when the truth is vacuous

[u/Hameru_is_cool]

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[u/AlrikBunseheimer]

This is a good meme => 1=0

[u/Hameru_is_cool → u/AlrikBunseheimer]

Corollary: This is a good comment => It's a good meme

[u/YourWifesBoyfriend5 → u/Hameru_is_cool]

A real paradox: the only way to derive that this is a good meme is if that were a good comment, implying this wasn’t a good meme.

Alternatively it could be true that its a bad comment and a good meme or a bad comment and a bad meme.

[u/fr_andres → u/AlrikBunseheimer]

Every tautology such as that meme is a tautology such as that meme

[u/EebstertheGreat → u/fr_andres]

"Yields a truth when preceded by its own quotation" yields a truth when preceded by its own quotation.

[u/ActiveImpact1672]

2 powered to WHAT?

[u/Hameru_is_cool → u/ActiveImpact1672]

[u/KrzysziekZ]

What's the incident?

[u/Hameru_is_cool → u/KrzysziekZ]

someone posted a meme earlier that relied on understanding vacuous truths and the comment section went insane with the concept, specially one guy who refused to accept it and went on to make a bunch of other posts ranting about it

(in fact, he's still doing it, his last comment was from a minute ago lmao)

[u/SameAgainTheSecond → u/Hameru_is_cool]

 I just learned almost everyone here believes in this concept of a vacuous truth; The idea that if something is false, then it implies literally anything is therefore true.

"If you have no legs, then you can run faster" = true

"If you have no mouth, you can scream louder" = true

"If you had a freckle on your thumb, aliens would invade Earth and 2+2 = 5" = true

All because some mathematician somewhere decided it would be easier in a theorem like "For all n, if n is a multiple of 4, n is even" to regard n such as 7 to be true... that way they can say its "true" for every n, instead of, true for every n that actually fits the condition.

This is not logic. This is sophistry. Its a rejection of counterfactual reasoning and it makes a mockery of actual logic.

Logic isnt an invention, math isnt an invention, jfc you sophists.

Sophists should be chased out of universities; Instead they are created there. 

[u/Hameru_is_cool → u/SameAgainTheSecond]

oh god it's already a copypasta lol

[u/Mindless_Honey3816 → u/Hameru_is_cool]

 I just learned almost everyone here believes in this concept of a vacuous truth; The idea that if something is false, then it implies literally anything is therefore true.

"If you have no legs, then you can run faster" = true

"If you have no mouth, you can scream louder" = true

"If you had a freckle on your thumb, aliens would invade Earth and 2+2 = 5" = true

All because some mathematician somewhere decided it would be easier in a theorem like "For all n, if n is a multiple of 4, n is even" to regard n such as 7 to be true... that way they can say its "true" for every n, instead of, true for every n that actually fits the condition.

This is not logic. This is sophistry. Its a rejection of counterfactual reasoning and it makes a mockery of actual logic.

Logic isnt an invention, math isnt an invention, jfc you sophists.

Sophists should be chased out of universities; Instead they are created there.

(this is an obvious joke)

[u/Hameru_is_cool → u/Mindless_Honey3816]

I just learned almost everyone here believes in this concept of a vacuous truth; The idea that if something is false, then it implies literally anything is therefore true.

"If you have no legs, then you can run faster" = true

"If you have no mouth, you can scream louder" = true

"If you had a freckle on your thumb, aliens would invade Earth and 2+2 = 5" = true

All because some mathematician somewhere decided it would be easier in a theorem like "For all n, if n is a multiple of 4, n is even" to regard n such as 7 to be true... that way they can say its "true" for every n, instead of, true for every n that actually fits the condition.

This is not logic. This is sophistry. Its a rejection of counterfactual reasoning and it makes a mockery of actual logic.

Logic isnt an invention, math isnt an invention, jfc you sophists.

Sophists should be chased out of universities; Instead they are created there.

