I'll pause there. I gather we agree at least that this is the account being scrutinised? — Banno

Yes. So the anti-realist responds by either noting that antirealism rejects classical logic or by accepting that the knowability principle as written is too broad, offering instead a restricted version such as (9) in my post above which does not allow for the substitution (p ∧ ¬Kp) → ◇K(p ∧ ¬Kp).

(9) is consistent with (1), and also (2), (3), and (4), and so does not entail (5). If (9) is still anti-realism then anti-realism is consistent with (1), (2), (3), and (4). Therefore realism must be saying more than just (1), (2), (3), or (4).

My suggestion is that realism is saying that there are unknowable Cartesian truths, where a Cartesian truth is a truth that it is not a contradiction to know, e.g. some instance of "the cat is in the box".

So the anti-realist is claiming that if something exists then it is possible to know that it exists, and that if it is doing something then it is possible to know that it is doing that thing, and that if it isn't doing something then it is possible to know that it isn't doing that thing, and that if something doesn't exist then it is possible to know that it doesn't exist. None of this entails that we actually know everything.

I think the distinction is more apparent when we consider counterfactuals and predictions. The realist, in accepting the principle of bivalence, will claim that all such propositions are either true or false, whereas the anti-realist will claim that if it is impossible in principle to know that a counterfactual or prediction is true or false then it is neither true nor false.

There seems to be a lot of ambiguous phrasing in this discussion and so I want to try to be as precise as possible:

1. For some p, p is true and unknown
∃p(p ∧ ¬Kp)

2. For some p, p is true and unknowable
∃p(p ∧ ¬◇Kp)

3. It is possible that for some p, p is true and unknown
◇∃p(p ∧ ¬Kp)

4. It is possible that for some p, p is true and unknowable
◇∃p(p ∧ ¬◇Kp)

5. For all p, if p is true then p is known
∀p(p → Kp)

6. For all p, if p is true then p is knowable
∀p(p → ◇Kp)

The realist accepts (1), (2), (3), and (4) and rejects (5) and (6).

The anti-realist accepts (1), (3) and (6) and rejects (2) and (5). They probably also reject (4), although strictly speaking (4) is consistent with (6).

The problematic proposition is:

7. For all p, it is possible that p is true and unknowable:
∀p(◇(p ∧ ¬◇Kp))

This entails radical scepticism:

8. For all p, if p is true then p is not known:
∀p(p → ¬Kp)

If the realist rejects (8) then they must reject (7). Note specifically the differences between (3), (4), and (7). (7) entails (3) and (4) but neither (3) nor (4) entail (7).

But we must ask whether or not (6) really is necessary for anti-realism, and so whether or not (2) really is sufficient for realism. As the SEP article mentions, some anti-realists offer a restricted knowability principle, perhaps such as the one I offered earlier:

9. For all p and all q, if p being true does not entail that q is an unknown truth then if p is true then p is knowable
∀p∀q((p ⊭ (q ∧ ¬Kq)) → (p → ◊Kp))

This is consistent with (2), avoiding Fitch's paradox even in classical logic, but is still sufficiently anti-realist, e.g. it still asserts that if some object exists then it is possible to know that it exists. It simply acknowledges that knowing that something is an unknown truth is a contradiction.

Given this, realism must be more than just (1), (2), (3), or (4). But if it isn't (7) then what is it? Perhaps the claim that there is at least one unknowable Cartesian truth (using Tennant's terminology), e.g. that there is at least one unknowably true "the object exists"?

And note the difference between "there is at least one unknowable Cartesian truth" and "it is possible that at least one Cartesian truth is not known". These are (2) and (3) respectively (restricted to Cartesian truths). Anti-realism is consistent with (3).

∀p(◊(p ∧ ¬◊Kp)) says "For all truths p, it is possible that p is true and it not be possible to know p"

I think that should be "For all truths p, it is possible that p is true and yet p is not known". That would be ∀p(◊(p ∧ ¬Kp)). — Banno

There are two different claims:

1. It is possible for the truth to be unknowable
2. It is possible for the truth to be unknown

These are represented as:

1. ∀p(◊(p ∧ ¬◊Kp))
2. ∀p(◊(p ∧ ¬Kp))

Certainly (2) is true, but at least according to that SEP article realists believe that (1) is also true, and as mentioned above, (1) entails that nothing is known.

