I think I see what the discussion is turning around. The matter with modus tollens is that q could be otherwise, while in reductio it is not the case, by definition. Then again, I don't think it is meaningful or interesting. I am quite happy with my natural language conversions.

My question then is whether we ever utilize (B∧¬B) without conceiving of it as a kind of P. — Leontiskos

If P can only be False, yes; otherwise, no.

So do we have a proof for ((a→(b∧¬b)) → ¬a)? — Leontiskos

Uh

Leo seems to think that choosing between ρ→~μ and μ→~ρ somehow involves an act of will that is outside formal logic. He concludes that somehow reductio is invalid. His is a mistaken view. Either inference, ρ→~μ or μ→~ρ, is valid.

Indeed, the "problem" is not with reduction, but with and-elimination. And-elimination has this form
ρ^μ ⊢ρ, or ρ^μ ⊢μ. We can choose which inference to use, but both are quite valid.

We can write RAA as inferring an and-sentence, a conjunct:

ρ,μ ⊢φ^~φ⊢ (ρ→~μ) ^ (μ→~ρ)

and see that the choice is not in the reductio but in choosing between the conjuncts.

Leo is quite wrong to assert that Reductio Ad Absurdum is invalid. — Banno

I think Leontiskos is talking about choosing between the conjuncts, while Banno is correctly stating that reduction ad absurdum is formally valid.

I think the only way we can utilize logical inference is by using the modus tollens — Leontiskos

Modus tollens is p→q is True, q is False, therefore p is False. «1»
While reductio would be:
p
p→absurd/contradictory
therefore not-p «2»

So I think that a→(b∧¬b)) → ¬a can indeed be proven by modus tollens:
a→(b∧¬b)) is True, (b∧¬b)) is False, therefore a is False (from «1»).

Proving a→(b∧¬b)) → ¬a by reductio would be:
a
a→absurd/contradictory
therefore not a (from «2»)

I don't see a meaningful difference.

Does classical logic not presuppose that such substitution is truth-preserving? — Leontiskos

Yes, it is truth preserving. That
(A∧(B∧¬B))↔(B∧¬B) is valid
But (A∧C)↔C is invalid
does not make me think rules of logic are conflicting, because the equivalence or not with the second term of an «and-operator» is not a rule of logic.

(B∧¬B) is ambiguous, and can be interpreted as p or as FALSE (i.e. always-false). — Leontiskos

As I replied to sime, interpreting (B∧¬B) as P is not a good move, for P can be True or False, (B∧¬B) cannot be True ever.

Why did you reject (1) and not (2) or (3)? The reductio is not formally valid in that tight sense. — Leontiskos

What about
1 A
2 A→¬B&B
3 ¬A

You would object why I rejected 1 instead of 2? I guess I see your point that it is not valid in a tight sense. After all, from A, A→¬B&B, everything follows, not just ¬A.

Edit: I would say that it is valid, but why choose one rather than the other, then I am not convinced that that is something that follows from logic.

Trump being almost shot is a borderline irrelevant event in the great scheme of things — despite the cool picture. It would be another matter had he actually been shot to death. Proof of that: the thing is already losing steam only 2 days after. Just a bunch of dorks thinking they are in a Hollywood drama meanwhile a cartoonishly evil literal cabal of p*d*philes envolving politicians and billionaires was found out in 2018 and people forgot about it one year later while Bill Clinton runs around free :rofl: A country of oestrogenated fist-shakers, despite the guns lying around. The attempt changes nothing besides give Trump an advantage in the elections.

Shinzo Abe's actual death, being no longer a PM and only a member of the chamber, seems more relevant in comparison, especially when the matter of Unification Church is taken into account.

What's wrong with Biden? — L'éléphant

Dementia.

Why is he being criticized so much? — L'éléphant

Dementia.

Aging is an issue for one of the most important jobs in the world and in Biden's case we're talking about dementia. He's a vegetable half of the time. — Benkei

Thanks. If you are interested, the "half of the time" is due to sundowning.

some rules of classical logic to come into conflict — Leontiskos

Such as?

