Is it possible for all truths to be known?

[lugita15]

People most often hear about paradoxes that challenge our notions of truth and falsity, like the Liar Paradox, Curry's Paradox, Russell's Paradox, Berry's Paradox, etc. But just as interesting are the paradoxes that challenges other notions we hold dear, the ones philosophers call "modal" notions: knowledge, possibility, morality. So let me present one of the most famous ones, called Fitch's Paradox of Knowability, and if people find that interesting I can talk about a few other favorites of mine.

The question we're dealing with is: Are all true statements knowable? To put it another way, is it possible for there to be some truth which can never be known, no matter how hard you try? Here's an argument that seems to answer this question. Obviously there are some unknown true statements out there; we don't know everything, do we? For instance, either "The Riemann Hypothesis is true" or "The Riemann Hypothesis is false" is one of these statements. In any case, let P be some unknown true statement. Then consider the statement Q, which says "P is an unknown truth." Then Q is obviously a truth. Is it possible for Q to be known? Well, suppose Q were known. Then we would be able to say "I know that Q is true" or equivalently "I know that P is an unknown truth" or in other words "I know that P is true and that P is unknown." But it's impossible for that to be true, isn't it? Because if you knew that P is true, then P would be known, so it would be impossible to know that P is unknown, because P is not unknown, and you can't know a false statement! Thus it's impossible to know Q, so in other words Q is an unknowable truth.

So to review, we started with the hypothesis that P is an unknown truth and we got to the conclusion that Q is an unknowable truth. So "there exists an unknown truth" implies "there exists an unknowable truth." Turning this around, "all truths are knowable" implies "all truths are known", which is crazy! Clearly it is possible for there to be some truths which we happen to be unknown right now, but might be discovered in the future. But Fitch's argument above seems to suggest that if you believe that any truth is within our grasp, you have to believe that we already know everything!

 

[alt]

People most often hear about paradoxes that challenge our notions of truth and falsity, like the Liar Paradox, Curry's Paradox, Russell's Paradox, Berry's Paradox, etc. But just as interesting are the paradoxes that challenges other notions we hold dear, the ones philosophers call "modal" notions: knowledge, possibility, morality. So let me present one of the most famous ones, called Fitch's Paradox of Knowability, and if people find that interesting I can talk about a few other favorites of mine.

The question we're dealing with is: Are all true statements knowable? To put it another way, is it possible for there to be some truth which can never be known, no matter how hard you try? Here's an argument that seems to answer this question. Obviously there are some unknown true statements out there; we don't know everything, do we? For instance, either "The Riemann Hypothesis is true" or "The Riemann Hypothesis is false" is one of these statements. In any case, let P be some unknown true statement. Then consider the statement Q, which says "P is an unknown truth." Then Q is obviously a truth. Is it possible for Q to be known? Well, suppose Q were known. Then we would be able to say "I know that Q is true" or equivalently "I know that P is an unknown truth" or in other words "I know that P is true and that P is unknown." But it's impossible for that to be true, isn't it? Because if you knew that P is true, then P would be known, so it would be impossible to know that P is unknown, because P is not unknown, and you can't know a false statement! Thus it's impossible to know Q, so in other words Q is an unknowable truth.

So to review, we started with the hypothesis that P is an unknown truth and we got to the conclusion that Q is an unknowable truth. So "there exists an unknown truth" implies "there exists an unknowable truth." Turning this around, "all truths are knowable" implies "all truths are known", which is crazy! Clearly it is possible for there to be some truths which we happen to be unknown right now, but might be discovered in the future. But Fitch's argument above seems to suggest that if you believe that any truth is within our grasp, you have to believe that we already know everything!

So to review, we started with the hypothesis that P is an unknown truth ..

But even at the start, that hypothesis seems a little shaky, and rather a play on, or fluid use of, wording.

How would you know it's a truth if it's unknown ?

Interested to hear more of your paradoxes though!

 

[lugita15]

So to review, we started with the hypothesis that P is an unknown truth ..