Our concern is whether or not truths are knowable not just whether or not truths are known.

Linda McMahon for Secretary of Education and Dr Oz for Centers for Medicare and Medicaid Services. :chin:

I address that here.

If "for all p, it is possible that p is unknowably true" is true then "for all p, if p is true then p is necessarily not known" is true.

I think something like "for all p, it is possible that p is unknowable".

So take any proposition at random, e.g. that there is a suitcase under my bed. Is it possible that this is unknowable? Given that the realist argues for "mind-independent" truths, or as Gaifman describes it "that there are no a priori epistemically derived constraints on reality", it would seem that the realist must answer in the affirmative. Which, under S5, entails that it is necessarily unknown.

Some more general musings.

From here, we have these sets of propositions:

1. "the cat is in the box" is true and justified (is known)
2. "the cat is in the box" is false and justified (is not known)
3. "the cat is in the box" is true and unjustified (is not known)
4. "the cat is in the box" is false and unjustified (is not known)

5. "the cat is in the box" is true and I have looked in the box and seen the cat
6. "the cat is in the box" is false and I have looked in the box and seen the cat
7. "the cat is in the box" is true and either I have not looked in the box or I have not seen the cat
8. "the cat is in the box" is false and either I have not looked in the box or I have not seen the cat

There is perhaps a reasonable argument that if (6) is possibly true then (5) does not entail (1); that if it is possible that I look in the box and see the cat even if the cat is not in the box then looking in the box and seeing the cat does not justify the belief that the cat is in the box.

This would seem to be skepticsm.

One response is to deny the possibility of (6), and so also (2), leaving us with:

1. "the cat is in the box" is true and justified (is known)
3. "the cat is in the box" is true and unjustified (is not known)
4. "the cat is in the box" is false and unjustified (is not known)

Which can be simplified to:

a. "the cat is in the box" is justified (is known)
b. "the cat is in the box" is true and unjustified (is not known)
c. "the cat is in the box" is false and unjustified (is not known)

Jp ⊨ Kp ⊨ p

If a proposition is justified then it is true.

This would seem to be a type of antirealism.

That applies to TKP rather than KP. I don't agree that we only know things that are not contradictory - cartesian truths. So while any particular truth might not have been known, it does not follow that every given truth is unknown. We do know things. That is, the "p" in your logic is all truths when it should be a particular truth. — Banno

So we have two propositions:

1. The realist believes that it is possible for truth to be unknowable in principle.
2. The realist believes that truth is unknowable in principle.

The article asserted (1), not (2).

The problem for the realist, however, is that (under S5), (1) entails (2):

∀p(◊(p ∧ ¬◊Kp)) ⊢ ∀p(p → □¬Kp))

Hence my earlier claim that one of these is true:

1. Realism is incorrect
2. S5 is incorrect
3. Nothing is known

Fitch’s Paradox of Knowability:

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. Fitch’s reasoning, at best, shows us that there is structural unknowability, that is, unknowability that is a function of logical considerations alone. But is there a more substantial kind of unknowability, for instance, unknowability that is a function of the recognition-transcendence of the non-logical subject-matter? A realist decrying the ad hoc nature of TKP (or DKP) fails to engage the knowability theorist at the heart of the realism debate.

You seem to think that a realist will say that nothing is knowable. — Banno

That follows from the claim, quoted from the SEP article, that "the realist believes that it is possible for truth to be unknowable in principle".

If it is possible that a true sentence is unknowable then it is possibly not possible that a true sentence is known, and if it is possibly not possible that a true sentence is known then it is necessarily not possible that a true sentence is known.

◇¬◇p→□¬◇p

What?!? — Banno

1. "the cat is in the box" is true and I have looked in the box and seen the cat
2. "the cat is in the box" is true and justified

If "the cat is in the box" is true then is it possible to look in the box and see the cat?
Does (1) entail (2)?

