
Quoting Olivier5
And your logic is flawed, as I have explained. You don't understand formal logic. You're also trading on the ambiguity as explained above. There is a difference between these:
1. If some chicken has not been eaten then it is not possible to (ever) eat it
2. It is not possible to eat a chicken and for that chicken to remain uneaten
The first is false, the second is true.
And that's precisely why the knowability principle fails, as Fitch's paradox shows. It isn't possible to know that p is true and that p is not known to be true, even though there is some p that is true and not known to be true. Therefore, some truths are unknowable.
Which isn't any different than saying knowledge is an interpretation that changes with new evidence - not that you never had it.
What qualifies as good evidence? Isn't there a chance that good counter-evidence emerges later? If yes, then you can never say that you possess knowledge. You would never know that you know or you would know something unknowable.
Quoting Andrew M
Yet we asserted that we did know and were wrong, which is good evidence that you could be wrong again, and again, and again - hence no such thing as knowledge unless we define knowledge as an interpretation that changes - not that you never had it. So, using your "good evidence" definition, you have good evidence that you can't ever possess good evidence. Your argument defeats itself.
Quoting Andrew M
As I pointed out, it is very possible that your good reason or evidence isn't actually a good reason or evidence, and you only find that out after you get good reason or evidence, yet it is very possible that your good reason or evidence isn't actually good reason or evidence, and you only find that out...,etc. It's an infinite regress.
Quoting Andrew M
No. It is you that assumes a standard of infallibility or Cartesian certainty by saying that "good evidence" is what is needed to possess knowledge. I'm simply asking you to define what that means, if not that "good evidence" is a state of infallibility (knowing the truth). I already pointed out that looking out the window is not good evidence because your brother could be spraying the window with a hose.
You keep using this term, "proposition" that you've you admitted to not knowing what they are. If you don't know what propositions are, then how can you even know what kind of relationship exists between them? You just continue to post scribbles on this screen and asserting that there is a relationship between them, but don't know what the members of that relationship actually are.
Is a proposition a relationship - a relationship between some scribbles or utterances and what those scribbles and utterances are about? So formal logic would be the relationship between one string of scribbles and what that string of scribbles is about and another string of scribbles and what that string of scribbles is about. It seems to me that you'd first have to determine what the scribbles are about (like an assertion of what is the case, like the cat being on the mat) before understanding the relationship between them.
I don't need to have some kind of in-depth metaphysical understanding of the nature of language and reasoning to make use of formal logic, just as I don't need to have some kind of in-depth metaphysical understanding of the nature of numbers to do maths.
I don't know what numbers are, but I know that 2 is a number, that a chicken isn't a number, and that 2 + 2 = 4.
I don't know what propositions are, but I know that "it is raining" is a proposition, that a chicken isn't a proposition, and that modus tollens is a valid rule of inference.
If you want an in-depth metaphysical discussion on the nature of numbers and logic and whatever then that's a topic for another discussion. It's not relevant to this one.
I wasn't asking for an in-depth metaphysical understanding of the nature of language. It's not necessary to answer a simple question. You said, "I don't know". I'm just asking for a simple definition of "proposition". What do you know, if anything, of what a proposition is? You have to have some understanding of the nature of numbers to do maths, or else what are you doing when you do maths?. :roll:
I can't give you any meaningful definition of "proposition", just as I can't give you any meaningful definition of "number". I can give you examples of things which are either numbers or not numbers, and examples of things which are either propositions or not propositions.
But, again, this has nothing to do with Fitch's paradox. If you want to talk about what propositions are then start another discussion.
That's not necessary. You've already shown that you have no idea what you're talking about, which is the point I was trying to make. Thanks. :smile:
I know exactly what I'm talking about, thanks.
How can you tell the difference between a proposition and a chicken if you don't know what a proposition is? How are a chicken and a proposition different? You said that you know that, so you should be able to answer that question.
I said I can't give you a definition of "proposition", just as I can't give you a definition of "number". But I know which things are numbers, which things are propositions, and which things are neither.
And I know that 2 + 2 = 4.
And I know that modus tollens is a valid rule of inference.
And I know that chickens are animals.
That's all that matters for this discussion.
Likewise, it isn't possible to eat a chicken and to have it remain uneaten, even though there are some chicken that remain uneaten. Therefore, some chicken cannot be eaten.
That doesn't follow at all.
There are five chickens in a cage. They haven't been eaten but they can all be eaten.
What is a definition if not the suggested, or commonly understood way of using the term? What you're saying is that you don't know how to use the term, proposition, so it doesn't follow that you can know how they relate using formal logic.
