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. — TheMadFool

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.

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). — Fafner

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.

If you cannot put 'know' in front of every sentence then there should be a principled explanation why — 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.

KP seems like a perfectly consistent thing to say — 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.

I assume you are referring to this:

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.

KP seems like a perfectly consistent thing to say — Fafner

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. — andrewk

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.

I don't know 'there's no such thing as a theory of types' means in this context. Here's the one I had in mind. — 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?

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. — andrewk

Ok, but what kind of justification is that?

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. — 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).

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? — Fafner

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.)

One difference is that formal languages are invented for particular purposes which are quite different from chatting with your friends or singing "happy birthday". There are also stipulative definitions in formal languages, but no such definitions exist for natural language, so it makes a big difference when we ask philosophical questions about natural languages as opposed to formal languages.

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.

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. — andrewk

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?

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. — Fafner

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.

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? — Fafner

It's a variable, because it's written Kp, preceded by a universal quantifier over p.

I'm not sure how well I can express this, but I think the problem is K itself. (I'd like to take a closer look at Dummett's response though.)

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.

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. — andrewk

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.

I tried explaining it to my son, and of course his reaction was that if you know p then nobody knows p is just false. When I explained how the conjunction is supposed to work, his feeling was that there's a problem here with the expression of human thought rather than with knowledge itself. (Admittedly, my son is not exactly the man on the street.)

My personal feeling is that the fishy bit occurs where we take the "P is unknown" part and ask whether it can be known in some possible world where P is the case. My feeling is that we simply aren't talking about the same P when we do this, but I'm struggling to formulate precisely what I have in mind.

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).

I'm still working on it. (I've been reading Dummett off & on for a while now, so I've acquired some sympathy for his program, and this is an interesting challenge.)

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.

The KP principle can be formulated as a claim about all truths (as it appears in the Stanford article) rather then all propositions — Fafner

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?

Yes, that seems to follow.

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?

To extend it then, are there truths which can never be known? — John

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 <insert here the infinitely long decimal expansion>. IF we count that as a truth - contrary to the usual convention that propositions must have finite length - then it is a truth that no finite organism can know.

If there's a God, then it can know it. Do we count God?

Also, would you be able to explain why you think that truths are a subset of propositions? — 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

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. — andrewk

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.

I was wondering whether your reason for counting truths as a subset of propositions is that there are both true and false propositions — John

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.

Wouldn't it be half the set of propositions, since half are true and half are false?

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.

On the contrary, it is precisely because I agree with that, that I see the set of propositions as containing exactly half true ones and half false ones (being the negations of the trues ones).

I suspect we must be at cross purposes here.

The issue of the infinitely long decimal line is actually finite. — WillowOfDarkness

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.

Oh frabjous day, calloo, Callay - I have found it!

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.

Here's the correct formulation of the paradox based on the Stanford article cited by Meta.

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. cannot obtain. 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). — Fafner

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.