-Any consequent which is false proves the antecedent
-(B∧¬B) is a consequent which is false
∴ (B∧¬B) proves the antecedent — Leontiskos

I think that is a valid way to frame it. The thing about (B∧¬B) is that, differently from other formulas, it is always False.

A→FALSE — Leontiskos

I don't think that is logically rigorous. As you say, it is not a term in classical logic, and for good reason.
If you want to say A always implies False, A→(B∧¬B) is good for that. While A→¬(B∧¬B) is "always implies True".

Another way to read the first argument, and the one I prefer*, is as follows:

A→ABSURD
∴ "A cannot be affirmed" — Leontiskos

If A implies a contradiction, not-A can be stated from LNC.
Dogs are fish. Fish, among other things, is defined as not-mammals. Dog is defined, among other things, as mammal. So we end up with "A mammal is not a mammal". Thus, "dogs are fish" has to be false, so "dogs are not fish" has to be true from LNC.

"... cannot be affirmed" does not stand to me as useful, as the LNC + LEM don't accept a third value.

I'd like to explore this idea next:

I think that "A does not imply B" can't even be put in terms of logic, because "A does not imply B" conveys no information.
— Lionino — Lionino

When we state "A does not imply B" as the first premise, we can't conclude B from {A as a second premise}, or A from {B as a second premise}, but we can conclude A from A, and B from B; we also can't conclude ¬A from ¬B, ¬B from ¬A, A from ¬B, B from ¬A, or anything at all besides the second premise. To be sure we are not getting into a contradiction, we must make sure we can't conclude ¬B from B and vice-versa either.
So, with that in mind, I think that "A does not imply B" could simply be put logically as: C. A dummy formula.
C, A does not entail B.
A does not imply B; A; B is not a conclusion.

C, A does not entail ¬A.
A does not imply B; A; ¬A is not a conclusion.

C, A does not entail ¬B.
A does not imply B; A; ¬B is not a conclusion.

C, A entails A.
A does not imply B; A; A is a conclusion.

C, ¬A does not entail B.
C, ¬A does not entail A.
C, ¬A does not entail ¬B.
C, ¬A entails ¬A.
A does not imply B; ¬A; ¬A is a conclusion.

Retracted. It is dumb.

The antecedent of a negated material conditional is always true, and this goes back to my point in the edit you may have missed above. — Leontiskos

More important than that

A negated material condition is only True when the antecedent is True and the consequent is False. When we know that ¬(A→B) is just (A∧¬B), it becomes obvious.

I'd like to explore this idea next:

I think that "A does not imply B" can't even be put in terms of logic, because "A does not imply B" conveys no information. — Lionino

And in each of the invalid cases if "B" could be made necessarily false they would presumably hold. — Leontiskos

A→(B∧¬B), ¬B does not entail anything besides the statements themselves.
(A→B)↔¬A, ¬B does entail however (A→B)↔¬A, even though (A→B)↔¬A is not True for any B, only when B is False.

Checking the natural language equivalent of logical terms.

Formula: A→B
Reading: A implies B
Checking: A→B, A |= B is valid
The reading seems fine.

Formula: ¬(A→B)
Attempted reading: A does not imply B.
Checking:
¬(A→B), A |= B is invalid.
¬(A→B), B |= A is valid.
¬(A→B), B |= ¬A is also valid.
That is because ¬(A→B) is always False when B is True, so taking ¬(A→B) as a true premise then taking B as a second true premise gives a contradiction, from where everything follows.
Same thing for ¬(A→B), ¬A |= ¬B and ¬(A→B), ¬A |= B, the formula ¬(A→B) is only ever True when A is True, so ¬A gives a contradiction here.
¬(A→B), ¬B |= A is valid.
A does not imply B, B is false, therefore A?
And that is not due to a contradiction, as ¬(A→B),¬B|=¬A is not valid. The argument ¬(A→B), ¬B |= A is valid by itself.
Therefore it seems to me that "A does not imply B" is also not a natural language interpretation of ¬(A→B), and I can't think of any interpretation for it. Likewise, I can't come up with any formula for "A does not imply B".
For ¬(A→B), Claude 3.5 gave me 'A more precise reading would be: "There's a case where A is true and B is false, and B is indeed false, therefore A must be true."', which makes sense as it is the same as A ∧ ¬B.
I think that "A does not imply B" can't even be put in terms of logic, because "A does not imply B" conveys no information.

