It's interesting that the proof of "if set theory is consistent then set theory does not prove that set theory is consistent" is not so much analogous to the logic of the liar or barber paradox but rather to a different paradox, viz. Curry's paradox aka 'the Santa Claus paradox'.

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.

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'?

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.

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.

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.

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.

Yes, and you are wrong that

A -> ~A

is a contradictory.

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.

There is no governing body in what to call it. — Hanover

So what? Modus ponens is well understood and defined in thousands of books and articles and your remarks about it are not consistent with the common definition. You can define it any way you like (though you haven't defined it but only given certain qualifications about it), but then you will be talking about something very different from other people - such as logicians, philosophers, mathematicians, and students of logic, philosophy or mathematics - are talking about. What is the point of that?

You only engender misinformation and confusion on this point. No one is stoppiing you from giving a different from giving it a name ('hodus honens' would be good) and defining it.

there are exactly zero citations so far found where someone other than us has analyzed whether the OP case belongs in mp. — Hanover

So what? I gave you over a dozen defintions where the variables range over formulas or statements, and such that it is trivial for us to deduce that the example is an instance of modus ponens.

You might as well say, "We haven't seen a citation in which is found an analysis that

54322995731999373272287 + 229699797833575592 is an instance of summation"

We haven't seen it since anyone can carry out the analysis for themself:

For ANY natural numbers x and y, x+y is a summation.

instantiate x to 54322995731999373272287
instantiate y to 229699797833575592
so 54322995731999373272287 + 229699797833575592 is an instance of summation.

For ANY statements P and Q, the inference of Q from P and P -> Q is modus ponens.

instantiate P to A
instantiate Q to ~A
so the inference of ~A from A and A -> ~A is an instance of modus ponens.

Where we have found debate over invalid mp formulations on the web — Hanover

You mean McGee?

(1) McGee says himself that the invalidity does not concern the material conditional.

(2) AGAIN, since you SKIP this point, that people may dissent from modus ponens doesn't affect what the definition of 'modus ponens' is.

I see no evidence supporting your usage. — Hanover

I've given you nearly two dozen textbook definitions from which anyone who has read the first chapter could deduce that the example is an instance of modus ponens.

The OP is not a problematic example of mp. It's not mp at all. — Hanover

You know nothing about it. Nothing at all.

here is no cite, no source, no reference that says such a thing.
— TonesInDeepFreeze
Except perhaps in your first citation above:
"if a conditional holds...
— TonesInDeepFreeze
and (a->~a) doesn't "hold." — tim wood

(1) Here is that definition in full:

"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.

That is wrong.

Modus ponens doesn't require that the premises hold. Modus ponens say nothing about the standalone truth or falsehood of the premises or the standalone truth or falsehood of the conclusion. Modus ponens only says that IF the premises are true then the conclusion is true. That is, there are no interpretations in which all the premises are true but the conclusion is false. That is:

For any formulas P and Q, there are no interpretations in which P is true and P -> Q is true and Q is false.

And that is verified by a truth table.

And it does not disallow P from being instantiated to A and Q from being instantiated to ~A:

If A is true and A -> ~A is true, then ~A is true.

That is an instance of modus ponens.

Again, there is nothing in the principle or rule that says P cannot be instantiated to A while Q is instantiated to ~A.

A is a formula.
~A is a formula.
Modus ponens is the principle that for any formulas P and Q, if P and P-> Q, then Q.
So, one instance of modus ponens is: if A and A -> ~A, then ~A.

(2) Not that it matters for the point above, but as a matter of fact, A -> ~A does hold when A is false.

In that post you wrote:

"If A is true, then A is false" is a necessarily false statement. — Hanover

That is wrong, as has been explained to you over and over and over.

my definition of MP — Hanover

Of course, anyone can stipulate their own definition. But your definition is not the one used in ordinary formal logic, or, as far as I know, in any treatment of logic.

And you haven't even stated a definition. You've incorrectly stated that modus ponens disallows certain premises such as those in the example. That's not a definition.

