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

I see, the earlier argument.

See my edit that I composed while you were posting.

A -> ~A is not contradictory.

A formula is contradictory if and only if the formula proves a contradiction. A contradiction is a formula of the form P & ~ P (or, sometimes we say, a pair of formulas of the form {P ~P}).

Using the propositional calculus, you cannot derive a contradiction from A -> ~A.

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

I don't see how it would not be natural to take you as first claiming that my remarks were non-cooperative and abusive. — TonesInDeepFreeze

Agreed, a natural reading, but my target was really someone who might present an argument in the OP's schema, as a perfectly respectable modus ponens. It's MP alright, but it's a degenerate case.

Similarly for the disjunction. My point was that disjunctions that amount to "heads I win, tails you lose" are disjunctive in form only, and we expect something more substantive.

In short, no attack was intended on you or any other poster, but only on the illicit use someone might make of legitimate argument form. To the extent that I was offering criticism, it was to say that we are not helpless when confronted with correct inference in form only, and can choose to block such deviant uses if we like.

And we can leave formal logic alone, as a study in its own right, but not import it wholesale when all we really need is the convenience of schematizing arguments.

A related example would be various attempts to deal with what many people find counterintuitive about the material conditional. There are several ways to block troublesome cases.

Hence my casual suggestion that we have very little practical use for "If grass is green then grass is green" or "If grass is green then grass is not green."

No, not right. The first premise is not necessarily false.

It's been correctly pointed out over and over and over, by different posters in this thread, that

A -> ~A

is true when A is false.

Truth tables have even been adduced.

Please look at those truth tables.

It's MP alright, but it's a degenerate case. — Srap Tasmaner

That's okay for me, as long as I take 'degenerate' in a non-pejorative sense as often in mathematics.

And we can leave formal logic alone, as a study in its own right, but not import it wholesale when all we really need is the convenience of schematizing arguments. — Srap Tasmaner

Of course, formal logic, or at least a particular formal logic, does not always apply in everyday and even in all philosophical contexts.

Hence my casual suggestion that we have very little practical use for "If grass is green then grass is green" or "If grass is green then grass is not green." — Srap Tasmaner

I would need to dig up documentation, but, I tend to think that P -> P does have importance in Boolean logic used along the way in switching theory, computation, etc. Just for starters, we use the Boolean 1-place function whose value is always 'true' (or '1') and it is definable propositionally as P -> P. Logic is a vast field of study, including all kinds of formal and informal contexts. I would not so sweepingly declare certain formulations otiose merely because one is not personally aware of its uses.

"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?

How are you getting A as a conclusion? — frank

I might have mistyped at some point.

The OP:

1. A->~A
2. A
3. Therefore ~A (1,2 mp)

A cab also be concluded from the second premise.

A (2)

I can also continue from the conclusion:

4 ~A v A (3, disjunctive introduction)
5. ~ (~A) (2, double negation)
6. Therfore A.

All grass is green
All grass is not green
Cows can bark.

You need to remedy your misunderstandings of this. I suggest starting with the first chapter of a good textbook in formal logic.

Yes, a conditional is false in all and only those interpretations in which the antecedent is true and the consequent is false.

And, yes , if in an interpretation, A is true, then A -> ~A is false.

But if, in an interpretation, A is false, then A -> ~A is true.

And a sentence is necessarily false if and only if it is false in all interpretations.

But A -> ~A is not false in all interpretations. So A -> ~A is not necessarily false.

Please read each of those lines again now carefully. And look at the truth table.

Yea, you're right.

Thank you.

Yes, so?

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.

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.

I'm not swapping any premises, and I'm not making a reinterpretation.

Let G stand for "grass is green".

Let C stand for "cows bark".

G
~G
therefore C

There are no interpretations in which both the premises G and ~G are true. Perforce, there are no interpretations in which both the premises are true and the conclusion is false. So the argument is valid.

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

I didn't change any premises. And I didn't make anything true or false. And there is no "first A" and "second A". There is only one A. I merely pointed out that A -> ~A is true in the interpretation in which A is false.

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

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 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.

You're mixing up 'premise' and 'antecedent'.

If the antecedent is false then the conditional is true.

As to premises, let's not mixup two things, and which argument are you talking about now?

(1)
A -> ~A.

That's not an argument and it has no premises. It is a formula that is true in an interpretation in which A is false, and it is false in an interpretation in which A is true.

(2)
~A
A
therefore A & ~A

That's an argument. It is valid since there are no interpretations in which all the premises are true and the conclusion is false.

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

That's an argument. It is valid since there are no interpretations in which all the premises are true and the conclusion is false.

It's not a matter of frustration. Rather, since you want to know about this, my sincere helpful suggestion is for you to get a book that explains this stuff methodically, step by step, starting at page 1.

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

If the hypothetical in the first premise is false, isn't the first premise trivially true? It doesn't say anything in that case. — frank

Which argument?

A conditional sometimes called a 'hypothetical'. Sometimes the antecedent is called 'the hypothesis'.

To avoid confusion between 'hypothetical' and 'hypothesis' let's stick with this terminology:

A conditional has an antecedent and a consequent.

For example:

P -> Q

is a conditional in which P is the antecedent and Q is the consequent.

If Jack is good then Jack reads Faulkner

is a conditional in which "Jack is good" is the antecedent and "Jack reads Faulker" is the consequent.

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

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

In more lax formulation:

If the antecedent is true then the conditional is false.

But with that lax formulation, do not forget that it is still implicit that truth and falsehood are relative to interpretations. That is, look at the truth table.

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

Vacuously true. Trivially true. Correct?

The term 'vacuously true' is used that way.

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

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

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

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?