Difference when the consequent is a contradiction:
(¬A→B)↔¬(A→B) is not valid.
(¬A→(B∧¬B))↔¬(A→(B∧¬B)) is valid.
So when the consequent is a contradiction, the ¬ may be pushed in. But when the consequent is a normal statement, you can't.

Does this support my claim that what is at stake is something other than a material conditional? The negation does not distribute to a material conditional in the way you are now distributing it. — Leontiskos

Well I forgot what was the stake of the discussion you all are having, but

So I guess that, in order to say "A does not imply a contradiction", we would have to say instead (A→¬(B∧¬B)). From there things start to make more sense.

Since ¬(A→(B∧¬B)) does not translate to "A does not imply B and not-B". I have to fix my post above. — Lionino

The idea here is that ¬(A→(B∧¬B)) can't become (A¬→(B∧¬B)), if such a thing were proper writing, it can only become (¬A→(B∧¬B)). When we want to deny the implication of the contradiction, we have to write (A→¬(B∧¬B)). However that is what happens when you have a contradiction as the consequent, I will see what happens when you have P (whatever) as a consequent.

From ~(A -> (B & ~B)) we infer that A implies no contradictions.

From (~A -> (B & ~B)) we infer that A implies no contradictions. — TonesInDeepFreeze

My conclusion was that, if we infer that, the logical sentence is at odds with common sense. Because ¬(A → (B ∧ ¬B)) (is true) entails A (is true). Well, if from ¬(A → (B ∧ ¬B)) we infer that A implies no contradiction, from the fact that A implies no contradiction we may conclude that A is true. When we think about it, prima facie just because something does not imply a contradiction it doesn't mean that it is true. So this inference is at odds with reason.

Because of that, and from the fact that ¬(A → (B ∧ ¬B)) ↔ (¬A → (B ∧ ¬B)), I would rather read both as "not-A implies a contradiction". From there, it is much more clear how either of them entails A is true. If the contrary of something is a contradiction, surely that something must be true (from LNC) — which is exactly along those lines that proofs by contradiction work in mathematics.

Therefore, I think a more intuitive reading of ¬(A → (B ∧ ¬B)) is "not-A implies a contradiction". While it is from A → ¬(B ∧ ¬B) that we may infer that A does not imply a contradiction. Truly, as A → ¬(B ∧ ¬B) does not entail A. Naturally, the natural language understanding of logical formulas does not change their validity or anything that matters, but an intuitive reading is essential to know in what contexts the formula may be applied.

A full breakdown of these relationships between the logical formulas and the natural language statements, with examples, is here in the quoted post:

From there things start to make more sense. — Lionino

Kazantzakis once said that if he were awarded the Nobel Prize in Literature, he would only accept it if he could share it with Sikelianos. — javi2541997

A noble soul. Rare nowadays, but being Greek it doesn't surprise me.

embraced aspects of the Dark Enlightenment, a movement — frank

LOL. Wikipedia and MSM truly are Cocomelon for politics-brained Millenials and Gen-Xers.

And yet now Catholics take the blood at Mass every week. — Count Timothy von Icarus

It seems it is up to the father if the blood is given. At least in my eucharisty, we, and all other 12 year olds, had wine. But in every other occasion the father only gave bread to the attendees.

Regardless. The crusade had political motivations too. If utraquism is orthodox or heterodox, I would give up the question to a priest instead, if the matter can even be categorised in either. Not all doctrines are equally essential after all.

Ontological assumptions are what foundations are made of — Metaphysician Undercover

Some would disagree. But it is quite possible.

and Platonism provides the assumptions required for formalism, the idea of pure form. — Metaphysician Undercover

That is not true for every formalist. If you want to know why, look it up.