Here are correct definitions (along with the many from standard textbooks I listed a few posts ago, where P and Q range over formulas):

(as an argument form:)
An argument is an instance of modus ponens
if and only if
the premises of the argument are P and P -> Q and the conclusion of the argument is Q

(as a rule of inference:)
The rule of modus ponens
is
from P and P -> Q, infer Q

symbolically:

{P, P -> Q} |- Q

(as a kind of entailment:)
modus ponens
is
the entailment of Q from P and P -> Q

symbolically:

{P, P -> Q} |= Q

this is a definitional debate — Hanover

There's no reasonable debate. That you fancy that you have your own definition has no bearing on the fact that in the field of study of formal logic, modus ponens is defined and your claims about it are inconsistent with the definition.

An odd lot we are. — Hanover

That would be true if you constituted a lot.

It violates the LNC — Benkei

What violates LNC?

1. not-G -> ( not (P->A) )
2. ( not (P->A) )
3. not-A
Therefore,
4. P — NotAristotle

More simply:

~(P -> A)
therefore P

lead to a conclusion that you can't in fact provide support for every mathematical assertion. Without reaching some paradox. — Cheshire

"provide support for" is vague in supposedly explaining the vague "has no foundation"

The main branches of classical mathematics are formalizable in set theory. However, if set theory is consistent then there are statements in the language of set theory such that neither the statement nor its negation is a theorem of set theory. Moreover, if set theory is consistent then set theory does not prove the consistency of set theory.

Thanks.

What I wrote eventually:

It turns out that his argument does not suppose that the conditionals mentioned are taken in the sense of the material conditional. He says that if the conditionals mentioned are taken in the sense of the material conditional then modus ponens is not impeached by his argument. — TonesInDeepFreeze

An example of Modus Ponen failure is presented in the Wiki article as the Vann Mcgee case — Hanover

If I recall, the Van McGee paper was the subject of a thread. And, if I recall, his argument hinged on adopting a different notion of the conditional.

Anyway, just to be clear, dissent from modus ponens doesn't change what the definition of 'modus ponens' is.

Of course, fair enough that it a tough and complicated matter for moderators.

But, in the meantime, I think it is appropriate for a poster to express exasperation when a poster plasters bot misinformation. Indeed, more appropriate to express it than to be quiet about it.

My own view is to not enforce censorship but on, the other hand, to be clear that it is not welcome.

So, while you ask for patience with the moderation, I suggest that the moderation have patience with justified exasperation in reaction to poster abuse of bot quoting, especially not to scold the poster who at least is providing correct info.

We might allow some space for them to learn. — Banno

Prime real estate was offered for free from the beginning.

Or put another way, the horse was offered the freshest, coolest, cleanest mountain spring water. He won't drink is his choice.

Logic is generally handled very badly here — Banno

It is deplorable the number of people who come into a philosophy forum without having read even page one of a book in logic or mathematics while spewing hyper-opinionated misinformation and nonsense on those subjects. It is utterly reasonable that one would become exasperated by that. Meanwhile, a moderator comes into scold the expression of exasperation while not a word that it is at least seriously frowned upon to cite bot misinformation and confusion, despite that (at least last I happened to read) the forum has said in general that that is not acceptable.

We haven't left Hanover any space to back down without loosing face. — Banno

It's not the job of the person who is giving correct information to provide a face saving escape hatch for the stubbornly irresponsible person who continues to spew misinformation no matter how many times he or she has been provided ample explanations and citations. If the person doesn't have the intellectual honesty to admit a glaring mistake then that's on the person entirely, especially after having been given copious explanation and citations. Also, one could be as conciliatory as pie to such a poster, and still he or she would not admit his or her error but rather on the contrary, he or she would persist even longer. That is the nature of Internet forums.

Nothing says we can — Hanover

The rule DOES imply we can since the rule quantifies over ALL formulas.

For that matter the rule doesn't explicitly mention any particular substitutions. [EDIT: replace 'substitutions' with 'instantiations', which is more strictly correct.] For example, the rule doesn't explicitly mention that:

(A & B) -> C
A & B
therefore C

is an instance of modus ponens. But it is an instance of modus ponens.

And

A -> ~A
A
therefore ~A

is another instance of modus ponens though it too is not explicitly mentioned in particular in the rule.

It is part of the POINT of being a rule that it can be applied to ANY formulas.

your definition of MP is not logically entailed — Hanover

It's a DEFINITION. It's not supposed to be "entailed".

After exhaustive explanations and citations, I'm waiting for someone to say to me, "You just argue with ad hominem".

But this is trivial stuff! — Banno

Calm down! You're making this emotive!

