From stackexchange:
"The phrase "vacuously true" is used informally for statements of the form ∀a∈X:P(a) that happen to be true because X is empty, or even for statements of the form ∀a∈X: P(a)→Q(a) that happen to be true because no a∈X satisfies P(a). In both cases, it is irrelevant what statement P(a) is."

"If the Baltic sea is salty, then the Eiffel Tower stands." According to material implication this is a perfectly good statement, but according to English it is foolish. There is nothing which surpasses this sort of statement according to material implication: the antecedent is true, the consequent is true, and therefore the implication is true. What more could we ask? But for the natural speaker what is lacking is a relation between the two things. What is lacking is a relation between the saltiness of the Baltic Sea and the standing-ness of the Eiffel Tower. — Leontiskos

Ok, I see. In the first page Flannel jesus brought up the same dilemma.

"First-order claims and second-order rules of discourse." — Leontiskos

But what is second-order rules of discourse?

Take a look at these examples from Russell. ϑ ⊧ ϑ and ϑ & ϒ ⊧ ϑ might seem to be candidates for logical laws one might expect to have complete generality.

Identity: ϑ therefore ϑ;: a statement implies itself. But consider "this is the first time I have used this sentence in this paragraph, therefore this is the first time I have used this sentence in this paragraph"

Elimination: ϑ and ϒ implies ϑ; But consider "ϑ is true only if it is part of a conjunction". — Banno

:gasp:

If self-transcendence can happen in the afterlife, why bother keeping us down on Earth to live horrors?

At the end of the day the English sense of implication simply isn't truth functional. It is counterfactual in a way that material implication is not. — Leontiskos

Elaborate.

metabasis eis allo genos — Leontiskos

Passing to another kind? What kind?

In English, on the other hand, we only say, "If P then Q," when we believe that the presence of P indicates the presence of Q. The English has to do with a relation between P and Q that transcends their discrete truth values. One way to see this is to note that an English speaker will be chastised if they use the phrase to represent a correlation that is neither causative nor indicative, but in the logic of material implication there is nothing at all wrong with this. — Leontiskos

Yes, that sounds about reasonable.

"Every instance in which A is true is an instance in which B is true"

equivalent with:

"There is no instance in which A is true and B is false."

If A is false in an instance, then that is an instance in which it is not the case that A is true and B is false. — TonesInDeepFreeze

Ok, that is true.

Negation is not at issue. — TonesInDeepFreeze

Let's go with "If A then B" if and only if "Every instance in which A is true is an instance in which B is true".

V: ¬(A→B)
X: It is not the case that if A then B.
Y: It is not the case that every instance in which A is true is an instance in which B is true.
Z: It is not the case there is no instance in which A is true and B is false.
W: There is an instance in which A is true and B is false.

Do we agree v, x, y, z, w are all the same?

If ¬(A→B) is correctly understood as "There is an instance in which A is true and B is false", that, in English, doesn't tell us anything about whether there is an instance in which A is true and B is true or all the other cases, only tells us that A=1 and B=0 returns 1.

By the truth table of ¬(A→B), every instance is false except when A is true and B is false. I think, if anything, that is better stated as "There is no instance in which A isn't true and B isn't false"; which, in English, I think is the same as "A is always true and B always false".

Someone else also gave the suggestion of A→B as "there is no A without B" and ¬(A→B) as "there is A without B".

Which is better and why? — I like sushi

Better for what?

"but no" does not work because bongo fury's suggestion for ¬(A→B) was "A without B", which I find pretty good, so the connective I had to use must've been "without".

I don't think that "It is not the case that" is usually ambiguous. — TonesInDeepFreeze

We are not in disagreement.

"If A then B" is understood differently by different people in different contexts.

So any ambiguity in "It is not the case that if A then B" stems from "If A then B".

So specify what you mean by "If A then B", then you will have specified what you mean by "It is not the case that if A then B". — TonesInDeepFreeze

Of course.

