This is the argument:

1. T("p") ↔ p (premise)
2. T("p") → ∃"p" (inference, existential introduction)
3. p → ∃"p" (inference, hypothetical syllogism)
4. ¬∃"p" → ¬p (inference, modus tollens)

Do you disagree with the premise, or with one or more of the inferences?

If the proposition "it is raining" does not exist then it is not raining — Michael

If it's not raining, the proposition "it is raining" exists. It's just false. — Tate

Your response has no bearing on the sentence you're responding to.

It's indigestible without some secret sauce. — Janus

The proposition "it is raining" is true if and only if it is raining
If the proposition "it is raining" is true then the proposition "it is raining" exists
If it is raining then the proposition "it is raining" exists
If the proposition "it is raining" does not exist then it is not raining

Some food for thought:

T(x) ≔ x is true

T("p") ↔ p
T("p") → ∃"p"
p → ∃"p"
¬∃"p" → ¬p

Burying the lede with that quote.

He flung his lunch across the room, smashing the plate in a fit of anger as ketchup dripped down the wall. He appeared to endorse supporters who wanted to hang his own vice president. And in a scene laid out by a former aide that seemed more out of a movie than real life, he tried to wrestle away the steering wheel of his presidential vehicle and lunged at his own Secret Service agent.

...

“You know, I don’t f-ing care that they have weapons,” Mr. Trump said in Ms. Hutchinson’s telling of the episode. “They’re not here to hurt me. Take the f-ing mags away. Let my people in. They can march to the Capitol from here. Let the people in. Take the f-ing mags away.”

So? Maybe I'm curious to know whether we have free will. — Luke

Then it's a topic for another discussion, not this one.

I find it epistemologically interesting that if we reject NonO then all truths are not only knowable but known, and if we reject KP then there is not only an unknown but an unknowable truth. These both follow from Fitch's argument, so I wouldn't say it has nothing to do with it. — Luke

This is where you're misunderstanding Fitch's paradox. It isn't showing that if we reject the non-omniscience principle then all truths are known or that if we reject the knowability principle then some truths are unknowable; it's showing that if we accept the knowability principle then all truths are known.

That a rejection of the non-omniscience principle entails that all truths are known is a truism, and that a rejection of the knowability principle entails that some truths are unknowable is a truism. This has nothing to do with Fitch's paradox.

But given the contradiction between KP and NonO, KP could also be denied. I am merely interested, for the sake of symmetry or completeness, to see what follows if KP is denied. — Luke

What follows from the knowability principle being denied has nothing to do with Fitch's paradox.

Assume that John argues that an omniscient God exists and that we have free will. Jane provides an argument to show that if an omniscient God exists then we don't have free will.

You then want to know what follows from an omniscient God not existing, which has nothing to do with Jane's argument.

And besides, I find it logically interesting to consider the rejection of each side. Not to mention that Janus raised a question about unknowability which follows from rejecting the KP side instead of the NonO side. — Luke

Then just reject the knowability principle. I don't understand the problem.

Also on this point, why does there appear to be the assumption that English grammar dictates (or at least reveals) facts about metaphysics? The realist will often say that the world doesn't depend on what we say about it. Well, the same applies to their own position: even if our language suggests that objects of perception have an existence independent of perception it doesn't then follow that they do.

There may indeed be a conceptual or semantic distinction between perception and the objects of perception, just as there is a conceptual or semantic distinction between ink on paper and the story being told, but nothing further follows from this. You need to do more than just play word games to argue for a more substantial distinction.

Do you know of any literature that speaks to the rejection of the KP side? — Luke

I think you misunderstand Fitch's paradox. It is a reductio ad absurdum against the knowability principle. So, Fitch's paradox is literature that speaks to the rejection of the KP side. Fitch is saying "if you accept the knowability principle then this implausible conclusion follows, therefore we must reject the knowability principle."

Yes, in a sense. It comes down to what 'real' is. To paint a Unicorn (if we're to take a painting as a kind of model) there has to be a Unicorn for you to paint (model). The question is then what kind of thing that Unicorn is. In this case, it's a figment of our collective imaginations. If you painted it with three horns, you'd have modelled it wrong. — Isaac

Then it's not clear what you mean by saying that if there is a model of a cup then there must be a cup. Are you saying that if there is a model of a cup then there must be an external material world and something in that world that "corresponds" to the model? Because that certainly doesn't follow.

Sure. And my car is real. — Banno

Neither idealism nor anti-realism deny this. It's a mistake to equate "real" with "part of an external material world."

So I don't understand what relevance this has to the discussion.

Is that so? If you dream of driving your car, you are not really driving your car, anymore than when you imagine driving your car.

But in each case, it's still your car. — Banno

And nothing in this says that my car is to be understand as being the mass of subatomic particles that is causally responsible for my experiences. — Michael

To borrow the way you like to argue, I don't drive subatomic particles, I drive a car.

