Grabbed on the fly:

Three equivalent variations:

"Valid: an argument is valid if and only if it is necessary that if all of the premises are true, then the conclusion is true; if all the premises are true, then the conclusion must be true; it is impossible that all the premises are true and the conclusion is false." [bold added]

https://web.stanford.edu/~bobonich/terms.concepts/valid.sound.html

Not just consistent, but equivalent with.

Quite not idiosyncratic.

And the argument is valid by Gensler, Enderton, SEP and Wikipedia.

Gensler:

"An argument is valid if it would be contradictory (impossible) to have the premises all true and conclusion false."

It is impossible to have both A -> ~A and A true. Perforce, it is impossible to have the premises all true and the conclusion false.

Wikipedia:

"an argument is valid if and only if it takes a form that makes it impossible for the premises to be true and the conclusion nevertheless to be false."

It is impossible to have both the premises true. Perforce, it is impossible for the premises to both be true and the conclusion nevertheless false.

SEP:

the argument is valid [if] the conclusion follows deductively from the premises... — Logical Consequence | SEP

That is true, though usually it is addressed as a theorem not a definition. It is the soundness theorem. Still:

The conclusion follows deductively from the premises:

1. A -> ~A ... premise
2. A ... premise
3. ~A {1 2 by modus ponens}

Enderton:

"the concept of validity turns out to be equivalent to another concept (deducibility)"

Again, that's not a definition of 'valid' but rather it mentions an equivalence with deducibility (as it applies to first order logic). Still

The conclusion is deducible from the premises:

1. A -> ~A ... premise
2. A ... premise
3. ~A {1 2 by modus ponens}

↪TonesInDeepFreeze - Wrong again:

An argument is valid if and only if there are no interpretations in which all the premises are true and the conclusion is false.

In this case there are no interpretations in which all the premises are true. Perforce, there are no interpretations in which all the premises are true and the conclusion is false. So the argument is valid.
— TonesInDeepFreeze — Leontiskos

You're wrong.

The definition of 'valid argument' there is standard.

And with the argument mentioned in the original post, it is the case that there is no interpretation in which all the premises are true. Perforce, there is no interpretation in which all the premises are true and the conclusion is false. So the argument is valid.

Here is Gensler speaking about validity in his introductory chapter:

An argument is valid if it would be contradictory (impossible) to have the premises all true and conclusion false. In calling an argument valid, we aren’t saying whether the premises are true. We’re just saying that the conclusion follows from the premises – that if the premises were all true, then the conclusion also would have to be true.
— Gensler, Introduction to Logic, Second Edition, p. 3 — Leontiskos

Indeed. Equivalent to the definition I've been stating.

In logic, specifically in deductive reasoning, an argument is valid if and only if it takes a form that makes it impossible for the premises to be true and the conclusion nevertheless to be false.[1] It is not required for a valid argument to have premises that are actually true,[2] but to have premises that, if they were true, would guarantee the truth of the argument's conclusion. — Validity | Wikipedia

Indeed. Yet another way of saying the definition.

Here is Enderton:

What is surprising is that the concept of validity turns out to be equivalent to another concept (deducibility)
— Enderton, A Mathematical Introduction to Logic, p. 89 — Leontiskos

Indeed. It is a point I've made many times. It is the completeness and soundness of first order logic.

TonesInDeepFreeze wants to say that an argument is definitionally/trivially valid if it its premises cannot all be true (i.e. if it is inconsistent). — Leontiskos

That is the third time in this thread that you've put words in my mouth.

tautology or contradiction — Count Timothy von Icarus

Just to note: tautology is semantic and contradiction is syntactic.

A premise is defined as an analytic truth. — Hanover

No, it's not.

An argument without premises is not a syllogism. — Hanover

Yes. So?

the OP is not a valid argument because it's not an argument at all. — Hanover

A rigorous definition:

An argument is an ordered pair such that the first coordinate is a set of statements* and the second coordinate is a statement*.

The members of the first coordinate are the premises. The second coordinate is the conclusion.

If purely symbolic, the premises and the conclusion are symbolic formulas. If in natural language, the premises and conclusion are natural language declarative sentences.