(1) "If A then B" if and only if "Every instance in which A is true is an instance in which B is true".(material conditional) — TonesInDeepFreeze

The issue is that the material conditional is not just that. A→B is also true whenever A is false. So by stating «"if A then B" if and only if "every instance in which A is true is an instance in which B is true"» you are not making "if A then B" equivalent to A→B. If you decide that «"if A then B" if and only "Every instance in which A is true is an instance in which B is true, or every instance in which A is false"», which would make "if A then B" equivalent to A→B, we are just back to the old problem. If A then B is ambiguous, as you yourself said:

"If A then B" is understood differently by different people in different contexts. — TonesInDeepFreeze

Saying A→B is "if A then B" does not provide a solution to the matter of unambiguously converting A→B to English.

First, that is not idiomatic — TonesInDeepFreeze

If I had said "There is rain without there being wetness" you'd have complained that "there being wetness" cannot work as a standalone proposition.

Your thoughts? — I like sushi

People can derive psychological comfort from determinism as well as from libertarianism. And?

The most popular argument against hardline determinism boils down to: "well if determinism is true why bother doing anything?", exempli gratia: the second post of this thread. It makes me think that the preference for libertarianism often stems not from a belief in the truth of the position but from the same place as a belief in the afterlife; while compatibilism is the in-between cope.

The resulting language is full of issues, collectively known as the foundational crisis in mathematics, which is clearly also a foundational crisis in logic. — Tarskian

lol

history amply shows, imo, that 'religion' is required only (or at least mostly) for herding sheep — 180 Proof

Modern affairs and lived experience are telling me that people are broadly still sheep that need herding.

When we speak to another we have certain expectations concerning the response, which will never be precisely fulfilled. — Joshs

That is true but I don't see the connection with my question.

Likewise when we think to ourselves we are communicating with an other, since the self returns to itself slightly differently moment to moment. — Joshs

Sure.

We always end up meaning something slightly other than what we intended to mean. — Joshs

I think that is somewhat paradoxical. But still that doesn't answer the question of what it is that we are communicating to ourselves.

Your post is all about things being lost in translation (trans-lation), but what is it that is being lost?

I don't get it, but I'm confident I could get it using natural English. Is there a substantial difference? — javi2541997

Here is the incongruence:

if ¬(A→B) is true and B is false, A is true. If we read it as such, we would have it "If A does not imply B, and B is false, A is true". Surely that can't be the case, otherwise obviously false sentences such as "An equation being quadratic does not imply it has real solutions, the equation does not have real solutions, therefore it is quadratic." would follow. So we can't read ¬(A→B) as "A does not imply B". — Lionino

A -> B is false in a given interpretation if and only if (A is true in the interpretation and B is false in the interpretation).

A |= B is true if and only if every interpretation in which A is true is an interpretation in which B is true.

A |- B iff and only if there is a derivation of B from A.

Example:

"If Grant was a Union general, then Grant was under Lincoln." True in the world of Civil War facts. But false in some other worlds in which Grant was a Union general but, for example, Lincoln was not president.

"Grant was a Union general" entails "Grant was under Lincoln". Not true, since there are worlds in which Grant was a Union general but, for example, Lincoln was not the president.

"Grant was a Union general" proves "Grant was under Lincoln". Not true, since there are not other premises along with "Grant was a Union general" to prove "Grant was under Lincoln". — TonesInDeepFreeze

That is helpful. I think it relates to and clarifies FJ's post:

In classic symbolic logic, a -> b is true, according to its truth table, if, for example, a is true and b is true.

(2+2=4) implies (Kamala Harris is a presidential nominee). These is true in classical logic. But it doesn't really match our intuition at all. — flannel jesus

If you are asking what is the most accurate English translation of the intended meanings in ordinary symbolic logic, just put in:

"it is not the case that" where '~" occurs
"if ____ then ____" where '____ -> ____' occurs
"and" where '&' occurs
"or" where 'v' occurs — TonesInDeepFreeze

Yes, I am asking that. I would only detail that ∨ is more appropriately called "__ or __ or both", while ⊻ is "either __ or __".

But let's go with that. Should we read ¬(A→B) as "it is not the case that if A then B"? If so, how should we understand "it is not the case that if A then B"? You said "A does not imply B" is ambiguous in English. Indeed. However, in plain English "it is not the case that if A then B" is also ambiguous:

[1] There are instances in which A is true but B is false.