Is that so? If you dream of driving your car, you are not really driving your car, anymore than when you imagine driving your car.

But in each case, it's still your car. — Banno

And nothing in this says that my car is to be understand as being the mass of subatomic particles that is causally responsible for my experiences.

What do you want to refer to? The dream-cup, perhaps, or the real cup that you are now dreaming of... there need be no "one right answer". — Banno

Dream-cups are to dreams as real-cups are to waking experiences: objects found only within the mental phenomena. Dream-cups aren't whatever physical stuff is responsible for the dream and real-cups aren't whatever physical stuff is responsible for the waking experience.

To have a model of a cup necessarily implies there's a cup. — Isaac

Does a painting of a unicorn necessarily imply that there's a unicorn?

But it's a model of a cup, so there must be a cup for it to be a model of, it is not this actual cup which directs our behaviour, it's the model (it must be, otherwise we couldn't account for those errors). So when I say "pass me the cup" I'm referring to the actual cup, but I'm using my model of the cup to do so. — Isaac

And I think this a very strong argument. — Banno

What am I referring to when I say "pass me the cup" when dreaming?

What we see is not the "emergent phenomena", but the cup... — Banno

This might be like saying "I'm not reading about words, I'm reading about wizards" in response to someone arguing that there's nothing more to stories about wizards than words.

As I see it though the proposition is disjointed because we don't know 1) we are merely stipulating it or imagining it is the case. And there would be no contradiction unless we make the mistake of thinking that we are not merely stipulating 1) but knowing it. — Janus

We know that one of these must be true, as per the law of excluded middle (and assuming for the sake of argument that we don't know whether or not the box is empty):

a) the box is empty and we don't know that it's empty
b) the box is not empty and we don't know that it's not empty

But we can never know either to be true because that would be a contradiction. We can't know that the box is empty and that we don't know that the box is empty, therefore we can't know a. We can't know that the box is not empty and that we don't know that the box is not empty, therefore we can't know b.

However, either a or b must be true. Therefore, either a or b is an unknowable truth. And if either a or b is an unknowable truth then the knowability principle is false.

is the truth of the proposition that there are unknowable propositions itself unknowable? — Janus

No. Both a) and b) are known to be unknowable propositions.

OK, assuming the knowability principle is itself true, the case doesn't contradict it anyway, because it says that ""the box is empty" is true and we don't know that it's true" not ""the box is empty" is true and we can't know that it's true". — Janus

But there are two parts to proposition p:

1) "the box is empty" is true
2) we don't know that "the box is empty" is true

If we know part 1) then we can't know part 2) and vice-versa. Therefore it's impossible to know that proposition p is true. But if it's impossible to know that proposition p is true then, according to the knowability principle, proposition p isn't true.

I'm sorry, but I don't see why "1.", if it is true, entails that it is possible to know that it is true. — Janus

Because that's what the knowability principle says. If some proposition p is true then it is possible to know that proposition p is true, and in this case:

p. "the box is empty" is true and we don't know that it's true

If "the box is empty is true" and we don't know that it is true it does not follow that it's possible to know that "the box is empty is true" and that we don't know that it's true, at the same time. — Janus

It does according to the knowability principle: if a proposition is true then it is possible to know that the proposition is true (p → ◊Kp).

a) "the box is empty" is true and we don't know that it's true

The above is a proposition which, if true, entails that it is possible to know that it's true (a → ◊Ka).

Is a map of the territory another "territory"? — Harry Hindu

It can be, e.g:

and yet S(M) is always more complex than M, S(M) can always be discarded via Occam's Razor. — hypericin

You should check out Boltzmann brains, because according to that M is much more complex than S(M):

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.

...

In Boltzmann brain scenarios, the ratio of Boltzmann brains to "normal observers" is astronomically large. Almost any relevant subset of Boltzmann brains, such as "brains embedded within functioning bodies", "observers who believe they are perceiving 3 K microwave background radiation through telescopes", "observers who have a memory of coherent experiences", or "observers who have the same series of experiences as me", also vastly outnumber "normal observers". Therefore, under most models of consciousness, it is unclear that one can reliably conclude that oneself is not such a "Boltzmann observer", in a case where Boltzmann brains dominate the Universe. Even under "content externalism" models of consciousness, Boltzmann observers living in a consistent Earth-sized fluctuation over the course of the past several years outnumber the "normal observers" spawned before a Universe's "heat death".

As stated earlier, most Boltzmann brains have "abnormal" experiences; Feynman has pointed out that, if one knows oneself to be a typical Boltzmann brain, one does not expect "normal" observations to continue in the future. In other words, in a Boltzmann-dominated Universe, most Boltzmann brains have "abnormal" experiences, but most observers with only "normal" experiences are Boltzmann brains, due to the overwhelming vastness of the population of Boltzmann brains in such a Universe.

See here for one such discussion we had.

Deflationary accounts of truth are apt to be anti-realist — Tate

Not according to many here.

