Is it possible for all truths to be known?

[Algren]

"This Statement is False"

Not being able to comprehend a truth does not mean its a paradox. We have some present phenomena whose origin cannot be comprehended by our current knowledge or are beyond the boundaries of human perception. We do have many examples of such.

One example of "Not Explainable Truths or Facts" can be of the various questions on life and existence on which physicists and this forum ponders on today.

The questions on cosmology & miracles (shooting stars, comets, etc.)which existed a millennium ago are considered as day-to-day fact nowadays.

 

[lugita15]

You disagree with 'a fact that has been verified' but you strongly AGREE (your caps) with 'proven by fact' (Webster's),

Sorry, out of the long list of definitions I didn't notice the one that said "something that has been proven by facts or sincerity" (which by the way is not from Webster's New College Dictionary as you said, but rather from LoveToKnow Corp). I disagree with this definition, but I think agree with the rest of the definitions outlined in post #98.

'the actual facts or information about something, rather than what people think, expect, or make up' (Macmillan)

I wholeheartedly agree with this definition. For instance, people may have "thought, expected, or made up" that the sun revolves around the earth, but that does not affect the "actual facts" about what revolves around what.

'correctness, accuracy' Wiktionary, etc.

I certainly agree with that as well. A statement can be correct or accurate even if no one knows that it is, and even if people believe that it is wrong. Wouldn't 1+1=2 even if everyone sincerely believed that 1+1=3?

So in your view, how do you get proven facts without verification ?

You can't get proven facts without verification, and I'm sorry I gave you the wrong impression about this earlier.

How do get actual facts without verification ?

I believe that a statement can reflect the actual state of the world even if no one has known, verified, or proven that it DOES reflect the actual state of the world. Do you disagree with this?

How do you get correctness and accuracy without verification ?

I think a statement can be correct and accurate even if no one has verified that it is correct and accurate. For instance, wasn't the statement "there are no humans in the universe" correct when there were no humans in the universe, and thus no humans to verify whether or not there were any humans in the universe? Or do you not believe in an objective reality, and do you think that reality is just whatever you happen to believe?

 

[alt]

Sorry, out of the long list of definitions I didn't notice the one that said "something that has been proven by facts or sincerity" (which by the way is not from Webster's New College Dictionary as you said, but rather from LoveToKnow Corp).

Click on the 'onelook' link I gave in post #98.

There it is there (presuming it comes up the same for you - I just did it again, and it did for me). Entry #10 on the main page "Webster's New World College Dictionary, 4th Ed."

Click on that and you get the exact definition I posted in #98. It must be, that 'Lovetoknow' are deriving it from Websters.

We seem to be going round in circles here. There is nothing in your post that we haven't gone through before - more than once. All you have to do is re-read this thread for my responses to the points you raised here.

 

[lugita15]

500 years ago it WAS their reality and their truth that the sun went round the earth.

OK, I think we're now getting to the heart of our disagreement. Do you believe that reality is subjective or objective? Subjective reality means that reality is just whatever you believe reality to be; for instance, if everyone believed that 1+1=3, then 1+1 would be equal to 3; if everyone believed that the moon does not exist, then it would really not exist. Objective reality means that there is a reality that is "out there", independent of what we believe about reality, so that even if humans thought the sun went around the earth, and even if humans didn't even exist, the fact of the matter would not change.

Worked for them - as far as it went.

Yes, it did. But just because you think you're right, and just because your belief "works" for you, does that mean that you are actually right? Or do you not believe that there is such a thing as being actually right?

Now it's our reality and truth that the Earth goes round the sun.

That is certainly our belief about reality, but I am not claiming that we know it for a fact. All I'm saying is that there IS a fact of the matter concerning it, whether we know it or not.

You have to allow for the possibility (probability I would even say) that your decendents will be possesed of a cosmology that will compel them to view your cosmology as partial, as ephemeral, as incomplete, as you view that of your antecedents.

I am certainly willing to allow for such a possibility, and for possibilities a million times more extreme than that. But what I believe is that either the Earth goes around the sun, or it doesn't. Regardless of which of these is correct, the important point is that one of these IS correct, even if we believe something contrary to what actually occurs. Don't you think that the Earth does what it does, and the sun does what it does, regardless of what we happen to believe about them?

 

[lugita15]

Click on the 'onelook' link I gave in post #98.

There it is there (presuming it comes up the same for you - I just did it again, and it did for me). Entry #10 on the main page "Webster's New World College Dictionary, 4th Ed."

Click on that and you get the exact definition I posted in #98. It must be, that 'Lovetoknow' are deriving it from Websters.

I hope you'll forgive me if I don't look into this, because I disagree with that particular definition regardless.