"If A implies B & ~B, then A implies a contradiction" is true, but it is a statement about the sentences, not a translation of them. — TonesInDeepFreeze

Yes, granted. I used the word "translation" wrong in basically all of my posts. I meant "is a true statement about..." instead.
Now, the conclusion that I arrived at is that "A does not imply a contradiction" is not an accurate statement about ¬(A→(B and ¬B)), it would be a true statement about (A→¬(B and ¬B)) instead. When it comes to ¬(A→(B and ¬B)), as it is the same as (¬A→(B and ¬B)), "not-A implies a contradiction" is a true statement about it.

I was playing the game, but people brought up philosophy of mathematics so I took the chance to talk about it — better than the pointless merry-go-rounds about ethics we often have here.

I'm not sure exactly what is "orthodox" here. — Count Timothy von Icarus

Whatever is given as doctrine by the Catholic Churches.

In any case, the doctrine was formally condemned in the first of the famous anathemas pronounced at the Council of Constantinople in 543: Ei tis ten teratode apokatastasis presbeuei anathema esto [See, also, Justinian, Liber adversus Originem, anathemas 7 and 9.] The doctrine was thenceforth looked on as heterodox by the Church. — Catholic Encyclopaedia

.

Because the goal is deification and once deification has occured the will is not corruptible. — Count Timothy von Icarus

Alright. Don't Mormons too believe they will become godlike themselves — leaving aside the whole planet thing?

Catholic theologians dance around this issue quite often — Count Timothy von Icarus

Though I was raised Roman Catholic, if I were today to choose between Churches, it would be Orthodox. The issue with filioque and Augustine's insufficient Greek skills don't please me. On the other hand, the matter is far from settled, and I personally feel the Eastern Orthodox feel some historical resentment towards Roman Catholics.

SO I supose one question is, can such an argument be constructed? — Banno

Keeping in mind that an argument broadly understood is not the same as a proof or syllogism, such an argument could be constructed, as whether a sound argument is persuasive is up to the receiver; the issue is that the argument(s) presented here is(/are) not sound.

Thinking a bit more about this, it does seem like discovering a category or theorem of a theory or its objects spells some trouble to someone who thinks math is purely invented, but under some conditions. For conceptualists, who think that mathematics is invented by the human mind, it is not clear how one could derive a theorem from existing facts without referencing something that was already set in place, independent of a mind — with that reference, it is no longer purely a mental activitity. Though they could make up arguments, like that mathematical truths are necessary from the axioms, but then their distinction from formalism/nominalism is not so clear cut. Even then, one would wonder, what is it that makes the fact necessarily follow from the axioms? Perhaps the laws of logic, but then we are back to Russell trying to reduce the foundations of mathematics to logical statements, and then Gödel...

No because you are losing information as to what C stands for. B and notB will always be either True and False, or False and True. B and C can be any combination.

notB is just ¬B
no ambiguity

https://www.nytimes.com/interactive/2023/climate/extreme-summer-heat.html?unlocked_article_code=1.000.e-Vm.9B61_NeAn8_Z&smid=url-share — Mikie

"There is a graph saying it is extremely hotter now compared to 70 years ago!"
There is a graph saying people with left-wing views have lower testosterone in average than right-wingers. The difference is that the latter is scientifically presented.

So you don't accept that 7=7? — Wayfarer

Outstanding.

My view is that numbers are real, but not physically existent. — Wayfarer

By itself, this could be framed within a conceptualist framework, where mathematics is reduced to psychology — not just platonism.

If you point to a number, '7', what you're indicating is a symbol, whereas the number itself is an intellectual act. — Wayfarer

The distinction between type and token.

And furthermore, it is an intellectual act which is the same for all who can count. — Wayfarer

Is it though? You may say this because you are a platonist, so you believe there is some unambiguous universal accessible to all.