What does it mean to know that 2+2=4 - that you've learned how to copy someone else's behavior typing that string of scribbles?
Chickens are animals and propositions are...? You didn't need to get in-depth and metaphysical with your description of a chicken, so why would you think I'd be asking for something different when describing a proposition? Seems like you just want to avoid the question by being purposely obtuse.
It's easy. Propositions and numbers are scribbles that refer to states of affairs. No metaphysics needed.
Read Wittgensteins Philosophical Investigations.
What is life? I know that Im alive and that a rock isnt. But theres no proper understanding of what life is, with over a hundred proposed definitions.
As if Wittgenstein is the prophet of propositions. :roll:
Will philosophy ever recover from the damage that Wittgenstein has dealt it?
Read a dictionary.
Quoting Michael
Which is to say that we have definitions of life that allow us to distinguish it from things that are not alive. All I'm asking is what those distinctions are. If you can't even answer that simple question then it does not follow that a chicken is not a proposition. A proposition could be anything, which makes your arguments non-sensical.
Philosophy has degenerated into a game of scribbles and utterances. Philosophers scribble and utter like they know what they are doing, but when you ask them what they are doing, they don't know.
That isn't what Fitch says. If the unknown truth is that "there is chicken in the fridge", then it becomes a known truth when you look in the fridge. Then you can eat the uneaten chicken.
But you can never come to know the truth that "there is chicken in the fridge and no-one knows there is". That's unknowable. The philosophical point is that Fitch's proof undermines antirealist theories that define truth in terms of knowability.
Now if you want to progress the analogy, you need a proof that not all chickens are edible. But, even if true, I'm not sure what theory it would undermine. Maybe that everything is a goat.
Quoting Banno
Undoubtedly. But I would further conjecture that the Great Goat is inedible.
That's the classical solution, that the Great Goat eats everything but is itself uneaten. Hence the heresy in the argument that:
...which apparently would have one conclude that the Great Goat is not a goat!
Using the convention C for the relation comeditur a , to be consumed, and G for being a goat,
There isn't an epistemic difference (i.e., either way, one is correct or mistaken about whether it is raining). However there is a semantic difference. With the "knowledge changes" position you can know it is raining when it isn't, on ordinary usage you can't.
Quoting Harry Hindu
If you want to know whether it is raining then looking out the window provides good evidence. You can say that you know it, but be mistaken, as with any claim. You can also know that you know. That's just how the logic of the usage plays out. As mentioned, the standard for claiming knowledge isn't Cartesian certainty. So its possible to think that you know that you know when you don't.
Quoting Harry Hindu
You could be wrong again and again. But that's unlikely for a given case, since you require good evidence for each iteration of the claim. The space of possibilities rapidly diminishes. Consider what it would take to be wrong that the Earth orbits the Sun.
Quoting Harry Hindu
It can be a good reason at the time. It may no longer be a good reason in the light of new evidence. Also there need be no infinite regress, as suggested by the orbit example. At some level of evidence you expect to converge on the truth.
Quoting Harry Hindu
It is good evidence. If it weren't, then essentially no knowledge claims could ever be made (as Descartes discovered). Yet we do have knowledge. However what constitutes good evidence at one time may no longer be sufficient in the light of new evidence. If you become aware that your brother sprayed the window, then you retract your former claim, since the fact that you looked out the window is no longer a good reason to believe it was raining (though it was a good reason before).
Quoting sime
Perhaps I'm just confused by: "K is...used in the future-tense in standing for "Eventually it will be known that ...".
Do you mean anything different to the SEP's definition of K: "it is known by someone at some time that?
Clearly an absurd conclusion. Thus the Great Goat is edible. Which raises the important dilemma of whether all goats partake of the Great Goat, or just the Great Goat itself.
I don't see how the conclusion follows. It seems to follow only that nothing eats goats.
EDIT: ah I see now.
The way I see it, Fitch is a joke of a paradox, and it debunks absolutely nothing. Just like the idea that one cannot eat an uneaten chicken is a joke, and debunks absolutely nothing. So to me, you guys are getting all hung up on a joke.
Have fun. :-)
Just another way of saying that it is a misuse of language.
Quoting Andrew M
Yet you did assert that you know when you didn't with ordinary usage. You just know something different now.
