Sure. If a statement is trivially true, it's not informative. For instance, a tautology is trivially true. The T-sentence rule is trivially true.

Under what definition of "valid" is the argument in the OP valid? I'm not being Socratic, I'm just asking.

an actual example — TonesInDeepFreeze

I agree with all that. The toy examples we're dealing with here are too transparent for anyone to get away with much.

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.

Yes, I do see the issue of identity versus inference, but that is solved by superfluous logical machinations and becomes a pedantic exercise to maintain the distinction between "identity" and "inference."

For example:

P1. A -> ~A
P2. . A

1. ~A (1,2 m.p.)
2. ~ A v A ( 1 and disjunctive introduction)
Therefore:. A (P2, 2 negation of ~ A)

Note I've not just reasserted P2 in my conclusion, but I've logically deduced that since not A could not be true based upon A being a given premise, by elimination, A must be true.

I'm sure there are more convoluted ways to go about it, but does that satisfy your objection?

It may seem bizarre that a valid argument could have at least one premise that is necessarily false at first glance, but I think it is fairly intuitive if one thinks in terms of truth-preservation. If the premises were true, it would preserve truth. But the "truth" of a false premise cannot be preserved.

And it's a good thing that it is valid because we often can reason from necessarily false conclusions in valid arguments to identifying false premises. This example is simple, but sometimes self-refutation is not simple.

I'm not sure what post you are responding to — Leontiskos

None, or .

Just trying to think of real world examples of a formula like "A → ~A", likely dressed up enough to be hard to spot. Excluding reductio, where the intent is to derive this form. What I want is an example where this conditional is actually false, but is relied upon as a sneaky way of just asserting ~A.

I suppose accusations of hypocrisy are nearby. "Your anti-racism is itself a form of racism." "Your anti-capitalism materially benefits you." "Your piety is actually vanity." Generalize those and instead of saying, hey here's a case where the claim is A but it's really ~A, you say, every A turns out to be ~A. Now it's a rule.

Still thinking.

The move is always to a meta-level. What is the game? What is the competition? What is logic? Our world has a remarkable tendency to try to avoid those questions altogether, usually for despair of finding an answer. — Leontiskos

With good reason, as you well know.

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

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.

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

Back to my truth by negation maneuver then.

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 because I would have shown F is T.

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.

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.

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.

Sorry Timothy your point there escapes me. I can't see how death, if we accept there is no afterlife, is not the negation of life from any point of view, commonsense or otherwise.

I'll need some more explanation.

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.

I'm sure there are more convoluted ways to go about it, but does that satisfy your objection? — Hanover

Your "disjunctive syllogism" is different than my A→A, so in that sense, sure. You are effectively saying that A flows or follows from the contradiction, not from itself.

So a second objection would be that nothing flows or follows from a contradiction (which is the flip side of saying that everything flows or follows from a contradiction).

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.

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

Oh. So then any argument that has no true premises is valid. That's weird.

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.

Just trying to think of real world examples of a formula like "A → ~A", likely dressed up enough to be hard to spot. Excluding reductio, where the intent is to derive this form. What I want is an example where this conditional is actually false, but is relied upon as a sneaky way of just asserting ~A.

I suppose accusations of hypocrisy are nearby. "Your anti-racism is itself a form of racism." "Your anti-capitalism materially benefits you." "Your piety is actually vanity." — Srap Tasmaner

Isn't that reductio?

I would say that, like argument, contradiction also requires a kind of middle term, and is therefore never direct. For example:

A→B
B→~A
A
∴ B — Leontiskos

People can only make this inference because they do not see that they are being inconsistent. When there is neglect we hold them responsible for the mistake.

So A→~A is never a self-conscious premise.

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.

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

Yes. I edited that post. It's just weird that any argument that can't have all true premises is going to be valid.

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

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

What you've done is imported the artificial truth-functionality of the material conditional into the consequence relation itself. You have contradicted 's "flows from." You are effectively saying, <Any "argument" with nonsense premises is "valid.">

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

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

You can retire to the blazes.

You can retire to the blazes. — TonesInDeepFreeze

Sounds uncomfortable.

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.

Yes. I edited that post. It's just weird that any argument that can't have all true premises is going to be valid. — frank

We could say with that if the conclusion flows from the premises then the argument is valid.

1. P→Q
2. P
3. ∴ Q

4. A→~A
5. A
6. ∴ B

Now one could say that (3) flows from (1) and (2); and that (6) flows from (4) and (5). But this latter use of "flows from" is very different from the former. 's contention that they are the same use is not "merely applying the definitions of ordinary formal logic."

As I said, in this particular regard, I'm merely applying the definitions of ordinary formal logic. — TonesInDeepFreeze

Ordinary formal logic does not define the consequence relation as identical to the material conditional.

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.

I think the error is yours though. Truth tables say nothing about the Truth of the statement but are about consistency of logical systems. Yes, there are different formal representations and truth tables but that's more about convention. With a false antecedent, the consequent is vacuously true. But that situation, where the antecedent is denied, is irrelevant because the second premise assumes A to be true. And it necessarily follows from the first premise that not-A is simultaneously true. This is self-contradictory and violates the LNC.