Please tell me where I am going wrong here:

The unknown truth that is the number of coins in the jar is expressed as: p ∧ ¬Kp

It is impossible to know the unknown truth: p ∧ ¬Kp

Therefore, it is impossible to know the unknown truth that is the number of coins in the jar. — Luke

Here are two propositions:

1. the cat is on the mat
2. the cat is on the mat and the mat was bought from Ikea

Both are true, and even though the first proposition doesn't express it, the mat was bought from Ikea (as explained by the second proposition). And it's possible that I (eventually) know that the cat is on the mat but not that the mat was bought from Ikea (so I know the first but not the second).

Similarly:

3. the cat is on the mat
4. the cat is on the mat and nobody knows that the cat is on the mat

Both are true. And even though the third proposition doesn't express it, nobody knows that the cat is on the mat (as explained by the fourth proposition).

The issue is that it is possible to (eventually) know 3 but it isn't possible to (eventually) know 4.

I'm asking you how else "p is unknown" could be expressed in logical notation - other than as "p ∧ ¬Kp", and other than as your mere assurance outside of logical notation that p is unknown. — Luke

p ∧ ¬Kp is how you express it.

The problem is that you seem to go from "p ∧ ¬Kp" is unknowable to "p" is unknowable. And that just doesn't follow.

It is not possible to know any proposition of the form "p & ~Kp", which means that all unknown truths (expressed in this way, at least) are unknowable. — Luke

This is where you have a fundamental misunderstanding that I don't know how to explain to you. Maybe like this?

a) p
b) a is not known to be true

Both a and b are true. Neither a nor b are known to be true. It is possible to know a but not possible to know b.

On the other hand, you do not accept the argument's implication that we cannot come to know mundane unknown truths such as the number of coins in a jar. — Luke

No it doesn't.

Look, smarter people than both of us have addressed Fitch's knowability paradox. None of them have argued that it somehow entails that all truths are unknowable; instead they accept that it shows that either the knowability principle is false or that every truth is known. Their solution to the problem (where they want to keep some form of the knowability principle) is to change the knowability principle. See Tennant's and Dummett's responses as detailed here.

2 (when expressed as "p & ~Kp") is unknowable, which means that so is the number of coins in the jar. — Luke

That p ∧ ¬Kp is unknowable isn't that p is unknowable. The number of coins in the jar is p. We can know p.

I asked how you would express (in logical notation) that it was unknown. — Luke

2 does that.

Then this should be able to be expressed in the argument. If it cannot be expressed in the argument, then it is not a failure of the knowability principle, but a failure of logic. Otherwise, accept the logic and the number of coins in the jar is unknowable. — Luke

I don't understand what you're asking for here. The argument simply shows that if you take the knowability principle and the non-omniscience principle as premises then it follows that the non-omniscience principle is false. It is then up to the reader to decide whether to accept that the non-omniscience principle is false or to reject the knowability principle.

So why can't you just accept that the knowability principle is wrong? Some truths are, in fact, unknowable.

Therefore, the number of coins in the jar remains unknowable. — Luke

It isn't. We can count the coins and then we will know how many coins are in the jar.

1. does not express that it is unknown — Luke

Which is why it is possible to know it.

2. expresses that it is unknown, but it is unknowable. — Luke

Which is why the knowability principle is wrong.

Presumably, the unknown truth of the number of coins in a jar is not expressed as "p & ~Kp", since this is unknowable. So how would you express the unknown truth about the number of coins in a jar? — Luke

1. p
2. p ∧ ¬Kp

Assume p is true. Both 1 and 2 are true. Neither 1 nor 2 are known to be true. 1 can be known to be true. 2 can't be known to be true.

You can't even prove propositions exist yet you used the term in your attempt to use rules of inference. — Harry Hindu

I'm not interested in proving that propositions exists. I am simply, for the sake of argument, taking as a premise that "p" is true iff p, or to use a specific example, that "the cat is on the mat" is true iff the cat is on the mat. I then show what follows from assuming this premise. You're welcome to reject the premise if you like, and then my argument just isn't directed at you anymore.

To borrow Andrew M's example:

Suppose there are 163 coins in the jar and no-one knows there is.