Quoting Andrew M
As I already pointed out, you being mistaken is good evidence that you can still be mistaken with any knowledge claim, which is to say that you can never know that you know. So thinking of knowledge as a changing interpretation based on new good evidence resolves the issue. There can be right and wrong interpretations. A wrong interpretation is not no interpretation, just a different one based on the good evidence one had at the time. Given that evidence you had at the time, it would be a valid interpretation. So either we make knowledge a synonym of interpretation or we just omit the word from usage because it would be useless. Using knowledge as a synonym for interpretation is how we use the word in ordinary usage anyway when we take into account how we used the term, "knowledge" in the past as well as now when we say we know but can't know that we know thanks to the good evidence that our interpretations have changed in the past.
The problem of induction is also good evidence that some observation is not good evidence to support an assertion of knowledge in that it seems to call into question observations as justification for forming knowledge.
Quoting Andrew M
Which addresses my question that I asked before about how many observations need to be made before we can claim knowledge which you responded:
Quoting Andrew M
How would you know that the space of possibilities "rapidly diminishes" without knowing how many observations need to be made? You are claiming to know something that you couldn't possibly know or else you would have made the correct interpretation in the beginning if you knew how many observations you needed to assert knowledge.
Our observations about the movement of the Earth took place on the Earth and out in space. What if we are able to move into another dimension and observe the movement of the Earth - could we say that it still orbits the Sun? "Orbit" might not make any sense when observed from another dimension. We keep trying dislocate ourselves from reality when making observations as if we can make an observation outside of reality. One QM interpretation is that observers have an impact on what they observe, so how do we know that the orbit of the Earth around the Sun is a product of just the Earth and the Sun, or also us as observers.
Quoting Andrew M
Which is to say that the interpretation we had was valid given the reasons we had at the time. Our interpretation can change, but that doesn't mean that we never had an interpretation in the past.
Quoting Andrew M
Which is the same as saying that it was a valid reason for arriving at that interpretation. Knowledge claims can be made if we define knowledge as an interpretation (which I have already shown that the ordinary usage of knowledge is a synonym for interpretation). So we do have interpretations/knowledge. What constitutes good reasons for one interpretation does not qualify as good reasons for a different interpretation. If you become aware of new evidence then you amend your interpretation. This doesn't disqualify that looking out the window is good evidence for interpreting that it is raining. Most of the time it is, and still is even though you were mistaken once before.
Will do!
Knowledge refers to the correct interpretations. One can incorrectly interpret something (like the planetary orbits or the weather), but one can't incorrectly know something. As ordinary language philosopher Gilbert Ryle pointed out (bold mine):
Quoting Harry Hindu
That's right (though I would use the term justifiable instead of valid). But if your interpretation changes and you believe that you have the correct interpretation now, then you should also believe that you had an incorrect interpretation in the past. But only a correct interpretation can be knowledge, per ordinary usage.
Ah, but one non-goat won't be sufficient if eating is ongoing.
Something like that, yes. If you make a funny face now, and no one sees you, and you don't even see yourself in the mirror or film yourself, the face you made will be lost to the world (tragically :razz: ). A temporary, transient signal that nobody picked up.
Seen this way, billions of things happen that are never picked up or recorded. What did young George Washington ate for breakfast on February 5, 1742? And the next day, etc.
But are these technically "truths"? It was said by someone here, wiser than me, that truth is a property of statements. Certain statements affirm that x or y is the case, and these statements can be true if indeed x or y is the case, that is, if the statement accurately describes something real, and false if it doesn't.
In the absence of a statement such as: "George Washington ate eggs for breakfast on the morning of February 5, 1742", there would be no truth about it. In the absence of a sentient being questioning what happened, there's no statement being made, no description that can be true or not in comparison to reality. There's only reality. A state of affairs. Things take certain shapes, stuff happen a certain way. A face is made, a young lad eats a breakfast.
This is what is lost: not really truth, technically, but information.
That said, it is quite possible that our assumption that the past is immutable speaks to nothing more than our own prejudice. Of course we do think that is how things must be...
In his History of Russia and its Empire, Michel Heller states that "Nothing changes faster than the past". He was speaking of historiography: of the way the past is told, of the stories that politicians feed to the people. And in Russia, this national pseudo history changes every decade, to fit today's ideology. So the past changes all the time over there.
Goats [s]shave[/s] eat all those, and only those, that do not eat themselves.
Im not sure that this is the same, but seems similar.
What about Gettier problems? Justifed, true, beliefs that we have doubts in re claims of knowing 'em!
What about (hyper)skepticism, the position that nothing is knowable in any sense of the word "knowable", argued to based on Agrippa's trilemma, etc.?
Then there's solipsism we have to deal with. The only person I'm certain exists is me! I clearly am not omniscient!