[2] It is not the case that A entails B (same as above).

[3] It is not the case that A implies B (where 'implies' means the material conditional).

[4] It is not the case that A implies B (where 'implies' means a connective other than the material conditional). — TonesInDeepFreeze

The English phrase "A does not imply B" typically means "There are instances in which A is true but B is false". By your list, that does not mean the same as the material conditional.
If 'it is not the case that if A then B' is to be understood as the third option, we are simply circling back. What is a phrase in English that unambiguously corresponds in meaning to ¬(A→B)?

'rain without wetness', 'wetness', 'rain' are not sentences. — TonesInDeepFreeze

Short for "There is rain without there is wetness". Does that work instead?

See algo bongo fury's proposal that ¬(A→B) can be read as "there is A without B".

Language isn’t simply a tool that we use to access concepts — Joshs

Well.

To think and perceive is to communicate with oneself by way of the world. — Joshs

What are we communicating to ourselves?

The constant comparison with Germany (and other countries) is not genuine because the comparisons are always illiterate. It is instead a sorry attempt from both sides of the political aisle to input culture and history to a country that has none.

Muh Hitler. Muh Germany. Not comparable. If Operation Paperclip hadn't happened, your "country" would be less relevant than Canada. And I don't know who Chris Hedges or Eric Voegelin are, but Chris Hedges' rendition of Voegelin is an idiotic idea. Germany was a diseased society during Weimar with prostitution of all ages and shemale bars. Hitler campaigned exactly against that. How can someone who campaigns against a disease embody the pathologies? It can't, even if Hitler later came to embody different diseases.

Ironically, everything that the "grotesque, marginal" Göbbels said about Burgerland 100 years ago still applies today, and the entire planet would agree:

One is never sure which of two characteristics is more prominent in the American national character and therefore of the greater significance: naivete or a superiority complex. When for example they say things about our region, our surprise at their ignorance is surpassed only by annoyance at their stupid insolence. The less they know about a matter, the more confidently they speak. They really believe that Europeans are eagerly waiting to hear from them and heed their advice.
[...]
They cannot believe that there are cultural values that are the result of centuries of historical development, which cannot simply be bought. It was no bad joke when, after the war, they bought the ruins of German castles and moved them stone by stone to the USA. They really thought that they had purchased a piece of national history embodied in stone, and were naive enough to think that mocking laughter from Europe was respect for the wealth that enabled them to buy what their own tradition and culture lacked.

If someone who is grotesque and marginal can reproduce correct moral judgement of you, there is a lot of soul searching you should, but yet:

We would not say anything if the USA were aware of its intellectual and moral defects and was trying to grow up.

Get off Reddit and pick up a book.

Are the guy that lost his mind and started punching himself in the head in their car on tik tok when Biden announced this? — AmadeusD

I would like to be sent the link of that. Same vibe:

The conclusion still always follows with p=1 and must in a way be "contained" in what one starts with. — Count Timothy von Icarus

I see now.

Everything contained in the conclusion must be contained in the premise; we learn nothing from deduction. — Count Timothy von Icarus

Hence, deduction is informative because it involves communication. — Count Timothy von Icarus

Perhaps mathematics is better thought of in terms of signs and relations instead of identity. — Count Timothy von Icarus

"2+2 has a relation with 4 that is informative."

Something that I can think of is that, for Peano, 4 is exactly S(S(0))+S(S(0)), which 2+2, which is S(S(S(S(0)))). I understand you will then reply that S(S(0))+S(S(0)) and S(S(S(S(0)))) are different things whose identity will be established by a computation.

Now, if I were to play the platonist, I would say that both S(S(0))+S(S(0)) and S(S(S(S(0)))) are symbols that point to the same thing. Just like "2" in Times New Roman and "2" in Arial are two symbols that, to an intelligent mind, reference the same universal. Though you will then reply that I must perform the computation to be able to reference the same universal. In that sense the two are not contradictory.

Were I to play the nominalist, S(S(0))+S(S(0)) and S(S(S(S(0)))) are both symbols that are constructed in relationship to each other (so in that sense your criticism that they are not quite the same is sensible) but that nevertheless are applied the same (in physical theories).