It's thus true that there's 163 coins in the jar and no-one knows there is.

That true statement is unknowable. Why? Because anyone coming to know that there's 163 coins in the jar (say, by counting) would render the statement false (since the second conjunct would be false). The statement doesn't change from an unknown truth to a known truth. It changes from an unknown truth to a known falsity. — Luke

These are two different propositions:

1. There are 163 coins in the jar
2. There are 163 coins in the jar and no-one knows there is

It is possible that both propositions are true. It is possible that neither proposition is known to be true. It is possible to know the first proposition. It is not possible to know the second proposition. Therefore, the knowability principle is false.

I mean that the unknown truth "p & ~Kp" of NonO cannot possibly become a known truth. If that is impossible from the outset, then so is knowability. — Luke

No it isn't. There are some things which are unknown truths which can become known, e.g. the number of coins in a jar.

But here S(M) does possess explanatory power above M. With M we wonder how this extraordinarily unlikely event happened. — hypericin

The same with S(M). As the article says, "the Boltzmann brain thought experiment suggests that it might be more likely for a single brain to spontaneously form in a void (complete with a memory of having existed in our universe) rather than for the entire universe to come about in the manner cosmologists think it actually did."

So both M (Boltzmann brains) and S(M) (common sense life) are extraordinarily unlikely, but given that S(M) is less likely than M, what greater explanatory power does it have?

Therefore, the starting suppositions make it impossible for an unknown truth to become a known truth. — Luke

No it doesn't.

"There are 163 coins in the jar" was an unknown truth before someone counted, and then it became a known truth.

But if "p & ~Kp" cannot possibly change from being unknown to being known, then of course it is unknowable: it's a rigged game from the outset. — Luke

I don't know what you mean by it being "rigged". It just shows that the knowability principle is wrong. Some truths are, in fact, unknowable.

But I never proposed that complexity be the sole criterion for choosing a theory. — hypericin

You said: "Since S(M) never possesses any explanatory power above M, and yet S(M) is always more complex than M, S(M) can always be discarded via Occam's Razor."

Replace S(M) with common sense life and M with Boltzmann brain.

Why don't we go back and see if we can define proposition. What forms do propositions take? If I were to look for a proposition where would I look? What would I see or hear? — Harry Hindu

I don't know, as I previously said.

Help me understand, why should a brain spontaneously materializing be more likely than one evolving naturally? — hypericin

This is what the Wikipedia article says:

The Boltzmann brain thought experiment suggests that it might be more likely for a single brain to spontaneously form in a void (complete with a memory of having existed in our universe) rather than for the entire universe to come about in the manner cosmologists think it actually did.

A brain evolving naturally requires a much larger ecosystem (a Star, a habitable planet, millions of years of reproduction and natural selection, etc., all of which have prior requirements of their own). That is far more complex than just a brain forming in a void.

As much as anyone streelight represented what is best in the forum. — T Clark

I have no representatives.

OK, I've done some further research and in classical logic ¬∃x(x=q) isn't allowed and in free logic T(q) ⊢ ∃x(x=q) is false.

I'll strike out the free logic stuff for now as the rest is still interesting and will come back to them when I've learnt more.

On that note though, what logic would you say ordinary language uses? Because in classical logic you can't say "if the cat does not exist then the cat is not on the mat" and in free logic you can't say "if the cat is on the mat then the cat exists."

It seems to me that in ordinary language we can say both, and so an ordinary language interpretation of my argument still holds.

Your proposition a. can likewise be true untill such a time when it becomes false. — Olivier5

According to the knowability principle, if a proposition is true then it is knowable. Therefore, if a proposition is not knowable then it is not true.

As Fitch's paradox shows, a isn't knowable. Therefore, according to the knowability principle, a isn't true.

In such a case, the sentence "the sentence p is an unknown truth" is true today; and, if all truths are knowable, it should be possible one day to learn that "p was an unknown truth" up untill that day. — Olivier5

a. "p" is an unknown truth
b. "p" was an unknown truth

These are not the same proposition.

According to the knowability principle, if a proposition is true then it is possible to know that proposition. Therefore, if a is true then it is possible to know a. You've only argued that we can know b. Knowing b is not the same as knowing a.

