Fitch's paradox of Knowability
p is an unknown truth = p is true & p is not known. F = p is true & p is unknown.
1. All truths are knowable.
2. F is true = p is an unknown truth (there are unknown truths) [Assume for reductio ad absurdum]
3. F is knowable [from 1]
4. Knowable that F [from 3]
5. Knowable that (p is true & p is unknown) [from 4]
6. Know p is true & Know p is unknown [possible if 3/5]
7. Know p is true [from 6 Simplification]
8. p is not an unknown truth [from 7]
9. p is an unknown truth & p is not an unknown truth [2, 8 Conj]
Ergo,
10. F is false = There are no unknown truths = All truths are known. [from 2 - 9 reductio ad absurdum]
OR
All truths are not knowable. Kurt Gödel (Incompleteness theorems)
1. All truths are knowable.
2. F is true = p is an unknown truth (there are unknown truths) [Assume for reductio ad absurdum]
3. F is knowable [from 1]
4. Knowable that F [from 3]
5. Knowable that (p is true & p is unknown) [from 4]
6. Know p is true & Know p is unknown [possible if 3/5]
7. Know p is true [from 6 Simplification]
8. p is not an unknown truth [from 7]
9. p is an unknown truth & p is not an unknown truth [2, 8 Conj]
Ergo,
10. F is false = There are no unknown truths = All truths are known. [from 2 - 9 reductio ad absurdum]
OR
All truths are not knowable. Kurt Gödel (Incompleteness theorems)
Comments (264)
How do you get "P is known" from "Assume proposition Q: P is unknown" and "Assume Q is known"?
So, if you know Q, you must know P
For example P: there is life on other worlds
Q: There is life on other worlds is unknown
So, knowing Q means you know P
If P is "there is life on other worlds" and if Q is "the truth of P is unknown" then to know that Q is true is not to know that P is true. In fact, to know that Q is true is to not know that P is true.
P is a truth: The USA has 1 president
Q: P is unknown: It is unknown that The USA has 1 president.
Now assume Q is known: We know It is unknown that The USA has 1 president.
Now we know that P is true AND that it was unknown.
1. x --> ?K(x) (All truths are knowable ie. it is possible that somebody knows x at some time.) (axiom)
2. P (Proposition P is a truth) (axiom)
3.a) ~K(P) (P is unknown) (axiom)
3.b) Q:=~K(P) (definition of Q)
4. ?K(Q) (Theorem from 1. and 3.)
5. K(Q) (Q is known) (axiom)
6. Theorem: K(P) (P is known)
Can you please clarify the proof of that theorem using this formal language? I don't think these axioms are enough to prove statement 6. Maybe your logic differs from this. In fact (6.) is not a theorem of the above system.
As long as I understand your premises are stronger than that of Fitch's paradox in existential terms.
While Fitch's paradox only assumes P and Q; you assume K(Q) also.
So first I don't see how we can prove 6. and even if we can get a contradiction from your axioms they may be stronger than the axioms of Fitch's paradox.
To be honest I don't really understand the paradox explained here:
https://plato.stanford.edu/entries/fitch-paradox/#ParKno
At (6) they get a contradiction and from that we can prove every statement (can't we?)
But they keep proving for 5 more steps for no apparent reason.
That doesn't follow. If we know that we don't know that the USA has 1 president then we don't know that the USA has 1 president, and so we don't know that P is true.
There is a good reason actually. It's because (6) is a contradiction that the premise of the argument must be discharged: that is, negated. That's how reductio proofs work. Step (7) just is the negation of the premise that logically led to the contradiction (assuming the validity of all the deductive principles that had been used in the previous steps).
(On edit: this was of course a comment regarding the proof as presented in the SEP article linked in Meta's post.)
That P is unknown, is known (Q). P being a part of Q, is, let's say, automatically known.
For example:
P: elephants are mammals
Q: P is unknown i.e it is unknown that elephants are mammals.
Then, if you know Q: it is known that it is unknown that elephants are mammals. That means we now know that elephants are mammals
No it doesn't.
Do you know what my hair colour is? No. Do you know that you don't know what my hair colour is? Yes. Therefore you know what my hair colour is? No.
If you know that you don't know X then you don't know X.
The goal of the argument was to show that {(KP),(NonO),(A),(B),(C),(D)} is inconsistent. To show that they looked for a contradiction. At step (6) they got a contradiction so for me that is the end of the proof. Their proof didn't end there and they found a contradiction at (9). This was weird for me; however the problem is not a big deal just a small technical detail.
Basically they prove {X,Y}?? by proving {X,Y}??=>~X by reductio; and then {X,~X}??
I think it wasn't the most elegant method but as I said it is not a big deal.
P: the color of your hair is red
Q: P is unknown: It is unknown that the color of your hair is red
Assume you know Q: It is known that it is unknown that the color of your hair is red.
By knowing Q, you now know the nested truth, which is: the color of your hair is red.
I think your mistake is not forming a proposition. Propositions like P have to be declarative sentences and not questions, as in your example i.e. ''what is the color of my hair?'' is a question
I think your grammar is confusing you (consider the apparent difference between "I don't know that aliens exist" and "I don't know if aliens exist").
The below formulation should make it clearer.
P: the colour of my hair is red
Q: the truth-value of P is unknown (the truth-value of "the colour of my hair is red" is unknown).
Assume you know Q: it is known that the truth-value of "the colour of my hair is red" is unknown.
By knowing Q, you don't know the truth-value of "the colour of my hair is red" (and so, if it's true, don't know that it's true).
Your formulation seems to conflate a meta-description (where we assert that P is true, and that some unspecified person doesn't know that P is true) with an ordinary description (where I assert that I don't know if P is true).
I think your mistake is in forming a proposition. It creates a grammatical confusion that can result in the contradiction given in 7). My wording here is far simpler, and shows that knowing that something is unknown doesn't entail knowing the unknown.
2. Proposition P is a truth
3. Assume proposition Q: P is unknown
If we take P to be "my hair is red" then we have:
2. "my hair is red" is true
3. Assume proposition Q: "my hair is red" is unknown
The problem is that 3 is ambiguous. Does it mean "the truth-value of 'my hair is red' is unknown" or does it mean "'my hair is red' is an unknown truth"? Note that the latter interpretation doesn't follow from the second premise. What you seem to have done is conflated 2 and P.
To better present Fitch's paradox, premise 3 should read "Assume proposition Q: P is an unknown truth".
Amounts to the same thing. Asserting P is the same as asserting P is true.
So, when I say Q: P is unknown, I mean P is an unknown truth. To show that that's exactly what I mean ''P is an unknown truth'' means ''P is a truth AND P and P is unknown''. There's no need to say ''P is a truth''. That's redundant.
Anyway, now that you agree there's a paradox, how do we solve it?
Then "P is false" would be a contradiction. So for any proposition P, P must be true. But that doesn't work. We're quite correct in saying "P is false", where P is the proposition that London is the capital city of France. You're conflating use and mention.
This isn't a case of asserting P, though. This is a case of saying something about P – it's mention, not use.
I think there is nothing paradoxical in the possibility that there are true sentences which are not knowable. The absolute conistency of arithmetic or set theory could be examples for this. The absolute consistency of these theories can be true but since we can only prove relative consistency we will never know if they are true or not.
edit: Another example: Let's say there are x particles in the universe. There are formulas which's length is 10^x^x^x^x^x. Some of them may be true. I don't think we would ever know the truth value of each of them.
Everyday experience evidences that we don't make statements like '' It is true that the sun is shining''. Rather we say ''the sun is shining''.
In formal logic too, an assertion doesn't have to qualify itself with the truth value ''true'' i.e. the proposition P means P is true.
However, falsity needs to be made explicit because simply stating P: ''the sun is shining'' is taken as a truth claim . Thus we say ''it is false that the sun is shining''. In logic this is ~P.
No contradiction.
''P is false'' is simply saying ''P is true is false''. So, P is false. There's no contradiction.
Fitch's argument proves that if one assumes that all truths are knowable in principle, then it follows that they must be knowable in actuality (that is, everyone is omniscient) - which is of course an absurdity. Here's how it goes.
First we assume the knowability principle:
KP. For every proposition P, it is possible to know P.
Now let us assume that the following conjunction is true (which by itself is a possible state of affairs):
a. P is true.
b. No one knows that P is true.
Now, according to the knowability principle, it is possible to know every P, so it follows that it is possible to know the conjunction of a. and b.. But it is impossible to know this conjunction: you cannot know that P and also know that nobody knows that P (since you do know it), so it follows that the conjunction of a. and b. is unknowable. But the conjunction says that there is a truth that no one knows, and this seems right, but the argument shows that it is impossible. So it follows that either we are omniscient or that the knowability principle is false.
And the paradox mainly consist in the fact that the knowability principle shouldn't entail such an absurd conclusion by a simple deductive argument (or so it seems - even if one doesn't accept the knowability principle, it still seems strange that it should entail such a conclusion).
That the language is second order can be seen from the implicit predicate in the statement
Quoting Fafner
This requires the existence of a unary predicate Know, which can take as argument any well-formed sentence in the language.
The paradox tells us about the problems of using unconstrained second-order languages, rather than telling us anything meaningful about knowledge.
If one of the standard ways of constraining higher-order logic to retain consistency (eg Russell's theory of Types), is invoked, the paradox disappears because some of the statements in the attempted proof cannot be made - they are syntax errors.
P: The statue of liberty exists: Statue of liberty exists is a truth
Q: P is an unknown truth: The statue of liberty exists is an unknown truth: The statue of liberty exists is a truth AND The statue of liberty exists is unknown.