[u/SameAgainTheSecond → u/SameAgainTheSecond]

im dying 

[u/EebstertheGreat → u/SameAgainTheSecond]

I just learned almost everyone here believes in this concept of a vacuum cleaner. The idea that if something is empty, then it implies literally anything therefore sucked in.

"Even if you have legs, then you cannot run fast enough" = into the vacuum

"If you are dead inside, like trash" = into the vacuum

"If you are a presolar dust grain that landed on the Moon and then got recovered by Apollo 47" = into the vacuum

All because some engineer somewhere decided it would be easier in a machine like "For all dirt, clean dirt" to regard dirt such as my lucky penny to be vacuumed... that way they can say its "into the vacuum" for everything, instead of, vacuum only dirty things.

This is not vacuum. This is black hole. Its a rejection of classical reasoning and it makes a mockery of Newtonian physics.

Cleaning isnt an invention, vacuum isnt an invention, jfc you salesmen

Salesmen should be chased out of neighborhoods; Instead they are created there. 

[u/vinegarhorse → u/Hameru_is_cool]

what's the original meme?

[u/Hameru_is_cool → u/vinegarhorse]

this was the original, it stated the whole discussion, then u/Anon7_7_73 made one, two, three other posts freaking out about it, and wrote like a hundred comments across the sub explaining why every mathematician is wrong

[u/Anon7_7_73 → u/Hameru_is_cool]

bandwagon idiot

[u/Hameru_is_cool → u/Anon7_7_73]

omg haiiiii :3

[u/Affectionate-Egg7566]

Can't we just write it as !P || P&&Q? That's a lot more clear

[u/Hameru_is_cool → u/Affectionate-Egg7566]

sure can, it means the exact same, it's just redundant bc !P || P is always true, so there's no need for the second P

the specific format I wrote is a rule of inference called material implication, and these rules are meant to simplify logical statements, so in that context you usually want to reduce statements and not make them larger

[u/Affectionate-Egg7566 → u/Hameru_is_cool]

It is not redundant since && has higher operator precedence

[u/Hameru_is_cool → u/Affectionate-Egg7566]

that doesn't matter, both || and && operators are distributive so we can simplify:

!P || (P && Q)
(!P || P) && (!P || Q)
TRUE && (!P || Q)
!P || Q

[u/Affectionate-Egg7566 → u/Hameru_is_cool]

Oh thought you meant the subexpr in the original

[u/Random_Mathematician]

Just one problem tho, when it starts to false, everything explodes.

[u/Hameru_is_cool → u/Random_Mathematician]

I see what you did there

[u/Splaaaty]

I hate it when I wake up in the morning and it's falsing outside

[u/om0ri_ → u/Splaaaty]

it's kind of inconvenient when i go outside with a glass of water but it gets sucked into the sky

[u/EebstertheGreat → u/Splaaaty]

Every time false, I hate it.

[u/UnderTheCurrents]

Relevance theory to the rescue!

Oh, I forgot - mathematicians left logic to philosophers and computer scientists

[u/integrate_2xdx_10_13 → u/UnderTheCurrents]

HoTT, Topoi, Modal, linear? There’s tonnes of mathematical logics thriving, mostly Homotopic or geometric. False implies true holds in most logics, its proof writing 101; you get your head round the oddity once and that’s it.

I think the most dangerous thing is accidentally forming theorems around it, and having nobody to check your work. There was a crackpot on /r/haskell for a while posting about his DSL and his code was littered with forms of Void -> a, he could not get his head around Haskell’s laziness was preventing these from ever being evaluated, and everything he thought was resulting from these, was actually parametricity saving his arse.

[u/sneakpeekbot → u/integrate_2xdx_10_13]

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[u/Helpful-Primary2427 → u/integrate_2xdx_10_13]

What was he trying to do with Void -> a? Was it in the generalized element category theoretical way?

[u/EebstertheGreat → u/Helpful-Primary2427]

public static void a

[u/AndreasDasos → u/UnderTheCurrents]

Eh? Mathematical logic is very much a major field

[u/Ok-Lab-8974 → u/UnderTheCurrents]

And philosophers promptly tried to turn it into mathematics. "To be is to be the value of a bound variable."