But even at the start, that hypothesis seems a little shaky, and rather a play on, or fluid use of, wording.

How would you know it's a truth if it's unknown ?

Sorry, maybe I was unclear. We start with the hypothesis that there EXISTS some unknown truth P. Presumably we don't know what that truth is.

 

[disregardthat]

IMO; For any proposition to be true, you need a criterion for its truth, and the criterion needs to be satisfied. And it is only upon verification we say that it is satisfied.

In this sense you can't have unknowable truth. "P is true, but I don't know it to be true" just doesn't make sense. "P is true" doesn't express more or less that "I know P is true". The paradox arise from abuse of language, just like any other.

 

[alt]

Sorry, maybe I was unclear. We start with the hypothesis that there EXISTS some unknown truth P. Presumably we don't know what that truth is.

No, you were clear. But I'm saying that the hypothesis is nosnensical, imo.

May as well start with the hypothesis that there exists a five legged tripod. It's a similar word play to say we have an unknown truth. You can't call it truth if it's unknown. To call it truth you would have to know it as being that.

 

[lugita15]

IMO; For any proposition to be true, you need a criterion for its truth, and the criterion needs to be satisfied. And it is only upon verification we say that it is satisfied.

In this sense you can't have unknowable truth.

That is a view called verificationism, which states that all truths are knowable. The whole point of Fitch's paradox of knowability is to disprove verificationism

"P is true, but I don't know it to be true" just doesn't make sense.

Knowledge is different than belief. You may believe one thing, but find out later you were wrong. On the other hand, if you know something then by definition it must be true. A common definition of knowledge used in philosophy is justified true belief. In other words, in order to know a statement P, the following three criteria must be met:
1. You believe that P is true.
2. P is true.
3. You are justified in believing that P is true, in the sense that you cannot possibly be wrong about it.

"P is true" doesn't express more or less that "I know P is true".

These two statements are very different. To say "P is true" is the same as saying "I believe P is true", but is very different from saying "I know P is true."

The paradox arise from abuse of language, just like any other.

No it doesn't, at least not in the straightforward way you're thinking.

 

[lugita15]

No, you were clear. But I'm saying that the hypothesis is nosnensical, imo.

May as well start with the hypothesis that there exists a five legged tripod. It's a similar word play to say we have an unknown truth. You can't call it truth if it's unknown. To call it truth you would have to know it as being that.

I think you still don't understand what I'm saying. I'm not saying that there is a particular truth which we know to be unknown. Rather, I'm saying that there EXISTS an unknown truth out there, even if we don't know what it is. Surely you agree that we don't know everything, don't you? Like we don't know whether the number of hairs on Obama's head is even or odd. Yet either "the number of Obama's hairs right now is even" or "the number of Obama's hairs right now is odd" must be true, and yet presumably no one knows which one. But one of these is surely an unknown truth, so we can at least say that there exists an unknown truth, can't we?

 

[disregardthat]

That is a view called verificationism, which states that all truths are knowable. The whole point of Fitch's paradox of knowability is to disprove verificationism

That's ridiculous. The whole point of requiring a criterion for truth is that one rejects the notion of true statements being true simply in virtue of their meaning. So there "existing unknown truth out there" is meaningless. Propositions require a well-defined criterion for truth. Fitch's paradox doesn't disprove anything in this regard, it is just playing around with words.

Knowledge is different than belief. You may believe one thing, but find out later you were wrong. On the other hand, if you know something then by definition it must be true.

The point is that by asserting a proposition, you can't deny that you believe it. Saying "P is true and I believe P is false" is simply meaningless. "P is true" and "I believe P is true" has no different criterion for truth, so it's impossible to assert one of them are deny the other. Many paradoxes arise from this kind of abuse. In the same fashion, asserting that "I know P to be true and P is false" is meaningless.

 

[alt]

I think you still don't understand what I'm saying. I'm not saying that there is a particular truth which we know to be unknown. Rather, I'm saying that there EXISTS an unknown truth out there, even if we don't know what it is. Surely you agree that we don't know everything, don't you? Like we don't know whether the number of hairs on Obama's head is even or odd. Yet either "the number of Obama's hairs right now is even" or "the number of Obama's hairs right now is odd" must be true, and yet presumably no one knows which one. But one of these is surely an unknown truth, so we can at least say that there exists an unknown truth, can't we?