We seem to be going round in circles here. There is nothing in your post that we haven't gone through before - more than once. All you have to do is re-read this thread for my responses to the points you raised here.

Well, I agree that I'm repeating myself, but that's because I haven't gotten a clear response from you on certain issues:

1. Do you believe that there is a reality independent of what humans happen to believe about reality?
2. Do you believe that every statement is either true or false?
3. Do you believe that for every statement P, either P is true or not P is true?
4. Do you believe that all true statements are known?
5. Do you believe that for every statement P, either P is known or not P is known?

 

[lugita15]

"This Statement is False"

This is not a modal paradox, which is what this thread is about. Modality involves attributes a sentence can have (except for truth/falsity, which all sentences presumably have). Examples of modal attributes include necessary, possible, good, known, etc. Here is a better place to discuss the liar paradox; you can see my preferred resolution in post #10.

Not being able to comprehend a truth does not mean its a paradox.

I agree with that, but what paradox on this thread do you think involves a truth or falsity beyond our comprehension?

The questions on cosmology & miracles (shooting stars, comets, etc.)which existed a millennium ago are considered as day-to-day fact nowadays.

OK, but what does this have to do with this thread?

 

[Algren]

After some surfing on Fitch's Paradox, imo: its not a paradox, its an argument / flaw pointed out in the Knowability Thesis. Why is it a Paradox?

And, Fitch says that humans are not omniscient. I.e. some truths are beyond the perception of the Current human mind. But since we have population, and we have script, we can gain and store all truths which are conceivable in our current state, without the need of our minds to store the truths, but only translate these truths into language and expression.

Anyway: As you said

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."

First of all you 'supposed' we know the P which is the unknown truth. Which is a contradictory statement in itself. A millennium ago, people 'Believed' that cosmological events were miracles.

Posted on July 5th, 997:

Comets are result of an event P.
We know that P is an unknown truth. (for now, well obviously, every event/occurence has a true event acting as an origin)
What is P? We attribute P to statement Miracle.
Hence, Comets are result of Miracles.
We know that these miracles are true, but these miracles are unkown.

But it's impossible for that to be true, isn't it?

We have to find P ourselves. Things called 'Gods' won't pop it out of thin air and tell us. Till then, we have to attribute P to a 'then' considered truth, and then see how things go. P can be false and can be true, but we arent sure of either, are we? With this, we aren't sure if anything is true until and unless we solve our assumptions right to the bottom. P=Miracle has the assumption of 'divine intervention' into the current state, which was then a truth, but know a debatable topic.

"A statement is a pure truth if all assumptions behind the statement have been proved as purely true."(purely true = no part of it is false)
- Me

and you can't know a false statement!

Obviously i can know a false statement. Knowing a false statement means that you are living in an ignorance. Such as, "i know that i have made my point" is a false statement, but i deeply believe in such statement means that i am pushing myself further into my own ignorance.

So, people in 997 were living in an ignorance which they built for themselves as they believed in god.

----------------------------------------

Another way to look at things:

Turning this around, "all truths are knowable" implies "all truths are known", [STRIKE]which is crazy![/STRIKE] 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!

Yes that's [STRIKE]true[/STRIKE] right. Let there be truths P1, P2, P3 ... Pn.

Then, we know that P(1-n) are all true. But we do not know what P(1-n) actually are. We continue to guess what they can be. But what i think would be in the end, that all truths will converge to a single assumption, an assumption which will prevent us from being omniscient, and until then, all truths are just a combination of assumptions. So techically, we know these truths exist, we just try to get 'near' to these truths over time. These statements are all true assuming my definition of a purely true statement is correct.

"A statement is a pure truth if all assumptions behind the statement have been proved as purely true."(purely true = no part of it is false)
- Me

I'm not sure if the following statement is correct, but an example to the above is: "Existence of Blackholes is 99.9999% true".

 

[alt]

Lugita15;

Rather than a debate on the nature of truth and reality, I keep trying to bring it back to what this thread is about. You opened the thread with a so called paradox, and you made the statement;

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

I rejected the proposition of an 'unknown truth' and on that basis, rejected that there was ever any paradox, calling it mere word play. I asserted that if we take the common meaning of the word, there is no paradox.

It then became necessary to see what the common meaning of the word was. I thought that a good way to do this, was to get a cross section of today's respected dictionaries, and see what they showed as being it's primary (as opposed to second, third, fourth, fifth rate) meaning.

At length, in post 98, I gave you the results of an internet search which showed TEN dictionaries, from which I had highlighted the primary definition of truth from each of those ten.

You agreed with some .. you didn't with others .. you changed your mind about a couple, having ostensibly misread them ..