This (and the comments by @Snakes Alive in the discussion you linked to) would suggest that ∃x(x = q) is valid in first-order logic, and doesn't require free logic? So I can do T("p") → ∃x(x = "p").

I've updated my original post accordingly.

Actually, looking at this, it does appear that steps 4 and 11 (x does not exist) of my argument depend on free logic:

Free logic allows such statements to be true despite the non-referring singular term. Indeed, it allows even statements of the form ∼∃x x=t (e.g., “the ether does not exist”) to be true, though in classical logic, which presumes that t refers to an object in the quantificational domain, they are self-contradictory.

I'm happy with this, as I would say that "the ether does not exist" is in fact true.

My one initial concern is with whether or not modus tollens applies to ∀p: p → ∃x(x=q). I'll do some digging.

I'm saying it's not allowed in the rules of classical logic. — Banno

Then what logic am I using when I say that if John is bald then John exists? Or that if the cat is on the mat then the cat exists? Because they seem like logical inferences to me. It would be strange to say that the cat is on the mat but there isn't a cat.

If your problem is with my (mis-)use of formal symbols then you can consider the argument in natural language as I started with here.

But then let's look at the argument using the complete form of existential introduction:

1. ∀p: T("p") ↔ p (premise)
2. ∀p: T("p") → ∃xT(x)
3. ∀p: p → ∃xT(x)
4. ∀p: ¬∃xT(x) → ¬p
5. ∀p: T("¬p") ↔ ¬p (premise)
6. ∀p: T("¬p") → ∃xT(x)
7. ∀p: ¬p → ∃xT(x)
8. ∀p: ∃xT(x)

The main conclusions being 4 (if there are no true truth-bearers then nothing is the case) and 8 (there is at least one true truth-bearer).

And so there is still the issue that either a) truth-bearers are dependent on thought and speech and so if something is the case then something true is thought or spoken or b) truth-bearers are independent of thought and speech.

If needed we can reformulate our initial premise and derive the slightly different conclusion that for all truth-bearers there is at least one true truth-bearer:

9. ∀"p": T("p") ↔ p
...
10: ∀"p": ∃xT(x)

Still working on how to properly formulate 11. Perhaps something like:

11. ∃p: p → ¬∃"p"T("p") (there is at least one case where if that thing is the case then there is no true proposition that it is the case).

But you have "if John is bald then John exists". That's invalid ill-formed. ∃(a) is not a formula in first order logic, unless you move to free logic. — Banno

How is it ill-formed? It makes perfect sense to me:

If John is bald then something exists which is bald
If John is bald then something exists

Are you saying that the second sentence doesn't make sense? Or is false?

It's "If John is bald then there is something that is bald". — Banno

Is there a difference between "if John is bald then there is something that is bald" and "if John is bald then there exists something that is bald"?

I'm pretty sure my comment above addresses that.

If John is bald then John exists
If the proposition "it is raining" is written in English then the proposition "it is raining" exists
If the proposition "it is raining" is true then the proposition "it is raining" exists

Looks like you've defined a fixed point of some function. But I doubt that is what you mean? — jgill

I'm just explaining what is meant by the predicate T. I could have written the argument as:

1. ∀p: "p" is true ↔ p

But that would require more typing.

According to existential introduction:

Q(a) → ∃xQ(x) (if John is bald then there exists at least one thing which is bald)

And surely:

∃xQ(x) → ∃x (if there exists at least one thing which is bald then there exists at least one thing)

And so:

Q(a) → ∃a (if John is bald then John exists)

Maybe my particular symbols aren't being used quite right, but surely the logic works? In ordinary language (and providing the complete account of existential introduction) it would be:

1. proposition "p" is true if and only if p
2. if proposition "p" is true then there exists some proposition which is true (existential introduction)
3. if proposition "p" is true then there exists some proposition
4. if proposition "p" is true then proposition "p" exists

4 must follow otherwise we would have the situation where, from 2, the truth of proposition "p" would entail that there exists some other (true) proposition. Or in the case of John being bald, that John being bald entails that there exists some other (bald) thing. Which seems absurd.