Why don't you already know this? — Banno

Indeed, at a certain point in discussions where a poster is flat out wrong about a matter that is not even a matter of opinion, and persists to insist despite copious explanations given him, then the pertinent question turns from the simple fact of the matter about the subject to what is wrong in the head of the stubbornly clueless poster.

Where pray tell do you find a definition of MP that takes into consideration a self referential contradictory conditional and asserts it satisfies the definition of MP?

All definitions I have located say otherwise, as do all Google and AI engines.

Provide to me your cite — Hanover

(1) There is no "self-reference".

(2) The conditional A -> ~A is not contradictory.

(3) Nowhere in the definition of 'modus ponens' is it disallowed to instantiate to P to A and Q to ~A.

(4) Where "pray tell" do you find a definition that says "except Q cannot be instantiated to the negation of what P is instantiated to"? Hint: You don't.

There is no cite, no source, no reference that says such a thing.

You just somehow got it stuck in your head that such a thing is implied by the definition. But it's not.

AGAIN you need to read and comprehend.

P and Q range over formulas.

From P and P -> Q, infer Q by the rule modus ponens

Since A and ~A are formulas, we have:

From A and A -> ~A infer ~A by the rule modus ponens

You cannot show any definition, explanation or argument in any logic book or reliable article that says, implies or insinuates that the definition of 'modus ponens' disallows:

From A and A -> ~A infer ~A by the rule modus ponens

But you many look up arbitrarily many logic books that do imply that

From A and A -> ~A infer ~A by the rule modus ponens

from the plain definition of 'modus ponens' such as:

From formulas Phi and Phi -> Psi, infer Psi

where 'Phi' and 'Psi' are variables ranging over formulas.

It's not a matter of opinion that

A -> ~A
A
therefore ~A

is an instance of modus ponens.

It is a plain fact.

It is quite impolite to continue to ignorantly insist on bad misinformation and to cite wildly erroneous and incoherent bot messages as if they are information.

Justifiably.

I thought this forum was going to warn against citing bot misinformation.

(1) That definition does not contradict that

A -> ~A
A
therefore ~A
is an instance of modus ponens

(2) Here are definitions of 'modus ponens':

"if a conditional holds and also its antecedent, then the consequent holds." (Beginning Logic - Lemmon)

"C is a direct consequence of B and B -> C." (Introduction To Mathematical Logic - Mendelson)

"From the formulas Alpha and Alpha -> Beta, we may infer Beta" (A Mathematical Introduction To Logic - Enderton)

"from P and P -> Q we may infer Q" (as the rule corresponding to the tautology (P & (P -> Q)) -> Q) (Introduction To Logic - Suppes)

"Psi is obtained from Phi and Phi -> Psi" (Mathematical Logic - Monk)

"A, A -> B |= B" (A Concise Introduction To Mathematical Logic - Rautenberg)

"the inference from A and A -> B to B" (Computability And Logic - Boolos, Burgess and Jeffrey)

"Gamma, Phi -> Psi and Gamma, Phi; therefore Gamma, Psi" (Mathematical Logic - Ebbinghaus, Flum and Thomas)

"passing from two formulas Alpha and Alpha -> Beta to the formula Beta" (A course in Mathematical Logic - Bell and Machover)

"Phi -> Psi, Phi; therefore Psi" (Logic: Techniques Of Formal Reasoning - Kalish, Montague and Mar)

"If P and P -> Q are proved, then one is entitled to infer that Q is proved" (Logic For Mathematicians - Rosser)

"A, A -> B |- B" (Introduction To Metamathematics - Kleene)

"p, p -> q |- q" (Foundations Of Mathematical Logic - Curry)

"from the premisses Phi -> Psi and Phi to Psi" (Mathematical Logic - Quine)

"From A -> B and A, to infer B" (Introduction To Mathematical Logic - Church)

"Psi may be entered on a line if Phi and Phi -> Psi appear on earlier lines" (Elementary Logic - Mates)

"From Psi and Psi -> Phi infer Phi" (Model Theory - Chang and Keisler)

"If p then q, p, conclude q" (Symbolic Logic - Copi)

And on and on in as many books on basic formal logic that you may look at.

All those definitions have in common that there is NO requirement that we may not instantiate the variables to A and ~A.

