It may seem bizarre that a valid argument could have at least one premise that is necessarily false at first glance, but I think it is fairly intuitive if one thinks in terms of truth-preservation. — Count Timothy von Icarus

Validity in propositional logic involves a relativization of truth-values with respect to inference-relations. Inference-relations are held steady, and if the truth-values cash out given the stable inference-relations, then we call it "valid." The inference relations are conceived as meaning-stable, and the variables are conceived as meaning-variable (i.e. truth-variable). But in this case what is at stake is the meaning and stability of the inference-relations themselves. The contentious move is to claim that the consequence-relation involved in the OP is the stable, familiar consequence relation of modus ponens. It isn't. That's that place to start.

To claim that it is involves:

...prioritizing truth-functional process over logical telos. — Leontiskos

Put differently, the notion of validity assumes a truth-functional context where truth and form are entirely separable. Yet when we think deeply about inferences themselves, such as modus ponens, truth and form turn out to be less separable than we initially thought. When we stop merely stipulating our inferences and ask whether they actually hold in truth, things become more complicated.

Probably unfortunately relevant to we Americans' immanent election (and the last one, and 2000... a pattern emerges). However, I think this would be an issue of unclear/disputed terms—equivocity re "winning"—not a case of A→~A.

Interestingly, it's another example involving stipulated rules, just as with conventional self-reference. In the thread on logical nihilism I was thinking out loud about the older distinction between formal and material logic. This goes back to Aristotle, who discusses pure form in the Prior Analytics and the "matter" of discourse in the Posterior Analytics. The latter discusses the ways in which subject matter shapes discourse.

I think it is at least possible that one might be able to ground the selection of different consequence relationships in the relevant subject matter in a way that is rigorous, preserving the intuitions of both pluralists and monists.

John Poinsot (John of St. Thomas) and later CS Peirce (who took a lot from the Scholastics) could lay the groundwork for this with their well developed theory of signs (including attention to the unique aspects of stipulated signs and sign systems). The sign relation is irreducibly triadic. There is always the object signified joined to an interpretant by a sign vehicle. Yet in cases of self-reference in stipulated signs systems the object is the sign vehicle.

Obviously, this isn't true in every sense. When we read "this sentence is false," there is a sense in which the paper or screen is the object, light acts as the sign vehicle, and we are the interpretant. Yet in the universe of the stipulated system, taken by itself, we have collapsed the necessarily triadic relationship into a dyadic one. The result, apparent "true contradictions."

And this would also bear out some of the intuitions of the post-modern semiotics that grew out of Sausser, which collapses the triadic relationship, while at the same time allowing us to at least plausibly overcome their more radical and destructive (destructive to notions of truth and meaning) theses by demonstrating how these are limited to a specific area of discourse and not all sign relations. Maybe.

Surely a hard sell, since "freeing the sign" has been bound up in notions of human freedom and flourishing in that tradition.
.

With a false antecedent, the consequent is vacuously true. — Benkei

No, with a false antecedent the conditional is true, sometimes described as 'vacuously true'. It's the conditional that is deemed true when the antecedent is false.

violates the LNC — Benkei

Yes, A -> ~A and A together are contradictory.

You've been busy...

You've been busy... — Banno

All he had to do is say that there aren't any cases where both premises are true, therefore it's valid.

All he had to do is say that there aren't any cases where both premises are true, therefore it's valid. — frank

I said it over and over and over for you.

All you had to do is read the replies given you. And that's hardly the only point I explained for you.

All you had to do is read the replies given you. — TonesInDeepFreeze

Yea, well...

Well, while I think Srap has a good point about our being able to live without A→~A in most situations, I think it is important that statements like "nothing is true," are able to entail their own negation—that logic captures how these claims refute themselves.

No, with a false antecedent the conditional is true, sometimes described as 'vacuously true'. It's the conditional that is deemed true when the antecedent is false. — TonesInDeepFreeze

:up: :100:

But that situation, where the antecedent is denied, is irrelevant because the second premise assumes A to be true. And it necessarily follows from the first premise that not-A is simultaneously true. This is self-contradictory and violates the LNC. — Benkei

What about my earlier example:

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 — Janus

Terms are ambiguous but it's also not modus ponens. More a play on words than logically interesting.

Neat.

Not quite propositional, though, using modus ponens on a mass noun.

Good to see that the number of folk who think that argument in the OP valid has dropped from a third to a quarter.

Still sad that it remains that high.

