So you think it is literally impossible to give argument 2 without implying argument 1? — Leontiskos

Yes. I'll rephrase the argument in propositional logic:

P1. P ∧ ¬P
P2. P ∧ ¬P ⊢ P (conjunction elimination)
P3. P ⊢ P ∨ Q (disjunction introduction)
P4. P ∧ ¬P ⊢ ¬P (conjunction elimination)
P5. (P ∨ Q) ∧ ¬P ⊢ Q (disjunctive syllogism)
C1. Q

Notice that P2 - P5 are all rules of inference; they are implicit in every argument (along with every rule I didn't write out) and so don't need to be said.

I don't know of any realists who believe in unknowable truths. Apparently you don't either. — Leontiskos

Try reading the article I directed you to:

Fitch’s paradox of knowability (aka the knowability paradox or Church-Fitch Paradox) concerns any theory committed to the thesis that all truths are knowable. Historical examples of such theories arguably include Michael Dummett’s semantic antirealism (i.e., the view that any truth is verifiable), mathematical constructivism (i.e., the view that the truth of a mathematical formula depends on the mental constructions mathematicians use to prove those formulas), Hilary Putnam’s internal realism (i.e., the view that truth is what we would believe in ideal epistemic circumstances), Charles Sanders Peirce’s pragmatic theory of truth (i.e., that truth is what we would agree to at the limit of inquiry), logical positivism (i.e., the view that meaning is giving by verification conditions), Kant’s transcendental idealism (i.e., that all knowledge is knowledge of appearances), and George Berkeley’s idealism (i.e., that to be is to be perceivable).

...

The middle way, what we might call moderate antirealism, can be characterized logically somewhere in the ballpark of the knowability principle:

∀p(p → ◊Kp).

which says, formally, for all propositions p, if p then it is possible to know that p.

Notice that antirealism is defined as being the position that believes that all truths are knowable. As realism rejects antirealism (and vice versa), in follows that realism asserts that some truths are unknowable.

But if you want something more explicit, the article continues:

He also points out that TKP, rather than the unrestricted KP, serves as the more interesting point of contention between the semantic realist and anti-realist. The realist believes that it is possible for truth to be unknowable in principle.

The "argument 1" sits in between the premise and the conclusion of "argument 2" to make a single argument:

P1. "I am a man and I am not a man" is true
P2. If "I am a man and I am not a man" is true then "I am a man" is true.
P3. If "I am a man" is true then "I am a man or I am rich" is true.
P4. If "I am a man and I am not a man" is true then "I am not a man" is true.
P5. If "I am a man or I am rich" is true and if "I am not a man" is true then "I am rich" is true.
C1. Therefore, "I am rich" is true

P2 through P5 are normally left implicit as they are simply stating the commonly accepted rules of inference.

Perhaps we we disagree about what may be considered a rule of inference. Unless you think an argument that is invalid only coincidentally doesn't follow? Or is it invalid because it does not follow? — NotAristotle

Logicians coined the term "valid argument" as a shorthand for "an argument with a conclusion that can be derived from the premises using a priori reasoning".

Logicians coined the term "sound argument" as a shorthand for "a valid argument with true premises".

I don't understand what misrepresentation you think I've made.

Are you denying that the realist rejects the knowability principle?

Are you denying what the IEP article says below?

One proposal is to construe metaphysical realism as the position that there are no a priori epistemically derived constraints on reality.

...

One virtue of this construal is that it defines metaphysical realism at a sufficient level of generality to apply to all philosophers who currently espouse metaphysical realism.

...

For there is a good argument to the effect that if metaphysical realism is true, then global skepticism is also true, that is, it is possible that all of our referential beliefs about the world are false.

No, they are two different arguments. One involves inferential reasoning and the other does not. — Leontiskos

It's one argument that uses deductive reasoning to derive the conclusion from the premise.

P2 - P5 simply make explicit the rules of inference and can normally be left unsaid.

In simple terms, given these two premises:

P1. If I am a man then I am mortal
P2. I am a man

You can use the rules of inference to derive the conclusion "I am mortal" using a priori reasoning, but you cannot use the rules of inference to derive the conclusion "I am English" using a priori reasoning.

It's one argument:

P1. "I am a man and I am not a man" is true
P2. If "I am a man and I am not a man" is true then "I am a man" is true.
P3. If "I am a man" is true then "I am a man or I am rich" is true.
P4. If "I am a man and I am not a man" is true then "I am not a man" is true.
P5. If "I am a man or I am rich" is true and if "I am not a man" is true then "I am rich" is true.
C1. Therefore, "I am rich" is true

P2 through P5 can be kept implicit as they simply express the commonly understood rules of inference.

