Per the definition of "valid":

An Argument is valid if and only if it would be contradictory for the conclusion to be false if all of the premises are true.[
— Hanover

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

The argument:

A -> ~A
A
therefore ~A
valid

Another argument:

A -> ~A
A
therefore A
valid

The fact that the premises are inconsistent doesn't vitiate that the argument is valid. Actually the fact that the premises are inconsistent entails that the argument is valid.

(2) A conclusion itself is valid if and only if it is true in all interpretations. An argument is valid if and only if there are no interpretations in which the premises are all true and the conclusion is false.

think you need to know what "trivially true" means. — frank

You think incorrectly.

I well know how logicians and mathematicians use the verbiage 'trivially true' and 'follows trivially' for statements and arguments respectively. Usually it means that the statement is very obviously true or the inference is very obviously valid. In logic and mathematics it doesn't mean that the statement or the argument is otiose (though, sometimes it does mean that the vacuous case needn't be considered since the generalization being proved is not claimed to include the vacuous case).

But 'true' has formal import while 'trivial' does not. There's nothing wrong with saying 'trivially true' or 'follows trivially' but it doesn't count toward evaluation of an argument.

That is, there is no definition:

P is trivially true if and only if [definens]

It's a personal choice for any author whether to use 'trivial' or not as part of the informal prose in which mathematics is often written.

And if we choose to say that "condition is true if the antecedent as false" is trivial, then we can say that about all the connectives:

It's trivially true that a conjunction is true if both conjuncts are true.

It's trivially true that a disjunction is true if at least one of the disjuncts is true.

It's trivially true that a negation is true if it is the negation of a false statement.

It's trivially true that a biconditional is true if either the left and right are both true or both false.

/

Moreover, see my previous post that explains in detail the fallaciousness and irrelevance of your argument.

So the argument is one in which the first premise doesn't say anything. — frank

A -> ~A
says
If A then it is not the case that A

That is equivalent with
it is not the case that A

In an interpretation in which A is false, "it is not the case that A" is true; in an interpretation in which A is true, "it is not the case that A" is false.

1. Trivial truth
2. A.

Conclusion: not-A.

That's not valid. — frank

Yes, that that is not a valid argument. But when you replace A -> ~A by "trivial truth" the new argument is not equivalent with the original.

(1) A -> ~A is not a truth simpliciter nor a falsehood simpliciter. As has been explained to you about 15 times today, it is true in an interpretation in which A is false, and it is false in an interpretation in which A is true.

(2) What you may say is trivial is not the truth of A -> ~A, since it is true or false depending on the interpretation of A, but rather that A -> ~A is true in any interpretation in which A is false, which, if you like, you may choose to call trivial.

(3) 'trivially true' is a term logicians and mathematicians usually use to describe a statement such that the truth of the statement is very obvious. It's not a technical term in a context such as this. So the judgement that a particular statement is trivially true does not count toward the validity of an argument. Validity is determined by the set of all interpretations, no matter whether one considered certain of the statements to be trivially true or not.

(4) A -> ~A has a deductive role in the argument. It's not the same argument if you take it out and replace it with "trivial truth".

(5) We actually do have a defined* constant called 't' such that t is interpreted as true in all interpretations (though 'trivial' is not part of this). In that case there is this argument:

t
A
therefore ~A

That argument is invalid.

Indeed, adding t as a premise to any argument has no effect on validity:

Let G be any set of premises. Let P be a statement. Let the argument be

(5a) From G, infer P.

Now add t. So we have:

(b) From Gu{t}, infer P.

Then (5a) is valid if and only if (5b) is valid.

* Usually, the definition is of the form, for some formula P, P -> P.

(6) So, even if we supposed that the premise A -> ~A were true in every interpretation, we can't just replace it with "trivial truth". That is, if we suppose that A -> ~A has as some property (such as being true, or trivially true, or having four symbols, or being a conditional, or having only one sentence letter, etc.) you can't just replace the formula with a mention of an adjective that applies to it. That would be blatantly fallacious.

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. — frank

That is wrong. You are plainly misusing the terminology.

Considering different interpretations doesn't change formulas.

And I'm not making "one A" false and not "the other A".

There is no sense of "another A" and thus no interpretation in which A is false but "another A" is true.

Again, you need to know what 'interpretation' means.

Do you intend for this to be a Socratic interview?

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

The term 'vacuously true' is used that way.

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 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.

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.

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.

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'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.

Yes, so?

Thank you.

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.

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.

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.

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.

~A is a negation but it is not a contradiction.

But wait, which argument are we talking about?