All those definitions have in common that there is NO requirement that the premises are not contradictory

Modus ponens doesn't require that a conditional is not contradictory, nor that the "major" premise (which must be a conditional) is not contradictory, nor that the "minor" premise (which might or might not itself be a conditional) is not contradictory, nor that the premises together are not contradictory
— TonesInDeepFreeze

What is your cite for this definition? — Hanover

It's not a definition! It's a comment about definitions. It is not itself a definition.

Meanwhile, you will find NO cite of a definition that requires that P can't be instantiated to A while Q is instantiated to ~A. And you will find NO cite of a definition that requires that the premises are not contradictory.

The absurdity is that you think this a question of logic and not definition. — Hanover

No, it's the DEFINITION of 'modus ponens'.

No meaningful logical conclusion can follow from a contradictory conditional that assets the proposition and its negation can occur simultaneously. — Hanover

Modus ponens doesn't require that a conditional is not contradictory, nor that the "major" premise (which must be a conditional) is not contradictory, nor that the "minor" premise (which might or might not itself be a conditional) is not contradictory, nor that the premises together are not contradictory.

Modus ponens "is the rule of logic stating that if a conditional statement (“if p then q ”) is accepted, and the antecedent ( p ) holds, then the consequent ( q ) may be inferred." — Hanover

What is your source of that quote?

If your conclusion is not true, you can't offer MP as the basis of it being true because it's not. — Hanover

You don't understand basic ordinary academic logic. You need the first chapter of a good textbook in print or online.

Meanwhile, you need to not litter a philosophy forum with confused, misinformational, and malformed bot garbage.

How pathetic that a philosophy forum is sidetracked by a poster not understanding the basic logic he is so adamantly making claims about.

It should not have to be said more than this:

The rule of modus ponens is:

If P and Q are ANY statements, then the following is modus ponens:

P -> Q
P
therefore Q

Since P and Q may be ANY statements, and A and ~A are statements we have:

A -> ~A
A
therefore ~A

is an instance of modus ponens.

Period.

And these also are instances of modus ponens, even though odd:

A -> A
A
therefore A

~A -> ~A
~A
therefore ~A

Period.

This is NOT the rule of modus ponens:

If P and Q are any statements except Q is not the negation of P, then the following is modus ponens:

P -> Q
P
therefore Q

And this is NOT the rule of modus ponens:

If P and Q are any statements and {P, P -> Q} is consistent then the following is modus ponens:

P -> Q
P
therefore Q

/

One more time, since the poster is presenting as seriously obtuse:

The rule of modus ponens is:

If P and Q are ANY statements, then the following is modus ponens:

P -> Q
P
therefore Q

Since P and Q may be ANY statements, and A and ~A are statements we have:

A -> ~A
A
therefore ~A

is an instance of modus ponens.

I don't usually say, "Please, let's move on" but I'm saying it this time.

That bit from the bot is atrocious confusion and misinformation. And it is intellectually shameful for a poster to post a confused and misinformational bot quote as if it is correct and settles a discussion about the logical matter. And risible that a poster quoting a bot doesn't understand that such bots compose text that seems like something that might be said, without fact checking itself that it is actually correct.

Get outta here with that bot garbage!

Amazing that someone would take the word of a bot on such a question. One could get bots to generate misinformation over and over again.

But not amazing that bots regularly get things quite wrong.

The bot does not understand substitution. And the bot can't even write formulas or English correctly: [EDIT: replace 'substitution' with 'instantiation', which is more strictly correct.]

"In your example, you have
→ ¬
A→¬A (If A, then not A), which leads to a contradiction when assuming"

(1) -> ~

is not well formed

(2) "assuming" doesn't end a sentence there.

(3) What the bot must mean is:

A -> ~A and A lead to a contradiction.

That is correct. But it doesn't show that

A -> ~A
A
therefore, ~A

is not an instance of modus ponens.

The bot incorrectly reasons that the inconsistency of the premises disallows the argument from being modus ponens. But it is not disallowed the premises of a modus ponens argument may be inconsistent

Here is correct information:

Let P and Q be meta-variables (read as 'phi' and 'psi' if you like) ranging over sentences.

Modus ponens is any argument:

P -> Q
P
therefore Q.

Let A be a sentence letter or any sentence.

Instantiate P to A. Instantiate Q to ~A. Those are perfectly legal instantiations. There is no rule that disallows them. Look in any logic book or ask any logician.

So
A -> ~A
A
therefore ~A

is an instance of modus ponens.

