Are there any introductory textbooks that talk about the principle of explosion? — NotAristotle

This is metalogic, but note that validity is meant to show how conclusions rightfully follow from premises. It is meant to provide us with a way to think correctly, and increase our knowledge.

Anything follows from an explosive system, and yet not anything follows with respect to correct thinking. This means that explosion is an aberration (along with the contradiction that it flows from). In propositional logic contradictions are supposed to be eliminated (via reductio), not utilized.

So is an explosive argument valid? In one sense it is, and in one sense it is not. It does not provide us with the thing that the notion of validity is meant to provide, but it is nevertheless valid in a certain (arguably degenerative) sense.

There are some logicians in these parts who view logic as mere symbol manipulation, without any relation to correct reasoning. For these logicians an explosive argument is uncontroversially valid.

It's not a matter of what I "regard" to be the case. — TonesInDeepFreeze

Sure it is, unless you are the Source of Truth Itself.

What textbook says that. If you can cite that statement I'll sell the farm. — NotAristotle

I'm sure you could find that in a textbook, but one must recognize that such textbooks presuppose that the premises are not inconsistent.

I think that is right, it is arbitrary. Although I would say that an argument can have inconsistent premises and still be valid as long as those premises do not do any "work" in the argument — NotAristotle

Sure, but a premise that is not doing any work in an argument is not a premise of the argument. It is an unrelated proposition. You want to say that if one half a contradictory pair is doing work, then the other half is implicated.

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. — Michael

You think it's valid because of explosion. It's that simple. Again:

TonesInDeepFreeze thinks that any argument with inconsistent premises is valid, and that the principle of explosion does not need to be presupposed in order to say this. Michael disagrees. He thinks that any argument with inconsistent premises is valid, and that the principle of explosion does need to be presupposed in order to say this. — Leontiskos

• Michael: Susie's argument is valid because of explosion.
• Tones: Susie's argument is valid, but not because of explosion.

The rule is: If P v Q is on a line, and ~P is on a line, then infer Q.

The rule is NOT: If P v Q is on a line, and ~P is on a line, and P is not on a line, then infer Q. — TonesInDeepFreeze

I think @NotAristotle is right insofar as the rule is ambiguous. There is no magical rule-book of logic that settles this issue, and in practice someone who contradicts themselves is responded to with a reductio.

But that doesn't work if A and not-A are both true. That's my point. The proof doesn't work. The proof only works if you ignore that A is also true. — NotAristotle

Sure, and that's the same puzzle of the OP. I see your point.

See:

In cases of inconsistent premises what happens is that the person arguing arbitrarily makes use of some premises while conveniently ignoring others. For example:... — Leontiskos

A or B
Not-A
Therefore, B (disjunctive syllogism)

- Of course we appeal to modus ponens.

I asked why you think Susie's argument is valid. You gave an argument which you admitted is a presentation of the principle of explosion. Clearly you think Susie's argument is valid because of the principle of explosion, just as we might think that a conclusion follows because of modus ponens. Someone who makes an inference based on a rule of inference is appealing to that rule of inference. This should not be so hard.

An inference is presupposed when it is interpreted as the tacit reasoning of an enthymeme. You are supplying Susie's argument with a rule of inference that she does not explicitly present. You are interpreting it as an enthymeme and supplying what you see as the implicit inferential steps.

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

Do you know if you are appealing to the principle of explosion? Because I asked if you are "appealing to or presupposing the principle of explosion."

And so you are appealing to or presupposing the principle of explosion when you claim that Susie's argument is valid, are you not?

- Which is a presentation of the principle of explosion, is it not?

It's valid. — Michael

Why?

No I'm not. — Michael

Susie gives an argument. Her premises are inconsistent. Is her argument valid? Do not presuppose the principle of explosion.

A. Yes, Susie's argument is valid.
B. No, Susie's argument is not valid.
C. We do not yet know whether Susie's argument is valid.

No I'm not. — Michael

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

Yes. — Michael

The issue is that if realism is true then supposition (1) is true. — Michael

No, the issue is that if (2) is true then no one can presuppose (1), because the proposition in question is justifiable. 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. This is the third of three separate reasons why your argument fails.

- No, you're wrong. I just had extended conversations with both of them. Tones is adamant that his claim does not presuppose explosion, and Michael is adamant that any such claim which does not explicitly rely on explosion is implicitly relying on explosion.

What I would be saying, analogously, is that we have taken the "you-do-you while I-do-me" principle too far: if your friend decides to go out and rape someone, then you have a duty to forceably impose your values on them insofar as they shouldn't be doing that. Similarly, a society has a duty to take over or at least subjugate another society to their values when the latter gets too immoral.

...

and [coercion] seems to be a valid resort to stop societal structures that are really immoral; and this entails some version of imperialism — Bob Ross

Well, in virtue of what do we have a duty to prevent immorality? Do we have a duty to perpetrators? Do we have a duty to victims? Do we have a duty to "friends"? Do we have a duty to strangers? Do we have a duty to strangers on the other side of the world?