(1)
~A
A
therefore A & ~A

(2)
~A
A
therefore ~(A & ~A)

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

In all three case, no premise is itself a contradiction. But in all three cases, the set of both premises together is contradictory (is inconsistent).

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

The first premise is:

~A

That is not necessarily false.

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

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

(2) I don't know what 'conclusion is false for validity' means.

(3) The conclusion is true in some interpretations and false in others.

You seem not to grasp the meanings, in the context of ordinary formal logic, of 'true', 'false', 'valid' and 'invalid'.

I think [@Hanover is] treating A -> ~A as if it's hypothetically true. They're just declaring it to be necessarily false. — frank

No, he's claiming that A -> ~A is necessarily false, and we are pointing out that it is true when A is false, so it is not necessarily false.

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

We are not restricted to looking only at the interpretations in which all of the premises are true.

If there is no interpretation in which all of the premises are true and the conclusion is false, then the argument is valid. If there is an interpretation in which all of the premises are true and the conclusion is false, then the argument is invalid.

Is it a problem that "not-(A and not-A)" is also a valid conclusion of the argument? — NotAristotle

'valid' has three senses:

(1) an argument is valid if and only if there are no interpretations in which all of the premises are true and the conclusion is false

(2) a formula is valid if and only if there is no interpretation and assignment for the free variables in which all of the premises are satisfied and the conclusion is not satisfied

(3) a sentence is valid if and only if there is no interpretation in which all of the premises are true and the conclusion is false

(3) reduces to a special case of (2).

~A
A
therefore A & ~A
valid

~A
A
therefore ~(A & ~A)

Of course, we recognize that that is problematic to many people regarding everyday reasoning and, more pertinently here, in different philosophical points of view.

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

I was looking at the argument schema presented in the OP. If you imagine this as the formal representation of a substantive argument, you would have to have serious doubts about what was going on in that argument. This was the "veneer" of logic I was talking about. Any argument that could be formalized in the schema presented would instantiate an accepted form in a deeply questionable way. Hence "sophistry". That wasn't intended to refer to you, to your explanations, to anyone in this thread, but to a hypothetical argument that would fit the schema under discussion. — Srap Tasmaner

I appreciate that you say that now.

Yet:

But what do you mean by 'abusive'?
— TonesInDeepFreeze

The basic idea is "formally correct but misleading". Akin to sophistry. Or to non-cooperative implicature, like saying "Everyone on the boat is okay" when it's only true because no one is left on the boat and all the dead and injured are in the water.

In this case, for instance, it is suggested that we conclude ~A by modus ponens. — Srap Tasmaner

I was the one who remarked that the argument whose conclusion is ~A is modus ponens. I'll take your word for it that you didn't mean that my remark was non-cooperative and abusive, but I don't see how it would not be natural to take you as first claiming that my remarks were non-cooperative and abusive.

Instead you could have first said what you say now: That some hypothetical argument, one not given in this thread, is abusive.

Also:

we get to ~A by noting that A→~A is materially equivalent to ~A v ~A. Now what kind of disjunction is that? It's a well-formed-formula ― no one can deny that ― but it's hardly what we usually have in mind as a disjunction. It's "heads I win, tails you lose." That's abusive.

There is, in this case, a veneer of logic over what could scarcely be considered rational argumentation. If this appearance of rationality serves any purpose, it must be to mislead, hence abusive, eristic, sophistical, non-cooperative. — Srap Tasmaner

The disjunction argument was given by another poster in this thread. I'll take your word for it that you didn't mean that his remarks were abusive, eristic, sophistical and non-cooperative, but I don't know how it would not be natural to first take you as claiming that his argument is abusive, eristic, sophistical and non-cooperative.

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

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

That is not a valid argument and it is not modus ponens.

2.
A->~A
~A
Therefore ~ A (2) — Hanover

in #1 A is true, but in #2 A is false — Hanover

The arguments themselves don't declare the truth or falsehood of A.

Test #2 for validity (which is really just a clearer restatement of #1 — Hanover

#2 is not a restatement of #1.

Premise #1 is logically equivalent to ~A. — Hanover

A -> ~A is equivalent with ~A.

~A
A
Therefore A.

Therefore ~A is also true.

This is not a valid argument. — Hanover

Strictly speaking an argument has only one conclusion. You have two conclusions there. But we can use conjunction:

~A
A
therefore A & ~A

That is a valid argument.

1

Tones think(s) logic is arbitrary symbol manipulation — Leontiskos

I have never believed that logic is arbitrary symbol manipulation. I have never posted that logic is arbitrary symbol manipulation. I have never posted anything that implies that I believe that logic is arbitrary symbol manipulation.