Well in that case, you can reduce a great many (perhaps all) things to your definition of unknown truth. The number of atoms making up your computer screen for instance. An unknown truth. The number of cells in your left ear. Same. The exact number of cents that flowed through the American economy between 9 AM and 10.29 AM today. The number of raindrops that fell on Tokyo between 1934 and 2011. All unknown truths. This however, is reduction to the ridiculous, as is your example of Obamas hairs.

So if reduction to the ridiculous is your thing, then I suppose Fitch's paradox is attracive.

 

[lugita15]

That's ridiculous. The whole point of requiring a criterion for truth is that one rejects the notion of true statements being true simply in virtue of their meaning. So there "existing unknown truth out there" is meaningless.

Don't you think that either "The number of hairs on Obama's head is even" or "The number of hairs on Obama's head is odd" is an unknown true statement?

Propositions require a well-defined criterion for truth. Fitch's paradox doesn't disprove anything in this regard, it is just playing around with words.

It's not just playing with words, at least not in the sense you're talking about, because it can be formalized symbolically using epistemic logic. See here. (That's a great article, and it has numerous proposed resolutions to Fitch's paradox. If anyone is interested I can discuss my preferred resolution.)

The point is that by asserting a proposition, you can't deny that you believe it.

I agree.

Saying "P is true and I believe P is false" is simply meaningless.

It's not meaningless, it's just wrong.

"P is true" and "I believe P is true" has no different criterion for truth, so it's impossible to assert one of them are deny the other.

I agree, they mean the same thing, so to assert one and deny the other would be wrong.

Many paradoxes arise from this kind of abuse.

As I said, Fitch's paradox does not arise from at least that kind of abuse of language, because it can be expressed in symbolic language which avoids all the ambiguities and vagaries of English.

In the same fashion, asserting that "I know P to be true and P is false" is meaningless.

It's not meaningless, again it's just contradictory and hence false.

 

[lugita15]

Well in that case, you can reduce a great many (perhaps all) things to your definition of unknown truth. The number of atoms making up your computer screen for instance. An unknown truth. The number of cells in your left ear. Same. The exact number of cents that flowed through the American economy between 9 AM and 10.29 AM today. The number of raindrops that fell on Tokyo between 1934 and 2011. All unknown truths.

Yes, we can find a lot of examples of unknown truths.

This however, is reduction to the ridiculous, as is your example of Obamas hairs.

I agree that these are silly examples, but there's nothing fundamentally wrong with them. They're just a way to illustrate that there are such things as unknown truths.

So if reduction to the ridiculous is your thing, then I suppose Fitch's paradox is attractive.

The reasoning in Fitch's paradox is not as ridiculous as you think. I suggest you examine Fitch's logic more closely.

 

[disregardthat]

Don't you think that either "The number of hairs on Obama's head is even" or "The number of hairs on Obama's head is odd" is an unknown true statement?

Absolutely not. I personally believe it is a very basic misconception of logic. Let me explain:

The logical conjunction "The number of hairs on Obama's head is even OR the number of hairs on Obama's head is odd" is true by virtue of being a logical tautology. There is no need for any criterion here.

But either of the statements P: "The number of hairs on Obama's head is even" and Q: "The number of hairs on Obama's head is odd" requires criteria for truthfulness, such as the result of counting the hairs being even or odd. The truth of P is realized by satisfying such a criterion.

It's tricky when it comes to time: If the criterion for a proposition P (which does not depend on time) is satisfied tomorrow, it doesn't make it correct to assert "P is true now" today. It would however be correct to assert "P was true yesterday" tomorrow. The statements have a different sense. So we could say "that the truth(-value) of P was unknown yesterday" tomorrow, but it wouldn't be correct to call it an unknown truth now.