Now the tricky part. I'll clarify as much as I can.
If I say ''Michael is a good man'', I mean to say, "Michael is a man'' AND ''Michael is good''. Decomposing compund statements into its parts is logically legitimate.
Consider now the statement: Q is known.
Q is known = it is known that P is an unknow truth = it is known that P is a truth AND P is unknown.
So, knowing Q, P is known to be true.
So the problemis with logic? Why is it then that it's relatively easy to understand the paradox using simple logic as I have?
(Y)
But there's no such thing as a theory of types, and there could be no "syntax errors" in a language (because every sentence in language can be potentially made sense of with the right interpretation).
It seems to me like a cheap trick to get out of a paradox by stipulating that you simply cannot say certain sentences. If you cannot put 'know' in front of every sentence then there should be a principled explanation why (because KP seems like a perfectly consistent thing to say, and some people even think that it is true), and so I do think that the paradox can teach us something interesting about knowledge.
Perhaps there are errors in my version of the argument. I'm only familiar with sentential logic and a little bit of predicate logic. May be you can do better. I'd really like to see that. Thanks
Maybe you got the right intuition but we need a logic to create a logical frame for your intuition.
If Q means (P and ~K(P)) ((Q means P is true and not knowable))
And we assume K(Q)
Then ~K(P) immediately follows.
That is a contradiction indeed; the similar to the Fitch paradox but not the same.
It's amazing that after so many explanations you can't get this.
By knowing Q you know that is is UNKNOWN whether the color of someone's hair is red. That doesn't imply that you know that P is true. It implies just the opposite.
It doesn't follow from that that you know the content of P. You're stipulating that you do not. You only know that it's true.
For P to be unknown, you either have to be saying that you don't know the content of P, or that you don't know whether P is true. Those are your only two choices, otherwise "unknown" wouldn't make any sense.
Sure, we can interpret a sentence to mean whatever we want it to mean. Thus, we can make sense of the famously uninterpretable sentence 'This sentence is false' by just interpreting it to mean 'Blue is a colour'. But I don't see how that is any way a useful thing to do.
I don't know what 'there's no such thing as a theory of types' means in this context. Here's the one I had in mind.
Quoting Fafner
That's not the way syntax rules work. Most syntax rules operate on a 'rule in' basis, not a 'rule out' basis. A positive justification is needed for a sequence of words being valid syntax - not just an absence of breaching any 'thou shalt not' laws.
Some people may not like that, but that's the way theories of language work. Without it, we'd be sitting around agonising over why we couldn't understand the sentence
'Unquestionably brick falafel entertain under'
I might sympathise with the 'cheap trick' complaint if the syntactic objection prevented us accepting a purported theorem that was highly intuitive. In such a case it would be natural to ask - is the problem with the theorem or with our syntax rules?
But in this case the purported theorem is completely contrary to our intuitions, and the syntax rules help us to understand why (ie because it is not a theorem at all). I would see that as case closed, with complete satisfaction, and intuition vindicated.
Quoting Fafner There's no problem with that statement, provided P is a constant, not a variable.
The restrictions to second-order logic that are needed to prevent inconsistency do not prohibit such a statement.
But if P is a variable, inconsistency will creep in, because we can then (unless prevented by other constraints) substitute a wff S containing KP for one or more instances of P in S, thereby generating circularity and in some cases infinite regress. It is often straightforward to generate a contradiction from such constructs.
That post omits the most controversial part, which is the assertion that all truths are knowable, so the post being simple does not mean that the purported proof as a whole is simple.
At most, the post proves that IF we know Q then we know P. But it does nothing to convince us that we do know Q.Quoting Fafner
I agree that this would not be very useful, but it is also the case that talk about ''syntax error" as a criterion for meaningfulness is also just as useless thing to say.
Quoting andrewk
Yes this is also what I had in mind. I said that there's no such thing as a 'theory of types' because in natural languages 'type distinctions' are constantly violated without rendering the sentences meaningless, so it's not clear what work logical type distinctions are supposed to do (e.g. we sometimes use names predicatively as in "he thinks he's Einstein" etc.).
Yes you can have a theory of types in formal language (maybe), but what's its use for explaining phenomena in natural languages?
Quoting andrewk
Ok, but what kind of justification is that?
Quoting andrewk
My point is simply that if you solve a certain problem in a constructed formal language, it doesn't by itself prove that you've solved the problem as it exists in natural language. Maybe it does, but it is something you have to demonstrate (and you cannot do this without taking a substantial philosophical position on whatever thing the problem is concerned with).
We say things like this, I say things like this, but don't forget Richard Montague, who swore up and down there's no principled distinction between formalized and natural languages. (There's a part of me that hopes he's right, but I can't even read him, yet.)
Sure. I just remind myself every time I say something like that that Montague was a helluva lot smarter than I am and he thought it was bollocks. I'm not in a position to argue on behalf of his view, just suggesting that it might not be wise to rely too heavily on the distinction. That's all.
Added: I still do it -- I used the distinction in another thread earlier today. I just feel a little less certain about it than I used to.
And how do you think 'P' is treated in the formulation of the paradox (say in the Stanford article) as a constant or a variable?
Sure. That's why I'm pointing out that the problem doesn't exist in natural language, because the proof is written in formal language. This isn't a case of a natural language statement that we all believe being unfairly torn down by formalism. It's a case of an attempt by Fitch to formally prove a natural language statement that nobody believes. So it is entirely pertinent to point out that the purported formal proof is syntactically invalid.
It's Fitch that chose to play by the rules of formal languages, not me.
If there's a natural language version of the purported proof, that a non-philosopher would accept as credible, we can discuss that but, so far as I'm aware, there isn't.
So, as far as I can see, there is no natural language problem to be solved.
Quoting Fafner It's a variable, because it's written Kp, preceded by a universal quantifier over p.
I think you cannot allow as a predicate anything that touches the logical constants or the syntax or semantics of your formal system. K has "true" in it, so you cannot let it run wild in a system that takes truth as a primitive. If you had a formal system that took colors as primitives, you could no doubt generate paradoxes by allowing "looks red" as a predicate.
Of course in ordinary English, there don't seem to be any restrictions on what you can say that might tame semantic (or logical or syntactic) predicates. I think there are two options for how to look at natural language paradoxes:
(1) It's down to the use you are making of the language whether something counts as a paradox. It's not a problem for the poet qua poet even to violate the law of contradiction.
(2) If a natural language is in fact an exceptionally complicated formal language, then the paradoxes tell you what the primitives of the language are, by showing you what leads to trouble.
(1) seems to undercut (2) but I'm not convinced it does. On the other hand, (1) still allows you to say that if your purpose in using language at the moment is reasoning, then certain predicates are off-limits.
I think the problem can be formulated without the use of formal language, that is, it doesn't arise merely because of some formal peculiarity of this or that notation. I don't think that we even have to use propositions as variables to formulate the problem. The KP principle can be formulated as a claim about all truths (as it appears in the Stanford article) rather then all propositions (as I formulated it), and then it can be easily shown that some truths must be unknowable.
Perhaps it has something to do with context: if we take a phrase such as "I know that P" and move it across different contexts, then there's no reason to expect that P should mean the same thing each time (think for example about Wittgenstein's "I know this is a tree" from OC 349).
My current, unfinished thoughts follow:
"Kp?p" is abhorrent. That's semantics intruding on syntax. "p" should only show up because it's assumed or derived (according to introduction and elimination rules), not because you have applied a specific predicate to p. That's insane.
So treat "K" as an operator, a primitive. We already have an elimination rule, so how about an introduction rule? No idea. But it still seems like an awful idea to me because it should be (a) shorthand for something else (the way "?" is, for instance), or (b) orthogonal to the other primitives. It is neither.
I think if you wanted a formal epistemic logic, you'd have to build that from scratch taking Known in place of True.
Maybe K is more like the modal operators and will only make sense there. So how does it interact with the others? I've played with that a little but I'm not even sure what the goal here is. And modal logic is not my strong suit anyway.
I think all of the writing about Fitch's paradox looks like it's taking place in some deductive system that includes the predicate "K" but it isn't really. I think it's just notation.
My understanding and, I would venture to guess, the understanding of the person in the street, is that the set of all truths is a subset of the set of all propositions.
Put simply: a pebble cannot be a truth, but the statement 'The object in my hand is a pebble' can.
So I can't see how framing the purported proof in terms of truths rather than propositions can help it to succeed.
If there are truths which have never been told, does it then follow, according to you, that there are propositions that have never been proposed?
It is possible that my interpretation of 'there are' in that sentence may be different from that of a hard-core Realist, but let's leave that aside for the moment, as it may not be important to the discussion.
To extend it then, are there truths which can never be known?
Also, would you be able to explain why you think that truths are a subset of propositions?
I expect there are truths that no human could ever know, because even the statement of the truth is too long to be held in a human brain. Further, for any organism of limited size, be it ever so much brainier than humans, there will be truths long and complex enough that the same restriction applies to them, just farther down the road.
As to whether there are exists a truth T such that we cannot imagine any finite organism - be it as large and brainy as we wish to imagine it - ever being able to understand it, I don't currently have an opinion on that.
But ... (thinks) ... what about truths that contain an infinite amount of information? For instance, consider the infinitely-long proposition that states that the decimal expansion of Pi is
If there's a God, then it can know it. Do we count God?
Quoting John That is simply my understanding of how the word is used. If there is a significant group of people that use it in some other way, I have yet to encounter them.