So does an instrumentalist have a model of the world which explains why their model of part of it works? — Isaac

Not that I know of.

A scientific realist will say that an electron is a mind-independent entity. A scientific instrumentalist will say that the mathematical model of an electron best describes and predicts the results of observation.

So a scientific realist will say that the Standard Model corresponds to the way the world is, whereas a scientific instrumentalist will just say that the Standard Models works.

3. (p ∧ ¬Kp) ∨ (¬p ∧ ¬K¬p) — Michael

In fact from this I'm pretty sure it follows that ∃q(q ∧ ¬Kq), so we're back to the initial formalism.

And more explicitly:

According to the idea of model-dependent realism introduced in Chapter 3, our brains interpret the input from our sensory organs by making a model of the outside world. We form mental concepts of our home, trees, other people, the electricity that flows from wall sockets, atoms, molecules and other universes. These mental concepts are the only reality we can know. There is no model-independent test of reality. — Hawkings

Hawking, if I recall correctly, also expressed quite a firm belief in model-dependent realism... If the views of our great scientists are anything to go by... — Isaac

Yes, which despite the term "realism" is instrumentalist (much like Putnam's "internal realism" is anti-realist).

I therefore take the view, which has been described as simple-minded or naïve, that a theory of physics is just a mathematical model that we use to describe the results of observations… Beyond that it makes no sense to ask if it corresponds to reality, because we do not know what reality is independent of theory.

...

According to model-dependent realism, it is pointless to ask whether a model is real, only whether it agrees with observation. If there are two models that both agree with observation, like the goldfish's picture and ours, then one cannot say that one is more real than another — Hawkings

(p or ~p) and ~Kp and ~K~p.

Can you work with that a little and see how it goes? — unenlightened

1. (p ∨ ¬p) ∧ ¬Kp ∧ ¬K¬p
2. (p ∧ ¬Kp ∧ ¬K¬p) ∨ (¬p ∧ ¬Kp ∧ ¬K¬p)
3. (p ∧ ¬Kp) ∨ (¬p ∧ ¬K¬p)

4. q → ◇Kq (knowability principle)

5. p ∧ ¬Kp → ◇K(p ∧ ¬Kp) (contradiction)
6. ¬p ∧ ¬K¬p → ◇K(¬p ∧ ¬K¬p) (contradicton)

I sometimes wonder if Kant, were he around and able to avail himself of our understanding of chemistry and physics. would puzzle that there were still folk who held to his surmise that there was unobservable stuff "behind" our observations. Would he just admit that he hadn't thought of the ways science has uncovered his noumena? — Banno

One can be a scientific realist but an anti-realist/idealist about everyday objects of perception. Electrons and protons and photons explain why we see what we do, but they are not the what we see. The what we see is an emergent phenomena, brought about by a causal chain involving a multitude of these subatomic particles.

Or perhaps Kant would be, like Hawkings was, a scientific instrumentalist.

I don't think so. I think the principle needs to be formalised differently, as I indicated. — unenlightened

That doesn't work. p ∨ ¬p just means "p is true or p is false" and says nothing about what we know.

For example: either my name is Michael or my name is not Michael. This statement is true, but doesn't say that I don't know my name.

I think it is a contradiction, because it asserts something and denies that it is known. "Either there is intelligent alien life or there isn't, but I don't know which." -- that makes sense. — unenlightened

We are able to assert things we don't know. We can make arguments with premises we either don't know or believe to be false, e.g.:

1. My name is Andrew
2. If my name is Andrew then my name is not Michael
3. My name is not Michael

The argument is valid.

1. There is intelligent alien life
2. If there is intelligent alien life then humans are not the only intelligent life
3. Humans are not the only intelligent life

The argument is valid.

It seems to be stating a contradiction by asserting p and claiming it to be unknown. — unenlightened

It's not a contradiction to say "there is intelligent alien life but I don't know that there is." Such a statement is possibly true.

I am questioning the legitimacy of that. — unenlightened

But that's the non-omniscience principle? Without it we must accept that every true proposition is known to be true – which is what Fitch's paradox shows follows from the knowability principle.

It seems to me that 'an unknown truth' cannot legitimately be formalised as p but only as (p or ~p) Is that right? does it make sense? — unenlightened

The formal definition is ∃p(p ∧ ¬Kp): there exists some proposition p that is true and not known to be true.

For example, either "the Riemann hypothesis is correct" is true and not known to be true or "the Riemann hypothesis is not correct" is true and not known to be true, and so either "the Riemann hypothesis is correct" is an unknown truth or "the Riemann hypothesis is not correct" is an unknown truth.

Can you lay out the argument clearly in plain English? — Janus

I've done so a couple of times: here and here.

The criterion is 'knowable' not 'known'. — Janus

Fitch's paradox shows that if all truths are knowable then all truths are known. Some truths aren't known, therefore some truths aren't knowable.

That's not how the rules of inference work.