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

Is that the right English translation of that?

Keep in mind that ¬(B and ¬B) is equivalent to B or ¬B - would you say A → (B or ¬B) can be worded as "Elvis is a man does not imply that Elvis is both mortal and immortal"?

Depending on circumstances, it might make more sense to say something like "A can't be proven to imply a contradiction" rather than "A doesn't imply a contradiction". Classic symbolic logic is very constraining, and it's hard to express certain things that seem simple in natural language.

the goal anyway is to translate "A does not imply a contradiction", not any other phrase. — Lionino

This might seem crazy to you, but I would translate that as just A, or in other words, A is true. If you are stating, for a fact, in classical logic, A definitely doesn't imply any contradictions, and we also know that false statements necessarily do imply contradictions, then the only way to say "A definitely implies no contradiction" is to say "A is true"

https://www.umsu.de/trees/#~3a~5(a~5~3(b~1~3b))


https://www.umsu.de/trees/#~3a~5(a~5(b~1~3b))

I would reword it from "a does not imply a contradiction" to "a implies this particular non-contradiction".

And when a is false, it implies everything, contradictions and non contradictions.

What's the general beliefs here regarding Trump's culpability for the infamous events of January 6?

The two main takes are:

he incited what happened
he didn't incite what happen

each of those takes usually has two subtakes, ...and it was good/acceptable, or ...and it was bad.

I have a kind of in-between take when it comes to the two main takes - I'm not convinced he did incite it, but I am convinced his actions after it started make him ethically culpable for it anyway. His actions after it started, reportedly, are he watched it unfold on TV, just sitting there watching it, and people around him kept asking him to make an announcement to stop it, an announcement that would hopefully reach the people at the capitol somehow, a request for them to stop and go home. And when he was requested to do that, repeatedly, he refused.

This makes me think that one of the two situations is the case:
1) he did deliberately incite what happened, even if his literal words allow for some plausible deniability
2) he didn't deliberately incite what happened, but *he was perfectly happy to see it unfold anyway*

There's not a good argument to be made that he didn't want it to happen, because he's a Leader, allegedly, and as a leader if he didn't want it to happen, and he didn't lift a finger to stop it, then... that's no leadership at all. If you want to excuse his lack of action, you cannot simultaneously believe he's a good leader. So either he wanted it to happen, or he's an exceptionally poor leader, not both. And it's clear to me that he IS able to get people to follow him, so if it was his will to stop what happened, he absolutely could have. He didn't want to stop it.

But is that criminal? If he has (a) plausible deniability in the words that led to the riot, and (b) just failed to do anything to stop it, is that criminal? Should it be?

Obviously if you think storming the capitol was good or ethically neutral, then that question doesn't really matter. But if you think it was bad, what happened at the capitol, then the obligation of a person to stop something bad from happening does matter. Some people don't beleive in obligations like that - some people believe you don't have to stop anything bad from happening if you didn't directly make it happen, and his plausible deniability in his words at the rally that preceded the storming arguably give the defense that he didn't directly make it happen.

So, how much plausible deniability does he have for what happened?
And, regardless of that deniability, was what happend *bad*?
And what kind of responsibility does he have given his refusal to lift a finger to stop it?

One way of approaching the problem might be thinking about the fact that wars are fought - some wars - ostensibly over freedom. Which means that a huge portion of humanity thinks it's sensible to take and give lives in exchange for freedom. So this seems to be an example of people valuing freedom over some other things.

So the challenge is, I guess, proving it.

I don't think the challenge is in principle possible to provably solve, because value doesn't seem to be a universally objective measure. In other words, I don't think there's an objective answer to "How many lives are worth exchanging for how many peoples freedom?" just like there's no objective answer to "How many scoops of chocolate are worth 5 scoops of vanilla?"