So, written in that form we have this argument:

<{A -> ~A, A} ~A>

The set of premises is {A -> ~A, A}, and the conclusion is ~A.

Written informally (where everything above 'therefore' is a premise and what follows the word 'therefore' is the conclusion.

A -> ~A
A
therefore ~A

* Or more generally, formulas.

A natural language example:

If the Great Pumpkin is orange, then Great Pumpkin is not orange.
The Great Pumpkin is orange.
therefore, the Great Pumpkin is not orange.

/

With your requirement, even the following would not be an argument (not just one of the premises is not analytic, but neither of the premises are analytic):

If Bob has poor eyesight, then Bob wears glasses.
Bob has poor eyesight.
Therefore Bob wears glasses.

Really, you want to disqualify that from being an argument because the premises are not analytic?

/

You've tried a few incorrect arguments, based on misconceptions, that the argument is not valid. And now another one.

So would it be fair to say that 'why' is more amenable to being answered when causality is involved? That goes to the point that in mathematics it is difficult to answer 'why' without finally resorting to showing the proof, which some people might consider to not be an answer to 'why'.

I can give you a story that comes to mind in which I'd assert something like that -- say I'm commiserating with a coworkers frustration about George not being as reliable as we'd like, even though he's a good enough fellow. — Moliere

Of course, I understand the basis of the sarcasm.

The substitution is there only because the OP starts with A -> ~A and asks for validity, so substitution seems to work as a model for the sarcastic talking. I agree that the person speaking sarcastically does not in any way mean these logical implications, though -- it's only an interpretation of everyday speech to try and give some sense to the original question that's not purely formal. — Moliere

The original post challenges the validity of
A -> ~A
A
therefore ~A

I don't see your substitution capturing irony.

When I say A sarcastically, I mean ~A, of course. And that is equivalent with A -> ~A. But I don't present it like that at all. I just say A and there is an implicit premise that when I say it, I mean its negation. I don't know how even modal logic could capture that. Or maybe, I am saying that A is true in an alternative world and false in the actual world, but even that seems far-flung.

Getting back to @Srap Tasmaner, he's looking for a use of A -> ~A in everyday discourse. I don't think your proposal works, since people don't acutually say things of the form A -> ~A to convey sarcasm. It seems to me that you followed an interesting idea, but it doesn't do the job here.

Though, related to a different kind of formula, people do say things like:

If 'Fear Factor' is great television then I'm the Queen of Roumania.

That is:

If P then Q (where Q is false)

/

And reductio ad absurdum may occur too (aside from mathematics where it is prevalent):

If Jack robbed the store, then has the loot in his car.
He does not have the loot in his car.
So, the claim that Jack robbed the store leads to a contradiction.
So Jack did not rob the store.

Of course, it can also be cast, more tersely, as modus tollens:

If Jack robbed the store, then he has the loot in his car.
He does not have the loot in his car.
So Jack did not rob the store.

Very good. Thank you.

In natural language, predication is often not totally univocal, but is also not totally equivocal. — Count Timothy von Icarus

I don't claim to have academic definitions of 'univocal' and 'equivocal', but at a naive level, as I'm merely winging it here, it seems to me that:

'totally univocal' is redundant. An expression is univocal if and only if it has one meaning. That's total.

'totally equivocal' is hard to conceive. An expression is equivocal if and only if it has more than one meaning. What would it mean to say it is totally equivocal?

For example, we might say that "lentils are healthy," or "running is healthy." These are true statements. And we might also say "Tones is healthy." Yet you would not be "healthy" in the same way that lentils are. — Count Timothy von Icarus

Right.

However, neither is the usage totally equivocal. We call lentils "healthy" precisely because (normally) they promote the health of human beings, i.e. the same "health" we refer to in "Tones is healthy." — Count Timothy von Icarus

Hmm, I'm not sold on that. That "Tones is healthy" and "This apple is healthy" are true two in different senses doesn't suggest to me that there's any matter of totality to consider.

/

By 'analogous predication' you mean as with the Tones/Apple example?

Do you mean that

This apple is healthy
is analogous predication with
Tones is healthy

I do understand that.