You appealed to the length of the list causing you not to notice one, yet you then went on to depreciate it somewhat, saying it was presented by some "Lovetoknow Corp''. Yet a little enquiry would have shown you that they we re-publishers, and that the definition came from highly respected Webster's New World College Dictionary, 4th Ed., as was clearly shown on the title page. Once I pointed this out to you however, you said won't be looking into it, because you disagree with it regardless.

Keeping your eye on the ball ? This thread is not about the nature of truth and reality - is about the common meaning of the word 'truth' in relation to your so called paradoxes.

So, you agree with the definitions of some (of my list of ten) but you disagree with others. But who are you to disagree with the framers of those respected dictionaries ? How are you more qualified than the authors of Webster's or the several others that you disagree with ? Have you spent a lifetime being involved in the evolution, the constant revision and updating of said respected, relied upon dictionaries ?

Those that you disagree with - they're fairly close to the ones you do agree with - but in any case, you said;

I'm saying the truth as commonly defined is consistent with there being both known truths and unknown truths

Now you would think - you would downright expect even, that if the framers, authors, publishers of the dictionaries - any of them, had in mind to commonly define truth as also being unknown, THAT THEY WOULD USE THAT WORD - UNKNOWN - at some point .. somewhere .. and preferably in the primary definition !

But they HAVEN'T.

You would think, that if they wanted to convey some idea that truth also had an unknown quality about it, they would DIRECTLY AND UNAMBIGUOULSY have done so. But they HAVEN'T. Not a skerrick of a direct reference to 'unknown' even in the inferior definitions, not to mention the primary. Do you think they're playing guessing games with the world ?

Now, I guess it's possible to go to each of the words used in the definition of the word truth, and thereof, twist this way and that to eventually find, perhaps, some 3rd, 4th, 5th rate meaning (of the words used to define the word truth) that might tend to the unknown .. but SURELY you don't think that that is the intention of a dictionary .. do you ?

But that is exactly what you are doing.

The common definition of the word truth, as given in the primary entries in any respected dictionary, has NO reference to 'unknown' about it. From that, any person is entitled to say that your paradox, being started on the hypothesis of an unknown truth is bunkum.

If you wish to continue to argue that ..

“truth as commonly defined is consistent with there being both known truths and unknown truths”

.. then you had better say why the dictionary folk (all of them) have thus far, failed miserably in their efforts to clearly define this, by actually using the word ‘unknown’.

But if you want to debate the nature of truth and reality, ‘unknown truth’, etc, well, that’s another thing altogether. Start a new thread on that, perhaps.

 

[Algren]

But they HAVEN'T.

You would think, that if they wanted to convey some idea that truth also had an unknown quality about it, they would DIRECTLY AND UNAMBIGUOULSY have done so. But they HAVEN'T. Not a skerrick of a direct reference to 'unknown' even in the inferior definitions, not to mention the primary. Do you think they're playing guessing games with the world ?

I found something you might have missed,Lets Recap:

My point on dictionaries is that a dictionary of English contains things appropriate for general communication in English. It will include definitions unrelated to the technical usages of terms, or even in direct conflict with them. Examples include

• the usual English meaning of "work" (e.g. sustained physical effort) versus the scientific meaning,
• the usual English meaning of the word "real" versus its usage in the mathematical term "real number".

Dictionaries will also omit many aspects or subtleties as well; it's just a dictionary, not an encyclopedia of human knowledge! For example, general English is a really, really bad language for working with many-valued logic.

So, according to you: "Work done in completing a circle is Zero" is word play.

Dictionaries are tools for English. Encyclopedias, references, and papers are tools for Science. I presume you are the ones who know better about which field does Fitch's Paradox lie in.

 

[lugita15]

Rather than a debate on the nature of truth and reality, I keep trying to bring it back to what this thread is about.

Well, it seems like your view of the nature of truth is what makes you insist that the phrase "unknown truth" is an oxymoron, so I'm trying to get at the root of our disagreement.

You opened the thread with a so called paradox, and you made the statement;

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

Yes, I did make such a statement. But let me reiterate, that I can formulate the paradox with no reference to the word "truth", so our discussion on the definition of this word is in some sense a red herring.

I rejected the proposition of an 'unknown truth' and on that basis, rejected that there was ever any paradox, calling it mere word play. I asserted that if we take the common meaning of the word, there is no paradox.

Yes, and I continue to maintain that my use of the word is in complete agreement with the common meaning, but there's really no point in arguing about this. First of all the common meaning of a word is not always what is used in philosophical discussions, and second of all the argument can be phrased without using the word at all.

It then became necessary to see what the common meaning of the word was. I thought that a good way to do this, was to get a cross section of today's respected dictionaries, and see what they showed as being it's primary (as opposed to second, third, fourth, fifth rate) meaning.

At length, in post 98, I gave you the results of an internet search which showed TEN dictionaries, from which I had highlighted the primary definition of truth from each of those ten.