This is a valid argument:

P1. If I am a man then I am mortal
P2. I am a man
C1. Therefore, I am mortal

This is an invalid argument:

P1. If I am a man then I am mortal
P2. I am a man
C1. Therefore, I am English

Both premises and both conclusions are true, but the second conclusion doesn't follow from the premises (whereas the first conclusion does).

Why not? It satisfies the definition, does it not? — NotAristotle

No, it doesn't.

Maybe you should try reading Fitch's paradox.

If I did live in Antartica it would have to be valid wouldn't it? — NotAristotle

No.

as premise 1 is faulty — NotAristotle

It's not. You might as well claim that mathematicians are wrong to define the "=" sign as meaning "is equal to".

you might as well argue "I am a human and it might snow this week, therefore I live in Antartica." — NotAristotle

But that argument isn't valid.

Even if conclusion and premise are all true i.e. the argument is sound, — NotAristotle

That is not what "sound" means.

It seems that that argument would be valid, but only if one accepts that an argument is valid iff there is no interpretation s.t. all premises are true and the conclusion is false per Tones' definition.

If it turned out that validity required more than what that definition suggests (I think it does), then the argument you stated may well turn out to not be valid, as I think is the case. — NotAristotle

You seem to be putting the cart before the horse.

It's not the case that the word "valid" means something and then we try to give a proper description of this meaning, and that we disagree on the proposed definition.

Rather, a bunch of logicians got in a room together and decided that if an argument's conclusion follows from its premises using the rules of inference then they will name this type of argument "valid". And that if the premises are also in fact true then they will name this type of argument "sound".

I don't know what you're talking about.

This recent argument started as a discussion on Fitch's paradox, which examines the anti-realist's knowability principle: ∀p(p → ◊Kp).

Realists reject this knowability principle, claiming that ∃p(p ∧ ¬◊Kp).

Assuming that knowledge is justified true belief, the realist's claim is that there is at least one proposition that is true but that either cannot be believed or cannot be justified (e.g. by seeing evidence that it is the case).

Given that the realist has no a priori reason to suggest that no use of "the cat is in the box" can be one such proposition, the realist accepts that it's possible that some use of "the cat is in the box" can be true but that either it cannot be believed or cannot be justified (e.g. by looking in the box and seeing the cat).

There's also the IEP article I quoted here. Although in that specific case they consider the proposition "I am a brain in a vat" rather than the proposition "the cat is in the box".

That ((P→Q)∧Q), therefore P is not valid, whereas ((A∧¬A)∧(P→Q)∧Q), therefore P is valid, does seem strange to me. Inconsistent premises don't seem to have anything to do with whether the argument "follows." Although I have a feeling that Tones will have something to say about that. — NotAristotle

I am a man and I am not a man. Therefore I am rich.

The argument is valid; the conclusion follows from the premise. We can show this in four parts:

1. If "I am a man and I am not a man" is true then "I am a man" is true.
2. If "I am a man" is true then "I am a man or I am rich" is true.
3. If "I am a man and I am not a man" is true then "I am not a man" is true.
4. If "I am a man or I am rich" is true and if "I am not a man" is true then "I am rich" is true.

But I don't think you're quite acknowledging the nuances of realism in that post. See here where I offer an example involving cats in boxes.

For your specific example, realism entails that even if we examine the entirety of the Solar System with the most sophisticated scientific instruments possible, and even if we fail to see a cup, it might still be the case that there is a cup orbiting the Sun.

Or on the other hand, even if we do (seem to) see a cup, it might still be the case that there isn't a cup (e.g. we're being deceived by someone or something casting an illusion).

These possibilities make justification of anything rather problematic.

I agree with your substitution. So what's the problem?

"to an adequate degree"?

This proposition is true:

1. We do not have evidence that we are brains in a vat

If realism is correct then this proposition is true:

2. It is possible that we are brains in a vat and that we cannot have evidence that we are brains in a vat

My suggestion is that if we cannot have evidence that we are brains in a vat then (1) does not sufficiently justify the claim that we are not brains in a vat.

I gave reasons above. — Leontiskos

But your reasoning is not directed at what I was claiming. You said "you are trying to claim that it follows from your premises that there are truths which are both known and unjustifiable".

I'm not trying to claim that, because that claim would be a contradiction.

You are conflating the possibility of skepticism with skepticism. — Leontiskos

I'm not sure what you think I'm arguing, or what you mean by the possibility of skepticism.

The skeptic doesn't say "we are brains in a vat"; the skeptic says "we might be brains in a vat". Something like "it is possible that we might be brains in a vat" is redundant, as, at least using S5 modal logic, ◇◇P↔◇P.

You are trying to claim that it follows from your premises that there are truths which are both known and unjustifiable — Leontiskos

I'm not.