This form of verificationism is very much alike the way we use ordinary language, and the way we treat scientific hypotheses and evidence. It is only in the platonic pits of formal logic or shaky metaphysics one end up with such silly paradoxes.

It's not meaningless, again it's just contradictory and hence false.

Contradictory, meaningless, useless. All the same to me. It isn't false in the sense of failing to satisfy its criterion, because there is no criterion, none can be given.

 

[lugita15]

But either of the statements P: "The number of hairs on Obama's head is even" and Q: "The number of hairs on Obama's head is odd" requires criteria for truthfulness, such as the result of counting the hairs being even or odd. The truth of P is realized by satisfying such a criterion.

It's tricky when it comes to time: If the criterion for a proposition P (which does not depend on time) is satisfied tomorrow, it doesn't make it correct to assert "P is true now" today. It would however be correct to assert "P was true yesterday" tomorrow. The statements have a different sense. So we could say "that the truth(-value) of P was unknown yesterday" tomorrow, but it wouldn't be correct to call it an unknown truth now..

OK, forget about truths that are unknown in general. Do you at least agree that there are truths that you do not know, but perhaps that other people do know? Because even with that assumption we can carry through Fitch's paradox, and use it to disprove the statement "Any truth can be known by you."

 

[disregardthat]

OK, forget about truths that are unknown in general. Do you at least agree that there are truths that you do not know, but perhaps that other people do know? Because even with that assumption we can carry through Fitch's paradox, and use it to disprove the statement "Any truth can be known by you."

What you are suggesting is that if P is a proposition known to be true by others, but not by me, and I realize that it is known to others and hence true (since it is supposed to be knowable by hypothesis), then upon realization (that it is known to others) I simultaneously can assert that it is unknown to me and known to me at the same time?

This time you can't deny you are playing with words, or more specifically you are ignoring the temporal aspect of the situation:

When I realize something I didn't know before, I am made aware of that I didn't know in the past. Not that I don't know now.

 

[lugita15]

What you are suggesting is that if P is a proposition known to be true by others, but not by me, and I realize that it is known to others and hence true (since it is supposed to be knowable by hypothesis), then upon realization (that it is known to others) I simultaneously can assert that it is unknown to me and known to me at the same time?

No, I'm suggesting something really obvious, namely that there is a statement P known to others and not to you, and that you do not know that P is known to others, but later you can come to know that P is true, at which point it will be simply be known to you, not known and unknown at the same time. Or if you prefer, you can later come to know that P is known tto others, at which which point you can conclude that P is true, so P will be known to you, not simultaneously known and unknown. What I'm saying is just trivial.

When I realize something I didn't know before, I am made aware of that I didn't know in the past. Not that I don't know now.

You and I are in complete agreement on that point.

 

[PlayingMonk]

disregardthat, do you believe there is such a thing as objective truths? Or do you think things can only be true to people? I'm having trouble understanding your objections.

 

[disregardthat]

disregardthat, do you believe there is such a thing as objective truths? Or do you think things can only be true to people?

Truth is a property of propositions, and propositions are constructs of language. Think about that for a moment.

You confuse objective truth with an objective reality.

 

[disregardthat]

No, I'm suggesting something really obvious, namely that there is a statement P known to others and not to you, and that you do not know that P is known to others, but later you can come to know that P is true, at which point it will be simply be known to you, not known and unknown at the same time. Or if you prefer, you can later come to know that P is known tto others, at which which point you can conclude that P is true, so P will be known to you, not simultaneously known and unknown. What I'm saying is just trivial.

I'm not following you here. In what sense is that a paradox?

We are supposing that a proposition P is known to some group of people. And that I don't know whether P is true of not. The paradox in this case starts out as follows:

"The sentence P is an unknown truth". I certainly cannot assert this. What would the criterion for the truth of this proposition be? What would incline me to accept this for any given proposition P? Nothing.

So only others can assert this of me. But this doesn't get us into a paradox.

 

[lugita15]

I'm not following you here. In what sense is that a paradox?

That statement you quoted isn't a paradox at all. It's just an assumption used in the paradox.