Yes, and I can now tell you that I agree with all the definitions in your post #98 other than the last one.

Keeping your eye on the ball ? This thread is not about the nature of truth and reality - is about the common meaning of the word 'truth' in relation to your so called paradoxes.

No, the thread is about the paradoxes themselves. Perhaps the biggest connection between Fitch's paradox and the word "truth" is just that I happened to use the word when I initially stated the paradox in this thread.

But who are you to disagree with the framers of those respected dictionaries?

Well, the definitions are not in perfect agreement with each other, so I have to disagree with something. Still, agreeing with 9 out of 10 of them isn't too bad, is it?

How are you more qualified than the authors of Webster's or the several others that you disagree with ?

That's the only one I disagree with in post #98. But let me turn the tables on you: do you disagree with, say, the Macmillan definition "the actual facts or information about something, rather than what people think, expect, or make up"? It seems like you would disagree with it, based on what you said in response to my Earth and sun example.

Now you would think - you would downright expect even, that if the framers, authors, publishers of the dictionaries - any of them, had in mind to commonly define truth as also being unknown, THAT THEY WOULD USE THAT WORD - UNKNOWN - at some point .. somewhere .. and preferably in the primary definition !

I don't think that the word "known" or "unknown" is necessary in defining the word "truth". Couldn't I similarly say that 9 out of the 10 definitions don't use the word "known" or "verified", so that your view that truth is always known can be ruled out?

You would think, that if they wanted to convey some idea that truth also had an unknown quality about it, they would DIRECTLY AND UNAMBIGUOULSY have done so.

But I'm not claiming that the definition of truth needs to have an "unknown quality about it". All I'm saying is that nothing in teh common definition and meaning of the word "truth" restricts it to what is known or contradicts the notion of an unknown truth.

Now, I guess it's possible to go to each of the words used in the definition of the word truth, and thereof, twist this way and that to eventually find, perhaps, some 3rd, 4th, 5th rate meaning (of the words used to define the word truth) that might tend to the unknown .. but SURELY you don't think that that is the intention of a dictionary .. do you ?

I think except for the last one, all the definitions you cite are exactly correct about the meaning of the word "truth". So I don't need to go to secondary or tertiary meanings.

The common definition of the word truth, as given in the primary entries in any respected dictionary, has NO reference to 'unknown' about it. From that, any person is entitled to say that your paradox, being started on the hypothesis of an unknown truth is bunkum.

That's ridiculous. The definition of "polar bear" in the dictionary may not say that it's in imminent danger of extinction, but it may not say anything contradicting it being in danger of extinction either. The definiion of "god" may not say that there are exactly three gods, but that doesn't preclude there from being exactly three gods, does it? So how can the mere fact that the dictionary definition does not say that some truths are unknown mean that the dictionary definition prevents there from being unknown truths?

If you wish to continue to argue that ..

“truth as commonly defined is consistent with there being both known truths and unknown truths”

.. then you had better say why the dictionary folk (all of them) have thus far, failed miserably in their efforts to clearly define this, by actually using the word ‘unknown’.

They have done perfectly well to define "truth", and if I were writing my own definition I wouldn't mention the word "unknown" either. The question of knowledge is entirely irrelevant to the question of truth (although the question of truth is of course highly relevant to the question of knowledge, just as rectangles are relevant to the meaning of square but squares are irrelevant to the meaning of rectangle).

But if you want to debate the nature of truth and reality, ‘unknown truth’, etc, well, that’s another thing altogether. Start a new thread on that, perhaps.

I don't feel any particular need to debate such things, although I can if you're interested in continuing to argue about whether "unknown truth" is an oxymoron. I am perfectly happy to debate Fitch's paradox without talking about truth at all.

 

[lugita15]

After some surfing on Fitch's Paradox, imo: its not a paradox, its an argument / flaw pointed out in the Knowability Thesis. Why is it a Paradox?

Because even if the Knowability Thesis is false, it seems like Fitch's argument disproves it too easily. It would seem that any disproof of the Knowability thesis must impose fundamental limitations on human knowledge, yet Fitch's argument seems to impose no such argument. Indeed, it does not even disprove the thesis "it is possible that all truths are known", and yet it disproves what seems like a much weaker claim. That is what makes it seem paradoxical, but you can see my preferred resolution to the paradox in post #33 of this thread.

And, Fitch says that humans are not omniscient. I.e. some truths are beyond the perception of the Current human mind. But since we have population, and we have script, we can gain and store all truths which are conceivable in our current state, without the need of our minds to store the truths, but only translate these truths into language and expression.

I'm not sure what you're trying to say here.

First of all you 'supposed' we know the P which is the unknown truth. Which is a contradictory statement in itself.

