Again here:

https://thephilosophyforum.com/discussion/comment/943647

Not exactly. You were saying that A → ¬A is necessarily false, which is saying that ¬A ∨ ¬A is necessarily false. But ¬A ∨ ¬A is true if ¬A is true, and so A → ¬A is true if ¬A is true.

Artistotle and Euclid use contradiction in reductio demonstrations all the time. If we have a valid argument with a conclusion we know to be false then we have warrant to reject a premise or assume that at least one is false.

However, I do agree that the common analysis that, if ~A is true, then A→~A doesn't sit well with common sense intuitions about consequence. The truth table also is liable to look confusing because it varies from how the premises are laid out, but I think that it isn't once properly understood.

The first premise isn't actually a contradiction — Michael

Nor is my first premise. I think my argument is valid and sound, and expresses the principle of explosion. Similar nonsense results when you divide by zero, and you can disguise that division in a similar way with a bit of algebra. Obviously the universe is the result of God accidentally contradicting Himself by making a mistake, which He cannot do, being infallible. Explains everything!

And from a contradiction, anything and everything follows. This is the principle of explosion.

That is to say, "1. A -> not-A" is impossible; when the impossible can happen, anything can happen. — unenlightened

Impossible cannot happen, therefore nothing can happen?

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."

Obviously the universe is the result of God accidentally contradicting Himself by making a mistake, which He cannot do, being infallible. Explains everything!

"Yes."

:rofl:

Fair point. Ok, I concede; "obviously" was a misspoken epithet on my part.

...but you may insist that it is as it is. — Hanover

Allowing substitution of any well-formed formula is not a personal foible. It is how propositional logic works. (φ, φ →ψ ⊨ ψ ) for any well-formed formula φ and ψ. Nothing says they must be different.

We're not debating what can be substituted and what the logical implications are of such substitutions.

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

There is no governing body in what to call it. My basis for excluding self contradictory versions has been stated.

As noted, there are exactly zero citations so far found where someone other than us has analyzed whether the OP case belongs in mp. Where we have found debate over invalid mp formulations on the web, exactly zero deal with the OP case.

The point being, should we guage term usage for meaning, I see no evidence supporting your usage.

As I've also repeatedly said, this is a definitions question, not a logic one. We both agree upon what entails what and what can be substituted in for what.

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

The Logic of the Diamond Sutra: A is not A, therefore it is A Shigenori Nagatomo

ABSTRACT This paper attempts to make intelligible the logic contained in the Diamond
Sutra. This `logic’ is called the `logic of not’. It is stated in a propositional form: `A is not A, therefore it is A’. Since this formulation is contradictory or paradoxical when it is read in light of Aristotelean logic, one might dismiss it as nonsensical. In order to show that it is neither nonsensical nor meaningless, the paper will articulate the philosophical reasons why the Sutra makes its position in this contradictory form. The thesis to be presented is that as long as one understands the `logic of not’ from a dualistic, either-or egological standpoint, it remains contradictory, but in order to properly understand it, one must effect a perspectival shift from the dualistic, egological stance to a non-dualistic, non-egological stance. This thesis is advanced with a broader concern in mind: to reexamine how the self understands itself, how it understands others, and how it understands its intra-ecological relationship with nature.

.pdf, 32 pages including footnotes.

The OP uses propositional logic. In propositional logic, the argument is valid.

It violates the LNC — Benkei

What violates LNC?

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 seems folk think A → ~A is a contradiction. It isn't.

Here is the truth table for the contradiction A and not-A:

Notice that the column under A and not-A is false for every assignment to A. That's why it is a contradiction: it is always false.

Here is the truth table for A→ ~A

Notice that the column for A→ ~A is not false if A is false. A → ~A is not a contradiction. Rather, it says that A is false.

If A is true, then A is false. Therefore A cannot be true.

, 's error, perhaps.

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.

The OP is a factual question, not an issues of opinion. The one-third of folk who think that the argument is invalid are wrong. As wrong as if they had asserted that 2+2=5.

One-third of folk who have at least enough interest in logic to respond to the OP do not have a basic understanding of validity.

That's pretty sad. On a philosophy forum, it's pathetic. That is, it arouses pity.

The OP uses propositional logic. In propositional logic, the argument is valid. — Banno

Let me test it.

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

MP has spoken. It doesn't use propositional logic

Right. The contradiction is 1. ~A, 2. A.

If the OP uses propositional logic, it doesn't use propositional logic. — Hanover

This is false. It corresponds to line two of the truth table given above.

The contradiction is 1. ~A, 2. A. — Hanover

Only line 1 is not, ~A. It's A→~A.

It's
1. A → ~A (assumption)
2. A (assumption)
3. ~A (1,2,MPP)

Not

1. ~A (assumption)
2. A (assumption)
3. ~A (1)

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.

It seems folk think A → ~A is a contradiction. It isn't. — Banno

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

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

The set {A->~A, A} is not a contradiction because it is not a formula, but a set. It is, however, inconsistent.

Would there be any harm in requiring that the conditional in a modus ponens have fresh variables on the right hand side? We would lose, in effect, only this one and A->A, which is either useless or the LEM, and thus innocent or tendentious, depending on how you look at it.

I mean, mathematicians always prefer the greatest generality, at the minor, to them, cost of letting in the degenerate case. If your project is automated reasoning, you'll go with the usual. But for doing philosophy, we don't have to let the mathematicians have the last word.

If the only question here is "How does formal logic work?" we know the answer to that. But around here we're more interested in the practical use of logic, and it seems to me letting mathematical logic have the last word is the tail wagging the dog.

Only line 1 is not, ~A. It's A→~A. — Banno

I mean, 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?

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)
5. 3 is an invalid conclusion derived from mp (3,4)

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

Yep.

Would there be any harm in requiring that the conditional in a modus ponens have fresh variables on the right hand side? — Srap Tasmaner

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

But around here we're more interested in the practical use of logic, — Srap Tasmaner

Too often this is an excuse for poor logic.

Back to the question: Do you, Srap, agree that the argument in the OP is valid?

Again, 1 is false, and your argument (1-3) valid but unsound.

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.

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. — TonesInDeepFreeze

Agreed. And earlier I posted per truth table, the argument is valid. But not to be confused (not ever) with the conclusion being true. And the "hold(s)" in your definition I take to be the qualification of being true. Thus the form, MP, will carry truth, but it requires truth for truth to be carried. Or in geek terms, GIGO.

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.

Thanks, that's an interesting one.



The first premise is false though. We are only affirming a contradiction if we affirm A and ~A.



In this one 2 is false. It is possible to have a valid argument that has some true premises and a true conclusion without all the premises necessarily being true.

It is indeed contrary to intuition that A →~A should be true if A is false however, that affirming that "if you are incorrect" then "your being correct implies that you are incorrect," is pretty tortured, I'd agree.


:up:

I cannot think of a way to frame this as a real example outside of self reference and removing A→~A would solve that.