We are supposing that a proposition P is known to some group of people. And that I don't know whether P is true of not. The paradox in this case starts out as follows:

"The sentence P is an unknown truth". I certainly cannot assert this. What would the criterion for the truth of this proposition be? What would incline me to accept this for any given proposition P? Nothing.

So only others can assert this of me. But this doesn't get us into a paradox.

We're not talking about you asserting "P is an unknown truth." Here is the logic of the paradox again.

We start with the assumption that there is some truth P which is unknown to you, but perhaps known to others. Now consider the statement Q, which says "P is a truth unknown to you." By assumption, Q is true. Now the question is, can Q be known to you? Well, suppose that Q were known to you. Then you would know the statement "P is a truth unknown to you". But if you knew that, you would know that P is true and that P is unknown to you, or in other words P would be both known to you and unknown to you, which is impossible. Thus the supposition that Q is known to you leads to a contradiction, and thus it is impossible for Q to be known to you, or in other words Q is unknowable to you. Thus we can disprove the thesis that all truths are knowable to you.

 

[disregardthat]

We start with the assumption that there is some truth P which is unknown to you, but perhaps known to others. Now consider the statement Q, which says "P is a truth unknown to you." By assumption, Q is true. Now the question is, can Q be known to you? Well, suppose that Q were known to you. .

What is the criterion for the truth of Q? Have you just ignored the premises for truth conditions I have posed? I can neither assert nor know Q. But this is not any more paradoxical than that I can't assert "P is true and I believe P is false". Because I simply have no criterion for its truth. Neither for its falsity.

 

[lugita15]

What is the criterion for the truth of Q? Have you just ignored the premises for truth conditions I have posed?

Q is true by assumption, because we are assuming P is a true statement unknown to you, and Q just states that assumption.

 

[disregardthat]

Q is true by assumption, because we are assuming P is a true statement unknown to you, and Q just states that assumption.

No, as I've said, this is just like the case of the proposition S "P is true and I don't believe P". If some group of people knows that P is true, but I don't yet believe P, "then S". But I have no criterion for S. S is a nonsensical proposition in my mouth in the same way Q is. So I cannot assert it. And if someone else but me asserts it of me, it won't get us into a paradox. But we're going in circles here, so I'd like to stress this point.

The paradox is a result of tricking oneself into believing that certain nonsensical propositions have sense.

When saying that a proposition is true, think of what criterion must have been satisfied. Most paradoxes of this nature simply ignores that, like this one.

Don't you find it odd how you would describe (given that I don't believe P) "P is true and I don't believe P" as an unattainable truth beyond my reach? It has no meaning for me, no use, and it doesn't express anything but nonsense.

EDIT: I didn't catch that you suggested that "P is true" is equivalent to "I believe P is true". That is certainly not the case. You can have a wide variety for believing the truth a proposition, as it is generally the expectation of the satisfaction of the criterion for truth. Correctly asserting "P is true" is not due to expectation, but the result of the realization of the criterion.

 

[Hurkyl]

Truth is a property of propositions...

Truth is a property of an interpretation of a proposition, not of the proposition itself. (and even then, only interpretations of certain types)

 

[disregardthat]

Truth is a property of an interpretation of a proposition, not of the proposition itself. (and even then, only interpretations of certain types)

Fine, I'd call it a property we assign to propositions. If an assigned property is a property or not is not something I'm going to argue about.

 

[Hurkyl]

The whole point of Fitch's paradox of knowability is to disprove verificationism

I hope it's flawed, because classical set theory proves verificationism in the form of "A statement is True if and only if there exists a proof of it" is consistent!

Specifically, given a formal language, one can define the truth of a proposition to be the set of all^* set-theoretic interpretations of the language in which the proposition holds.

*: Well, one has to deal with technical size issues here. There's probably a conservative way to do so, but I'm just going to invoke a large cardinal axiom, and say I mean "all small" rather than "all".

This notion of truth value forms a Boolean algebra, so it's a model of classical logic. "True", in this algebra, is the set of all interpretations.

