If the OP uses propositional logic, it doesn't use propositional logic.
It uses prepositional logic
Therefore it doesn't use propositional logic.

MP has spoken. It doesn't use prepositional logic — Hanover

No. Because the premise "If the OP uses propositional logic, it doesn't use propositional logic" is false.

At least read the first chapter of a book on the subject.

Yes, and you are wrong that

A -> ~A

is a contradictory.

But (A->~A) & A is a contradiction.

If you assert A->~A, and then go on to assert A, then you have contradicted yourself. — Srap Tasmaner

Right. But there is a person here who claims that A -> ~A is necessarily false. That is what is being addressed.

Also, the person claims that if the premises are contradictory, then it is not an instance of modus ponens. That also is being addressed.

A->A, which is either useless — Srap Tasmaner

As I recall, that is a needed theorem along the path to proving the important sentential theorems from certain axiomatizations. Indeed, it must be a theorem for the sentential calculus to be complete.

it seems to me letting mathematical logic have the last word is the tail wagging the dog. — Srap Tasmaner

Whatever the merits of that view, the most recent discussion is not how logic should be set up but as to what is the case with the way ordinary propositional logic is set up.

True, but if, pace Frege, we assume assertoric force, then to claim "A → ~A" along with "A" is to contradict oneself, and therein I think lies the confusion.

Well for a start you would no longer be dealing with a complete version of propositional calculus... — Banno

1. Meaning what exactly?
2. Is the answer to (1) something I should care about?

Do you, Srap, agree that the argument in the OP is valid? — Banno

I don't really care. It's abusive.

I cannot think of a way to frame this as a real example — Count Timothy von Icarus

I'll come up with one. I think you see it around the forum and elsewhere in the wild pretty regularly. Informally, we're looking for a case where you try to disagree with me, but that attempt misfires because in stating your position you have to tacitly agree with me.

So you claim A, but I show that still leads, by some chain of reasoning or argument to my claim ~A. Introducing A⊃~A is just icing on the cake, because it's still just an extravagant way of saying ~A.

I get that MP requires a "->", but (A->~A)<->~A, so I'm puzzled by insisting on this nicety. In classical logic it's materially equivalent to the disjunctive syllogism, isn't it? — Srap Tasmaner

I don't know what question you are raising.

In ordinary formal logic:

(A -> ~A) <-> ~A
(A -> ~A) <-> ~A v ~A
(~A v ~A) <-> ~A
etc.

A -> ~A
A
therefore, ~A

is an instance of modus ponens.

Whether that should be the logic for certain reasoning is another question. But one can't properly address that question if one doesn't at least understand how that logic does operate and not have misconceptions about it.

1. If Hanover is correct, Hanover is not correct
2. Hanover is correct
3. Hanover is not correct (1,2 mp)

4. Hanover is not correct or 3 is an invalid conclusion derived from mp.(3, introduction) — Hanover

No, you need to know the difference between truth and validity.

That difference has been explained in this forum at least a hundred times. It is fundamental to formal logic.

Or read a logic book.

1. Meaning what exactly? — Srap Tasmaner

Meaning that it wouldn't be the case that all tautologies are theorems.

2. Is the answer to (1) something I should care about? — Srap Tasmaner

If you are interested in the basics of ordinary formal logic, then it would be a question that would naturally occur to you. But I don't see why you couldn't study other branches of philosophy without understanding the completeness of the propositional calculus.

Do you, Srap, agree that the argument in the OP is valid?
— Banno

I don't really care. It's abusive. — Srap Tasmaner

It's incorrect. But what do you mean by 'abusive'?

Yeah, I actually thought of a more concrete one we see on this forum: "it is true that nothing is true."

Normally this is just the claim "nothing is true," made with assertoric force. Same for "it is true (I know) that knowledge is impossible."

This is still self-reference though I guess.

1. Meaning what exactly? — Srap Tasmaner

Exactly that. If you modify the substitution rule to remove substitution of the same variable on both sides of a function, can you demonstrate that the resulting calculus will be complete? Can you prove A→A, for example?

Yep.

Is the answer to (1) something I should care about? — Srap Tasmaner

That's entirely up to you. But you are on this thread, so forgive my presumption. Failing to see that the argument in the OP is valid is an indication of a lack of understanding of basic logic. Refusing to give an opinion says something else.

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

Some folk think that pointing out an error os abusive. Odd, sad, but true

to claim "A → ~A" along with "A" is to contradict oneself, and therein I think lies the confusion. — Count Timothy von Icarus

Yes, A -> ~A with A is contradictory.

I don't know anyone who has said otherwise.

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", other than that those people have never read the first page of a book or article on the subject and are stuck with misconceptions that they adopted blindly.

Meanwhile, such ignoramuses don't understand even the most basic distinction between truth and validity while they promulgate that ignorance on a supposed philosophy forum.

Here's an example in ordinary langauge with the same form.:

1.Life therefore death
2.Life
Therefore
3.Death.

Both valid and sound it seems.

Also famously:

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

:up:

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.

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.

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

It doesn't seem that hard to understand to me. If people don't use formal logic often, then the most common thing to do is to translate into natural language. In natural language, we don't say that a falsehood implies anything. There is a relevance condition to consequence. Thus, to say "if my dog is alive then my dog is not-alive," is to say something that seems necessarily false—false regardless of if the dog is alive, dead, or even if the dog never existed. And the "necessarily false" is in the same mental bucket as "contradiction."


Or, if something like the OP is framed in terms of self-reference, e.g. "nothing is true," with assertoric force, it would be a self-refuting statement, which also is in the same vein. A Catch-22 would be similar, and people might even call it contradictory in common parlance.

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.

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.

I mean, obviously people do confuse this quite often, I can recall several threads, so I figure it's something like that, similar mental buckets.

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.

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.

can you demonstrate that the resulting calculus will be complete? — Banno

If you are interested in the basics of ordinary formal logic, then it would be a question that would naturally occur to you. But I don't see why you couldn't study other branches of philosophy without understanding the completeness of the propositional calculus. — TonesInDeepFreeze

Quite. I worked through some of the usual metatheorems years ago when I was studying formal logic. If you're interested in the properties of these formal systems, such results are just what you're interested in. And I'm sure there are issues that come up in philosophy that depend substantively on such results.

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. The cash value of completeness for such applications is nil.

Can you prove A→A, for example? — Banno

As Tones suggested, it might be necessary for proving certain metatheorems, but of course in real applications of logic ― such as on TPF ― "A→A" usually only appears as an accusation of question-begging. It's not something anyone would have any reason to argue for, and it's not a premise anyone would intentionally rely on. ― Hence my suggestion that we could usually get along without it.

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. The form is indeed instantiated ― I'm not contesting that ― but the first premise is materially equivalent to ~A. 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. Who needed modus ponens?

(Besides, you are effectively arguing from the set {A, ~A}. You could as well conclude A from that ― or any B you like ― so in what sense should this count as a "demonstration" that ~A? In what sense is the relationship of A and ~A revealed or clarified? It may be modus ponens in form, but hardly in spirit.)

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.

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.

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.

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.

these threads — Count Timothy von Icarus

FTFY

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.

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.

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.

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

Folk who understand that the argument is valid yet not sound will make no such conclusion.

The argument presented in the OP is valid, and has the form MPP. Pointing out the error of thinking otherwise has more of disabuse than abuse.

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.