If "yes" to both then if "the cat is in the box" is true then it is knowable.

If "the cat is in the box" being true is not knowable then either (1) does not entail (2) or it is not possible to look in the box and see the cat.

Do you disagree with any of this?

If the existence of objects is mind-independent then the truth of “the object exists” is mind-independent such that it could be true even if it is not possible, in principle, to know that it’s true.

There’s a reason that Dummett, the man who coined the term “antirealism”, framed the dispute between realism and antirealism as a dispute about the logic of truth.

Read further in the article you posted, under “6. Views Opposing the Independence Dimension (I): Semantic Realism”.

I'm going to try to summarise the reasoning. I'm taking it for granted that knowledge is justified true belief.

Given that the proposition "the cat is in the box" is believed to be true, there are prima facie four possible scenarios:

1. "the cat is in the box" is true and justified (is known)
2. "the cat is in the box" is false and justified (is not known)
3. "the cat is in the box" is true and unjustified (is not known)
4. "the cat is in the box" is false and unjustified (is not known)

In more specific terms:

5. "the cat is in the box" is true and I have looked in the box and seen the cat
6. "the cat is in the box" is false and I have looked in the box and seen the cat
7. "the cat is in the box" is true and either I have not looked in the box or I have not seen the cat
8. "the cat is in the box" is false and either I have not looked in the box or I have not seen the cat

The anti-realist claims that (5) entails (1) and that if "the cat is in the box" is true then it is possible in principle to look in the box and see the cat. If both of these claims are true then if "the cat is in the box" is true then it is knowable.

Whereas, as explained here, "the realist believes that it is possible for truth to be unknowable in principle."

Which means that the realist believes either that (5) does not entail (1) or that it if "the cat is in the box" is true then it is possibly not possible to look in the box and see the cat. Either entails that if "the cat is in the box" is true then it is unknowable¹.

¹ _{In S5, ◇¬◇p ⊢ □¬p. Technically the realist could reject S5, but as mentioned here, "this result suggests that S5 is the correct way to formulate a logic of necessity."}

Addendum: In fact, ◇¬◇p ⊢ □¬p can be applied to the very claim that "it is possible for truth to be unknowable in principle": if it is possibly not possible to know the truth then the truth is necessarily unknown.

Therefore, one of these is true:

1. Realism is incorrect
2. S5 is incorrect
3. Nothing can be known

It is true that there is gold in Boorara. If all life disappeared from the universe, but everything else is undisturbed, then it would still be true that there is gold in Boorara. — Banno

Let's take mathematical antirealism; we might say that a mathematical proposition is true if it is provable from the axioms. The mathematical antirealist doesn't then claim that if everyone were to die then mathematical propositions are no longer true; they continue to be true because they continue to be provable – there's just nobody around to prove them anymore, which is irrelevant.

Sure, something might be (as yet) unjustified and yet could be justified. In which case, since it could be justified, there is something which counts as it's justification. — Banno

What does it mean that something counts as its justification? Are you just repeating the claim "p can be justified"? What is the difference between (1) and (3)?

It woudl help considerably if you explained what you think a justification might be. I've already pointed out that mere logical entailment will not do. — Banno

It's whatever distinguishes knowledge from a mere true belief.

As a specific example, if "the cat is in the box" is true then perhaps the strongest justification is looking in the box and seeing the cat. That's an ordinary reason that we can be said to know that the cat is in the box.

Given that looking in the box and seeing the cat is always possible in principle, every case of "the cat is in the box" being true is justifiable, even if it hasn't yet been justified (i.e. we haven't yet looked in the box) – and even if it never is justified (i.e. we never look in the box).

So, at the very least, we should be antirealists about cats in boxes.

You want (1) not to entail (3). — Banno

I don't even know what (3) is. You won't explain it.

Again, I suspect you are equivocating. First you treat (2) and (3) as meaning different things, allowing you to say that (3) follows from (1) without saying that (2) follows from (1), and then you treat (2) and (3) as meaning the same thing, allowing you to say that if (2) is false then (3) is false.