I know Farrell Williams says 'Happiness is a Truth', but I think he is speaking poetically, not literally. :D
The word is used in several different senses, one of which is also applicable to the word 'fact'. That is both are sometimes used more or less synonymously with 'actuality', as distinct from 'proposition'. I was wondering whether your reason for counting truths as a subset of propositions is that there are both true and false propositions, but you haven't explained your reason, so I can't tell if that is correct.
That isn't my reason. My reason, which I did give, is that my observation is that most people use the phrase 'a truth' in that way and, in order to facilitate communication with others, I adopted what I judged to be their practice.
But if it helps advance the discussion - which I am finding jolly interesting - we can assume for the sake of argument that I accept the reason you gave (which sounds OK to me) and see where it leads us.
OK, great, my reply would then have been that for every proposition that is a falsity there is an equivalent proposition that is a truth, namely the proposition's negation.
Also, in regard to you saying that the word is generally used in the sense of being a subset of 'proposition' I would have thought that was only in the sense that there are understood to be false propositions. Now, in light of my qualification of that it seems to me that the set of truths just is the set of propositions.
Then you don't agree that for every false proposition there is an exactly equivalent true one? If so, then perhaps a counterexample is in order?
The issue of the infinitely long decimal line is actually finite. Our problem is we just can't think enough of the discrete numbers at once to name the all-- we might say that we actually know an infinite here (the neverending nature of decimal numbers), but we lack knowledge of the finite (the many discrete numbers in the infinite set).
Thinking like this changes the question of knowablity. The infinite doesn't present any problem to knowledge. Limitations on knowledge are drawn from that any instance of it discrete and finite.
Someone cannot know everything because any instance of knowledge poses a logical distinction that excludes all else-- if I know "1", I do not know "2" in that moment. This limitation even applies to any existing omniscient being.
Even if someone managed to know everything, all at the same given "point" (a bit like Dr Manhattan experiencing all times of his life at once), each instance of knowledge would still be a seperate logical entity. Despite knowing everything all at once, the omniscient being would still not know everything in any one moment.
I suspect we must be at cross purposes here.
[/quote]
What makes you you think that?
Here is the proposition that states the decimal expansion of e.
sum (k=1 to infinity) 1/k! = 2 + 7 * 10^-1 + 1 * 10^-2 + 7 * 10^-3 + 2 * 10^-4 + ........
the sum of terms on the RHS goes on forever, and cannot be condensed.
Here is the nice clear demonstration that I knew I had read, but could not find, of how unconstrained second-order logic is inconsistent. I highly recommend it.
The relevance to this thread is that, since unconstrained 2nd-order logic is inconsistent, ANY proposition can be proven in that system, and so can its negation (by the Principle of Explosion - cf Bertrand Russell's proof that he is the Pope).
Hence, since Fitch's Paradox uses unconstrained 2nd-order logic, with some modal quantifiers thrown in, we can conclude nothing meaningful from any contradiction it may derive.
Perhaps we are at cross purposes. The way I see it all propostions are both true and false depending on which way you look at them. So true and false are both functions of either truth or falsity, and there is just one set of propositions, really.
Re this: "you cannot know that P and also know that nobody knows that P (since you do know it), so it follows that the conjunction of a. and b. cannot obtain."
That's not the case. It's only the case that both KP and a&b can't obtain. a&b would be fine on its own.
At any rate, this is easily solvable under my epistemology. There are no propositions that someone doesn't know. The idea of that is nonsensical. Propositions only obtain, and truth-value only obtains, when someone has the proposition or the truth-value judgment in mind.
Which is exactly what I said... You can't know the conjunction of a&b.
Quoting Terrapin Station
This is not a solution because you change the subject. The paradox is directed at someone who believes that both a) there are unknown truths and b) all truths are knowable in principle, and the challenge is to show how both can be true at the same time.
You wrote: " so it follows that the conjunction of a. and b. cannot obtain." That's false. The conjunction of a and b can obtain. The conjunction of KP and a&b is what can't obtain.
Quoting Fafner
It's not changing the subject, it's just saying that "there are unknown truths" is false. That's the same subject. It's just disagreeing with the premise. Otherwise you'd have to say that any response (to anything) that rejects a premise is "changing the subject."
Oh I see what you mean, yes I made a mistake in my formulation, I'll fix it.
Quoting Terrapin Station
As they say in the Stanford article, the paradox is interesting because (a) and (b) don't seem to be mutually inconsistent (and thus it is surprising if they are), and this is something that people who don't accept one of the premises can agree about.
Yeah, I agree with that. It's not obvious that buying both (a) and (b) would be a problem, and the paradox, especially in light of buying (a) and (b) doesn't seem to have any obvious problem either.
( @Banno tells me in another thread that it's material for a doctoral thesis; those interested, I just ask that you quote TPF as your source of inspiration :smile: )
The seeming paradox is due to adopting a point of view that lays outside of the world of human experience, outside of time and space, the POV of God. If you take time and the human condition into consideration, the Fitch's paradox simply disappears.
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. Maybe that's what Fitch tried to prove?
Note - this is a four year old thread. It would be helpful if you would point that out when you post. That being said, here is my response:
Lamest paradox ever.
Quoting Olivier5
I agree it makes no sense to say that a proposition about something unknown is true; the most that can be said is that it could be true.
2. All truths are knowable
3. Possible to know F [from 2]
4. We know F = Known that (p is true & p is unknown (true/false?)) [assume for reductio ad absurdum]
5. Known that p is true & Known that p is unknown [knowing a conjunction is to know each conjunct]
6. Known that p is true [5 Sim] = p is known
7. Known that p is unknown [5 Simp]
8. p is unknown [7, knowing the earth is round implies the earth is round i.e. knowing q implies q]
9. p is known & p is unknown [6, 8 Conj]
10. We can't know F [4 - 9 reductio ad absurdum]
11. p is true & p is unknown [assume for reductio ad absurdum]
12. Knowable that p is true & p is unknown [from 2]
13. Knowable that F (from 1, 2)
14. We can know F (from 13)
15. We can know F & We can't know F [10, 14 Conj]
16. ~(p is true & p is unknown) [11 - 15 reductio ad absurdum]
17. ~p v p is known [16 DeM]
18. p implies p is known [17 Imp]
19. All truths are known! [from 18]
An example!
1. All truths are knowable
2. p & ~Kp = F = there's an unknown truth
3. F is knowable [from 1]
4. K(p & ~Kp) [assume for reductio ad absurdum, from 3]
5. Kp & K~Kp [knowing a conjunction implies knowing the conjuncts]
6. Kp [from 5 Simp]
7. K~Kp [from 5 Simp]
8. ~Kp [Kq implies q]
9. Kp & ~Kp [6, 8 Conj]
10. ~K(p & ~Kp) [4 - 9 reductio ad absurdum]
11. ~LK(p & ~Kp) [from 10, ~Kq implies ~LKp]
12. p & ~Kp [assume "there's an unknown truth" for reductio ad absurdum]
13. LK(p & ~Kp) [from 12, 1]
14. LK(p & ~Kp) & ~LK(p & ~Kp) [11, 13 Conj]
15. ~(p & ~Kp) ["there's no unknown truth" 12 - 14 reductio ad absurdum]
16. ~p v ~~Kp [15 DeM]
17. ~p v Kp [16 DN]
18. p -> Kp [17 Imp]
19. If a proposition (p) then p is known
Legend:
L = possible
K = know
Kp = known p
How?
Do it then. Post a picture and show me a truth.
Incorrect assumption. Some truths are beyond the knowability by humans, by way of complexity or escaping detection.
Prove it!
Quoting god must be atheist
Why?
If truth must be in the form of a proposition, then there is no unknown truth because there's no such thing as a realm of already formed English sentences waiting to be discovered. A proposition must be proposed by someone before it can exist.
It has to if you're right.
Quoting Olivier5
1. All truths are propositions [you disagree but haven't been able to make your case]
2. All propositions are known [Fitch's argument]
Ergo,
3. All truths are known. [from 1, 2]
Quoting TheMadFool
Nope. The procedure only makes sense if truth can only be expressed in words. It's begging the question.
Quoting TheMadFool
You haven't been able to make yours either.
Quoting TheMadFool
Only because there's no such thing as an "unknown proposition".
You denied that truth has to be propositional. I asked you to prove it. You said a picture would do the trick. I pointed out that every object in the picture you posted has a word assigned to it - demonstrating my position on the issue or, as it matters to you, disproving your assertion that truth can be non-propositional.
In short, I refuted your argument because I assumed your position not mine. So, if there's any petitio principii, it's not me.
Name a truth that's not propositional. We're going round in circles.
[quote=Wikipedia]Truth is the property of being in accord with fact or reality.[1] In everyday language, truth is typically ascribed to things that aim to represent reality or otherwise correspond to it, such as beliefs, propositions, and declarative sentences.[/quote]
If I name it, I make it propositional. But okay, maybe you are right. Truths are descriptions of some state of affairs, therefore there is not such thing as an unknown truth.
Funny.
This can only be proven by an example. And if I know the example, then it is impossible to use that example.
On the other hand, without proof it is acceptable, that the human mind is only capable of some complexity but not of all complexity. For instance, religionists will tell you that god is so complex, that we can't fathom his thoughts. This is an example which has no proof value, but enough creative force to make you see the point.
The escaping from detection is easier to see. We sense the world and create our thoughts based on our senses. For truth we have to rely on a model of the world which model we built relying on our senses. However, we can't trust our senses. Maybe they relate to use reality, maybe they don't. Thus, all the knowledge and truth we have accumulated about the world and its truhts, may be misguided, and completely off. Again, how does one prove this? It is completely unprovable but totally conceivable.