Since one entails the other but other does not entail one, we may say that everytime «A implies a contradiction» is false, «A does not imply a contradiction» is true; but it is not everytime «A does not imply a contradiction» is true that «A implies a contradiction» will be false. Therefore there is an assymetrical relationship between the two statements quoted.
The prover confirms my intuition:
(a→¬(b∧¬b)) does not entail ¬(a→(b∧¬b))
¬(a→(b∧¬b)) entails (a→¬(b∧¬b)) — Lionino

Which one of the above phrases are you saying is the english translation of (a→¬(b∧¬b))?

It's either «A implies a contradiction» is false
or «A does not imply a contradiction» is true, right?

But consider this:
((a → ¬(b∧¬b)) ∧ ¬a) → (a → (b∧¬b)) is valid

(a→¬(b∧¬b)) doesn't actually stop a from implying a contradiction - it can be assumed true, and still be the case that a implies a contradiction.

¬(b∧¬b) just means (b v ¬b)

It seems plausible that:

(φ^~φ) takes on the meaning of <explosion> as the antecedent of a modus ponens
(φ^~φ) takes on the meaning of <reductio-rejecton> as the penultimate step of a reductio
(φ^~φ) takes on the meaning of <false> as the consequent of a modus tollens

Weren't you mocking me earlier in the thread for relating this side of logic to the principle of explosion? Or was that someone else?

Assuming contraposition and MP is the same as assuming MT. — Lionino

This of course makes the argument you brought up for MT circular. That's fine, we can move past that and find one that obeys the rules presumably.

so there are more rules to the game then, apparently.

Rule 1. Don't assume mt.
Rule 2. Don't simultaneously assume contraposition and MP

Can I assume MP if I don't also assume contraposition? Can I assume contraposition if I don't also assume MP? Are there any more rules you haven't explicitly stated yet?

Are there any proofs of MT that obey the rules of the game we're playing? Obviously the one you've been talking about doesn't obey .

My proof did not assume mt, it did assume contraposition and MP.

If you believe that's the same as assuming mt, then that means the proof of mt that uses those two assumptions is circular.

yes, using MP, not mt.

If it's circular, fine, give me one that isn't circular. I assumed, perhaps wrongly, that when I asked you for a proof of mt that you like, that you wouldn't like an explicitly circular one.

Give me a proof of mt you like that isn't circular.

so you believe the MP+contraposition argument is circular? It's just using mt to prove mt?

You want a proof of some argument Y that doesn't assume modus tollens.

You presumably have a proof of Modus tollens that you like, that doesn't itself assume modus tollens.

Whatever format of argument that proof takes - that argument that doesn't assume modus tollens, but proves Modus tollens - I can use that exact same format of argument to prove Y similarly without assuming Modus tollens.

I've already done it above, but I can do it again

if you give it to me I will show you how it makes perfect sense

you want a proof of ρ→(φ^~φ) , therefore ~ρ that doesn't assume modus tollens, if you give me your preferred proof of Modus tollens I can give that to you.

didn't really talk about proving MT at any point, so I don't know why you are asking that. — Lionino

You don't know why I'm asking what?

MT can be derived from MP and contraposition — Lionino

Ok, I'm going to assume you mean this proof (the one wikipedia lists as "Via contraposition"):

1
P→ Q (Given)
2
¬ Q (Given)
3
¬Q →¬P (Contraposition (1))
4
¬ P (Modus ponens (2,3))

This is the proof of modus tollens that you like - it proves modus tollens without assuming it, correct?


So it's pretty straight forward to use the same format, I'll take my previous argument which assumes Modus Tollens:

ρ→(φ^~φ) (premise)
~(φ^~φ) (law of non contradiction)
:. ~ρ (modus tollens)

And reformat it to be in the style above, the proof that you like of modus tollens that doesn't assume modus tollens:

ρ→(φ^~φ) (premise)
~(φ^~φ) (law of non contradiction)
~(φ^~φ)→ρ (contraposition)
:. ~ρ (modus ponens)

I'm asking you what proof you like - that's not a claim that mine is a proof of that. What proof of Modus tollens do you like?