Even if the negative consequences were very low (or non-existent), are you saying that the West would not be justified in taking over North Korea by force? — Bob Ross

If there were no negative consequences then we would be justified. But even something as simple as resource allocation is a negative consequence, so there will always be negative consequences.

What I gave is a simple reductio:

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

(Your modus tollens was addressed in my first sentence.)

@TonesInDeepFreeze thinks that any argument with inconsistent premises is valid, and that the principle of explosion does not need to be presupposed in order to say this. @Michael disagrees. He thinks that any argument with inconsistent premises is valid, and that the principle of explosion does need to be presupposed in order to say this. Michael does not apprehend the difference here, but his overall argument is nevertheless stronger than Tones'.

At this point it's obvious that both of you are more interested in being contrarians than actually meting this out, so I'll leave you to it.

I posted to make clear that a particular quote of him is correct and not inconsistent with anything I've said — TonesInDeepFreeze

Yes, you cherry-picked a highly tailored and dialogically misrepresentative quote in order to go out of your way in affirming an idea that no one was even contesting.

I laid it all out here: go look:

If you don't want to be honest and reckon with the actual object of the conversation, I'm sure no one will be surprised.

You cited me in your disputes (and without linking my name). So I exercised the prerogative to make clear that that particular argument is not incompatible with anything I've said. — TonesInDeepFreeze

You failed to understand the point I was making to Michael. I invite you again, for the third time, to go back and reckon with that point. If you can't see it by now then you are surely closing your eyes to what is plainly obvious.

(It is fascinating that in the last few hours I disagreed with Michael twice, and then both you and Banno attempted to agree with Michael in order to disagree with me, despite the fact that you ultimately disagree with Michael. Michael is aware that Banno disagrees with him, even if he is not yet aware that you do. But the whole thing is a bit comical. You and Banno are not doing philosophy, you are doing a gossipy "Contradict Leontiskos" game. :grin:)

There are a lot of things to untangle in a discussion. I didn't purport to vindicate everything the poster has said. I mentioned one particular thing he posted and I said it is correct. — TonesInDeepFreeze

You focused on something which no one contested. I am inviting you to focus on the point at issue.

(Further, the reason NotAristotle is so confused is because Michael is failing to recognize that he is justifying validity in a different way than you are; and you are aiding and abetting his failure. NotAristotle made an argument against your view and Michael defended your view, falsely believing himself to be holding the same view. NotAristotle therefore ends up with constant ignoratio elenchus from Michael.)

That is what I said is correct. — TonesInDeepFreeze

You came in and said, "Michael's argument is valid," but you haven't at all reckoned with what was really said. I invite you to do that. NotAristotle made an observation about your construal of validity, and Michael defended your construal of validity with recourse to the principle of explosion. Reckon with that. — Leontiskos

(Michael thinks your construal of validity is true in virtue of the principle of explosion. You explicitly say that it is not. You are obviously disagreeing with Michael. Stop being disingenuous.)

Edit: And Michael's interpretation is not unreasonable (except for the fact that it is a misrepresentation that has been addressed earlier in the thread). Your interpretation is irrational without recourse to the principle of explosion. Michael's general approach is much more rational than your own:

if an argument's conclusion follows from its premises using the rules of inference then they will name this type of argument "valid". — Michael

It's not a definition of validty! It's not supposed to be definition of validity! — TonesInDeepFreeze

"If the premises are inconsistent then the argument is valid by definition," does not mean that the definition of validity is equivalent to the premises being inconsistent. It only means that any argument whose premises are inconsistent is valid in virtue of the definition of validity. Your boat is being swamped by your quibbles. :roll:

Here is what Michael said:

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

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

• NotAristotle: Tones' claim that inconsistent premises make an argument valid by definition seems strange to me.
• Michael: It is valid because of explosion.
• Leontiskos: Tones is giving a different explanation for why it is valid.

You came in and said, "Michael's argument is valid," but you haven't at all reckoned with what was really said. I invite you to do that. NotAristotle made an observation about your construal of validity, and Michael defended your construal of validity with recourse to the principle of explosion. Reckon with that.

We have a definition of validity. Then we show that that definition entails the principle of explosion. — TonesInDeepFreeze

Well, if your strange interpretation of your definition is not based on explosion, then we are back to square one, and it is simply wrong. "If the premises are inconsistent then the argument is valid by definition (and this does not presuppose the principle of explosion)," is just a terrible interpretation of the definition of validity.

Arguments are not valid in virtue of being inconsistent (lol). You obviously won't admit this, but most TPFers are able to recognize its truth. Indeed, there is only one person who has agreed with you in this.

If you link to my quote, time permitting I will address it. — TonesInDeepFreeze

Did you see the colon at the end of that sentence?

I haven't said anything about 'presuppose'. Rather, I have shown that the definition of validity entails the principle of explosion. — TonesInDeepFreeze

It only "entails" it because it has presupposed it. Else you do disagree with Michael, who thinks that your construal of your definition is nothing other than a tacit appeal to the principle of explosion.

But you are stuck in your quibbles again. When you figure out how exactly the principle of explosion relates to your definition, feel free to get back to me.

And the definition is not based on the principle of explosion. Rather, the definition implies the defintion of explosion. You have what I said backward. — TonesInDeepFreeze

Your interpretation of your definition presupposes explosion.