But even, and especially, for the nominalist, I think 2+2=4 can be informative while maintaning an identity. 2+2 and 4 are indeed not the same object, but as soon as you establish 2+2=4, you are giving a hint as to what kind of mathematics you are doing. That is, 2+2=4 doesn't give you all the information you need, as there are different arithmetics where 2+2=4, but 2+2=4 rules out many arithmetics; but that is why we have formulations of those different arithmetics and the person using them needs to know how to use them. Peano arithmetic is the one we use the most in everyday life — doing taxes and splitting a bill. But we use module-60 arithmetic when adding up minutes of time. Within Peano arithmetic, 2+2=4 is the same, so it is indeed not informative within Peano arithmetic, but it is informative when Peano arithmetic is applied in real life: pushing two rocks against two rocks gives you four rocks. Peano arithmetic is applicable to this structure, thus the application gives us information.

The less obvious computations are $\int_{0}^{2}\frac{3x^2}{2}dx$ that gives us 4, but $\int_{0}^{2}\frac{3x^2}{2}dx$ is descriptive exactly of the speed of an accelerating car and knowing that a certain arithmetic also applies to the situation will give us the information of the total traveled distance.

Of course, this might be circular on the application level:

PA applies to a structure.
PA gives information about the structure.
The information we get is assumed in the first premise.

But I know everything you said about computation is still going to apply:

It perhaps depends on how much you embrace pancomputationalism in physics and information theoretic explanations of the other special sciences. If we think number only appears in nature in terms of computation and process then that seems suggestive at the very least. — Count Timothy von Icarus

I don't know anything about either of those theories, so I can't say. But it doesn't feel like this conversation applies to mathematics per se.

His argument is that intuitionist assumptions — Count Timothy von Icarus

Like the denial of LNC?

"Thinking" is a very flexible concept. — Ludwig V

Yes, I think that is key. To establish the relation between thought and language we need a good anatomy of thought, and that is no easy task.

Maybe that is a naïve point, but, since rats and other languageless animals are known to be able to perform mental simulations, doesn't that show in a straightforward manner that language is not necessary for thought?

Yep. Even if you add the irrelevant and contradictory P2, which is going to make everything true anyway. — bongo fury

That was also a mistake, that was supposed to be "B is false" like the first quote, but your point stands.

Rain without wetness
Wetness
Therefore rain. — bongo fury

Ok, so your "A without B" is not that "it is possible to have A without B", but that "there is A without B". I guess that can make sense as ¬(A→B) ↔ (A∧¬B).
See the link on the OP. It says "A→B" means B is a necessary condition for A (this doesn't need to be interpreted in a causal one-directional flow of time sense). How would you put ¬(A→B) in terms of conditions?

No, not because they are in the desert. Because they don't need to give out handouts in order to expand. — Tarskian

What?

And where are they going to grow the food, if they keep expanding? — Tarskian

The food that the city consumes does not come from the fields immediately next to it...

No, no, there is definitely some logic to the madness over there. People like it a lot. They really like the zero-tax strategy, especially when they have a lot of money. — Tarskian

The logic is that UAE accounts for 3% of the world oil supply despite having less than one thousand of a percentage of the world population. And Dubai isn't zero tax. It has import/export taxes, property taxes, service taxes, etc. It just doesn't cut money out of your income directly. If it had zero tax it would collapse, it is not an ancap State.

Most people don't like that shit (pun intended). — Tarskian

So I wonder why every Instagram/Youtube clown wants to move to Dubai, but not engineers and philosophers.

We'll see. Nayib Bukele is trying the same thing in El Salvador. — Tarskian

Nayib Bukele is a successful statesman and El Salvador does not have infinite money (for 150 years) like Dubai. No connection.

On that specifically, here is the exact continuation of the quote I posted:

[...] and then at a later phase, claim to have been conversant with the wider philosophical tradition all along / rediscover the wider philosophical tradition, but still subordinate that tradition to goofy analyticisms and still somehow retain the decontextualized feel of early analytic philosophy.
[...]

