No. It doesn't say that Q being true depends on P being true. Q can be true whether P is true or false. — Michael

But in this case, they're the same variable. They're both A.

The logic is explained in that link I posted. — Michael

I read it, thanks. It just looks like that if the A in the antecedent is false, the A in the consequent should be false too. I think you were only making the antecedent A false.

That's cool. But if the antecedent is negated, why wouldn't it be:

¬A ∨ ¬(¬A)

In other words, why wouldn't you negate both A's?

A → ¬A does not mean A ∧ ¬A. It means ¬A ∨ ¬A — Michael

How do you figure that?

Any argument with inconsistent premises is valid, according to Tones — Leontiskos

Inconsistent? If you look at the argument in the OP, there can never be a case where both premises are true. According to the definition of validity in the SEP article on propositional logic, the argument in the OP is valid. It's odd at first glance.

If there is no assignment in which all the premises are true, then the argument is valid.

That is very different from what you mentioned. — TonesInDeepFreeze

Yes, I understand.

Just because they could be wrong doesn't mean they are wrong.

If climate change is not making the weather worse and adverse here, what is the main cause then — javi2541997

We just went through an El Nino phase. Don't conclude climate change just because you can't think of anything else. Scientists use super computers to model the climate.

All you had to do is read the replies given you. — TonesInDeepFreeze

Yea, well...

You've been busy... — Banno

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

You can retire to the blazes. — TonesInDeepFreeze

Sounds uncomfortable.

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

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.

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.

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.

"Trivial" has a clear meaning in analytical philosophy.

Sometimes it's A and sometimes it's not A. — Hanover

Correct

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.

Sure. I would encourage you to write out in English the only case where both premises are true, and see if you think not-A makes sense as the conclusion. If it does, great. Bon Voyage.

Assuming all premises in the OP true, the conclusion of not A is shown to be false because a valid conclusion of A was shown. — Hanover

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

I think you need to know what "trivially true" means.

Do you intend for this to be a Socratic interview? — TonesInDeepFreeze

No. It's that if A is false, the first premise is trivially true.

So the argument is one in which the first premise doesn't say anything. The argument would be:

1. Trivial truth
2. A.

Conclusion: not-A.

That's not valid.

It's up to you whether you want to say it is trivially true. 'trivially true' is not a formal notion. — TonesInDeepFreeze

If I gave you a quote from a respected authority advising that if the antecedent of a conditional is false, the conditional is trivially true, would you believe it?

The term 'vacuously true' is used that way. — TonesInDeepFreeze

If the antecedent is false, the conditional is trivially true, right?

If, in an interpretation, the antecedent is false, then, in that interpretation, the conditional is true. — TonesInDeepFreeze

Vacuously true. Trivially true. Correct?

Ok. If the antecedent of a conditional is false, the conditional is vacuously true. Right?

If the hypothetical of a conditional is false, the conditional is trivially true. Is this correct?

I suspect you don't know what is meant by 'interpretation'. — TonesInDeepFreeze

I get that you're frustrated. Thanks for hanging in there. If the hypothetical in the first premise is false, isn't the first premise trivially true? It doesn't say anything in that case.

I didn't change any premises. And I didn't make anything true or false. I merely pointed out that A -> ~A is true in the interpretation in which A is false. — TonesInDeepFreeze

Ok. So with a false premise, the conditional is true by default.

That means the first premise is actually not-A, right?

Wait, no, the first premise doesn't say anything at all if A is false. It's trivially true.

So the conclusion to the argument should be the 2nd premise. It should be A.

I think you did swap out the first premise when you made the first A false, but not the second one. Is that wrong?

Wait a minute. If A is false, then the first premise is:

If not-A, then not (not-A)

You can't change one of the A's to false and not the other one. If A is false, they both have to be false.

The conclusion always comes out as not-A. Tones is basically swapping the first premise out with a different one by considering an "interpretation" where A is false.

In other words, you can interpret that cows can bark if you want to.

Yea, you're right.

"A conditional statement is false if hypothesis is true and the conclusion is false.".

here

And if A is true, we can't have not-A as the conclusion, so the conditional in premise 1 is false.

How would you be warranted to examine what happens when A is false?

I think Hanover was talking about the argument in the OP. It can't be valid because the first premise is necessarily false, right?

Isn't the first premise: If A, then not-A? That's what it looks like

(1) The first premise in that argument is not necessarily false. — TonesInDeepFreeze

Is it not? It's expressing a contradiction. Contradictions are necessarily false, right?

I think the first premise is necessarily false in propositional logic.

So we know the first premise is necessarily false. That means the conclusion has to be false for validity. Is the conclusion false?

If a premise is necessarily false, then the argument is valid. — TonesInDeepFreeze

But with validity, aren't we looking at what happens when all the premises are true? If a premise is necessarily false, can we still look at the argument in terms of validity?