7 is invalid reasoning, because you drop off an assumption that can't be dropped. You use the effect of this "drop" in the argument later. However, the knowledge that p is true, does not affect whatsoever the fact that p is not known. The two are independent. Not related, yet both apply. Therefore you can't drop one of the two (and you also can't drop both of them).
Right, but oddly I believe it is still controversial, at least in certain circles. Perhaps that's a testament to how it is possible to become confused by predicate logic in ways that you wouldn't if you thought in plain langauge.
Hmmm... time... okay:
[hide="Some Python code"][/hide]
Its output:
The current time is 2021-10-01 20:03:44.341319.
1. "IHLLVJCU" is lexically prior to "VDTSHSGB" but successive to "EPOOTTLS".
2. "MACVLKEG" is lexically prior to "YGDUUBSU" but successive to "KCQCCFJB".
3. "RGTVFTLM" is lexically prior to "UHVHSFPG" but successive to "HYQSJOIO".
4. "MVEXIWWB" is lexically prior to "WWZISWGD" but successive to "HEDIMULP".
5. "MCCRNUUP" is lexically prior to "RLDYLGBP" but successive to "EMJAPWVJ".
6. "QMCVNBOO" is lexically prior to "SHXCBJYN" but successive to "BJJZBYPU".
7. "TDIVFGHM" is lexically prior to "UMJXUXXY" but successive to "JBIVRFWT".
8. "KEJBOCEO" is lexically prior to "WQKQFLJC" but successive to "HBSLGRPO".
9. "LIUJYHWQ" is lexically prior to "OUAJAFZK" but successive to "CROCJGNY".
10. "VCINWCVZ" is lexically prior to "YRLIFYUF" but successive to "NVWXWPXE".
There are 10 labeled "things" here. I'll pick, I don't know, 6 to talk about. I think 6 is a proposition. I also think it's true. In fact, I'll make a blanket claim. Every time this program is run, if you read what is generated by it, it will be a set of true propositions.
And yes, I have a mind. And yes, I wrote this program. But I did not write proposition 6.
Quoting Olivier5
Who wrote proposition 6?
Quoting Olivier5
Proposition 6 was generated more or less around 10:03:44pm local time on this day October 1, 2021. But nobody proposed it at 10:03:44pm. In fact, nobody read it until it least 10:04:44pm.
Quoting Olivier5
Why not?
Quoting Olivier5
Not sure what you mean by exist. There is some code that executed at 10:03:44pm. Is that when proposition 6 began to exist?
Quoting Olivier5
At 10:03:44pm proposition 6 was an encoding of a true statement whose physical form was that of particular stable states of a set of bistable mechanisms. At 10:04:44pm the states began to modulate particular areas of a 4K LG monitor in such a way that a mind belonging to a native English speaker, for the first time, could read it.
Quoting Olivier5
I could tell that proposition 6 would be true prior to running the program. I can ask the question of whether proposition 6 would be true of a future run of the program right now. And yes, it will be.
There are two meanings / usages of the words "true" or "truth." @TheMadFool gave one definition from Wikipedia a few posts back, I'll repeat it:
"Truth is the property of being in accord with fact or reality.[1] In everyday language, truth is typically ascribed to things that aim to represent reality or otherwise correspond to it, such as beliefs, propositions, and declarative sentences."
This is how the word "truth" is used in the legal system in USA (and I assume most countries). When a witness says that they will speak the truth, the whole truth, and nothing but the truth, the witness is swearing that their statements will correspond to reality to the best of their abilities.
But there is another definition/usage of the words "true" or "truth" - and that is within mathematics / logic. Statements / propositions are true if they can be derived according to the basic axioms of the particular mathematical/logical framework under which they are generated (the basic axioms are defined as true). A particular proposition may be true in one framework and false in another.
I know that there are some very smart people who believe that mathematics is "real" in some sense of the word, but I'm ignoring that for purposes of this particular discussion.
So. When I look at the output of the Python program, these lexical strings can be converted into numbers. So when the program prints the output that string "A" is lexically prior to "B", this is simply another way of saying:
For all integers x & n (where n > 0), x - n is always less than n
and likewise, when the program prints that "C' is lexically successive to "B", this is
x + n is always greater than n.
It seems like all this program is doing is generating random numbers and ordering them according to the the rules of standard arithmetic; i.e., this is within the context of a math framework and is not about the real world.
But maybe I'm not getting the point (happens on a regular basis)
= = = = = = = = =
Meanwhile, how does all this relate to the OP (Fitch's Paradox)? I'm not sure. I'm an amateur at this stuff - but I tried plowing through the Stanford discussion. It's very dense - and truth be told my eyes glazed over fairly quickly. The thing that jumped out at me is that it Fitch seems to mix both definitions of the word "truth": it introduces an "epistemic operator" K which means that ‘it is known by someone at some time". I can't help but be suspicious of this "epistemic operator" since it entails knowledge of the real world. The article points out various objections to this usage but does not draw any conclusions one way or the other.
But beyond that, I would disagree with the statement that all "truths" are knowable - i.e. all sentences that correspond with reality are knowable. Given the inflation that happened during the beginnings of the big bang, portions of the physical universe are outside our event horizon and are not knowable. Of course future scientific discoveries could that statement.
The point here is... well, phrased as a challenge, but really... to get Olivier5 to clarify some of his claims about when propositions exist, where they could possibly come from, and whether or not they really do need a "proposer".
When we talk about truths. there has to be something which is about reality, that something then checked if it corresponds to reality - if it does, that something is said to be true or is a truth.
With propositions, we have that something viz. propositions.
If someone claims truths can be nonpropositional, I'm at a loss as to what it is (the something) that can be true.
So you have complaints about the natural deduction rule simplification. Care to expand on that a bit.
1. The sun is hot & Grass is green.
Ergo,
2. The sun is hot. [1 Simplification]
Now,
3. Know p is true & Known p is unknown
Ergo,
4. Know p is true [3 Simplification]
Quoting TheMadFool
S1 = {x: x is a true proposition} = Set S1 is the set of all possible true propositions.
S1 contains those propositions that have already been made and those that are yet to be made.
S2 = {x: x is a true proposition that's already made} = Set S2 is the set of already made true propositions.
Fitch's argument is made with set S1 in mind. In fact, according to Fitch, S1 = S2. There can be no true proposition that's such that it is unknown; in other words, every true proposition is already made and known.
You think your machine proposed this proposition? But how can it be, when the machine has no clue what it proposes? The machine just organizes ink on paper or pixels on a screen according to your instructions. It cannot understand what it 'writes'. It doesn't even know what a proposition is. So it ain't proposing anything. YOU, when you read the output, understands it a certain way, to mean a certain thing. You then create the proposition -- understood as a meaningful sentence -- based on what is for the computer just dots on a screen or on paper.
I do? Why then would I write this?:
Quoting InPitzotl
Quoting Olivier5
Yes. At 10:04:44pm.
Quoting Olivier5
That's a funny use of the word "create". Incidentally, you also have funny uses of the word "author", "stated", and "phrased":
Quoting Olivier5
I did not state proposition 6 at 10:04:44pm. I did not author proposition 6 at 10:04:44pm. I did not phrase proposition 6 at 10:04:44pm. Now, we need not actually interpret the things you say in this quote as being correct, such that we're forced to say the program stated, authored, or phrased proposition 6. You could just be wrong.
If Bob typed this up:
11. "PJZVOWMW" is lexically prior to "YEMRBVGD" but successive to "KFJZTEOI"
...at 6:51:03am, and I read that at 6:52:03am, I did not create proposition 11.
What I did at 10:04:44pm is what I would have hypothetically done at 6:52:03am were I to read Bob's proposition 11.
Quoting Olivier5
It makes sense to say that if the program is run at 7:05:00am, it will generate true propositions that no one knows about until 7:06:00am. It makes sense to say this program will generate only true propositions, as opposed to false propositions, as opposed furthermore to all sorts of non-propositions including gibberish.
It certainly makes a lot more sense than saying that one can author propositions by reading them.
ETA: Here is roughly what I think I'm doing. You're generally proposing that there's a time relationship here: First, a proposition is proposed by a proposer (and thereby understood). Then, we can ask whether it's true or not. Finally, we can answer it.
I've arranged a scenario where this is flipped around. First, we can say the propositions will be true. Then, the propositions are created. Only after that, they are read and for the first time understood.
No, it does not. An unproposed statement cannot be a proposition; at best it is an unproposition.
You did take what was a bunch of dots on paper and you did make a proposition out of it by assigning some meaning to it.
p = a proposition (truth is implied)
Kp = p is known
Let's go over this together. The conclusion of Fitch's argument in formal logic is: [math]p \rightarrow Kp[/math] which in plain English means: all true propositions are known propositions. In other words, all true propositions exist as fully formulated propositions in some mind capable of making propositions and they are known to be true.
The set of true propositions = The set of made propositions.
There was no paper. As mentioned, it was a 4K LG monitor. This actually happened; it was not a thought experiment.
Quoting Olivier5
Yes, but more than that. I didn't just read gibberish and just say, you know what, let's call that a label, and attach this meaning to it. I read natural English sentences and interpreted their meaning as I would if Bob himself wrote it.
If I see an animal walking by, I might look at it and say, "awww, what a cute little fox!" When I do that, I do not create a fox. Nor do I, on saying this, make an animal be a fox. Likewise, when I read proposition 6, I could look at it and say, "awww, what a cute little grammatically correct true English sentence!"
[math]p \rightarrow Kp[/math]
p = a true proposition
Kp = p is known
I'm afraid, you'll have to figure this out for yourself. Good luck.