Now, invoke Gödel's completeness theorem to say that any statement evaluates to "True" under this truth valuation if and only if there is a proof of that statement.Note that, for any statement P, "P is true or P is false" is "true" under the truth valuation I describe above! To wit, the set of interpretations where "P is true" holds is precisely the complement of the set of interpretations where "P is false" holds. The "or" of those truth values, therefore, is "True".
Where your argument fails in this setting is very easy to spot, of course: there is no proposition "P is known". You would have to do more work to set up a logic that can talk about knowability. (But, we knew that you have to do more work anyways, since you want both knowability and truth to be temporal)

Putting explicit references to things like "truth" into your language is always a huge can of worms anyways. e.g. Tarski's theorem on the undefinability of truth...

 

[alt]

Yes, we can find a lot of examples of unknown truths.

No, not just a lot - a minute after I responded in my earlier post, I realized that in keeping with your example of Obamas hairs, there is actually an infinite number of such examples. I can concoct any any number I want. Infinitely. If you can't think this through (though I'm sure you can), let me know and I'll give you some more examples. But they are all TRIVIAL, like your Obamas hairs example.

I agree that these are silly examples, but there's nothing fundamentally wrong with them.

There is ! They are REDUCTIO AD ABSURDUM. In fact, what happens to any logical process or equation when you multiply it by infinity ?

They're just a way to illustrate that there are such things as unknown truths.

Nothing in our dialouge so far has given me any evidence of unknown truths.

The reasoning in Fitch's paradox is not as ridiculous as you think. I suggest you examine Fitch's logic more closely.

I have examined it more closely, and am even more of the opinion that is a fluid word play.

 

[lugita15]

I hope it's flawed, because classical set theory proves verificationism in the form of "A statement is True if and only if there exists a proof of it" is consistent!

Indeed, one form of Fitch's paradox can be formulated via the Hilbert-Bernays derivability conditions, and it deals with claims like "all truths are provable" in a well-defined sense.

*: Well, one has to deal with technical size issues here. There's probably a conservative way to do so, but I'm just going to invoke a large cardinal axiom, and say I mean "all small" rather than "all".

Yes, there is a conservative way to do this: just restrict yourself to computable models.

Where your argument fails in this setting is very easy to spot, of course: there is no proposition "P is known". You would have to do more work to set up a logic that can talk about knowability.

Have you heard of modal logic? Among the different modal logics are alethic modal logic, which deals with possibility and necessity, and epistemic logic, which deals with knowledge. Fitch's paradox emerges as a quite easy theorem ito prove if you combine the standard systems of alethic and epistemic logic.

(But, we knew that you have to do more work anyways, since you want both knowability and truth to be temporal)

No I don't, I was just speaking informally. Formal derivations of Fitch's paradox do not require temporal logic at all.

Putting explicit references to things like "truth" into your language is always a huge can of worms anyways. e.g. Tarski's theorem on the undefinability of truth...

There is no reason to actually refer to truth, again I was just speaking informally. The verificationist thesis Fitch's paradox is concerned with can simply be formulated as "If P, then ⋄KP", where ⋄ means "it is possible that" and K means "it is known that".

 

[lugita15]

No, not just a lot - a minute after I responded in my earlier post, I realized that in keeping with your example of Obamas hairs, there is actually an infinite number of such examples. I can concoct any any number I want. Infinitely. If you can't think this through (though I'm sure you can), let me know and I'll give you some more examples.

Yes, we can clearly come up with an infinite number.

But they are all TRIVIAL, like your Obamas hairs example.

I don't know what you mean by trivial. Do you mean unimportant? For an example of a profound unknown truth, consider either "the Riemann hypothesis is true" or "the Riemann hypothesis is false".

There is ! They are REDUCTIO AD ABSURDUM.

In logic, reductio ad absurdum doesn't just mean you reduce to anything you think is silly or absurd. In order to prove a statement is false using reductio ad absurdum, you have to prove that the statement leads to an actual contradiction or falsehood.

In fact, what happens to any logical process or equation when you multiply it by infinity ?