So
The animal runs
is analogous predication with
The refrigerator runs

It's sort of like how, as far as I am aware, there is no popular formalization of the distinction between quia vs. propter quid demonstrations (i.e. demonstrating "that something is the case," vs. demonstration "why it is the case.") I don't think most people would deny that they're different (although some would), but rather it seems that the difference should be entirely reliant on the arguments' content, not their form (i.e. an issue of material logic). — Count Timothy von Icarus

I would think
'that it is the case' is a matter of giving an argument
but
'why it is the case' is a matter of exposition, not argument

It's interesting that in mathematics, some people demand to know "Why is that theorem true?" And I can't think of an answer other than "Because there is a valid argument for it from true premises, and here it is ..." That is, I can show you the proof, which, at least for me, does answer "why?". I may be able to give real world examples, and abstract analogies, and point to coherency. But those don't fully answer "why" in the same definitive way that proof does.

Can we substitute salva veritate into an ironic statement? — Banno

It occured to me, when I thought of the irony operator, that we have the problem that we can't substitute salva veritae within a belief operator. That even makes negations tough, since we can't assume the ironic speaker knows double negation. I glossed over it. So we ould have to formulate further:

Let P be not a negation:

Df. i(s P) <-> (T(s P) & B(s ~P))

Df. i(s ~P) <-> (T(s ~P) & B(s P))

But I'm just fooling around. And probably someone has already worked out an irony logic.

[EDIT: I should have seen right away that my "irony operator" is not adequate since it does not distinguish between speaking ironically and lying in general.]

A = "George is going to open the store tomorrow"

So, by substitution:

George is going to open the store tomorrow implies George is not going to open the store tomorrow. — Moliere

You mean substitute "George will open the store" with "If George will open the store then George will not open the store"?

Why make that substitution? I don't see how that is what the ironic speaker is saying.

I edited my post to mention that, if I understand your main point, I agree with it.

And, if I understand, I agree with your disjunction:

Either make the predicate encompassing, in which case the first premise is false.

Or have two separate predicates, in which case the argument is invalid.

/

analogous predication — Count Timothy von Icarus

I don't know what you mean.

Everything that runs is an animal.
My refrigerator is running.
Therefore, my refrigerator is an animal.

Works great formally if you're allowed to us "R" for "that which runs" in both premises. — Count Timothy von Icarus

If I understand your point, I agree with it, and I think it is incisive and apropos.

Let Rx mean x runs (where 'runs' includes both 'moving quickly on feet' and 'operates')
Let Mx mean x is an animal
Let t mean the refrigerator

1 Ax(Rx -> Mx) ... false premise
2 Rt ... true premise
therefore Mt ... {1 2} ... false conclusion

valid but unsound

No absurdity is a validity.

What is the conditional?

/

Maybe there could be an intensional logic with a defined irony operator 'i'.

s is a speaker
P is a statement
B(s P) iff s believes P
T(s P) iff s states P

Df. i(s P) <-> (T(s P) & B(s ~P))

All he had to do is say that there aren't any cases where both premises are true, therefore it's valid. — frank

I said it over and over and over for you.

All you had to do is read the replies given you. And that's hardly the only point I explained for you.

With a false antecedent, the consequent is vacuously true. — Benkei

No, with a false antecedent the conditional is true, sometimes described as 'vacuously true'. It's the conditional that is deemed true when the antecedent is false.

violates the LNC — Benkei

Yes, A -> ~A and A together are contradictory.

What is the definition of 'flows'?

TonesInDeepFreeze contention that they are the same — Leontiskos

I didn't say they are the same. They are very different. This the second time, in this thread alone, that you've put words in my mouth.

As I said, in this particular regard, I'm merely applying the definitions of ordinary formal logic. As I said, I don't claim that those definitions have dominion over all other contexts.

I imagine you finally had to retire to the insane asylum. Enjoy the rocking chair. — frank

You can retire to the blazes.

I very much appreciate that it may be quite counter-intuitive to many people.

You can't be serious. For the umpteenth time:

An argument is valid if and only if there is no interpretation in which all the premises are true and the conclusion is false.

In this case, there is no interpretation in which all the premises are true. Perforce, there is no interpretation in which all the premises are true and the conclusion is false. So the argument is valid.