No, I'm not saying that. Let me explain what I'm doing. I am first supposing that P is an unknown truth (note to alt: I'm using the word truth out of convenience, not necessity). I am not supposing that it is known which truth is unknown. Rather, I am saying, IF it were known that P is an unknown truth, you would get a contradiction, so it's impossible for it to be known that P is an unknown truth. To put it another way, it is unknowable that P is an unknown truth, even though it's true that P is an unknown truth. So there exists an unknowable true statement.

A millennium ago, people 'Believed' that cosmological events were miracles.

What does this have to do with anything?

We have to find P ourselves. Things called 'Gods' won't pop it out of thin air and tell us. Till then, we have to attribute P to a 'then' considered truth, and then see how things go. P can be false and can be true, but we arent sure of either, are we? With this, we aren't sure if anything is true until and unless we solve our assumptions right to the bottom. P=Miracle has the assumption of 'divine intervention' into the current state, which was then a truth, but know a debatable topic.

Again, I have no idea what you're talking about.

"A statement is a pure truth if all assumptions behind the statement have been proved as purely true."(purely true = no part of it is false)
- Me

What does it mean for an assumption to be behind a statement? Does it mean that the assumption implies the statement, or that the assumption is required to conclude the statement? In any case, what does this have to do with Fitch's paradox?

Obviously i can know a false statement. Knowing a false statement means that you are living in an ignorance.

You can believe a false statement, and you can even believe that you know the false statement. But you cannot KNOW a false statement. The traditional definition of knowledge used in philosophy is "justified true belief". In other words, in order to know a statement, you must believe it, your belief must be true, and you must be justified in believing it, in the sense that your reason for believing it cannot possibly lead you to believe something false.

Such as, "i know that i have made my point" is a false statement, but i deeply believe in such statement means that i am pushing myself further into my own ignorance.

Anything that is believed to be known does not have to be true, but anything that is known must be true.

 

[Algren]

Again, I have no idea what you're talking about.

I'll stay out of this, you guys use your extensive research and terminology on the topic, i just used some logic and hardly any 'research'. I was trying to state that Its possible all truths P are known, but we don't know what these P are. Referring these truths as 'P' does not meaning we actually know all these P. Thats why i took an example. Its an Example. Its not referring to any 'real' events etc. We continue guessing what P could be, but we don't know if these P are true yet.

In other words, in order to know a statement, you must believe it, your belief must be true, and you must be justified in believing it, in the sense that your reason for believing it cannot possibly lead you to believe something false.

"Belief must be true"? If its true for you, then you know a false statement.

Look, I believe in P, my belief is true and is justified (according to current society). I have no idea P is false or true. (Well, P IS false) Hence, I know a false statement P.

Lets go back to Year 0, and how many "false statements" did they know? (-an example)

How about Boh'rs model of atom, they thought it was full proof truth that day, but now we know its incomplete. (- another example)

 

[lugita15]

"Belief must be true"? If its true for you, then you know a false statement.

. I'm not talking about "true for you", whatever that means. In order for a statement to constitute knowledge, one of the requirements is that it must actually be true, independent of belief, not merely believed to be true or "true for you".

Look, I believe in P, my belief is true and is justified (according to current society). I have no idea P is false or true. (Well, P IS false) Hence, I know a false statement P.

It's not good enough if your belief is true and justified according to current society. It must actually be the case, regardless of what you or society thinks, that it is true and justified.

Lets go back to Year 0, and how many "false statements" did they know? (-an example)

They knew zero false statements, because by definition you can't know a false statement. They may have believed in any number of false statements, and may have even believed that their beliefs were justified, but that does not constitute knowledge.

How about Boh'rs model of atom, they thought it was full proof truth that day, but now we know its incomplete. (- another example)

They may have believed that Bohr's model was fully proven truth, but that doesn't mean it WAS fully proven truth.

 

[lugita15]

Does anyone have any thoughts on the Paradox of the Gentle Murderer I outlined in post #91?

 

[dcpo]

Because even if the Knowability Thesis is false, it seems like Fitch's argument disproves it too easily. It would seem that any disproof of the Knowability thesis must impose fundamental limitations on human knowledge, yet Fitch's argument seems to impose no such argument. Indeed, it does not even disprove the thesis "it is possible that all truths are known", and yet it disproves what seems like a much weaker claim. That is what makes it seem paradoxical, but you can see my preferred resolution to the paradox in post #33 of this thread.

I understand that you want to move away from the Fitch paradox but I think this aspect is interesting. I see no reason to assume that a disproof of the knowability thesis should place any more fundamental limit on human knowledge than the 'paradox' provides. That the disproof is not the kind of thing that may have been expected would indicate to me a problem with the conception of knowability. I admit I am not familiar with this branch of philosophy, but what a fundamental limit to human knowledge should look like is rather obscure to me.

