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
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.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
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.If you cannot put 'know' in front of every sentence then there should be a principled explanation why — Fafner
There's no problem with that statement, provided P is a constant, not a variable.KP seems like a perfectly consistent thing to say — Fafner
KP seems like a perfectly consistent thing to say — Fafner
I agree that this would not be very useful, but it is also the case that talk about ''syntax error" as a criterion for meaningfulness is also just as useless thing to say.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
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.).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
Ok, but what kind of justification is that?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
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
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
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
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.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
It's a variable, because it's written Kp, preceded by a universal quantifier over p.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
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
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.The KP principle can be formulated as a claim about all truths (as it appears in the Stanford article) rather then all propositions — Fafner
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.To extend it then, are there truths which can never be known? — 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.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. — andrewk
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.I was wondering whether your reason for counting truths as a subset of propositions is that there are both true and false propositions — John
What makes you you think that?The issue of the infinitely long decimal line is actually finite. — WillowOfDarkness
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
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