So spell it out for me. What does (3) mean? How does it differ from (1) and (2)?

As it stands, anti-realism simply says that (1) is always true and (2) is sometimes false. And that's it. There is no additional proposition (3).

(1) entails (3). (2) entails (3). — Banno

So you have:

P1. If (1) then (3)
P2. If (2) then (3)

And then you seem to go:

C1. If not (2) then not (3).

That's denying the antecedent.

It would still really help if you explain what (3) means, and how it differs from (1) and (2).

This is your game. you get to decide, I supose. — Banno

You're the one who has introduced new grammar, so you need to explain it.

1. p can be justified
2. p is justified
3. p has a justification

I can't help but think that you're equivocating. You say that (1) entails (3), say that (2) is false, somehow use that to conclude that (3) is false, and so use that to conclude that (1) is false.

Sure. But "can be justified" entails "has a justification". — Banno

What is the difference between "is justified" and "has a justification"?

Only if you do not wish to allow for justifications in other possible worlds. — Banno

What? It doesn't follow because it doesn't follow, just as the spouse example doesn't follow.

Your other analogs do not work. — Banno

They work perfectly. They show that your re-phrasing of the claim has changed the meaning of the claim.

Do you really want to say that if a proposition is true than in some possible world there is a justification? Fine, then for you every truth has a justification. — Banno

That simply does not follow.

If some entity is a person then in some possible world it has a spouse. Therefore, every person has a spouse?

No, obviously not.

If every truth is justifiable, then for every truth there is some justification. — Banno

Just no.
"can be justified" does not entail "is justified".
"can be killed" does not entail "is killed".
"can be broken" does not entail "is broken".

It's honestly crazy that I have to explain this to you.

The point of that was to show that there is a meaningful difference between these two propositions:

1. If A is B then it can C
2. A can be B only if it has C

Banno is repeatedly misinterpreting/misrepresenting (1) as (2).

Sure you do. If you want to deny A→B then you must give an example of A^~B. — Leontiskos

Firstly, I don't. One approach is to show that A→B is a contradiction or is in some other sense incoherent. Antirealists often do this by addressing the meaning of the word "true" and explain that this meaning is inconsistent with unknowability.

Secondly, it is the realist who denies p → ◇Kp, and so if you follow your own reasoning you must provide an example of an unknowable truth.

Were Michael to disagree with this, he would have to show us a justifiable truth with no justifcation. — Leontiskos

No I don't. Just as the realist doesn't have to show us an unknowable truth.

Another example:

1. If the vase is fragile then it can break
2. The vase can be fragile only if it has a break

These do not mean the same thing. (1) is true and (2) is false.

If, for antirealists, as you say, all truths are believable and justifiable, you can drop the modality. p→Jp — Banno

No we can't. Dropping modality changes meaning.

These mean different things:

1. All truths are believed and justified
2. All truths are believable and justifiable

↪Michael It says that if something is mortal, then there is an something which is the death of that thing. Pretty plain. — Banno

This still doesn't explain what that means.

In ordinary English we say that if something is mortal then it can die; we don't say that if something is mortal then something it its death. I understand the former, not the latter.

No. I'm saying that somethign can be mortal only if it has a death — Banno

What does that mean? It doesn't even appear to be grammatically correct.

The set of true propositions is on your account a proper subset of the set of propositions with a justification. — Banno

No it's not. That would be:

1. p → Jp

That's not what is argued. What is argued is:

2. p → ◊Jp

Do you understand the difference?

To better explain it, I am saying:

1. If something is mortal then it can die

You are misinterpreting/misrepresenting this as:

2. Something can only be mortal if it’s dead

If, for antirealists, as you say, all truths are believable and justifiable, then for any given truth there is some justification. On your account, a proposition can only be true if it has a justification. — Banno

You’re making the same mistake. I explained it clearly above:

Note that these mean different things:

1. If a proposition is true then it can be justified
2. A proposition can only be true if justified

In propositional logic:

3. p → ◊Jp
4. ◊p → Jp

At least, (4) is my best attempt at formulating (2). The position of the "can" is a little confusing. — Michael

Do you understand what (1) and (3) mean?