You can believe a false statement, and you can even believe that you know the false statement. But you cannot KNOW a false statement. The traditional definition of knowledge used in philosophy is "justified true belief". In other words, in order to know a statement, you must believe it, your belief must be true, and you must be justified in believing it, in the sense that your reason for believing it cannot possibly lead you to believe something false.Anything that is believed to be known does not have to be true, but anything that is known must be true.

I'll take issue with this if I may. I realize that what you describe is an established philosophical position, but it is not an objective truth. An individual may wish to use the word 'knowledge' it in a different way. Certainly you can explain that when you talk about knowledge you demand truth, but another person may, with good reason, wish to use it another way. It should be no barrier to communication if the difference is acknowledged.

I apologize in advance if I am not able to reply in this thread for a few days; I'll be without an internet connection.

 

[lugita15]

I understand that you want to move away from the Fitch paradox but I think this aspect is interesting.

There's no need to move away from Fitch's paradox. We can discuss multiple paradoxes at once.

I see no reason to assume that a disproof of the knowability thesis should place any more fundamental limit on human knowledge than the 'paradox' provides.

If you heard there was a proof that proved that there were some truths that were unknowable by Man, wouldn't you assume that that meant that human knowledge has limits? Well, strangely Fitch's paradox says that some truths cannot be known and yet it does not contradict the statement that it is possible to be omniscient, i.e. know all truths! That's what's neatly resolved, in my opinion, in my post #33.

That the disproof is not the kind of thing that may have been expected would indicate to me a problem with the conception of knowability.

What problem is that? The definition of knowability is straightforward: P is said to be knowable if it is possible that P is known.

I admit I am not familiar with this branch of philosophy, but what a fundamental limit to human knowledge should look like is rather obscure to me.

A fundamental limitation on human knowledge would not only say that there are some true statements that cannot be known, it would also say at the very least that there are some statements whose truth value cannot be known. Yet Fitch's paradox does not imposes any such limitation.

I'll take issue with this if I may. I realize that what you describe is an established philosophical position, but it is not an objective truth.

It's not a philosophical position, it's just the standard definition used in philosophy.

An individual may wish to use the word 'knowledge' it in a different way. Certainly you can explain that when you talk about knowledge you demand truth, but another person may, with good reason, wish to use it another way. It should be no barrier to communication if the difference is acknowledged.

Of course people may choose to use words in all sorts of nonstandard ways. But I'm using knowledge with the standard philosophical meaning.

Anyway, do you agree with the resolution I present in my post #33? Also, have you taken a look at the other paradoxes I have presented, in posts #35 and #91?

 

[dcpo]

I think I’m starting to see what you’re getting at here. So the paradoxical nature comes from the fact that if all truths are known the problem statement cannot be constructed. This is certainly a noteworthy quirk, but I don’t find it overly worrying at first glance. This is what I mean when I say it points to a problem with the concept of knowability. Clearly knowability is a simple concept when applied to concrete statements, but the paradox hinges on a rather abstract construction. I would not expect such a superficially simple, naive even, concept to extend without issue to a sphere where we discuss `all truths’. I would expect to be able to generate paradoxical statements out of the acceptance of a totality of truths. If anything I’m surprised that more damaging paradoxes have not been discovered.

Am I correct in understanding that the resolution to the paradox is that while the truth of Q is unknowable, it is possible to stop Q being a truth (by establishing the truth of P)? If so then that seems a reasonable resolution to me.

I’ve looked at the other paradoxes but I’ve not had many thoughts on them yet. I am happy with the simple resolution to the second proposed by Hurkyl.

 

[dcpo]

To expand on the above, if we allow there to be a class of all truths then for each subclass of this class the statement that every member of that subclass is true is a truth, and thus the class of all truths cannot be in 1-1 correspondence with itself.

It strikes me that any discussion of knowability that does not put some limit on what truths are included in the discussion has more serious problems to overcome than Fitch's paradox.

 

[lugita15]

I think I’m starting to see what you’re getting at here. So the paradoxical nature comes from the fact that if all truths are known the problem statement cannot be constructed.

The paradoxical part is that the proof rules out the weaker claim "all truths are knowable", but it does not rule out the stronger claim that "it is possible to know all truths". In other words, it somehow says that certain truths cannot be known, and yet it is still possible to know everything (i.e. all true statements). That seems very strange.

This is what I mean when I say it points to a problem with the concept of knowability. Clearly knowability is a simple concept when applied to concrete statements, but the paradox hinges on a rather abstract construction. I would not expect such a superficially simple, naive even, concept to extend without issue to a sphere where we discuss `all truths’.