It just means that analytic philosophy is less special than it thinks it is. — Ludwig V

The fact that analytic philosophy (vaguely understood because the term really means little) has dumped itself into the same problems that philosophers were talking about 400 years ago with p-zombies (mind-body dualism), supertasks (Zeno), how it feels to be a bat (solipsism) and else, tells us enough; they are not solving any problems, just eating sand on the playground, let them have their fun. Maybe in 120 years they will reinvent Kant.

every new philosophical approach thinks that — Ludwig V

I am not sure if that arrogance is universal. Wittgenstein had quite the temper.

The global poor want to move there because they can get handouts, i .e. free housing, free healthcare, free education, welfare benefits, and so on. — Tarskian

Getting benefits as non-EU in Italy is quite difficult, as I recall. In Denmark especially it is not a thing.

Dubai can keep expanding. — Tarskian

Because it is in the desert. So what?

Leiden or Bologna? I don't think so. — Tarskian

They can... they are both surrounded by fields.

When they land a good job in Dubai, they are "da man" back home. Everyone admires them, because"They did it!"

What you call "slave labor" is very prestigious in places like Bangladesh. — Tarskian

None of that is true. Anyway, Dubai is a bubble built on luck, Bologna and Leiden have 1000 and 500 years of successful history respectively. The model is obvious.

We all know what the rich arabs of Dubai pay Instagram models to do on their bodies. Dubai is much more degenerate than it meets the eye.

If they had found no oil, there is no universe in which Dubai would have been more than a fishing village, like it was 50 years ago.

Thus for ~(p -> q) I'd say, "It is not the case that p implies q." — tim wood

It is not the case that p implies q
Not q
p
That is valid
Yet it doesn't agree with our intuitions.

Or second, relying on the equivalence of (p -> q) <=> (~p v q), I'd say, "Either it is not the case that p, or (it is the case that) q." — tim wood

That is viable as well. But I think that the latter phrase just ends up meaning "P implies Q", which is really "Everytime you see P you also see Q", which is essentially "Either it is not the case that P, or it is the case that Q".

In any case, going back and forth between "logical formulas" and natural language" is always going to be problematic. — tim wood

It is, but proof checkers and logic have helped us check the validity and consistency of many arguments that would be otherwise extremely difficult to verify. So I think it is very much worthwhile to look into how we can bring language into logic.

To be fair, if ¬(A→B) is true and A is false, anything is true. — bongo fury

Good catch. The premises ¬(A→B) and ¬A together are explosive. But ¬(A→B) and ¬B aren't, yet ¬(A→B) and ¬B entail A. It A does not imply B and B is false, can we really infer that A is true?
I have updated the thread to remove the explosion :)

¬(A→B) means A without B. — bongo fury

We can go with that.
P1: ¬(A→B)
P2: B is true
Concl.: A is true

¬(A→B) means A without B
B is true
Therefore A is true

Does that make intuitive sense to you?

What about the following example?
Rain without wetness
Wetness
Therefore rain.

Dubai is a city built on oil money and Indian slave work, not on competence and scientific breakthrough. If every city were like Dubai it would be horrible news for civilisation. If every city were like Leiden or Bologna we would be reaching the stars.

I am talking about primaries as well. The appeal of those candidates over others (for many) was exactly that they were not white men.

The only points that are clear to me are 1 and 4. I can't fathom what 2 would mean, especially. It would be helpful if you could give an example.

It is not about "representation", a sorry concept that is never applied when whites are a minority in a country. It is about that many, perhaps millions, voted on Hillary and Obama because he/she is black/female.

To give some input, I do not understand.

What "ontology of rules"?

This post is so riddled with political and historical ignorance that any recommendation I could give would sound patronising.

If Kamala steps up, she has to reflect deeply about who her VP will be. It must be someone with experience, connections in congress, someone who has been elected before and who is comfortable working under a person of colour president. Someone comes to mind.

Trump will not debate her. She would shred him. Go Kamala. — creativesoul

:rofl:
Coming from the same people who until last week denied that JB has dementia

There used to be something called scholarpedia http://www.scholarpedia.org/article/Main_Page
There still is, but it is dead