Indeed, first you define a class of propositions of the type X > Y. This implies the assumption or definition of an ordinal (classified, indexed) set, two different elements if which can always be attributed one and only one of the following propositions: A > B or B> A. Like the set of natural numbers.
Then you create some code, i.e. a text, that translates all this in a set of instruction a machine can process and chew on. The machine produces an output of the type: X > Y after generating X and Y randomly.
What the machine does is follow your instructions to generate particular examples of an arbitrary humanly-created ordinal classification system. The machine doesn't even know it's doing that, not anymore than a mechanical clock knows what its needles mean for us.
Quoting InPitzotl
That's a stretch. Your machine-generated "sentences" would strike an odd chord in a natural conversation between people.
Quoting InPitzotl
That makes no difference to the argument. The machine doesn't know the meaning of what it writes. It is just arranging pixels on a screen the way you told it to.
I still feel like you're playing catch up from your poor reading comprehension skills. You misunderstand even the basic nature of the problem. You keep trying to tell me what the machine isn't doing, as if it solves the problem before you. What you seem to have failed to grasp is that the fact that machine isn't doing things is the problem before you.
Quoting Olivier5
You quoted, and therefore are allegedly objecting to, this:
Quoting InPitzotl
But proposition 6 (a) is grammatically correct, (b) is English, (c) is true.
We're still left with a problem. During the minute from 10:03:44 and 10:04:44, there is something that:
Quoting Olivier5
...it makes sense to say exists (like the fox, before anyone sees it):
Quoting Olivier5
...that is a proposition, and that is true. This is demonstrated by my ability to meaningfully say that this program generates only true propositions.
Right. So when my little sister's doll used to say "Maman" and "J'ai faim", it was not just playing a recording? It was actually stating the proposition: "I am hungry"? I need to call my sister. It never occured to her that the doll was meaning it.
And when I use a pen to write on paper, my pen and my paper are the ones doing the writing.
And when you greet a friend over the telephone, you are actually greeting your telephone.
A recording plays back something that happened in the past. Proposition 6 didn't "exist" in any form at all until 10:03:44pm October 1; unless we're appealing to some mathematical sense of existence in which it's a set of the possible set of strings of a certain length of something like that.
Quoting Olivier5
"QMCVNBOO" is lexically prior to "SHXCBJYN"; and likewise "QMCVNBOO" is lexically successive to "BJJZBYPU". But your sister's doll isn't hungry.
You are not even remotely in the ball park of replying to me; you seem obsessed with pinning me on something I explicitly denied in the same post I put the program in. Go back and reread that post.
In the mean time, let me phrase it this way. At 10:03:44pm, I did not know what proposition 6 was. Nevertheless, I knew it was a proposition, and I knew it was true. So at 10:03:44pm, I can call it a true proposition. None of the things I'm saying in this paragraph have anything to do with your sister's hungry doll or my Dell Inspiron 3847 appreciating English sentences.
If my sister's doll was not saying "j'ai faim" but instead "This is a random noise" and would then make a random noise (as she was found to do), it would be exactly like your computer. It would have said things like:
"This is a random noise: WOOOEEZKREW."
"This is a random noise: RATABOOM."
"This is a random noise: POOPOOPIDOO."
And it would have been right all the f.....g time! Why? Because someone programmed it to produce random noises and say "this is a random noise" in rapid succession.
The argument of the recording doesn't hold either, because you did record the phrase "is lexically prior to" in your code, and it's the only meaningful part of the output sentences, just like for the doll...
This must be some new meaning of "exactly like" that I have been previously unaware of. The way I read "exactly like", it means something like "like in all respects". Given there are six permutations of strings in the template I have, only one of which would formulate a true proposition; and there's only one form in your template; these two things clearly are not "exactly alike". The means by which the one permutation of substitutions applied to the template turns out to be the requisite one to form the true proposition is the call to Python's sorted method. But surely if you recognize that the program doesn't know what the proposition is, you should recognize that the program doesn't know it's sorting those strings.
But the program doesn't have to know that, because the program isn't calling this a true proposition. I am.
Quoting Olivier5
That is a red herring. No amount of waffling on about meaningful parts of phrases, including these particular ones, and whether those parts are "recordings" has anything to do with the fact that your sister's doll isn't hungry or the fact that "QMCVNBOO" is lexically prior to "SHXCBJYN".
It's not. It's the core of my argument that by coding in this phrase "is lexically prior to", you created a pattern your computer would follow to compose sentences that have nothing new in them, that teach us absolutely nothing new. These sentences are mere recordings, and that's all there is to it, in this particular case.
Even in a more sophisticated case of a program writing poetry, the sentences produced are not actually understood by the machine, and therefore it is hard to say they are proposed by the machine. Rather they are produced mechanically.
Not all sentences are propositions. Maybe your computer does not really mean it.
And even if, for the sake of the argument, I accept that my sister's doll really meant what she said and that your computer really proposes something, what problem does that pose exactly?
But that still has nothing to do with whether propositions are true or false. Consider that we humans repeat things humans say all of the time, at the word and the phrase level; it has nothing to do with whether the thing we're saying is true or false. I'm sorry for you that it's your core argument, because this "argument by reusing parts" shtick is DOA.
Quoting Olivier5
It is unlikely that proposition 6 has been assigned a truth value in human history prior to 10:03:44pm on October 1.
Quoting Olivier5
Proposition 6 is a statement about the relative ordering of the strings "QMCVNBOO", "SHXCBJYN", and "BJJZBYPU". The first time these three strings were lexically compared in human history is very likely on October 1, 10:03:44pm.
Quoting Olivier5
That sounds like a you problem, not a me problem. I didn't say propositions need to be proposed by a proposer. You did.
Quoting Olivier5
Sure. But out of the 10 propositions displayed by the program's output, 10 out of 10 of them permute the strings in the requisite 1 out of 6 ways for each statement to read as a true proposition, and 0 out of 10 of them permute the strings in the 5 out of 6 ways to read as a false proposition. I can be sure before running the program that this would be the case.
Insofar as I can be sure these would read as propositions, I don't have to wait until I see them to call them propositions; think of this as my having sufficient reason to categorize these "things" as propositions without direct examination. Insofar as I can be sure they will read as true propositions, I do not have to wait until I see them to call them true propositions. Insofar as I can be sure of both of these things, I can generically say of the program that it will generate true propositions. And that conflicts with your claim that there's no sense in which I can say that (at least by one reading, but it seems to be the one you insist on defending).
But we mean it, when we do so. A mere recording or mechanical production of a sentence cannot invest meaning in that sentence. And a sentence without meaning or intention is not a proposition.
Quoting InPitzotl
Ok, for the sake of the argument... So what?
Sure.
Quoting Olivier5
Mmmm... sort of.
Person A can write an English sentence on a sheet of paper in the form of a string of some length of English letters, spaces, and punctuation; and "slip this under the door" to Person B in such a way as to communicate meaning to Person B. On doing so, the strings on the slip of paper do not have any inherent meaning; rather, the strings encode meaning in the form of a language... in this case, English.
Machines can generate strings that have no inherent meaning. Of the strings that have no inherent meaning that machines can generate, some of those are strings that encode meaning in the form of a language. Machines don't need to understand the language or be persons or be proposers or be actors or whatever to produce said strings. The fact that certain strings "are English sentences", some of them "are propositions", and some of them "are true propositions" builds absolutely no fence of any sort preventing machines from generating strings in these categories.
Quoting Olivier5
Sure.
But it does not follow that if a machine produces a string that the string does not convey meaning (e.g., that the string is not an encoding of meaning using the encoding scheme of the English language). Given a particular such string, I can test if it has meaning in English by looking at it and attempting to read it. But this is not the only method I can employ to tell that strings encode meanings in English. Likewise, I can read some statements in English that propose a particular thing to be the case, and on understanding what is being proposed I can test whether the statement actually is true. But once again, this is not the only method I can employ to tell if strings are propositions that are true.
I've demonstrated another method of doing exactly these two things.
Monkeys hitting randomly at a typewriter could produce English sentences too. So fucking what?
This is what I meant. You drop an assumption that can't be dropped. If you don't drop the same assumption, then you can't arrive at 9 and at 10. Hence, your drop is invalid.
Let me give you an example which is not your ill-put together attempt at proof.
George is 6 feet tall, and George is a boy.
Know that George is 6 feet tall, and know that George is a boy.
Know that George is 6 feet tall. (By simplification.)
Therefore George is not a boy.
This what I wrote about George follows the same structure that you employed. Find the logical mistake in it, and you found the logical mistake in your argument.
Quoting TheMadFool
... but you claim in your parallel structure of the original argument, that the Grass is NOT green. That you can only claim because you drop the assumption that the grass is green.
Quoting TheMadFool
In this parallel example of the original argument, you drop the bold faced assumption in order to arrive at the bold-faced italics result. But you contradict your own assumption in this move, so the logic is ill, it is faulty. The fault with the logic is that in the simplification process you drop something that may not be dropped -- not allowed to be dropped.
If a proposition has an element that is essential to the proposition, you can't drop it. Yet you do that precisely.
Sorry, Mad Fool, this is the extent of my capability to explain your error. If you still don't understand my argument, that's fatal, because I can't provide a better one. So please don't ask for a better one.
It doesn't lead me anywhere. I already knew all of this stuff. But it implies that a proposition does not need a "proposer". It also implies that a proposition does not need a "proposer" to be a true proposition. All a proposition needs to be created is to be some string that something creates.
Quoting Olivier5
Sure. But they're highly unlikely to do so. By contrast, the program that I wrote is certain to produce true propositions.