I'm trying to explain this:

One proposal is to construe metaphysical realism as the position that there are no a priori epistemically derived constraints on reality (Gaifman, 1993). By stating the thesis negatively, the realist sidesteps the thorny problems concerning correspondence or a “ready made” world, and shifts the burden of proof on the challenger to refute the thesis. One virtue of this construal is that it defines metaphysical realism at a sufficient level of generality to apply to all philosophers who currently espouse metaphysical realism. For Putnam’s metaphysical realist will also agree that truth and reality cannot be subject to “epistemically derived constraints.” This general characterization of metaphysical realism is enough to provide a target for the Brains in a Vat argument. For there is a good argument to the effect that if metaphysical realism is true, then global skepticism is also true, that is, it is possible that all of our referential beliefs about the world are false. As Thomas Nagel puts it, “realism makes skepticism intelligible,” (1986, 73) because once we open the gap between truth and epistemology, we must countenance the possibility that all of our beliefs, no matter how well justified, nevertheless fail to accurately depict the world as it really is. Donald Davidson also emphasizes this aspect of metaphysical realism: “metaphysical realism is skepticism in one of its traditional garbs. It asks: why couldn’t all my beliefs hang together and yet be comprehensively false about the actual world?” (1986, 309)

I try to prove an even stronger version of this using propositional logic here.

So you think you need antirealism to avoid being a vatted brain. Right. — Banno

These mean different things:

1. We are brains in a vat
2. It is possible that we are brains in a vat

In propositional logic:

1. P
2. ◊P

If realism is correct then (2) is true. Putnam argues that (2) is false and so that therefore realism is incorrect.

Repeatedly you either fail to understand or intentionally ignore the distinction between what is said to be true (or known, or proved) and what is said to be possibly true (or knowable, or provable).

Sure. And they do this by rejecting classical logic. — Banno

Well, yes. That’s how Dummett defined the distinction between realism and antirealism; realists commit to the classical logic of bivalence and antirealists reject it.

Realism does not commit to vat brains. — Banno

It entails their possibility. Antirealism provides an opportunity to dismiss the notion as nonsense.

and from which follows that that all truths are known. — Mww

Well, no.

The antirealist argues that if "the cat is in the box" is true then it's possible for someone to know that it's true, e.g. by looking in the box and seeing the cat. The fact that it's possible for someone to look in the box and see the cat does not entail that someone already knows that the cat is in the box: perhaps nobody knows because nobody’s looked.

The realist argues that "the cat is in the box" can be true even if it's not possible for someone to look in the box and see the cat.

Which is more reasonable? I say the former. The latter arguably doesn't even make sense. What does it mean for it to be not possible for someone to look in the box and see the cat (despite the cat being in the box)? Looking in the box and seeing the cat is certainly not a contradiction, so it can't be that.

Some musings.

Let's take the knowability principle from Fitch's paradox: ∀p(p → ◊Kp).

According to this, "a marker of necessity functions as a universal quantifier: it indicates that the basic proposition is true in all possible states of affairs. A marker of possibility functions as an existential quantifier: it indicates that there is at least one state of affairs in which the basic proposition is true."

So the knowability principle can be rephrased as:

□(p → ◊Kp)

The realist rejects this knowability principle.

But under S5 ¬□(p → ◇Kp) → □¬Kp.

So if the knowability principle is false and if S5 is correct then nothing is knowable (and so nothing is known)?

What is it about antirealism that you have to say? — Banno

The first thing is that it isn't any of your strawmen. Antirealism doesn't claim, and nor do antirealists acknowledge that it entails, that all truths are known.

The second thing is that it is consistent with a deflationary account of truth.

The third thing is that it avoids certain absurdities that realism allows for, e.g. that it is possible that we are unknowably brains in a vat.

The fourth thing, albeit directed at Janus, is that it is not obviously wrong.

It seems to me that you ignore most of what I've writ, preferring to nit pick a few near-irrelevancies. — Banno

I’m sorry but this is quite the ironic thing to say given the rest of your response to my post. Pot, meet kettle.

And I don’t think I’ve ignored anything?

I think a better example is: “Banno has stopped beating his wife”.

Assuming, I hope, that you have never beaten your wife.

The choice is between saying that there are unknown mathematical truths and saying that there are unknown physical truths. I'd entertain Kripke's approach to truth for maths but not for physics. So we can usefully say that Goldbach's conjecture so far has no truth value but that there is water on Miranda is either true or it is false. — Banno

How do you get from “there are unknown mathematical truths” to “Goldbach's conjecture so far has no truth value”? And who has denied that that there is water on Miranda is either true of false?

Again, you don’t seem to fully acknowledge the distinction between being knowable and being known.

If Goldbach’s conjecture is provable then even if it hasn’t yet been proven it is true.

If water on Miranda is provable then even if it hasn’t yet been proven there is water on Miranda.

Perhaps the best way to understand antirealism is to rephrase Putnam’s argument against metaphysical realism: if there are unknowable truths then it is possible that “we are brains in a vat” is an unknowable truth. It is not possible that we are brains in a vat. Therefore, there are no unknowable truths.