AGAIN: There is no rule of logic that prohibits substituting A for P and ~A for Q. [EDIT: replace 'substituting' with 'instantiating', which is more strictly correct.]

I'm not inclined to compose a post about it.

Of course LNC and LEM are different.

I can't find the post about the liar paradox; my own point was merely the technical one that the contradiction of the liar does not require LEM.

My statements are in context of ordinary symbolic logic. Things may be different depending on alterative logics:

Isn't a tautology as much a contradiction as anything? (p or ~p) — Cheshire

No. A tautology is a formula that is satisfied by every interpretation. No contradiction is satisfied by any interpretation. Therefore, no tautology is a contradiction.

And

P v ~P

is not a contradiction.

Godel concluded that no system really has a foundation — Cheshire

What specific remarks by Godel are you referring to?

if we follow the evidence it suggest that self-reference isn't a reliable source of truth, in the sense the system breaks down per Russell and Godel. — Cheshire

From Godel-Rosser we have certain systems that have self-reference and are (if consistent) incomplete. What do you mean by "break down"?

/

I've always wondered if Russell's paradox is coming from the foundations of set theory: the contradiction of fencing in infinity. — frank

The paradox pertains to any 2-place relation, not just the 'member of' relation.

Irrespective of set theory:

For any 2-place relation R, there is no x such that x bears R to y if and only if y does not bear R to y.

/

I don't recall the post, but in this thread (or another?) someone mentioned LEM in relation to the liar paradox. We don't need to refer to LEM for the liar's paradox. The contradiction is obtained even without LEM.

We don't need to argue this way:

L -> ~L
~L -> L
L v ~L
therefore L & ~L ... contradiction

Rather we can argue this way:

L -> ~L
so ~L
~L -> L
so L
therefore L & ~L ... contradiction

Or, intuitionistically:

L -> ~L
so ~L
~L -> L
so ~~L
therefore ~L & ~~L ... contradiction

Yes, that is the truth table.

You need to reevaluate your mistaken notion about substitutions. [EDIT: replace 'substitutions' with 'instantiations', which is more strictly correct.]

Let P and Q be meta-variables (read as 'phi' and 'psi' if you like) ranging over sentences.

Modus ponens is any argument:

P -> Q
P
therefore Q.

Let A be a sentence letter or any sentence.

Instantiate P to A. Instantiate Q to ~A. Those are perfectly legal instantiations. There is no rule that disallows them. Look in any logic book or ask any logician.

So
A -> ~A
A
therefore ~A

is an instance of modus ponens.

/

And look at the truth table for

A -> (A -> ~A)

to see that it is true when A is false.

/

And, you said, "If A is true, then A is false" is a necessarily false statement.

That's incorrect.

If "A is false" is true, then "If A is true, then A is false" is true.

The antecedent is "If A is true" and the consequent is "A is false".

If "A is false" is true, then the antecedent is false and the consequent is true, so the conditional is true.

So "If A is true, then A is false" is not necessarily false, since there is an interpretation (viz. when ""A is false" is true" in which it is true.

The correct statements are:

(If A is true then A is false) then A is false.

If A is false then (If A is true then A is false).

A is false if and only if (If A is true then A is false).

You cannot substitute P and Q to be — Hanover

You can instantiate P and Q to whatever formulas you want.

You somehow got in your head a wrong notion.

maintain logical equivalence — Hanover

I said nothing about logical equivalence.

Read what I wrote.

A -> ~A = ~A — Hanover

Actually, the notation is:

(A -> ~A) <-> ~A

(A -> ~A) is not equal to ~A. They are not the same formula. Rather they are materially equivalent. Equality and material equivalence are not the same. The distinction is important.

A->~B is not reducible to ~A. — Hanover

I said nothing about "reducible".

Again, read what I wrote.

A-->~A is not logically equivalent to A --~B. — Hanover

I didn't say they are. Read what I wrote.

It's like saying A+A = 4 and since it's generic — Hanover

No, it's nothing like that. I made no such argument.

We're in disagreement that P--> Q = P --> P. The former is a conditional, the latter a tautology. — Hanover

You're confused.

I did not say "P--> Q = P --> P". I said that

A -> ~A is an instance of P -> Q.

Nothing about equality, only instancehood.

And both P -> Q and P -> P are conditionals. The fact that P -> P is a tautology doesn't make it not a conditional.