Yeah, but if you affirm that "death" is equivalent with "not-life," you'll be stuck affirming Plato's argument for the immortality of the soul in the Phaedo, which in turn implies that you may be reincarnated for innumerable lifetimes where you have to debate these same topics before finally achieving henosis and completing the process of exitus and reditus. That's a pretty rough commitment to have to make. — Count Timothy von Icarus

I'd have thought it is the opposite. If death is equivalent to not-life it means no afterlife.

opprobrium — TonesInDeepFreeze

I'll just say that no opprobrium was intended. I too have gained, I believe, from my study of logic and mathematics, and I have found formal methods intensely interesting, so neither am I disparaging that as an interest.

Some people are interested in formal logic full-stop. Some people are only interested in formal logic as a help meet to argumentation and analysis. For the latter, results about logic ― the completeness theorems and such ― are not only of less interest, but less everyday use. All I was saying.

Nor was I accusing you or the original poster, or anyone in this thread, of engaging in sophistry, if that's what you thought. Maybe it seemed to you I was engaging in the current controversy on the other side, but I was not. You are, of course, correct about the formal question.

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. ― You'll note that @TonesInDeepFreeze and I were trying to think of a genuine example of such an argument, and I'll now be getting back to that.

I hope my "position" is clearer now.

Were debating whether to call certain formulations "modus ponens." — Hanover

I figured this would be an interesting thread. This is the standard set piece where Banno and Tones think logic is arbitrary symbol manipulation and others think it has to do with correct reasoning, but this thread brings it out quickly.

For my money the question here is whether modus ponens is arbitrary or non-arbitrary. (Whether what is at stake is a mere matter of definition.)

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. — Srap Tasmaner

Yep. :up:

Thanks, but no thanks.
Thanks.
No thanks.

Common sense wise, yes, but Plato has Socrates make an argument that relies on notions contrariety and us accepting death as the polar opposite of life (e.g. as darkness is the absence of light).

A lot of scholars think this argument is meant to be bad and to have this hole in it (i.e. that death is not a straightforward negation of life). I tend to agree. Socrates chains several arguments and only one is really good (more or less the same argument used against forms of reductionism to this day), and then suggestively breaks into an interlude where he tells his interlocutors that they shouldn't give up on reason if they happen to discover that arguments they once embraced turn out to be bad ones. I honestly don't love the philosophy of the dialogue as far as Plato goes, but the execution is brilliant.

"Argument is valid if and only if it would be contradictory for the conclusion to be false if all of the premises are true.[3] Validity does not require the truth of the premises, instead it merely necessitates that conclusion follows from the premises without violating the correctness of the logical form. If also the premises of a valid argument are proven true, this is said to be sound.[3]

https://en.m.wikipedia.org/wiki/Validity_(logic)#:~:text=An%20argument%20is%20valid%20if%20and%20only%20if%20it%20would,correctness%20of%20the%20logical%20form.

From the same wiki article:

"A Formula of a formal language is a valid formula if and only if it is true under every possible interpretation of the language. In propositional logic, they are tautologies."

So:

1.

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

But

2.

A->~A
~A
Therefore ~ A (2)

Test 1 for validity: It is valid if it would "be contradictory for the conclusion to be false if all of the premises are true."

So, #1, could A be false if the premises true? Yes, see #2. Same premises, yet in #1 A is true, but in #2 A is false.

Test #2 for validity (which is really just a clearer restatement of #1): "A formula of a formal language is a valid formula if and only if it is true under every possible interpretation of the language."

Note "every possible interpretation of langauge."

Premise #1 is logically equivalent to ~A. That is, a possible interpretation of this syllogism:

~A
A
Therefore A.

Therefore ~A is also true.

This is not a valid argument.

It is an interesting problem to me because according to this website -- https://www.umsu.de/trees/#(A~1~3A)~5(A~1~3A) -- the argument is apparently valid. Even though you and I can plainly see that such an argument can never be true. It is an obviously bad argument in a way that:

not (P->Q)
Therefore, not-P

is not an obviously bad argument (bad argument though it is).

I would also note that the argument "A->not-A, Therefore not-A", though it is apparently a valid argument, does not make much sense in natural language; it would be like saying "if it is raining then it is not raining." Maybe someone could infer from that statement that it is not raining, but the statement seems more like a contradiction then a "valid" logical statement.

I think you're treating A -> ~A as if it's hypothetically true. They're just declaring it to be necessarily false.

I now wonder which way of writing the truth table for MP is correct, 1 or 2?

1. ((p -> q) ^ p) -> q
or
2. ((p -> q) ^ p) ^ q

Makes a difference!

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.

1

Can anyone think of an invalid argument where the conclusion does not contradict one of the premises?

think you're treating A -> ~A as if it's hypothetically true. They're just declaring it to be necessarily false. — frank

No, I get the distinction between a deductive conditional premise, and a linguustic counterfactual. I'm just engaging in the pedantry of determining whether the OP satisfies a hyper analyzed definition of "valid."

As @Banno notes, validity is determined by asking if the conclusion flows from the premises, and so he argues under mp, it does, so it is valid.

The wiki cite adds criteria, namely (1) that the negation of the conclusion cannot also flow from the premises for validity and (2) the premises under any formulation must also reach the same conclusion.

The OP falls under those criteria because: (1) both A and ~A can be derived from the premises, and (2) when Premise 1 is changed from a conditional format to a disjunctive one, it reduces simply to ~A, clearly contradicting the second premise A, and further violating criterion 1 that prohibits the negation and assertion to consistently flow from the same premises.

This is to say, if I were reviewing a contract, and it said "you get $1,000,000 if the OP is a valid syllogism," I'm saying no if I'm the guy who has to pay. Does the other side have a colorable argument? Maybe, but it must argue validity despite contradiction and accept the absudity that follows.

I do think @Benkei's comment regarding the necessity of acknowledging the LNC as foundational is correct.

Nevermind, "A and B Therefore C" would be an invalid argument where the conclusion does not contradict the premises.

(1) both A and ~A can be derived from the premises, — Hanover

How are you getting A as a conclusion?

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.

Is it a problem that "not-(A and not-A)" is also a valid conclusion of the argument? According to the definition proffered by Hanover, it would seem to be a problem given that "the negation of the conclusion flows from the premises."

Can you provide a citation for that criterion of validity? I did not find it in the wiki article.

Also, Hanover, thanks for articulating an argument against validity as I was not sure how to do so.

But what if you say the first premise is necessarily false? It can't be true. Then what do you get?

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.

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

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?

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.

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.

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

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.

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

The first premise is:

~A

That is not necessarily false.

(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?

~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).