¬(a→(b∧¬b)) |= a
A not implying a contradiction does not mean that A.
So ¬(a→(b∧¬b)) can't be read as A not implying a contradiction

But (a→¬(b∧¬b)) can be read as such, and it does not entail A. Thus "A does not imply a contradiction" is (a→¬(b∧¬b)), not ¬(a→(b∧¬b))

Elvis is not a man – ¬A
Elvis is a man does not imply that Elvis is both mortal and immortal – ¬(A → (B and ¬B))
Therefore Elvis is a man – A
¬A, ¬(A → (B∧ ¬B)) entails A. That doesn't make sense — Lionino

So ¬(A → (B∧ ¬B)) is the same as (¬A) → (B∧ ¬B), which may be read as "Not-A implies a contradiction", it can't read as "A does not imply a contradiction". — Lionino

Elvis is not a man – ¬A
Elvis is not a man implies that Elvis is both mortal and immortal – ¬(A → (B and ¬B))
Therefore Elvis is a man – A
¬A, ¬(A → (B∧ ¬B)) entails A, from contradiction everything follows. — Lionino

Elvis is not a man – ¬A
Elvis is a man does not imply that Elvis is both mortal and immortal – (A → ¬(B and ¬B))
These two do not entail that Elvis is a man. — Lionino

¬◇(a → (b∧¬b)) entails □a — Lionino

Something about this is that the more general ¬◇(a ↔ (b∧¬b)) |= □a is also true. If we read ¬◇(a↔(b∧¬b)) as "It is not possible that A is False", ¬◇(a → (b∧¬b)) |= □a starts to make a bit more sense.

□¬(a→(b∧¬b)) and □(¬a→(b∧¬b)) are the same formula
https://www.umsu.de/trees/#~8~3(a~5(b~1~3b))~4~8(~3a~5(b~1~3b))
So at least my reading is correct.
The issue with "It is necessarily not the case that A implies a contradiction" is that, if we remove the □ from □¬(a→(b∧¬b)), we end up with ¬(a→(b∧¬b)), and this can't be read as "It is not the case that a implies a contradiction".

But I guess it can be solved in a similar way:
¬◇(a→(b∧¬b))↔□¬(a→(b∧¬b)) is valid
¬◇(a→(b∧¬b))↔□(¬a→(b∧¬b)) is also valid
Since ¬◇(a→(b∧¬b)) is the same as □(¬a→(b∧¬b)), it can be read as "It is necessary that not-A implies a contradiction". From that alone I think we can accept that it follows that necessarily A.
So, since ¬◇(a→(b∧¬b)) would be read by many as "It is not possible that A implies a contradiction", is that the same thing as "It is necessary that not-A implies a contradiction"? If not, "It is not possible that A implies a contradiction" is not a correct reading of ¬◇(a→(b∧¬b)).

I would guess so. But then the issue has come back, just because something can't possibly imply a contradiction, does that make it necessarily true?
Besides, ¬◇(a→(b∧¬b))↔◇(¬a→(b∧¬b)) is invalid, so the issue can't be solved like the original one was.

But with that in mind
¬◇(a → (b∧¬b)) entails □a
How should we read this in English? Because "{It is not possible that A implies a contradiction} entails A is necessary" is not obviously right.

Let's hope your interpretation is the correct one, though I think they left it ambiguous on purpose.

I gather you worked through this? Nice. — Banno

Yea, a→(b∧¬b) can be read as "A implies a contradiction" but ¬(a→(b∧¬b)) cannot be read as "A does not imply a contradiction", it is read instead as "not-A implies a contradiction". "A does not imply a contradiction" would rather be (a→¬(b∧¬b)). So the opposite in natural language is not the same as the opposite in logical language, in this case.

For a religious conservative? Yes. — RogueAI

Odd, Mikey doesn't come across as one.

It is strange that Discord would just purge the account.

https://www.independent.ie/irish-news/its-too-many-people-in-too-small-an-area-row-brewing-over-plans-to-house-280-migrants-in-tipperary-village-of-165-people/a953110517.html

When I discovered the conditions under which infinite compositions of parabolic transformations converge to their fixed points that was a discovery based upon a creation. — jgill

Though I am completely out of my depth when it comes to the example you brought up, it seems that is a case of one discovering a consequence of one's invention — the consequence was invented, just unintentionally.

But "discovery" and "invention" can be sifted by asking if the system in question would still be true had humans never put in paper. If so, discovery; if not, invention.

https://www.theverge.com/2024/7/15/24198953/donald-trump-assassination-attempt-shooter-discord-social-media

Strange