I can't put it more starkly than that.

P1. A -> ~ A
P2. A

1. A&~A (1,2)
2. ~A (1)
3. ~A v A ( 2 disjunctive introduction)

Still not valid, considering the contradiction allows me to prove anything I want, even that T is F. — Hanover

1. A -> ~A ... premise
2. A ... premise
3. A & ~A {1 2}
4. ~A {1}
5. ~A v A {}

That is correct. Each inference is valid.

No invalidity has been shown.

It would be nice to have a specification of your logic so that other people could determine for themselves what obtains and does not obtain in it, without having to take you as the arbiter in each particular case.

Negating the conclusion and showing it leads to a contradiction from the same premises proves invalidity. — Hanover

I can better comment on that if you provide the specific arguments you have in mind.

The way to prove the invalidity of an argument is to show that there is an interpretation in which all the premises are true but the conclusion is false.

The premises are
A -> ~A
A
The conclusion is
~A

So provide an interpretation in which both A -> ~A and A are true, and ~A is false.

Hint: There is no such interpretation, since there is no interpretation in which both A -> ~ A and A are true. I'll do it for you:

There are only two interpretations:

(M1) A maps to true (i.e."A is true")

(M2) A maps to false (i.e. "A is false")

In M1, A -> ~A is false

In M2, A is false

So there is no interpretation in which both A -> ~A and A are true.

lright, so you're substituting the conclusion of the OP from A to A &~A, which can simply be represented by an F, for false. — Hanover

No, I am not doing that.

The original argument is one thing. I don't substitute anything in it.

But you claimed that the premises are consistent.

So I gave a proof that they are inconsistent.

The opposite of (A & ~ A) is (A v ~ A), which is a tautology

So, if I can prove from the OP that (A v ~ A) flows, then the argument is invalid — Hanover

(1) Whatever you mean by "flows", I say "is entailed by", or "follows from" or is "implied by".

(2) In classical logic, A v ~A follows from any set of premises whatsoever. Proving A v ~A from a set of premises does not in and of itself tell us anything about the premises nor does it show that a particular argument is invalid.

I'll do it for you:

(3) A -> ~A
A
therefore A v ~A
valid

Indeed, we can explain its validity in at least two different ways: Valid since there is no interpretation in which all the premises are true. Valid since there is no interpretation in which the conclusion is false.

meanwhile, still

(4) A -> ~A
A
therefore ~A
valid

Valid in at least two ways: Valid since there is no interpretation in which all the premises are true. Valid since it is modus ponens, which is proven to be a valid form.

That argument has been addressed extensively in another thread.

In ordinary formal logic, the argument forms mentioned are valid. "ignoring premises" has nothing to do with it. Indeed, the logic is monotonic. And this pertains even to certain natural language situations.

And my saying that does not at all entail the decidedly and outrageously false representation that I take logic to be just symbol manipulation or that my view of logic is confined to truth-functionality.

Under what definition of "valid" is the argument in the OP valid? — frank

You can't be serious. I've given the definition probably at least fifteen times already.

An argument is valid if and only if there are no interpretations in which all the premises are true and the conclusion is false.

The key word there is "at first glance". Upon consideration, it is seen that the first premise is not contradictory.

You are confusing consequence or inference with identity. — Leontiskos

I said nothing about identity.

Certain posters are disputing the validity of an argument.

I am only remarking about what happens to be the case in ordinary formal logic.

It is fine if posters wish to provide a logic and definition of 'validity' in which the argument is invalid. And fine also to point out that, in many contexts, ordinary formal logic is not used or could be misleading. That is not in dispute.

But when critiquing ordinary formal logic, one should at least not be confused and self-misinformed as to how it does go.

In ordinary formal logic:

A -> ~A
A
therefore ~A
is valid

Again, that does not dispute that that may be quite counter-intuitive to many people, nor that there are many other formal and informal logics, notions of validity, understanding of the conditional, and all kinds of other everyday and academic contexts.

The premises are consistent and the conclusions are not.

