READ exactly what I wrote:

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.
— Michael

The difference between an argument from the definition of validity and an argument from explosion has been explained multiple times throughout this thread. Tones himself recognized it.
— Leontiskos

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

I referred exactly to this:

"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."
— Michael

That is what I said is correct.

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

You said

a terrible interpretation of the definition of validity. — Leontiskos

It's not an interpretation of the definition (as if I intepret it to be the definition of as if I interpret the defintion in some strange way. Rather, it is simple inference from the definition. That is a critical point not a "quibble".

There are a lot of things to untangle in a discussion. I didn't purport to vindicate everything the poster has said. And I am not, at least at the present, interested in untangling whether the way you represented what he said is correct or not. In that post by me, I mentioned a particular thing he posted and I said it is correct.

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.)

No, YOU quoted a certain argument by him and followed with a comment about me, as if that argument is not compatible with what I said. So I made clear that that argument is not incompatible with what I said.

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.

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:)

Oh, for Pete's sake. I just wrote that I am not, at least at this time, interested in sorting out the disagreement between you and him. Rather, at that juncture, I posted to make clear that a particular quote of him (quoted by you followed by mention of me and something I wrote) is correct and not inconsistent with anything I've said. That's it. That's all I posted. Get it now? [Edit: Also, I said, "The definition of validity entails that the principle of explosion is valid." And that is correct.]

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.

I didn't "cherry pick" anything. And I didn't "misrepresent" anything. YOU picked out a quote by him, and cited ME in connection with that quote by him. So I quite rightly exercised my prerogative to make clear that what he wrote is correct and is not in contradiction with anything I've written.

I laid it all out here; go look: https://thephilosophyforum.com/discussion/comment/948275

you and Banno attempted to agree with Michael in order to disagree with me, despite the fact that you ultimately disagree with Michael. — Leontiskos

I don't know what passages you're referring to. You've not shown that anything I've said is incorrect. Your picture of this thread as some kind of tag team match doesn't interest me.

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. — Leontiskos

I've addressed the subject of this thread in detail.

My intellectual credibility does not require that I sort through your own disagreements with a poster, nor even that I sort through your notions in and of themselves.

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 — Leontiskos

Again, I'm not interested, at this juncture, in untangling Michael's role. I am especially not interested in commenting on his posts vis-a-vis you in between with your characterizations of his posts and your characterizations of my posts and your characterizations of how they compare.

NotAristotle is confused because he knows virtually nothing about the subject, not even chapter one of any book or first material in an article, whether standard or alternative - doesn't even know the differences among connectives, sentences and arguments. And talk about cherry picking, that one by NotAristotle from Wikipedia is a doozy! It was a terrible misrepresentation and failure to even read the context cited by the article, resulting in him flirting with the very point of view he claims to oppose! Talk about what is NOT philosophy!

Your interpretation is irrational without recourse to the principle of explosion. — Leontiskos

I didn't give an "interpretation" of the definition. I stated the standard definition. The definition doesn't make "recourse" to the principle of explosion. Rather, from the definition, it is simple to show that the principle of explosion is valid.

That can't be more clear: (1) State the definition. (2) From the definition, infer the principle of explosion. In that order: (1) then (2). How cannot someone not understand that?

Including the prior steps:

(1) we define an 'interpretation'

(2) we define 'true in an interpretation'

(3) we define 'argument'

(4) we define 'valid argument' (mentioning only truth, falsehood and interpretations)

(5) we define 'inconsistent'

(6) we define 'unsatisfiable'

(7) we easily show that an inconsistent set is unsatisfiable

(8) we easily show that the definition of 'valid argument' implies the principle of explosion (that is, that the principle of explosion is valid)

The principle of explosion was not mentioned, assumed, invoked or recoursed to. Only that it was proven to be valid from the definition of 'valid argument'.

I gave details in an earlier post.

I don't see how this cannot be understood.

From the definition of validity, we show that the principle of explosion is valid.

Then, if an argument has an unsatisfiable set of premises, we thereby show that it is valid, no matter what the conclusion is. (i.e. an instance of the principle of explosion).

Also, from the definition of validity, we show that modus ponens is a valid argument form.