Knowability is a simple concept in general. It is just knowledge plus possibility. Knowledge is well studied in epistemic logic, and possibility is well studied in alethic modal logic. Yet somehow when we combine epistemic and alethic modal logic, we get this seemingly paradoxical result.

I would expect to be able to generate paradoxical statements out of the acceptance of a totality of truths. If anything I’m surprised that more damaging paradoxes have not been discovered.

We can actually carry through this paradox without using the phrase "all truths" at all.

Am I correct in understanding that the resolution to the paradox is that while the truth of Q is unknowable, it is possible to stop Q being a truth (by establishing the truth of P)? If so then that seems a reasonable resolution to me.

That is precisely the resolution I outline in post #33. By knowing P, you make Q false, so you can't know Q. However, although it impossible to know that Q is true, you can very easily know that Q is false. So the correct way to say that there are no limits to human knowledge is not to say "For all truths P, P can be known to be true." Instead, you should say "For all statements P, the truth value of P can be known." (The truth value of a statement means whether it is true or false.) So you can replace "all truths are knowable" with "all statements are decidable", and this latter claim does not lead to analogus paradoxes.

I’ve looked at the other paradoxes but I’ve not had many thoughts on them yet.

If you like, I can explain my preferred resolution to the paradox of the gentle murderer, or I can wait a little while to see whether anyone else would like to wrestle with it.

I am happy with the simple resolution to the second proposed by Hurkyl.

Yes, that is indeed the standard resolution to the inventor of bifocals paradox.

 

[lugita15]

To expand on the above, if we allow there to be a class of all truths then for each subclass of this class the statement that every member of that subclass is true is a truth, and thus the class of all truths cannot be in 1-1 correspondence with itself.

But it is not the case that for each subclass of the class of truths there is a statement saying that every member of that subclass is a truth! That's because you can't talk about most of the subclasses of the class of truths using statements. A statement must be a finite length, so there are only countably many statements, and thus countably many true statements. Thus there are more classes of statements then there are statements, so most classes of statements cannot be described by statements. There are uncountably many classes of statements, but only countably many definable classes of statements. Since the class of truths and the class of definable subclasses of the class of truths are 1-1 correspondence, there is no contradiction.

It strikes me that any discussion of knowability that does not put some limit on what truths are included in the discussion has more serious problems to overcome than Fitch's paradox.

No, there is no need to limit the truths under discussion.

 

[Ken G]

If you like, I can explain my preferred resolution to the paradox of the gentle murderer, or I can wait a little while to see whether anyone else would like to wrestle with it.

I looked up Forrester's paradox, but I don't know the exact version you gave above. I saw a symbolic version of it at http://rationalhunter.typepad.com/close_range/2004/05/forresters_para.html,
but I didn't find it convincing as a paradox, because it seemed to me that assumption #2 was not a good assumption at all. There the structure of "obligation" is symbolically built to be bimodal (either you are obliged to do something, or you are not obliged to do it, there are no "levels of obligation" built into the symbolism, and putting in such levels, like O1 and O2 where O1 > O2 would seem to fix the paradox), but when this is translated into human sensibility, the language gets mauled, and we end up with a hierarchy of obligation (we are obliged not to murder, call that O1, but if we do murder, we are obliged to murder gently, call that O2). Mistaking a bimodel obligation (used in the logic of the paradox) with a hierarchical version (used in real life) seems to be the source of the paradox.

Incidentally, this seems to connect with your other thread about what mathematics is. We use logical structures to make proofs, but the logical structures must be very closely connected (some, not I, might dare to say identically connected) to things that we experience in our daily lives, if we want the theorems to make sense in our daily lives. So if we want to prove things about obligations that check with how we use that term in daily life, we must tailor the axioms around obligations to fit with how we use the term colloquially. That creates a kind of "back door" through which paradox can creep, some (perhaps even me) might say, through which paradox inevitably creeps. This is a cautionary tale about limits for using mathematical proofs to know truths, the most celebrated example being Godel's theorems.

 

[dcpo]

But it is not the case that for each subclass of the class of truths there is a statement saying that every member of that subclass is a truth! That's because you can't talk about most of the subclasses of the class of truths using statements. A statement must be a finite length, so there are only countably many statements, and thus countably many true statements. Thus there are more classes of statements then there are statements, so most classes of statements cannot be described by statements. There are uncountably many classes of statements, but only countably many definable classes of statements. Since the class of truths and the class of definable subclasses of the class of truths are 1-1 correspondence, there is no contradiction.