I am saying no such thing. — TonesInDeepFreeze

So you would say that someone who does not understand the principle of explosion can apply the definition in the way you prefer?

I submit to you, that you should accept a sense of nationalism in two respects. The first, in the sense that whatever nation you belong to you must have a vested interest in its flourishing and protection against other nations—or move to a different one (if you can). The second, in the sense that, if your country has substantially better politics than other ones, you should have a pride in it and want to expand its values to the more inferior ones (which leads to imperialism).

For those who are upset at my rhetoric (and perhaps the lens by which I am analyzing this), I challenge you to try to justify, in your response to this OP, e.g., why Western, democratic values should not be forcibly imposed on obviously degenerate, inferior societies at least in principle—like Talibanian Afghanistan, North Korea, Iran, China, India, etc. Some societies are so obviously structured in a way antithetical to the human good, that it is virtually impossible to justify leaving them be in the name of anti-imperialism. E.g., if we could take over North Korea right now without grave consequences (such as nuclear war), then it is obviously in our duty to do so—and this is a form of imperialism. Why would you not be a Western supremacist? — Bob Ross

I would say that cultures interact in much the same way individuals do. In both cases there are things like exchange, mutual cooperation, conflict, argument, persuasion, and coercion.

Cultural relativism is a silly idea, but the opposed idea of imposing values is also rather silly. One cannot impose values. When it comes to individuals, we will try to convince opponents and we will coerce them if they become too dangerous. It is the same with cultures. The values of persuasion and reason are really Western values themselves. In the West coercion is a last resort.

Some societies are so obviously structured in a way antithetical to the human good, that it is virtually impossible to justify leaving them be in the name of anti-imperialism. — Bob Ross

Anti-imperialism is a very limited justification in the first place. But the disorderedness of a society is not in itself a sufficient reason for intervention. Should we intervene in North Korea out of compassion? Perhaps, but perhaps not. Compassion can be a motive, but it is seldom a sufficient condition for action.

Michael's reasoning is correct there and doesn't contradict anything I've said.

The definition of validity entails that the principle of explosion is valid. — TonesInDeepFreeze

That's fine if you want to say that the strange way you want to apply your definition is based on explosion, but this is a new claim. Earlier in the thread you said that the two are "similar," not that one presupposed the other:

So explosion and "any argument with an inconsistent set of premises is valid" are similar. — TonesInDeepFreeze

What you are apparently saying now is that someone who does not understand the principle of explosion cannot apply the definition in the way you prefer.

My point here has been that validity has to do with "follows from." If you think that your idiosyncratic application of your definition of validity is permissible because "anything follows from a contradiction" (i.e. explosion), then you have not disagreed with me that validity has to do with "follows from."

(I won't belabor the point of how strange it is to count on explosion enthymeme to understand a foundational definition.)

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". — Michael

1. If metaphysical realism is true, then global skepticism is possible
2. If global skepticism is possible, then we can be brains in a vat
3. But we cannot be brains in a vat
4. Thus, metaphysical realism is false (1,2,3) — Brain in a Vat Argument | IEP

The simplest answer here is that (3) is false. The second answer is that even if we grant (3), it then follows that (3) is not unjustifiable. That is, if it is possible to justify a (necessary!) claim about brains in a vat, then the matter of brains in a vat is not unjustifiable. I would guess that the average realist is fine with either approach, depending on how (3) is presented.

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

The names appear, but where is the claim that they are realists? To take an example at random: Mackie. Here is what your quote says about Mackie:

Mackie (1980) and Routley (1981), among others at the time, point to difficulties with this general position but 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. — SEP

How in the world do you construe this as, "Mackie is a realist who falls into my criteria for realism"?

↪Leontiskos So you don't think unjustifiable truths are unjustified? How odd.

Or is that just pedantry in the service of your confusion? — Banno

"Unjustified" and "unjustifiable" are two different things. Michael's post would be entirely innocuous if we misread "unjustified" for "unjustifiable," as you did. You are being disingenuous (again).

- You quoted the word "unjustified" in response to a quote from Michael that does not contain that word. Instead it contains the word "unjustifiable." Trouble reading, you have.

I've responded to that quote already:

Or, "...then a proposition not being justified is not a good reason to reject it." There is an equivocation here on 'reject'. If 'reject' means falsify, then this strikes me as uncontroversial. If 'reject' means "abstain from affirming," then the consequent is false but it does not in fact follow from your premises. — Leontiskos

I'll leave you and Michael to it, and await your concession.

- If Banno is a realist who fits your characterization then that would be exceedingly helpful. If he had better reading comprehension his affirmation would be more promising.

That there are "unjustified" truths is pretty obvious. — Banno

Unjustifiable.

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

I asked you for an example of a realist who holds to your misrepresentation. You didn't give one. If you did, then what is his/her name?

It's right there, explicitly saying "the realist believes that it is possible for truth to be unknowable in principle." — Michael

You fished out a single sentence in an SEP article? Who cares? Find a new god to put your faith in. I am asking about realists, not SEP. You need to start arguing with real people, instead of merely making arguments from authority.