The conclusion is not true under all interpretations. Sometimes it's A and sometimes it's not A. — Hanover

1 A -> ~A ... premise
2 A ... premise
3 ~A ... {1 2}
4 A & ~A {2 3}

So the premises are not consistent.

A -> ~A
A
therefore ~A

Let A be false. That is an interpretation in which the conclusion is true. So the conclusion is a consistent statement, or put as a set: {~A} is consistent.

I would encourage you to write out in English the only case where both premises are true — frank

There are no interpretations in which both premises are true.

↪TonesInDeepFreeze
"Trivial" has a clear meaning in analytical philosophy. — frank

I'm interested in what that definition is.

Meanwhile, the context here is examination of a particular formal argument. In that context, you fallaciously used the notion of triviality, as I detailed for you.

A -> ~A
A
therefore ~A

There is no interpretation in which both the premises are true.
— TonesInDeepFreeze

If the antecedent in the conditional is false, then the first premise is true. Now say the second premise is true. Then the conclusion does not follow.

If you insult me one more time, we're done. I'm satisfied with ending this discussion. — frank

It's hardly an insult to say that I don't understand why you are not minding the definitions and explanations. I've patiently given you information and explanations, repeated as has been needed. I am sincere when I wonder why you, in one post, seem to understand, but then blow right past again. As to whether you choose to reply to me, of course, that is entirely your choice

Here we go again:

If, in an interpretation, the antecedent is false, then, in that interpretation, the first is premise true, and the second premise is false not true. In that interpretation, the second premise is false.

You need to learn what an interpretation is. That's not an insult. It is good advice, given lagniappe in addition to all the information and explanations I'm giving you.

I'm restating in my own manner some of the points you've made.

in argumentation on degenerate cases is often inadvertent or deceptive — Srap Tasmaner

I agree that a pedantically correct application of notions in formal logic could be abusive sophistry in a context in which they are not understood But, at least at the moment, an actual example from public discourse doesn't leap to my mind. On the contrary, for example, if you were interviewed by a news outlet about the sunken boat and you tried to pull the stunt you mentioned, you would be pilloried. Or if you tried to make a vacuous argument among your non-logician friends, the best you'd get would be "Huh?" Meanwhile, of course, contentious, especially polemical, public discourse is rife with hideous, obnoxious, downright sneaky and pernicious use of all kinds of informal fallacies.

"begging the question," generally considered a fatal problem for an argument. That conditional is legitimate in form, and is generally a theorem, but it is fatal if relied on to make a substantive point or demonstrate a claim. It will only happen inadvertently ― in which case, a good-faith discussant will admit their error ― or with an intent to mislead by sophistry. — Srap Tasmaner

Question begging happens a lot. But, again, I can't think of an instance in public discourse in which the speaker appealed to P -> P as a tautology in order to convince anyone about anything.

As to complaints about formal logic, people who don't know about the subject often miss the point about such things as axioms proving themselves. In formal logic it is completely open that that one doesn't assert that a non-logical P holds in and of itself without respect to being either an axiom or derivable from the axioms. That is, when we say "Put P on the line as P is an axiom", we don't hide what we are doing. It's a kind of "benign" question begging. On the other hand, informal question begging is malign when, as is usual, it tries to hide the nature of the inference.

Fundamentally, all we're talking about in this case is arguing from a set a premises which are inconsistent — Srap Tasmaner

It is such a case. It is also a case of modus ponens. I've mentioned why I pointed that out.

necessarily inconsistent. — Srap Tasmaner

I don't know what that means.

But people arguing from inconsistent premises often make inferences that, while in themselves correct, continue to hide their inconsistency. — Srap Tasmaner

That is very very common in public discourse. Maybe it's the norm! It's maddening, frightening and ultimately depressing.

PS. 'vacuous' is more informative for the conditional than 'trivial', since 'vacuous' is specfic while 'trivial' is not. That is, vacuousness is a certain kind of triviality.

Another twist. A conditional may turn out to be vacuously true, but it might be quite nontrivial to prove that the antecedent is false.

In the only interpretation where both premises are true, there's no way to conclude not-A — frank

A -> ~A
A
therefore ~A

There is no interpretation in which both the premises are true.

I don't know why you continue to ignore the definitions and explanations given you.