Well then, that thing is the proposer. If your computer is proposing a proposition, it is the proposer of this proposition.
Quoting InPitzotl
That's only because you limit it to very simple arbitrary statements mechanically derivable from arithmetic. So computers can sort letters alphabetically. Big deal. Try and have your Inspiron 3847 answer questions about real states of affairs, like elephants and castles for a change.
Siri? How many castles can fit in an elephant?
Sure; I'm fine with that too, so long as we don't suppose proposers understand things.
Quoting Olivier5
Of course. I programmed it to generate true propositions.
Quoting Olivier5
Computers can do lots of things.
Quoting Olivier5
The point is to correct you, not to impress you.
Quoting Olivier5
Not to be rude, but my voluntary role in this forum isn't to do tricks for your amusement; especially if you're going to ask for something so banally trivial it's pointless like a coding of a math equation or something complex like a 3d packing problem solver using irregular shapes.
This is your take. Mine is that we have to suppose understanding for there being a proposer. But irrespective of which perspective you take, whether machines can be 'proposers' or not, it makes no difference to the argument that a proposition must be proposed by a proposer in order to exist.
I'm confused. You're now saying my program understands things?
I think we're done. Clarify your position, then get back to me if you want to engage me.
K = known
L = possible
LK = knowable
1. (p & p is unknown) = p is an unknown truth = p & ~Kp (this is a truth)
2. All truths are knowable
3. K(p & ~Kp) = p is an unknown truth is known [from 2, assume]
4. Kp & K~Kp [from 2, knowing a conjunction is to know the conjuncts]
5. Kp [4 Simp]
6. K~Kp [4 Simp]
7. ~Kp [from 6, [math]Kq \rightarrow q[/math]]
8. Kp & ~Kp [5, 7 Conj]
9. ~K(p & ~Kp) [3 - 8 reductio as absurdum]
10. p & ~Kp [assume]
11. LK(p & ~Kp) [2, 10, if true, knowable that true]
12. ~LK(p & ~Kp) [9, if not knowable, not possible that knowable]
13. LK( p & ~Kp) & ~LK(p & ~Kp) [11, 12 Conj]
14. ~(p & ~Kp) [10 - 13 reductio ad absurdum] {there are no unknown truths}
15. ~p v ~~Kp [14, DeM]
16. ~p v Kp [15 DN]
17. [math]p \rightarrow Kp[/math] [16 Imp]
18. If there is a proposition then, that proposition is known
QED
Where does Fitch commit an error, make a boo boo?
F**k!
In assuming that this applies only to true propositions. In fact it applies to any proposition, true or not. An unknown proposition is an unproposed proposition. It's like an unthought thought: a contradiction in terms.
The propositions that are known are those that exist. They exist because they are known. There is no reservoir of unknown propositions out there, waiting for us to discover them and propose them. We make them propositions, or our computer surrogates.
When one makes a proposition or knows a proposition, truth is implied.
p (a propisition) = p (is true)
A proposition can be false, yes but when you state it, it means you're claiming it to be true.
Thanks @TheMadFool!
Yes, but Fitch's paradox is about true propositions. Restrict the domain of discourse to true propositions and you'll get it, hopefully.
I explained it three times, I think. One gets tired of pointing out the same mistake. If you don't get my reasoning, and stick to your guns, it does not make Fitch's proof valid.
This is the point in reasoning when one would want an independent judge who is reasonable. Without that, it seems there is no way of convincing you of Fitch's error. And even the judgment of an independent thinker (who is not biased for you or for me) would not make a difference in your conviction. So why are we doing this? If someone makes an error in reasoning, and showing him his error makes no difference to anything in the world, then the debate is fucked. I tried to convince myself of that, and I stayed away for a while, but I get sucked in by incorrect reasoning and feel compelled to point out the mistake in it. Unfortunately nobody on this site is receptive to criticism, and if they can't fight it with logic, then they start to hate those who nailed them.
In this sense I am proud to be one of the most hated persons on this site.
That's another reason to quit here. No emotional support... yes, it is needed, if one encounters one failure after another of logically convincing others of the truth. It gets to you after a while, you feel like you are running in a ferret's wheel.
Enough if this already. It is really impossible to make you see your mistake, mad fool. And you are not the only one... all people who make propositions on this site are like you.
Enough of charging windmills. Enough of wasted brain cells. Enough of fanatic thinkers.
I know. I'm extending it to false propositions as well. Sue me.
Quoting Olivier5
[math]p \rightarrow Kp[/math]
To what end, may I ask?
You cannot.
¬p?K¬p
Let q=¬p. Then ¬p?K¬p is simply q?Kq, which is the same as p?Kp (under a change in labels).
.
I must agree. If there was a proposition that was not known, what would make it a proposition? The idea of propositions in general, not created, hence not known, is fine, but the idea cannot be its own object.
Furthermore, it is true all propositions are known, iff the negation....
1.) is a contradiction, in the form all propositions are not known. “All propositions are known” is itself a known proposition, therefore the contradiction holds, or,
2.) is an impossibility, in the form not all propositions are known. In which case, some proposition not known must be proved, and the proof of it necessarily manifests as that proposition, which is then known, therefore the impossible unknown holds.
Another way to look at it is, the truth of P as such, relates a conception to itself. Any proposition in which the subject and predicate are subsumed under the principle of identity, cannot be falsified. If there is P, or when there is P, it is analytically true P is. Here too, the negation is also true, insofar as if or when there is no P, then P isn’t. It follows that to suppose it cannot be known whether or not P is or isn’t, is patently irrational, bordering on the pathologically stupid.
Mike drop, exit stage right.......
(Or maybe.....enter giant hook, yank speaker by the neck stage right)
We just finished this. Proposition 6 was a proposition on October 1, 2021, at 10:03:44pm. At that time, nobody knew what proposition 6 was. But at that time, I knew that it was a proposition. To know S is a proposition, it is not necessary to know S.
Why is this so difficult? And what is with this obsession to demand that propositions cannot be propositions if you don't know them?
My computer hasn't the ability to distinguish a proposition from garbage.
What do you mean by "then"? The hidden premise here is that in order for the computer to create a proposition, the computer needs to distinguish propositions from garbage. Why would you hold that premise?
A mother fox and a father fox can make a baby fox. Not one of these things know they are foxes. A computer can generate displays of the mandelbrot set. Computers don't know what mandelbrot sets are. Why should propositions be special?
On October 1, 2021, I caused a computer to generate statements that are accurate representations of states of affairs. The computer generated those statements at 10:03:44pm on that day.
Quoting Olivier5
You have the same problem classifying strings as sentences... either you don't know these words mean or you're special pleading.
There are strings I would interpret as propositions, and strings I would not interpret as propositions. This implies that there's a classification of strings I would interpret as propositions:
Quoting Olivier5
1. It "makes sense to say" that strings that fall into the class of strings I would interpret as propositions, are propositions.
2. I have sufficient justification to say, before I interpret the strings produced by this program (i.e., at 10:03:44pm), that they are propositions in the sense introduced in 1.
3. Likewise, there are some propositions which, should I understand them and assign truth values to, I will assign the truth value of "true" to.
4. It "makes sense to say" (see above) that such propositions are true propositions.
5. I have sufficient justification to say, before I interpret the strings produced by this program (i.e., at 10:03:44pm), that the "propositions" (see 2) are "true propositions".
That you interpret as such. I don't.
That's fine, and I have no problem with that per se, except that you did explicitly appeal to the "makes no sense to say" criteria (which you even bolded, FTR), and it's that which I'm demonstrating. If you can prove it does not genuinely make sense to say what I'm saying, that would be relevant. Otherwise, you cannot appeal to the "makes no sense to say" criteria to defend your own interpretation.
Because on the face of it, the sentence is a ridiculous contradiction, for to say “to know S is a proposition, it is not necessary to know S (is a proposition). Only when understood that the S known as a proposition is not the S it is not necessary to know, does the difficulty disappear. But that understanding is not implicit in the sentence itself, it must be deduced from it, in order to reconcile the contradiction. It follows that the only relevant deduction can be that the S to know is a form, the S not necessary to know, is a content. What remains is, to know S is a proposition, it is not necessary to know what is contained in S.
Some people, not difficult; most people, irrelevant.
Also not the difficulty to which I directed my comment.
No, Olivier5, we haven't been through "this", because "this" refers to what you just linked to. That "this" is a post where I pointed out your bolded "makes no sense to say that" criteria. Not only did I point that out in the reply you're pretending to reply to, but that was the entire point of the post you're pretending to reply to!
And whereas you are not even trying to employ the "makes no sense to say that" criteria, you're not even going over "this" in your reply.
Quoting Olivier5
None of these are in the form "it makes no sense to say that". What is the thing you're claiming it makes no sense to say? Without that thing, you're not even going over "this" in this reply.
I'm not in any realistic sense the one who proposed proposition 6. I did technically meet the criteria you called out for in that post, but not in any way you're spinning your wheels arguing... I can call what the computer generated propositions because I know "about" them ("it makes no sense to say that a proposition no one knows about is true"), not because I "proposed" them, or because my computer "knew" them, or even because I knew what they were. I didn't propose prop 6. My computer didn't understand prop 6. My computer didn't know prop 6. My program didn't know prop 6. I didn't know what prop 6 was at 10:03:44pm. But I did know "about" prop 6, at that time, because I wrote the program.
So who is doing the proposing then?
Not my concern. I'm not bound by your theories that propositions require a proposer, so I don't have to name one. If you can't find one, once again, that's a you problem, not a me problem. If you can figure out an answer, knock yourself out. If you can't, maybe consider giving that up. I don't require it; so I'm all good.