Well, yes, but nevertheless it is true that all the statements in any subclass of the class of all true statements are true. My point is that you cannot get a complete handle on general truth using the formal machinery of logic. I guess this is rather trivial, as was my example, and Fitch's construction is more interesting in that it produces a seemingly paradoxical, though easily resolved, result using only ideas that can be easily translated into familiar formal systems. What I'm trying to say is that it's interesting as an investigation into the possible limitations of our formal machinery in this respect, but that I remain somewhat unconvinced that this kind of reasoning gets us very close to understanding the actual limits of human knowledge.

No, there is no need to limit the truths under discussion.

But you've just described how you are limiting them; by having 'true' things for which there is no statement in the formal language.

 

[elegysix]

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.

Sorry if someone already touched on this, but my favorite question like this is:

How ignorant are we?

In order to answer it, you'd need to know all absolute truths. And if you knew all absolute truths, you wouldn't be ignorant.

This means that we can never know how ignorant we are.

 

[Bill_McEnaney]

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.

PlayingMonk, what do you mean by "objective?" A truth can be objective in more than one way at the same time.

In your post, "true to" seems to mean what many relativists about truth seem to mean "true for" when they say, "That may be true for you, but it's not true for me." Well, their comment merely another way to say, "You may believe that, but I don't believe it." Am I telling you an objective truth when I tell you that for each proposition p, someone or other may or may not believe that-p?" If that claim is true, it's objective, too, in one of three relevant senses of the word "objective." If the claim is true, it's true about each person. But relativists about truth deny that there's any proposition that's true about each person. They want to have their cake and eat it, too.

A relativist about truth told me that since every truth depended on some context or other. But here opinion was absolute in one sense of "absolute." It was an opinion about every context.

 

[Ken G]

A relativist about truth told me that since every truth depended on some context or other. But here opinion was absolute in one sense of "absolute." It was an opinion about every context.

When I teach, I always say "I must warn you that nothing I say will be absolutely or completely true. Including that."

 

[sigurdW]

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.

Whats this? "Fitch's Paradox". Are we talking about THE Fredric Fitch!? Oh my...I see it is so.

He is one of the sharpest logicians I've encountered. I found he was extremely short in the amount of selected words but a few lines of his compares with chapters from lesser logicians.

But ...awww...Its modal logic! What shall I do? All my life I refused to take part in it!

I always assumed no modal logic to be consistent, so why bother? Well Ill start reading now and it will take me some time to catch up, if ever I will since modalities disgust me. But I can't ignore Dear old Fredric...Here I go :)

 

[dcpo]

^^ Why would you assume no modal logic could be consistent?

 

[sigurdW]

^^ Why would you assume no modal logic could be consistent?

It was originally Frege that made me suspicious of modal concepts: You know the case of the Morning star and The Evening star being both identical with the "star" Venus,and if Mr X knows that the Morning star is Venus then he should also know that the Evening star is Venus... but it need not be so. (If I remember the argument correctly.)
Other things, the Liar Paradox in particular, kept occupying my attention and I never started a search for a way to "defuse" modal concepts. By the way what are the "^^" supposed to add to your question?

 

[dcpo]

I must say that seems a rather flimsy reason to doubt the consistency of modal logic, especially considering that it has been rigorously studied for 100 years, and has its own well developed proof and model theories. Maybe you mean something non-standard when you talk about consistency.

p.s. The '^^' indicates I'm replying to the post immediately above mine.

 

[sigurdW]

I must say that seems a rather flimsy reason to doubt the consistency of modal logic, especially considering that it has been rigorously studied for 100 years, and has its own well developed proof and model theories. Maybe you mean something non-standard when you talk about consistency.

p.s. The '^^' indicates I'm replying to the post immediately above mine.

Flimsy reason or not...I rely rather heavily on my intuition :) But I don't trust my life on it,I check. I now want to start a research into modal logic...Im not fast and I have to defend my solution of the Liar paradox meanwhile: https://www.physicsforums.com/showthread.php?p=3980342#post3980342 So any "deep" comments have to wait...perhaps forever ;)
By "inconsistent" I mean simply that a contradiction follows from the axioms of the theory in question.And I think I did not CLAIM there was inconsistency... I just reported my suspicion: So what interesting results are there after the hundred years of rigorous research? Quines claim that Modal Logic was conceived in the sin of confusing use and mention is refuted? Modal Logic is now wholly without sorrows? I haven't given it much thought the last thirty years but I suppose my stand still is that Modal Matters matter but formalizing its logic seemed to me just a fun game for formalists bored by ordinary logic. (No disrespect intended.)

 

[lugita15]

If anyone is still interested, I can present my preferred resolution to the Paradox of the Gentle Murderer I introduced in post #91. (The previous two paradoxes, Fitch's paradox of Knowability and the Inventor of Bifocals paradox, are stated in posts 1 and 35 and resolved in posts 33 and 61). Then I can introduce yet another modal paradox.

 

[sigurdW]

I always enjoy reading your posts.