The problem with realism is that it entails this kind of global skepticism. If there are unknowable truths then there are unjustifiable truths, and if there are unjustifiable truths then a proposition not being justified is not a good reason to reject it.

In fact, for all the realist knows, perhaps almost all truths are unknowable and so unjustifiable, which arguably gives them even less reason to reject an unjustified proposition.

"P" is true IFF P. — Banno

The above sentence is true because the sentence fragment on the left hand side (“‘P’ is true”) means the same thing as the sentence fragment on the right hand side (“P”).

You appear to be trying to conflate deflationism and disquotationalism. They are not the same thing. One can be a non-deflationary disquotationalist (in fact I seem to recall Tarski as thinking of his theory as a type of correspondence theory).

Well, no. That's far too vague. One is about the weather, the other is about a sentence. But (1) and (2) are arguably truth- functionally equivalent. — Banno

I’m confused. Are you a truth deflationist or not? A truth deflationist will accept that (1) and (2) mean the same thing. But now you say that they don’t mean the same thing and are only “truth-functionally” equivalent. That strikes me as being decidedly non-deflationary.

Ok, so are you agreeing with Dummett? — Banno

I’m just offering an example of what I think is a deflationary anti-realism.

Yeah, they can reject it all they like. It doesn't follow that they are right. ∀p(p→◊Kp)⊢∀p(p→Kp — Banno

As the SEP article says, “Fitch’s proof is not a refutation of anti-realism, but rather a reason for the anti-realist to accept intuitionistic logic. … without double negation elimination one cannot derive Fitch’s conclusion ‘all truths are known’”.

If you think this is wrong, tell me why. — Banno

I don’t think it’s wrong. I just don’t understand why you think that antirealism about mathematics doesn’t entail that all mathematical truths are known/attitudes but that antirealism about the weather entails that all truths about the weather are known/attitudes.

We are sometimes surprised by things that are unexpected. How is this possible if all that is true is already known to be true? — Banno

The claim isn’t that all truths are known. The claim is that all truths are knowable. Remember, anti-realists reject Fitch’s conclusion.

Overwhelmingly, you and I agree as to what is true. How is that explainable if all there is to being true is attitudes? How to explain why we share the same attitude? — Banno

The claim isn’t that all truths are attitudes. The claim is that all truths are knowable.

We sometimes are wrong about how things are. How can this be possible if all that there is to a statement's being true is our attitude towards it? — Banno

The claim isn’t that all truths are attitudes. The claim is that all truths are knowable.

I honestly don’t understand what you think antirealism is.

Remember, you claim to be a mathematical antirealist. Presumably you accept that mathematical truths aren’t attitudes and that there are unknown mathematical truths. So simply extend your understanding of propositions about numbers to propositions about medium sized dry goods.

it remains unclear how this helps the topic, or relates to it any more than bringing in intension. — Banno

So we have this:

Sentences (1) and (2) mean the same thing.

Dummett then argues that (2) is only meaningful if it is verifiable. This is his Language Acquisition Argument.

If (1) and (2) mean the same thing and if (2) is only meaningful if it is verifiable then (1) is only meaningful if it is verifiable.

If (1) is only meaningful if it is verifiable then unknowable truths make no sense.

One of us is clearly misunderstanding the other. I’ll try to rephrase what I was saying more clearly:

1. “It is raining” is true
2. It is raining
3. “It is not raining” is true
4. It is not raining

Sentences (1) and (2) mean the same thing.
Sentences (3) and (4) mean the same thing.

This is precisely the deflationary account that you claimed to support above.

The problem is that no conjecture can be proven to be true or false, so on the antirealist view, assuming you have correctly outlined it, no conjecture could be either true or false. — Janus

I’m not sure what you mean.

The conjecture “alien life exists on Pluto” can be proven true or false by going to Pluto, looking everywhere for life, and then either finding it or not finding it.

But "'it is raining' is true" means that it is raining, not "it is raining". — Banno

Take these two sentences:

1. “It is raining” is true
2. “It is not raining” is true

According to your reasoning, (1) means that it is raining and (2) means that it is not raining.

It’s not clear what you mean by “means” here. Do you mean “entails”? If so then there are two issues:

The first issue is that when I say “means” I don’t mean “entails”; I mean “is semantically equivalent to”. So, (1) is semantically equivalent to “it is raining” and (2) is semantically equivalent to “it is not raining”.

The second issue is that (1) doesn’t entail that it is raining; rather, (1) being true entails that it is raining. You appear to be mixing up your use and mention.

And it’s important to remember that, at least if we accept the premises of Fitch’s paradox, we’re dealing with modal possibility, i.e that if it is not possible to know something then it is necessarily unknown, e.g that in no possible world does anybody know it.