I suppose accusations of hypocrisy are nearby. "Your anti-racism is itself a form of racism." "Your anti-capitalism materially benefits you." "Your piety is actually vanity." Generalize those and instead of saying, hey here's a case where the claim is A but it's really ~A, you say, every A turns out to be ~A. Now it's a rule.

I am not sure about this one. The person is not arguing that A is actually ~A. Presumably, they believe real piety exists, just that this person doesn't possess it. They are arguing that what the person claims is A is actually just B. Perhaps, "anti-racism is racist," is closer to the mark, but again, I think this is still more of the same, a claim that what is presented under the term A is actually B. Presumably the person who earnestly makes such claims normally believes that one can be actually anti-racist without being racist (normally by being "colorblind.")

And on second thought, about the first use case, I think that often, when people argue that the other party is accidentally implying the falsehood of their own position, the issue will also be unclear terms. Not always, sometimes people do refute themselves. But such arguments might not settle the issue even if both sides are acting in good faith, not because one party rejects the form, but rather the content.

On a related note, although not the case here, I think a lot of the "gotcha" puzzles that involve presenting good formal arguments alongside what appear to be faithful natural language translations of them, which are nonetheless either clearly wrong or at least not obviously right, involve equivocation. I think these are particularly disarming because, at least in my experience, the basics of form is taught while ignoring the possibility of vagueness, which is a problem because arguments can fail in three ways, invalid form, false premises, or unclear terms.

I didn't really get a straightforward introduction to this risk until being led to the ol' "three acts of the mind," in historical treatments of logic curricula in the past. Although perhaps my experience is not typical.

Edit: Just for an obvious example:

Everything that runs is an animal.
My refrigerator is running.
Therefore, my refrigerator is an animal.

Works great formally if you're allowed to us "R" for "that which runs" in both premises.

It violates the LNC, which is foundational and introduced by Aristotle before modus ponens so he certainly didn't intend that the inference can work. — Benkei

Perhaps I misunderstood you. I had taken "it" and "the inference" to be the argument in the OP. Hence it appeared you were saying the argument in the OP was invalid.

What I want is an example where this conditional is actually false, but is relied upon as a sneaky way of just asserting ~A. — Srap Tasmaner

A thought I have is sarcasm, but in the context of asserting a falsehood mistakenly.

So I can sarcastically say "George is going to open the store tomorrow" to mean that George is usually late and we are the ones who open the store on the regular. But if George opens the store tomorrow then the conditional was false because I asserted A to imply not-A, but in fact A is true so the conditional is false.

Well, while I think Srap has a good point about our being able to live without A→~A in most situations, I think it is important that statements like "nothing is true," are able to entail their own negation—that logic captures how these claims refute themselves. — Count Timothy von Icarus

But does logic really capture how these claims refute themselves? I don't think so. It merely defines a formal notion of contradiction and shows that a contradiction has occurred. The how/why question is beyond the logic (as is the reductio-remedy), and I believe you yourself pointed earlier to the logical simplification of 'contradiction' (i.e. an all-false truth table).

It may seem bizarre that a valid argument could have at least one premise that is necessarily false at first glance, but I think it is fairly intuitive if one thinks in terms of truth-preservation. If the premises were true, it would preserve truth. But the "truth" of a false premise cannot be preserved.

And it's a good thing that it is valid because we often can reason from necessarily false conclusions in valid arguments to identifying false premises. — Count Timothy von Icarus

Why would it be a good thing? It is good that we can reason from non-necessarily false conclusions in valid arguments to identifying false premises. An argument from a necessarily false conclusion is a reductio, and the question of whether an absurdity is valid is part of the very question at hand.

:wink:

What is the conditional?

/

Maybe there could be an intensional logic with a defined irony operator 'i'.

s is a speaker
P is a statement
B(s P) iff s believes P
T(s P) iff s states P

Df. i(s P) <-> (T(s P) & B(s ~P))

No absurdity is a validity.

Everything that runs is an animal.
My refrigerator is running.
Therefore, my refrigerator is an animal.

Works great formally if you're allowed to us "R" for "that which runs" in both premises. — Count Timothy von Icarus

If I understand your point, I agree with it, and I think it is incisive and apropos.

Let Rx mean x runs (where 'runs' includes both 'moving quickly on feet' and 'operates')
Let Mx mean x is an animal
Let t mean the refrigerator

1 Ax(Rx -> Mx) ... false premise
2 Rt ... true premise
therefore Mt ... {1 2} ... false conclusion

valid but unsound

What is the conditional? — TonesInDeepFreeze

A = "George is going to open the store tomorrow"

So, by substitution:

George is going to open the store tomorrow implies George is not going to open the store tomorrow.