Who said anything about multiplication by infinity? We're just talking about there being an infinite number of examples of something, which particularly problematic; there are an infinite number of examples of prime numbers, so does that mean that there is something wrong the notion of prime number?

Nothing in our dialouge so far has given me any evidence of unknown truths.

Didn't you admit that there were infinitely many examples of unknown truths?

I have examined it more closely, and am even more of the opinion that is a fluid word play.

How can it be word play, if it can be formulated using symbolic language? See here.

 

[alt]

Yes, we can clearly come up with an infinite number.

Yes, clearly - I'm glad you appreciate this.

I don't know what you mean by trivial. Do you mean unimportant?

I mean absurd;
1550s, from M.Fr. absurde (16c.), from L. absurdus "out of tune; foolish" (see absurdity). The main modern sense (also present in Latin) is a figurative one, "out of harmony with reason or propriety." Related: Absurdly; absurdness.
http://www.etymonline.com/index.php?allowed_in_frame=0&search=absurd&searchmode=none

To have a philosophical discussion and introduce as your evidence the possible numbers of hairs on Obamas head, let alone the other 7 billion heads, or the number of grains of sand in the world, fits the above, and renders the discussion nonsensical.

For an example of a profound unknown truth, consider either "the Riemann hypothesis is true" or "the Riemann hypothesis is false".

Sorry - I can't see any definite unknown thruth here. Can you show it ?

In logic, reductio ad absurdum doesn't just mean you reduce to anything you think is silly or absurd. In order to prove a statement is false using reductio ad absurdum, you have to prove that the statement leads to an actual contradiction or falsehood.

It leads to absurdity. OK, how many grains of sand .. ? - and prove the truth of this.

Who said anything about multiplication by infinity?

You did. You accepted my proposition that there is an infinite number of unknown truths, according to your definition. Multiply infinity .. add infinity .. same, same ..

We're just talking about there being an infinite number of examples of something, which particularly problematic; there are an infinite number of examples of prime numbers, so does that mean that there is something wrong the notion of prime number?Didn't you admit that there were infinitely many examples of unknown truths?

Nothing wrong with the notion of a (presumably) infinite number of prime numbers. It is a known truth, as far as it goes .. although even here if one thinks about it, if they haven't all been counted, then I suppose there must be a minute space, however vanishingly small it might be, allowed for doubt of this truth.

How can it be word play, if it can be formulated using symbolic language? See here.

I went to your link ... ARGHHH ! I know nothing about this. But why defer to symbolic language, when we haven't yet sorted out the, ummm, 'bolic' ? Try that first.

 

[Hurkyl]

Formal derivations of Fitch's paradox do not require temporal logic at all.

The English notion "is known" is an inherently temporal quality; whether something is known or not changes over time.

The interpretation of the formal statement in your link is something different: it is about "known at some time". (though the prose can't seem to decide if it wants to talk about "known" or "known at some time in the future or past")


The Knowability Principle from your link:

$$\forall p: (p \to \diamond Kp)$$​

is highly implausible:

• If we assume two-valued truth semantics, then this seriously opposes the various incompleteness results of logic.
• If we allow other Boolean algebras like the one I suggested, then this statement fails to discuss the notion "p is a truth".

(In fact, $\square p$ already works fairly well for "p is a truth", under the interpretation that of U is a truth value, then $\square U$ is true iff U is true, otherwise it is false)

 

[disregardthat]

I think the term "knowable" is more relevant than "known" here. But it doesn't change the fact that nonsensical propositions cannot be knowable.

 

[alt]

Now that we're done on that one, I wonder where lugita15 got to. Interested to hear more of his/her modal paradoxes

 

[lugita15]

Now that we're done on that one, I wonder where lugita15 got to. Interested to hear more of his/her modal paradoxes

Sure, I can go on to other ones. (And it's a he by the way.) But let me first explain my preferred resolution to Fitch's paradox. I came up with it on my own soon after reading about the paradox a while back, but then I later found out that Joseph Melia thought of more or less the same solution in 1991; see the attached paper. The key idea is this: Fitch's "argument presupposes that we can discover a statement's truth value without affecting that statement's truth value. But this is not so: there exist statements which are true, yet which would have been false had we performed the procedures necessary to discover that statement's truth value."