Okay so you're not quite sure whether it's true or not that "QMCVNBOO" is lexically prior to "SHXCBJYN" now?
As formulated, the statement is a bit unclear because "lexical" means "relating to words or the vocabulary of a language as distinguished from its grammar and construction". Nothing to see with indexing nonsensical strings of uppercased letters...
Assuming you mean something like "comes in alphabetic order before", then the statement could be interpreted as a true proposition.
Lexical has another sense: "relating to or of the nature of a lexicon or dictionary". That's closer to what is meant. "Lexically" in this particular sense refers to "how" the strings are prior/successive to each other, i.e., in what sense they are; it's referring to a lexical ordering.
A lexical ordering is the same as dictionary ordering, and refers to the type of ordering words have in a dictionary.
https://en.wikipedia.org/wiki/Lexicographic_order
It's slightly distinct from "alphabetical order", in that it formalizes the concept of the ordering and generalizes it.
Quoting Olivier5
It's slightly more precise to say "lexical", since that describes what the sorted function does with strings. "alphabetical" works because I'm limiting this to strings containing only capital letters and 8 characters, but then, so does "numerical" with your prior mapping given this description, which is why I didn't bother commenting on it then.
So it's computer language. That's why is sounds odd in regular English.
Your program is generating arithmetic truths: 2 > 1, 3 > 2, etc.
Put differently, Fitch is talking about apples and your program is doing math. So not even apples & oranges.
But that aside, suppose your program were to write each line out to a file and then delete that file before generating the next line? Would you still consider your program to be generating propositions?
Please note that I'm not saying you're wrong. Much of this discussion relates to the question of how we define/use the words "proposition", "truth", "knowability", etc.
for all p we have p -> Kp.
Rather, the import is that
if for all p we have p -> LKp, then for all p we have Kp,
but since it is not the case that for all p we have Kp, it is not the case that for all p we have p -> LKp, so we reject that for all p we have p -> LKp.
(2) SEP gives the argument with quantifiers ranging over sentences, but I'm not sure that is necessary, and so I am not sure that necessarily the argument is second order. I don't see why the argument can't be as proof schema for a language of sentential modal logic with two primitive modal operators 'L' and 'K', letting 'p' and 'q' and 'r' be metavariables ranging over sentences:
Axiom schemata:
(a) Kq -> q
(b) K(q & r) -> (Kq & Kr)
(c) q -> LKq [this is the axiom presumably to be rejected]
Inference rule:
(d) from |- ~q infer |- ~Lq
Proof:
1. K(p & ~ Kp) assume toward RAA
2. Kp & K~Kp from 1 and (b)
3. ~Kp from 2 and (a)
4. ~K(p & ~Kp) from 1 by RAA
5. ~LK(p & ~Kp) from 4 and (d)
6. p & ~Kp assume toward RAA
7. LK(p & ~Kp) from 6 and (c)
8. ~(p & ~Kp) from 6 by RAA
9. p -> Kp from 8 [note that non-intuitionistic logic is used here]
So from the rules (a) - (d) we proved for arbitrary sentence p that p -> Kp. But we don't believe that for all sentences p we have p -> Kp, while (a), (b) and (d) seem eminently reasonable, so we reject (c).
(3)
The SEP article explains some of the arguments against Fitch's argument.
Also, this book looks like a great overview:
Salerno
I'm fine with that. Here, I've analyzed TMP's latest proofs, and don't have any particular issues with them, outside of the fact that they could probably be made a bit clearer by organizing it a little better.
Quoting EricH
Understood; the point of the program was just to test certain ideas about what a proposition is. It kind of had to be mathematical, because I wanted it in the post, this makes it short, and that would make it much easier to explain to someone who might not be familiar with programming if need be. I also felt it worthwhile to make the examples concrete rather than hypothetical.
Quoting EricH
Sure. I'm perfectly happy with counterfactuals, and it suffices for me that "were I to read it, I would classify it as a proposition". I think I apply the same criteria to other things; "were I to see this animal, I would classify it as a fox" suffices for me to call the thing a fox; "were I to see this object, I would call it an ice cube" suffices for me to call it an ice cube, etc.
Quoting Olivier5
Quoting InPitzotl
To both of you
Incorrect!
[quote=Wikipedia]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.[/quote]
What's wrong with it?
:flower:
[math]\neg p[/math] is true
I don't think you're quite following this.
1. Let q=¬p.
2. Then ¬p?K¬p is simply q?Kq
3. q?Kq is the same as p?Kp with change of labels.
1: I'm just defining another variable. Surely I can do that?
2: When you see "¬p", you can replace it with "q" (per 1). That's just substitution. Do you have a problem with substitution?
3: Specifically, we're relabeling q as p; what was p, you can call anything else. Do you have a problem relabeling? I seem to recall you actually relabeling for me just earlier today!
Sorry for the confusion but
1. If q = [math] \neg p[/math], q is true. @Olivier5 claims that Fitch's paradox can be extended to falsehoods but [math] \neg q[/math] is not false.
I think you mean that if p is a falsehood, and q = ¬p, then q is true. So you have a falsehood p, and a truth q. So if there's logic requiring q to be true, you can put your falsehood into p.
Yes, but for @Olivier5's argument to work [math] \neg p [/math] should be false. It isn't.
Which argument are you referring to?
Quoting Olivier5
Axiom schemata:
(a) Kq -> q
(b) K(q & r) -> (Kq & Kr)
(c) q -> LKq [this is the axiom presumably to be rejected]
Inference rule:
(d) from |- ~q infer |- ~Lq
Proof:
1. K(p & ~ Kp) assume toward RAA
2. Kp & K~Kp from 1 and (b)
3. ~Kp from 2 and (a)
4. ~K(p & ~Kp) from 1 by RAA
5. ~LK(p & ~Kp) from 4 and (d)
6. p & ~Kp assume toward RAA
7. LK(p & ~Kp) from 6 and (c)
8. ~(p & ~Kp) from 6 by RAA
9. p -> Kp from 8 [note that non-intuitionistic logic is used here]
Charitably, you take a false proposition p. You extend that by building a true proposition ¬p from it. You then use ¬p as you would any true proposition. I'm not sure your interpretation is charitable.
Notice that in your example: q = ~p , you used q (~p is true) and not p (p is false). So, you didn't expand the scope of Fitch's paradox to include falsehoods.
How can we embiggen the scope of Fitch's paradox to cover falsehoods?
An attempt was made by saying [math]\neg p \rightarrow K \neg p[/math] but
1. Isn't actually doesn't cut it because [math] \neg p[/math] is true.
and
2. Knowing also implies truth. We cannot know a falsehood.
p: <- the false proposition.
q: <- the true proposition.
q = ~p <- makes true proposition q out of false proposition p.
Does your proof need a true proposition? Use q.
What's the problem?
Quoting TheMadFool
p is true here, right?
Let's change labels from p to q. q?Kq. Now q must be true (because we changed labels), right?
So let's take the case where q = ~p, where p is false. q is still true, right? What did q have to be? True? Okay, well it is true. What did p have to be to extend to falsehoods? False? Okay, well, it's false.
Now, we can talk about p's that are false. And when you do your proof, you use q's for where you used to use p's. What's wrong?
We can do the same thing in reverse. Just take p, where p is true, and that's your typical application. To do falsehoods, take p where p is false. But we can't do the typical, we have to convert that to a true proposition. No problem; add a complement; if p is false, ~p. But the proof only works when p is a true proposition. Okay, no problem; relabel p's in the proof as q's, and say q=~p. Now we have a false p, and a proof that uses the fact that q is true. What's wrong?
'p' is a meta-variable ranging over sentences in the object language.
The proof is syntactically correct, as far as I can tell.
Whether any given sentence is true or false in any given semantics for a modal language with two modal primitives, I don't see what that has to do with the correctness of the proof.
And, especially, I don't know what "cover falsehoods' is supposed to mean. Whether 'p' stands for a sentence that is true or is false in any given model, the proof is correct as far as I can tell, thus the conclusion:
It is not the case that for every sentence p we have p -> LKp.
It seems we've been talking past each other.
1. What you're saying is true of course. If p is false then, ~p is true and indeed, [math] \neg p \rightarrow K \neg p[/math]. If then q = ~p, it follows that [math]q \rightarrow Kq[/math]. If by Fitch's paradox being extended over falsehoods means this to you, you and @Olivier5 are absolutely right. Negations of falsehoods, truth, are being utilized.
2. To me, if Fitch's paradox includes falsehoods within its scope, it can only be so if falsehoods themselves, not their negations which are true (see 1 above), can be part of Fitch's argument.
A very difficult question to answer but let's just stick to the basics: Fitch's paradox is about unknown truths.
That seems a different case than that of an unknown proposition: a known proposition whose truth value is yet unknown. There are many many examples of this kind of proposition.
q = ¬p
q?Kq
therefore
¬p?K¬p = all false propositions are known propositions.
This is elementary, really. Reason for which I did not write it down, not wanting to insult people's intelligence. @InPitzotl got it immediately. So make an effort, calm your contrarian demons and for once, TRY and understand these ultra basic logical steps above.
You need to work on your logic.
The import is:
It is not the case that for all sentences p, we have p -> LKp.
From that, it does follow that there are true sentences such that it is not possible that they are known to be true.
Moreover, that vitiates certain philosophical views.
q -> Kq
being stated?
No one believes that as a generalization for all q.
I do believe that.
Fitch's paradox is about unknown truths. That's the essence of the argument.