I pointed out that the particular argument in the original post happens to be an instance of modus ponens, so the argument is valid on that basis too. That doesn't contradict showing that the argument is valid on account of explosion. Both are correct: it is valid on account of modus ponens and it is valid on account of explosion. And it valid on account of both of those because both of those are valid on account of the definition of validity.

That modus ponens is a valid argument form is shown from the definition of 'valid argument'; and that explosion is a valid argument form is also shown from the definition of 'valid argument'. The validity of modus ponens and the validity of explosion are both consequences of the definition of 'valid argument'.

And that is the case no matter what I say, no matter what Michael says, and no matter how you characterize any relationship between what Michael says and what I say.

/

Again, since people say things like "Tones's view":

When I report how it goes in ordinary formal logic with the standard definition of 'valid argument', I have not thereby claimed that ordinary formal logic (with the usual definition of 'valid') is the only reasonable logic, or the best one in all situations, or the best one for a philosophical understanding of logic and validity, or that it accords with all of everyday reasoning, or even that is more than hardly found in everyday reasoning, or that is the only one worthy of study or adopting. Alternative formal logics abound and are, in my opinion, worthy of study and adoption in the contexts they are suited for. But ordinary formal logic is appreciated by, studied, applied and discussed greatly among mathematicians, logicians, computer scientists, and philosophers. It is at the heart of mathematical logic that axiomatizes the branches of mathematics, including computability, including the very invention of, and improvements to, the digital computer. But, meanwhile, ordinary formal logic is fair game to critique. But critiques of any logic are mindless, and not even close to philosophy, when they are premised in ignorance and confusion about how it actually goes in the logic. And corrections given to such ignorance and confusion are not themselves a form of endorsement of the logic nor do they constitute claims that it is "right", unless one does go on to endorse the logic. For one to explain what actually happens in the logic is not in and of itself to say that people should adopt the logic - but rather, at least that they should not attack it on false bases. By analogy, one doesn't have to agree with the constitution of a country just to study it and report what it actually says. Moreover, one may rebut arguments against a logic without endorsing the logic, as one's rebuttals may be merely that the arguments are not good, without opining on whether the conclusions are correct or incorrect.

There is repetition in the above, but it is there to drive these points that keep getting overlooked no matter how many times they are stated and especially to, hopefully, foreclose against someone yet again putting words in my mouth to make it appear that I've adopted a position that in fact I have not adopted.

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".

@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.

Tones is just super precise in his expressions and Michael is a little more intuitive. Once you understand the issue, you'll see that they're saying the same thing.

- 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.

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. — Leontiskos

Do you understand why the argument is valid?

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 I'm not. — Michael

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

Yes. — Michael

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.

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.

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

(1) Contrary to Leontiskos, I am not "disingenous" in this discussion. No statement of mine, about Michael or anything else, has been shown to be dishonest and none to be materially incorrect.

(2) As to validity, I said that the standard definition of 'valid argument' implies that any argument with an inconsistent set of premises is valid. That it is correct: The standard definition implies that any argument with an inconsistent of premises is valid.

(3) I said that the standard definition of 'valid argument' does not mention the principle of explosion, and I listed the definitions and points that the definition of 'valid argument' does rely on. Then from the standard definition of 'valid argument' we easily show that the principle of explosion is valid. That is correct.

(4) There is no "contrarianism" in any of that.

Again, below is what obtains in ordinary formal logic, regardless of any inclinations I might have or not have toward what the definitions or results "should be or should not be". That point stands throughout this discussion.

Semantic: Explosion as an argument form is valid. It is easy to show that explosion as an argument form is valid by noting that the definition of 'valid argument' implies that explosion is a valid argument form.

Syntactic: Explosion as an inference rule, depending on the system, can be either a primitive rule or a derived rule.

" ¬∃x(P∧Q) "

where x is an interpretation, P is "all premises are true" and Q is "the conclusion is false."

Is there something problematic about writing the definition of validity that way?

P5. (P ∨ Q) ∧ ¬P ⊢ Q (disjunctive syllogism) I do not understand the move from P5 to C1 using disjunctive syllogism. Would you mind explaining?

It's valid. — Michael

Why?