I've said before, and hopefully won't need to say again:

When questions about formal logic come up, it is appropriate to check that the claims about formal logic are correct and coherent. And if one wishes to regard the questions in a context other than formal logic or other than the ordinary versions of formal logic, then of course that is fine, but it helps if the person makes clear that they are not mixing contexts, or if they are comparing contexts or even critiquing the ordinary versions of formal logic, then at least they should not misstate, misrepresent or mangle what actually happens in the formal logic, which is abusive.

A --> B = ~ A v B.
A --> ~A = ~A v ~A
~A v ~ A = ~A — Hanover

Then you'd argue incorrectly
— TonesInDeepFreeze

This is where we disgree.

A --> ~A <> A --> ~ B because A-->~A = ~A, yet A-->~B <> ~A. — Hanover

All purely symbolic. Get outta here with this [paraphrase:] "Oh but the poor boy was just trying to get in a bit of common sense everyday logic; not about formal logic" stuff.

There is, in this case, a veneer of logic over what could scarcely be considered rational argumentation. If this appearance of rationality serves any purpose, it must be to mislead, hence abusive, eristic, sophistical, non-cooperative.

Heh, describes how these threads normally go. — Count Timothy von Icarus

Exactly. As cranks and people ignorant of the subject inject their confusions and ignorance, harbored in such forums on the Internet.

But for the everyday use of logic just to schematize and clarify arguments, you get a lot more mileage out of de Morgan's laws, contrapositives, a solid understanding of quantifiers, and such. — Srap Tasmaner

Actually, I have personally gained a lot from study of mathematical logic, in use outside of mathematics, in practical applications, in organizing ideas, and in appreciation of rigor and clarity in practical applications.

If one happens to be interested in the subject of symbolic logic, then it is an eminently natural and wise question to ask: Does this symbolic calculus prove all and only the valid formulas? I wondered about that and had no idea that mathematicians had answered the question with proofs about it. When I saw the proofs, I found the intellectual curiosity acted on, the intellectual honesty and the intellectual creativity to be tremendously inspiring. If you don't, then so be it, but also, so what.

/

Some mathematicians have a solid understanding of quantification naturally without studying symbolic logic. But personally, I found that studying symbolic logic brought me up to speed with what is natural for others.

And, for example, consider the most economically important question in mathematics: P v NP with its economic implications thus its million dollar prize. Work on that presupposes mathematical logic. If one is not interested, then fine, but what is the point then in carping about it?

But what do you mean by 'abusive'?
— TonesInDeepFreeze

The basic idea is "formally correct but misleading or akin to sophistry". Akin to sophistry. — Srap Tasmaner

You used the pronoun 'it' right after the argument of the first post was mentioned. So I didn't know whether you meant 'it' to refer to indeed its antecedent. Now I see that you didn't.

Anyway, the counterargument to the original argument is not misleading or like sophistry.

People worry over the sense in which the conclusion of a deductive argument is "contained" in the premises ― here it is one of the premises. — Srap Tasmaner

It is equivalent with one of the premises.

The original post asked about a symbolic argument. The matter was not whether such things are useful in your life. At least speaking for myself, I have no interest in convincing you that you should regard such matters as useful to you. But that does not vitiate my interest in technical matters in formal logic, even if for the mere mathematical/intellectual pleasure, and especially as that mathematical/intellectual pleasure is attached to a field of study that provides core context for the mathematics for the sciences, engineering, and computing, and especially since I also enjoy philosophical aspects of the subject and having a solid technical founding greatly contributes to both appreciation and understanding of the philosophical discussions.

In what sense is the relationship of A and ~A revealed or clarified? It may be modus ponens in form, but hardly in spirit — Srap Tasmaner

The argument was:

A -> ~A
A
therefore ~A

There was discussion that this is an example of aspects of negation and contradiction. My point was that we can see that the argument is valid in another, quite immediate way, viz. that it is staring us right in the face as modus ponens.

I am not declaring any profundity. It's just a correct observation, with a nice reminder, and pertinent to the context of the discussion.

And if we step back and look at the offending premise, we get to ~A by noting that A→~A is materially equivalent to ~A v ~A. Now what kind of disjunction is that? It's a well-formed-formula ― no one can deny that ― but it's hardly what we usually have in mind as a disjunction. It's "heads I win, tails you lose." That's abusive. — Srap Tasmaner

Another poster took that path. It's correct and pertinent too to the formal matter at hand. To call it "abusive" is to abuse the word 'abusive'.