I don't understand what you're trying to get at. Either there are rules of inference or there aren't. If there are then my argument is valid. If there aren't then I guess anything goes and we can say anything we like and we abandon all talk of reason or contradiction. I don't even understand how you expect us to engage in argument unless you accept the reality of logic.

I don't know. Regardless, unless you want to reject the accepted rules of logic, you have to accept that my argument is valid (and as you accepted the premise, that my argument is sound).

I take issue with 2 and 4. — Harry Hindu

2 is an application of existential introduction. 4 is modus tollens. They're valid rules of inference.

2 and 3 seem to be saying the same thing. — Harry Hindu

2 is saying that if the proposition "it is raining" is true then the proposition "it is raining" exists.
3 is saying that if it is raining then the proposition "it is raining" exists.

A true or false propsition is not synonymous with an existing or non-existing proposition. A false proposition is just as real as a true one. — Harry Hindu

I'm not saying otherwise. You appear to be denying the antecedent when looking at 2.

Did propositions exist prior to humans existing? If the answer is no, then propositions depend on our existence. — Harry Hindu

Then we run into this issue:

T(x) ≔ x is true (definition)

1. ∀p: T("p") ↔ p (premise)
2. ∀p: T("p") → ∃"p" (from 1, by existential introduction)
3. ∀p: p → ∃"p" (from 1 and 2, by hypothetical syllogism)
4. ∀p: ¬∃"p" → ¬p (from 3, by modus tollens)

Or using a specific example in ordinary language:

1. The proposition "it is raining" is true if and only if it is raining
2. If the proposition "it is raining" is true then the proposition "it is raining" exists
3. If it is raining then the proposition "it is raining" exists
4. If the proposition "it is raining" does not exist then it is not raining

Propositions are a causal relation just like everything else in the universe. Any particular thing does not exist independent of the causes that led to its existence. — Harry Hindu

OK, but do propositions exist when nothing is said? Do propositions exist when nothing is thought? Does the existence of a proposition depend in some sense on us?

Platonic realism isn't obviously absurd. It's just not fashionable. — Tate

The same could be said of idealism/anti-realism.

They exist as abstract objects. The set of all non-penguins exists whether anybody ever refers to it or not. I guess it's part of a logical landscape. They don't exist in time, in other words. They don't age. — Tate

This sounds like Platonic realism.

I personally wouldn't argue for something Platonic. I would say they're residents of human thought. They're part of the way we interact with our environment. — Tate

But then if they're "residents of human thought" then presumably they don't exist when not thought?

It's usually thought of as an abstract object, which just means a proposition is "beyond" any particular person. I can be wrong about the status of a proposition, so it's not just a resident of my noggin. Mathematical entities are also abstract, so you can compare propositions to things like numbers.

Propositions are the things people assert or agree to. If you adopt an ontology that rules them out, you're headed for some type of behaviorism. — Tate

I'm not ruling out propositions, I'm questioning what it means for a proposition to exist. Do propositions exist when nothing is said? Do propositions exist when nothing is thought? If they do then it strikes me as Platonic realism. Is that what you're arguing for?

You're not providing a meaningful account of what a proposition/truth-bearer is. Is it a physical entity? Is it a mental concept? Is it a Platonic Idea? Is it some magical substance that is able to "attach" to concrete utterances?

Truth bearer. — Tate

And what's a truth-bearer? A sentence, e.g. an utterance?

"The term ‘proposition’ has a broad use in contemporary philosophy. It is used to refer to some or all of the following: the primary bearers of truth-value, the objects of belief and other “propositional attitudes” (i.e., what is believed, doubted, etc.[1]), the referents of that-clauses, and the meanings of sentences.". SEP — Tate

So which of them are you saying exist(s) when it is not raining?

If it is not raining, then the proposition "it is raining" exists. — Tate

I think that this is certainly questionable. What is a proposition? Is it a sentence, e.g. an utterance?

So, if it is raining then the phrase "it is raining" is spoken?

Or is a proposition something else?

The conclusion follows from the premise via valid rules of inference. If the premise is true then the argument is sound.

The last inference is wrong. — Tate

It's a valid inference from step 3. It's called modus tollens.

p → q
¬q → ¬p

In this case, q is ∃"p".