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.

Michael thinks your construal of validity is true in virtue of the principle of explosion. — Leontiskos

No I don't. I'm saying that P ∧ ¬P ⊢ Q is valid and that P ∧ ¬P ⊢ Q is called "the principle of explosion".

Much like P → Q, P ⊢ Q is valid and that P → Q, P ⊢ Q is called "modus ponens".

Specifically, you want to say that realism entails that <"We are brains in vats" can be true even if it is not possible to justify such a proposition>. You then go on to attempt to justify the proposition, <"We are brains in vats" is false>. Even supposing you succeed, your success would show that the putatively unjustifiable proposition is in fact justifiable, which moots the criterion of realism (per your strange/exaggerated definition). — Leontiskos

It's a simple modus tollens:

1. If realism is true then it is possible that we are (unknowably) brains in a vat
2. It is (knowably) not possible that we are brains in a vat
3. Therefore, realism is false

I don't see that you can substitute □(p → ◊Kp) for ∀p(p → ◊Kp). — Banno

Yeah, going over it again I think I misunderstood what the article was saying about the relationship between the two.

The "principle" of explosion directly infringes the law of non-contradiction. It's silly to even call it a principle. — NotAristotle

The principle of explosion is simply the acknowledgement that if we apply the rules of inference to a contradiction then we can derive any conclusion we like. That is simply an a priori fact about propositional logic.

Obviously almost nobody will accept that a contradiction can be true.

Although there are dialetheists like Graham Priest who argue that they can.

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

But that's not what has been said. What has been said is that if a proposition is true then it is justifiable.

So, what? — Banno

So it entail global skepticism, which might be seen as undesirable. And if Putnam's argument is a sound refutation of global skepticism then it's a sound argument against realism.

And if the reasoning here is correct then it entails that nothing is known; my counterpart to Fitch's paradox.

Every unjustifiable truth is unjustified but not every unjustified truth is unjustifiable. Realism allows for unjustifiable truths, and that is a problem.

The wikipedia article you cited literally says the principle of explosion is "disastrous" and "trivializes truth and falsity." — NotAristotle

It says "the existence of a contradiction (inconsistency) in a formal axiomatic system is disastrous", which it is.

Given that P ∧ ¬P entails Q, we must be sure that we never allow for P ∧ ¬P to be true.

This has already been explained to you. — Leontiskos

He's talking about something else. I'm taking about this:

P1. P ∧ ¬P
C1. Q

This is literally the principle of explosion:

In symbolic logic, the principle of explosion can be expressed schematically in the following way:

P ∧ ¬P ⊢ Q For any statements P and Q, if P and not-P are both true, then it logically follows that Q is true.

---

Your contention that argument 2 cannot ever exist without argument 1 is magical, ad hoc thinking. There is nothing serious about it. — Leontiskos

See the section titled "Proof" which includes all my additional steps P2 - P5. They don't need to be made explicit because they are inherent rules of inference.

There's also Banno.

It's in that post above. There's Hart, McGinn, Mackie, and Routley. And of course Fitch himself.

You ask me for an example and then complain that it's just one example?

And you accuse me of sophistry and not being serious... :roll:

This is evidence of your sophistry. — Leontiskos

How is it sophistry? I have provided the source. It's right there, explicitly saying "the realist believes that it is possible for truth to be unknowable in principle."

All I've asked is for you to give me an example of a realist who holds to your strange version of realism. — Leontiskos

Again, read the article:

Rediscovered in Hart and McGinn (1976) and Hart (1979), the result was taken to be a refutation of verificationism, the view that all meaningful statements (and so all truths) are verifiable.

...

Mackie (1980) and Routley (1981), among others at the time, ... ultimately agree that Fitch’s result is a refutation of the claim that all truths are knowable, and that various forms of verificationism are imperiled for related reasons.

I am going to limit myself to serious interlocutors. — Leontiskos

I am being serious. Read up on the principle of explosion.

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".