This should not be so hard. — Leontiskos

You’re right, it shouldn’t. Which is why I don’t understand why you are taking issue with what I am saying.

It is simply an a priori fact that from “p and not p” one can derive any conclusion, and so any argument with “p” and “not p” as premises is valid.

I don’t know what you mean by “appealing” to the principle of explosion.

It’s like saying that we “appeal” to modus ponens.

We use modus ponens to derive some conclusion and we use the principle of explosion to derive some conclusion.

This is all a priori reasoning based on logical axioms, not some a posteriori proposition that is possibly false.

I don’t know what you mean by “presupposing” the principle of explosion.

See the “⊢ Q” at the end? That means that Q follows from the bit before.

We’ve already established that “(P ∨ Q) ∧ ¬P” is true, so therefore “Q” is true.

Yes

Why? — Leontiskos

Because of the reasoning explained here.

And again, if that is no more than that it can be made the consequent of a material implication, that is trivially right. So again, what is it to be "justified"? — Banno

Yes. And if B then A→B, for any A or B. So if we take justification as being the consequent of a material implication then that any truth is justified is trivial.

But is that what you mean? — Banno

I don't really understand what you're asking.

We have realists claiming that some truths are unknowable and antirealists claiming that all truths are knowable.

Assuming that knowledge is justified true belief, this reduces to the realist claiming that some truths are either unbelievable or unjustifiable and antirealists claiming that all truths are believable and justifiable.

What counts as justification is certainly an issue worth considering at some point, but I don't think it's particularly relevant to the current issue being discussed.

We can tentatively just say that it’s whatever distinguishes knowledge from a mere true belief.

(do I need to add that if it is justified, then it is by that very fact justifiable?) — Banno

Sure, justified entails justifiable. But justifiable does not entail justified. Antirealists are only claiming that if a proposition is true then it is knowable (justifiable), not that if a proposition is true then it is known (justified).

Returning back to a previous comment of yours:

Just to be clear, my target here is the idea that a proposition can only be true if justified. — Banno

Note that these mean different things:

1. If a proposition is true then it can be justified
2. A proposition can only be true if justified

In propositional logic:

3. p → ◊Jp
4. ◊p → Jp

At least, (4) is my best attempt at formulating (2). The position of the "can" is a little confusing.

It's valid. I don't even know what you mean by "not presupposing the principle of explosion".

As I said, there aren't two arguments; there is one argument:

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

But we don't have to write out P2 - P5 because they are all necessarily true; they are some of the rules of inference. We can leave it as:

P1. P ∧ ¬P
C1. Q

I haven't said that this has something to do with every kind of valid argument. It has nothing (necessarily) to do with modus tollens or modus ponens, for example.

Michael is adamant that any such claim which does not explicitly rely on explosion is implicitly relying on explosion. — Leontiskos

No I'm not.

No, the issue is that if (2) is true then no one can presuppose (1), because the proposition in question is justifiable. — Leontiskos

Anyone can suppose anything.

1. Suppose that I am a woman.
2. I am not a woman.
3. Therefore, supposition (1) is false.

If you don't believe that (2) is true then you might assert (1), perhaps because someone has tricked you into believing that I am a woman.

The issue is that if (2) is true then "We are brains in vats" is not representative of global skepticism at all. (2) does not invalidate global skepticism, it invalidates the idea that "We are brains in vats" is representative of global skepticism. — Leontiskos

I don't understand what you're saying here at all. All I can do to correct is you is to re-quote the IEP article on the brains in a vat argument:

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)

1. Suppose, "We are brains in vats" can be true even if it is not possible to justify such a proposition
2. "We are brains in vats" is (justifiably) false
3. Therefore, supposition (1) is false — Leontiskos

That's basically what I said.

The issue is that if realism is true then supposition (1) is true. Given that supposition (1) is false, realism is false. That's Putnam's reasoning.