There is, in this case, a veneer of logic over what could scarcely be considered rational argumentation. If this appearance of rationality serves any purpose, it must be to mislead, hence abusive, eristic, sophistical, non-cooperative. ― Again, I am only talking about how logic is used as an aid to ordinary philosophizing, not what people get up to in a logic lab. — Srap Tasmaner

The purpose is to be clear about the formal logic. That is a good purpose. One my wish not to talk about formal logic, or to disagree with it, but it is a good purpose to at least be correct and clear about what does happen in formal logic, especially when the first post regards a symbolic argument.

And "logic lab". The original post regarded a symbolic argument. So what, you gonna sue a bunch of people for taking it in that very context? And, by the way, "the logic lab" is part of philosophy too, as formal logic and symbolic logic are regular undergraduate philosophy courses, and sometimes even set theory and mathematical logic too.

And it is not a "veneer".

You might choose to heap on more opprobrium against posters who are interested in the subject enough to strive to be correct about it, but you've not at all successfully impeached them, not even touched them.

3 follows from 1 and 2 by modus ponens.
— TonesInDeepFreeze

1 means "If A is true, A is false." This means A can never be true, despite it being true. It's a walking contradiction. This in itself can be taken to mean A is false because, as noted A -> ~A is logically equivalent to ~A or ~A as a disjunction of the conditional ( A --> B = ~A or B). 1 therefore means ~A.

This can be reduced to:

1. ~ A
2. A
Therefore ~A.

The conclusion is a restatement of #1. 2 is a contradiction of 1.. — Hanover

There the poster is (in horrible confusion) discussing truth values per formulas in a context of formal logic (a response to my own mention of modus ponens). Whatever is in the slop bucket of his mind about everyday reasoning, he's just plainly confused and wrong about the formal logic at hand.

Well, if you've been taught that a contradiction has a truth table that is always false and you think you have identified something that is necessarily/always false, it seems possible to conflate the two. — Count Timothy von Icarus

Only because the poster refuses not to conflate.

mental buckets — Count Timothy von Icarus

'buckets' is a good word choice.

In his first post:

1. A -> not-A
2. A
Therefore,
3. not-A.
— NotAristotle

#1 is a contradiction, reducible to ~ A or ~A. — Hanover

That misconception is not explained merely as a contrast with everyday reasoning. I have never heard everyday reasoning say ""~ A or ~A" is a contradiction".

Moreover the poster is using symbolizations that at least suggest a formal, not everyday, context.

But I can't say what is the source of the mental block in people who don't understand that "A -> ~A with A is contradictory

That cuts a crucial part of the sentence:

But I can't say what is the source of the mental block in people who don't understand that "A -> ~A with A is contradictory" doesn't entail "A -> ~A is contradictory" — TonesInDeepFreeze

Of course I understand that many people refer to everyday senses. My point though is that after it is made clear that the context is formal logic, those people persist to incorrectly declare what is the case in formal logic.

Also, I don't say "from a falsehood anything follows". It is better to say, "from a contradiction anything follows". And we don't need that principle in this case anyway, but only the more special instance: from a conditional whose consequent is the negation of the antecedent, that consequent follows.

Modus ponendo ponens is the principle that, if a conditional holds and also its antecedent, then its consequent holds." (Beginning Logic - Lemmon)

Perhaps your argument is based on taking that to mean this?:

If a conditional holds and also its antecedent, then modus ponedo ponens is the principle that then its consequent holds.
— TonesInDeepFreeze

Be good enough to make clear the difference between these two. — tim wood

The first states modus ponens as a principle not an argument form. But as an argument form it is such that the premises are a conditional and its antecedent, and the conclusion is the consequent of the conditional, so that if both the conditional and its antecedent are true then the consequent of the conditional is true.

The second says that if a conditional and its antecedent are true then the argument is modus ponens if the premises are the conditional and its antecedent, and the conclusion is the consequent of the conditional.

The first is a correct definition of 'modus ponens'. The second is not a correct definition of 'modus ponens'.

The first does not require that the conditional and its antecedent are true; only that IF they are true then the consequent of the conditional is true.

The second allows us to take an argument as an instance of modus ponens if and only if the premises are a conditional and its antecedent, and the conclusion is the consequent of the conditional, and the conditional and its antecedent are true.

These also are not correct definitions of modus ponens:

A modus ponens argument is one in which the premises are a true conditional and its true antecedent, and in which the conclusion is the consequent of the conditional.

That's wrong, since modus ponens does not require that the premises are true.

A modus ponens argument is one in which the premises are a conditional and its antecedent, and they are not together contradictory, and the conclusion is the consequent of the conditional.

That's wrong, since modus ponens does not require that the premises are consistent.

Also famously:

Life then Death and Taxes.
Life.
therefore Death and Taxes.