If it turns out, extensionally at least, that George opens the store tomorrow then the implication is false -- and I don't think that sarcasm means to invoke material implication, but this seems an example of everyday communication which material implication seems to capture. George opens the store tomorrow, so tho I state one and believe another it turns out that my belief is false and the assertion true (attempting to use your intensional definition here) -- so the implication turns out to be false. I'm thinking more baby logic here:

A -> ~A

Put it in a truth table and if A is true then the implication is false.

I like the idea of an irony operator :D

I mean sure, if you want to collapse "moving quickly on feet" and "operates" into a single term. You could cover other equivocations of "run" as well and have a single term cover "flows," "seeks elected office," "is a candidate for winning," etc.

But if you want to use your terms in any sort of a broad fashion, or if you want to make things simple, you can simply demand that the terms be disambiguated.

Plus, these are just obvious examples, relying on equivocity. When it comes to analogous predication it will not be so simple to use such a solution.

And at any rate, prior to recent advances in robotics, it was true that only animals run in the proper (proper here) sense.

I edited my post to mention that, if I understand your main point, I agree with it.

And, if I understand, I agree with your disjunction:

Either make the predicate encompassing, in which case the first premise is false.

Or have two separate predicates, in which case the argument is invalid.

/

analogous predication — Count Timothy von Icarus

I don't know what you mean.

"George is opening tomorrow, and we all know what that means."
"George isn't opening tomorrow."

The conditional here is actually true, because George never opens.

Interesting. Can we substitute salva veritate into an ironic statement? Seems to work.

If George is George Atkins then "George is going to open the store tomorrow" is true iff "Atkins is going to open the store tomorrow".

A = "George is going to open the store tomorrow"

So, by substitution:

George is going to open the store tomorrow implies George is not going to open the store tomorrow. — Moliere

You mean substitute "George will open the store" with "If George will open the store then George will not open the store"?

Why make that substitution? I don't see how that is what the ironic speaker is saying.

Can we substitute salva veritate into an ironic statement? — Banno

It occured to me, when I thought of the irony operator, that we have the problem that we can't substitute salva veritae within a belief operator. That even makes negations tough, since we can't assume the ironic speaker knows double negation. I glossed over it. So we ould have to formulate further:

Let P be not a negation:

Df. i(s P) <-> (T(s P) & B(s ~P))

Df. i(s ~P) <-> (T(s ~P) & B(s P))

But I'm just fooling around. And probably someone has already worked out an irony logic.

I don't know what you mean.

In natural language, predication is often not totally univocal, but is also not totally equivocal. There is a vagueness problem. For example, we might say that "lentils are healthy," or "running is healthy." These are true statements. And we might also say "Tones is healthy." Yet you would not be "healthy" in the same way that lentils are. However, neither is the usage totally equivocal. We call lentils "healthy" precisely because (normally) they promote the health of human beings, i.e. the same "health" we refer to in "Tones is healthy."

Perhaps we could dismiss this as just a case of equivocation in disguise, but I don't think so.

I know that people have tried to formalize this sort of thing; I am not particularly well-versed in how though. My understanding is that no attempt has proven particularly popular because they do not seem to fully capture how analogous predication is used.

It's sort of like how, as far as I am aware, there is no popular formalization of the distinction between quia vs. propter quid demonstrations (i.e. demonstrating "that something is the case," vs. demonstration "why it is the case.") I don't think most people would deny that they're different (although some would), but rather it seems that the difference should be entirely reliant on the arguments' content, not their form (i.e. an issue of material logic).

True. The one time he did we all know it's because his wife grilled him the night before and he felt guilty but ever since the divorce it's been same old George: He only opens when he feels worried and ever since the divorce the man is never worried!

:D

This seems the easier approach to making sense of A -> ~A in a commonsense setting.

You mean substitute "George will open the store" with "If George will open the store then George will not open the store"?

Why make that substitution? I don't see how that is what the ironic speaker is saying. — Moliere

I can give you a story that comes to mind in which I'd assert something like that -- say I'm commiserating with a coworkers frustration about George not being as reliable as we'd like, even though he's a good enough fellow.

The substitution is there only because the OP starts with A -> ~A and asks for validity, so substitution seems to work as a model for the sarcastic talking. I agree that the person speaking sarcastically does not in any way mean these logical implications, though -- it's only an interpretation of everyday speech to try and give some sense to the original question that's not purely formal.