I didn't say mine was, are you reading the words I'm posting?

what proof of Modus tollens do you like? We can prove φ→(ψ^~ψ)⊢~φ without assuming Modus tollens is the case, but by instead directly using the proof of Modus tollens.

if not by modus tollens? I don't think we do. — Lionino

Ok so we're playing a game (I don't mean that pejoratively, I like games) where we have to prove the conclusion without using modus tollens, is that right?

What are the rules of the game? Are we allowed to use the rule of non contradiction?

You said before that the proof for modus tollens is easy - does that proof obey the rules of this game? If so, which proof of Modus tollens do you like? There are multiple, I want to make sure I'm using the right one.

I will play this game, if you answer my questions then we can have a solution.

This is perhaps my favorite proof for the modus tollens thus far. The question is whether that second step justifies the modus tollens. — Leontiskos

This isn't a proof of Modus tollens. This is a use of Modus tollens.

You've been asking for 12 pages for a proof of Modus tollens?

The problem is that modus tollens can be proven syllogistically quite easily, but how do you prove that you may derive ~ρ from ρ→(φ^~φ)? — Lionino

ρ→(φ^~φ) (premise)
~(φ^~φ) (law of non contradiction)
:. ~ρ (modus tollens)

1. a → (b ∧ ~b)
2. If b is true (b ∧ ~b) is false. If b is false (b ∧ ~b) is false, so (b ∧ ~b) is false.
3.~a → ~(b ∧ ~b) - contraposition (1)
4. ~a - modus ponens (2,3) — Count Timothy von Icarus

That's not how contraposition works.

Edit. I see you corrected yourself already, nevermind.

Hey you got it!

didn't, you are referring to this ↪Lionino, which I already said was a copypaste mistake, it has been edited. I don't see what the issue is. — Lionino

The issue is you said you never wrote it, but you did write it. I understand it's a mistake. Therefore it's not correct to say you never wrote it, it's correct to say you wrote it by mistake.

"Any operator" is not any mathematical operator you want. — Lionino

I don't know the rules of that game, my bad

It is like 2(x*y)=2x*y, but 2(x*y)≠x*2y — Lionino

This doesn't make sense if * is "any operator" either. Replace * with + and 2(x*y)=2x*y is not true

I don't see where I did that. — Lionino

You wrote
¬(A → B) is the same thing as ¬A→B

It is like 2(x*y)=2x*y, but 2(x*y)≠x*2y — Lionino

Is * multiplication here? I don't think this is right either.

I never replied positively to the question — Lionino

Well you gave what certainly looked like an affirmation. If I ask you "is lemonade your favourite flavour", and you say "lemonade is the same as my favourite flavour", most people are gonna think that's pretty much a "yes" to the question.

understand that you'd think that B∧¬B should be able to be replaced by any proposition P — Lionino

Me? You understand that I think that?

But what just happened is that you did that, and I told you it's incorrect...

¬(A → B) is the same thing as ¬A→B — Lionino

That's what I was asking, thank you.

I don't believe that's correct.

I was misplacing the associativity of the → operator.
So ¬(A → (B∧ ¬B)) is the same as (¬A) → (B∧ ¬B) — Lionino

Do you believe for, for all statements (A -> B), you can do ¬(A -> B) and transform that into ¬A -> B?

material implication is an example of the principle of explosion — Leontiskos

I don't think I claimed that. But as you're eager to reject basic reason, I'm not going to be one to stop you.

Another way to think about it is, "The only way you can be CERTAIN that A doesn't apply a contradiction is if you know A is true."

We are very far beyond Wikipedia at this point. At this point one can no longer simply appeal to authorities and logic machines. — Leontiskos

No, you asked for the rule of inference from classical logic - it's right there, common knowledge in wikipedia. I don't see any good reason why my answer should be considered unacceptable, other than you just don't want to accept it. You asked for the rule, that's it.

What rule of inference in classical logic are we appealing to? — Leontiskos

Funnily enough, the rules of inference we're appealing to are in fact the very first ones listed on the Wikipedia page:

https://en.wikipedia.org/wiki/List_of_rules_of_inference
Reductio ad absurdum