To illustrate this point, suppose for sake of argument that it were possible for someone to be omniscient (i.e. knowing literally everything) but that no one was actually omniscient. Now consider the statement "No one is omniscient." That would be a true statement. But could it be known? Well, since we're assuming that omniscience is possible, by definition it would be possible for someone to know literally all true statements. But in that case "no one is omniscient" would not be a true statement, so it obviously wouldn't be known. So the thesis "all truths are knowable" doesn't make much sense, only because if the truth value of some true statements were found out they would no longer be true, and thus no longer be in the set of statements people can know (because you can't know a false statement).

So how do we remedy that? Surely "all truths are knowable" does try to capture some sensible and debatable sentiment, namely the belief that there are no limits to human knowledge. A better way of expressing that sentiment, one that does not fall victim to Fitch's paradox, is to say, "all truths are verifiable" or to put it another way "the truth value of any statement is knowable". To put it in more formal language, "For all statements P, either P is knowable or not P is knowable." You might think that that's equivalent to "For true statements P, P is knowable and for all false statement P, not P is knowable." But that's not true. Because your knowledge of the truth value of P may change the truth value of P (e.g. "the statement "there is no light in the room" becomes false if you turn on the light to test whether there's any light in the room!). But the important point is that Fitch's paradox allows for the possibility that you can find out the truth value of any statement, and if that's the case then surely it does not put any limitations on human knowledge.

Does that make sense to everyone? If not, look at the attached paper, and if you still have questions I'm happy to try and clarify matters.

 

[lugita15]

The Knowability Principle from your link:

$$\forall p: (p \to \diamond Kp)$$​

is highly implausible:

• If we assume two-valued truth semantics, then this seriously opposes the various incompleteness results of logic.

• No, contrary to popular belief Godel's theorem does NOT say there is a single statement that is undecidable in all sufficiently strong axiomatic systems. Rather, it says that for each sufficiently strong axiomatic system, there exists a statement in that system, but easily decidable in other systems. To conclude that there exist absolutely undecidable statements requires more work and assumptions, as Godel outlined in his famous Gibbs lecture; see here or the attached paper. (Personally I think the attached Shapiro paper is a more interesting read than the linked Feferman paper.)

 

[lugita15]

Now for another modal paradox. This one is pretty simple and unlike Fitch's paradox, where I heavily insisted that it wasn't just a simple case of word play, this one can more justifiably be called playing with words (although it can still be expressed in symbolic form). It goes as follows: Benjamin Franklin was the inventor of bifocals, glasses that correct for both near-sightedness and far-sightedness. And since he was the inventor of bifocals, e.g. Albert Einstein was not the inventor of bifocals. But we can readily imagine alternate histories in which all kinds of things happened, like the Confederates winning the civil war or Japan not attacking us on Pearl Harbor. Similarly, we can say that although Benjamin Franklin invented bifocals, he did not have to be; someone else could have done it instead. So we can say "It is possible that Benjamin Franklin did not invent bifocals."

But there's nothing special about Ben, is there? It's also true that we can think up alternative histories in which Albert Einstein did not invent bifocals; in fact, we don't even need to go to alternate histories, because in our actual world Einstein didn't invent them! So it's fair to say "For all persons X, it is possible that X did not invent bifocals." Particular instances of that general thesis are "It is possible that William Shakespeare did not invent bifocals" or "It is possible that the discoverer of general relativity did not invent bifocals." So far so good? But let's say we make the particular substitution X="the inventor of bifocals." Then our statement reads "It is possible that the inventor of bifocals did not invent bifocals." But that seems absurd, doesn't it? Surely the inventor of bifocals invented bifocals, so what's going on here?

As I said, this is a much simpler paradox to resolve. For a hint, try solving it along similar lines as my preferred resolution to Fitch's paradox, outlined above.