False that all truths are knowable. For if it is, Fitch is right, all true propositions are known. Gödel's Incompleteness Theorems.
For all q, we have q -> Kq
is extraordinarily contrarian.
It should not be overlooked that 'Kq' does not stand for 'q is knowable' but rather for 'q is known'.
What he shows is that
If all truths are knowable then all truths are known. But since not all truths are known, we infer that it is not the case that all truths are knowable.
In Fitch's argument, p, whereever it appears, stands for true propositions. (p & ~Kp) = an unknown truth.
Regarding some members who want to say Fitch is talking about falsehoods, yes but only in the sense of their negations which are, well, truths.
It is an extraordinarily outlandish view that every truth is known.
One may go back and read the exact expositions to see that.
A proposition is a form of knowledge. And there can be no such thing as unknown knowledge, even though there's plenty we don't know.
What we don't know is not neatly set in the form of propositions yet. This work still has to be done.
The earth is round. True
The earth is flat. False
What's the differenc between:
1. Knowing the earth is round [sounds ok]
and
2. Knowing the earth is flat [believe seems a better fit]
and
3. Not knowing the earth is round [looks, feels right]
and
4. Not knowing the earth is flat [something's off]
?
To cut to the chase, it doesn't look like we can know a falsehood.
If one conflates 'it is known that the sentence exists' with 'it is known that the sentence is true', then of course the whole discussion falls apart.
Also, in this context, it is not just sentences that have known to have been expressed but sentences in general (in a formal context, that would be all the sentences of the formal language).
and
"We believe that the earth is flat"?
The differences are so easy to point out that I don't see the sense in asking about it.
Indeed, it doesn't look like you can.
Ad hominem?! I'll ignore that for the time being. Let's focus on the issue at hand. You claim that we can know falsehoods. Expand and elaborate (if you can :wink: ).
I did pay attention and you're evading the question. Why? I wonder if it's because you can't prove your case.
It's not about me, it's about you. Also, why are we discussing this like little children? You made a claim: Fitch's argument can be extended to false proposition. A philosopher would justify that claim. I'm waiting...
Quoting Olivier5
Bitchin'? Me? :lol:
By the way, I'm waiting for you to make an argument.
You remark is such an obvious way to make a mistake that I don't see the sense in pointing it out.
https://thephilosophyforum.com/discussion/comment/603520
Quoting Olivier5
[math] \neg p \rightarrow K \neg p \neq all \space false \space propositions \space are \space known \space propositions[/math]. Sorry.
I'm sorry about your unsound argument. Again, let's stay focused on the problem at hand, shall we?
I'll try and help you if that's even possible since you seem so confident about your position.
Your claim: All false propositions are known propositions.
All false propositions. I need one example of a false proposition:
1. The earth is flat
This proposition, you claim, is known. Ok, let's put it down in words: I know the earth is flat. Does that make sense to you? It doesn't to me. If it does to you, how? please and thanks in advance.
You know the proposition "the earth is flat". Otherwise you couldn't talk about it...
Knowing that there is a false proposition is not the same as knowing a false proposition.
This :point: Quoting Olivier5
Kp= Know that p.
So,
1. if p = the earth is round, Kp = know that (the earth is round) [No problem]
2. If p = the earth is flat Kp = know that (the earth is flat) [Problem]
Because p -> Kp was stated.
Quoting TonesInDeepFreeze
Apparently some people do. It's an antirealist position; the p doesn't exist until it's proposed, and it isn't true until you say it is, or some such thing.
On the contrary, it would be unrealistic to assume that there can exist English sentences that nobody speaking any English has ever composed or crafted... A realist view of the world does not imply that ideas nobody ever thought of exist already in some Platonic realm, waiting to be discovered.
Antirealistic != unrealistic
You're confusing "antirealist"/"realist" with "unrealistic"/"realistic"... the terms convey completely different things. A realist (in this particular sense) is someone who accepts the reality of something, usually external. An antirealist denies the reality of something. The "something" in this case is English sentences nobody has mentioned. You're objecting to an accurate term describing what you're doing, on the basis of the ill-conceived notion that it was commentary.
What does "p is known" mean?
1. p = The earth is round
p is known = The earth is round is known (all ok)
2. p = the earth is flat
p is known = the earth is flat is known (? not ok)
As explained: some people know about the existence of proposition p.
You're equivocating. I'm out. Thanks for the interesting conversation. I learnt a lot from the exchange. Good day.
Wrong battle. I already know your position; you're now proselytizing.
To think of knowledge as a commodity, like salt or copper, a thing that can exist irrespective of whether human beings pay attention to it or not, is the wrong way to think about it. Knowledge is not a commodity, it is an activity.
As a consequence, the unknown is NOT a commodity either. And it does not come already prepackaged in neat little English sentences called 'propositions'.
You have no choice.
Quoting Olivier5
Have to be proposed in order to... what?
There is no mistake in that.
Who stated it? To be clear, Fitch does not hold that p -> Kp.
Quoting InPitzotl
What specific quotation or reference is given by anyone (other than a flagrantly errant poster) that p -> Kp? The supposed antirealist notion is p -> LKp.
(1) 'Kq' stands for "q is known to be true" and it does not stand for "q is known to be a sentence (or proposition)".
(2) Fitch is not arguing that for all p, we have p -> Kp. Fitch is arguing that if for all p, we have p -> LKp, then for all p, we have p -> Kp, but since it is not acceptable that for all p, we have p -> Kp, it is not acceptable that for all p, we have p -> LKp.
It's about propositions. Let's not change the goal post.
One needs to read the expositions such as at SEP.
That's probably a fair point. The SEP is not very clear on this but I trust you.
Let me start by noting a residual ambiguity here. I can see two very different ideas of an "unknown truth":
1. An hypothesis or proposition that has been actually proposed as true by someone, which is in fact true but unbeknown of most, and on which truth value there is no consensus whatsoever yet. For instance: the CIA killed the Kenedy's (if it's in fact true).
2. An hypothesis or proposition that has not yet been proposed by anyone, that is in fact true. For instance: Charlie Chaplin killed the Kenedy's (if it were in fact true that he did but my money is on #1, and if I had not just now proposed it of course...).
While the first meaning is not problematic, I have explained at length why #2 is for me a logical impossibility. Sentences not yet said are not yet said. They don't sit out there in Plato's realm or some similar otherwordly place, waiting for us to say them.
Probably none; this was stated by TMF, and the view appears to be held by Olivier5 (haven't caught up here; you've likely already met).
Which is exactly why I am asking why it is being stated.
It is not the case that for all propositions p that have been expressed we have p -> Kp. Therefore, as Fitch shows in the proof, it is not the case that for all propositions p that have been expressed we have p -> LKp.
Understood, but the best I can do is link to context. @TheMadFool mentioned it, but there was something silly there with Fitch only applying to true propositions (follow it to see why I'm saying it's silly).
The furthest I can take you is there; I cannot go inside TMF!
That is indeed correct and I understand it is at least one of the 'traditional' readings. Not all propositions are necessarily decidable and we have good reason to think some are indeed not decidable.
In short: we have more questions than answers.
My beef was more with the other idea. The illogical one, which treats knowledge as a commodity.
Apophatic Theology
Apophatic theology (via negativans): To deny of God all posssible predicates. So,
God is material. No! God is a male. No! God is x (any predicate). No!
In other words, where Q stands for any predicate and g is God, Qg is always false or ~Qg is always true.
Knowing something, say apples, is nearly always in terms of positive predicates i.e. I know an apple in terms of what it is and not in terms of what it is not.
In apophatic theology, we're asked to know God from what God is not i.e. all we know about God is what is false about God.
Apophatic anything: Knowing not by knowing what is true about that thing but by "knowing" what is false about that thing.
The Justified True Belief Theory Of knowledge asserts that,
S (a person) KNOWS P (a proposition) IFF
1. P is true
2. P is justified
3. S believes P
This looks like a perfectly good definition of knowledge.
Consider now the following 4 paragraphs:
Paragraph 1: Apples are a fruit. Apples are sweet when ripe and they come in different varieties. Apples grow in temperate regions of the world and are widely consumed. Apples are rich in vitamins and antioxidants.
Paragraph 2: Apples are dogs. Apples grow in the desert and apples are found on Venus. Too, apples are cars and they eat meat.
Paragraph 3: Apples are not dogs. Apples don't grow in the desert and apples are not found on Venus. Too, apples are not cars and they do not eat meat.
Paragraph 4: Apples are dogs is false. Apples grow in the desert is false and apples are found on Venus is false. Too, apples are cars is false and they eat meat is false
Paragraphs 1, 3 and 4 contain all true propositions but paragraph 2 consists of all false propositions.
A) If S believes all true propositions in paragraph 1, it can be said that S knows these propositions and also that S has knowledge of apples.
B) If S believes all propositions in paragraph 2, we can say for sure that S doesn't know anything about apples.
C) If S believes all propositions in paragraphs 3 and 4, S knows these propositions BUT it feels odd to say that S has knowledge of apples.
D) Paragraph 3 and 4 are identical as NOT = is false.
Issues:
(i) The JTB theory of knowledge needs reviewing (see C).
(ii) If a proposition p is false, we can't know p (see B) but we can know that p is false, this is a different proposition, ~p (see C)
How does all this matter to Fitch's paradox?
Some have claimed that it's possible to know false propositions.
To clarify that assertion, consider apophatic theology: We can know God via negativans (by way of denying).
The basic idea behind apophatic theology is that we can know by [b]not knowing which can be translated as knowing falsehoods (about God). All I know of God is what is false about God which is a very clever, roundabout, elliptical, way of saying I don't know anything about God.