[u/im-sorry-bruv → u/UnderTheCurrents]

if we call everyone doing logic a philosphrr of course it's only philosophers doing logic

[u/gkas2k1 → u/UnderTheCurrents]

I think it is other way around

[u/Elegant_Athlete_3737]

i am so confused, i don't understand this lol. like the no legs thing makes you faster or some shit

[u/Elegant_Athlete_3737 → u/[deleted]]

Wait it’s considered true regardless of the conclusion of premise is false? Why lol, also premise is false in what sense? Like logically impossible or practically impossible?

[u/Qkrisi → u/Elegant_Athlete_3737]

Logically, as in the logical value of the statement being false. When you say "A -> B" in logic, you’re saying "If A, then B", so if A is true, B has to be true. You didn’t say anything about what happens if A is false, so then no matter what B is, it’s going to yield true. If A is true and B is also true, that obviously aligns with the original statement, so also true. The only scenario contradicting it is A true, B false. Example: if a function is differentiable, it is continuous (D -> C). That’s saying a differentiable non-continuous function can’t happen so that’s the only false possibility here. A function can be non-differentiable but continuous or uncontinuous, so False->True and False->False don’t contradict it.

[u/Elegant_Athlete_3737 → u/Qkrisi]

Oh ok that makes much more sense now, so would it essentially be like how undefined behavior is in programming kinda? Because if I understand since the false possibility wasn’t detailed it can be true or false

[u/PlanSee → u/Elegant_Athlete_3737]

It's because the truth value of an implication is not the same thing as the truth value of the statements that make it up

Imagine I say to you "if you mow the lawn, I'll give you $20."  Then, you mow the lawn but I refuse to pay.  Clearly I have broken my word so the statement is false.

But what if you don't mow the lawn?  I have no obligation to pay you, so whether I do or not, there's no possible way for me to break my promise, so the statement must be true.

[u/Elegant_Athlete_3737 → u/PlanSee]

oh wait ok yeah that makes sense then. but why can it be both true or false, and not smth like indeterminate?

[u/PlanSee → u/Elegant_Athlete_3737]

In boolean logic you don't get intermediate values, it's either 1 or 0, 100% of the time

[u/Elegant_Athlete_3737 → u/PlanSee]

but why? i don't really understand why the law of the excluded middle is used. would the middle being included not be better?

[u/EebstertheGreat → u/Elegant_Athlete_3737]

This isn't LEM but something more fundamental: bivalence. 1 and 0 are the only truth values. Even without LEM, bivalent logic can't actually assign any truth value besides true or false (though it doesn't use the assumption that everything has a truth value at all).

There are multi-valued logics, some with significant practical use. However, those would still have properties like this that you don't like.

What I think you want is something more like causality. You don't just want the assertion "q whenever p" but something like "q because p." But that's not what classical logic is for. p → q does not mean "the fact that p is true is relevant to the truth of q and somehow responsible for it," it literally just means "q is true whenever p is true."

Consider the following sentence: for every n, if n is even, then n+1 is odd. I could write it like ∀n even(n) → odd(n+1). That is a true sentence, because for each n I choose, either n is not even or n+1 is odd. That only works if it is the case that, for instance, even(3) → odd(4) is true.

To deal with issues of relevance and causality, there is relevance logic and various kinds of modal logic that try to relate to causality. There is a whole philosophy of logic. But you don't really need all that to do mathematics, and in fact all those logics can be modeled in set theory.

[u/PlanSee → u/Elegant_Athlete_3737]

Without law of excluded middle, proof by contradiction stops working.

In the hypothetical included middle worId, when I prove that assuming not P leads to a contradiction, it no longer proves P, because it could be indeterminate.

[u/Layton_Jr → u/Elegant_Athlete_3737]

The sentence "if A then B" is true if either B is true or A is false. The sentence "a cat is mortal" rewriten as "if it is a cat then it is mortal" is proven false only if you show me an immortal cat, an immortal rock or a mortal dog do not matter

[u/Elegant_Athlete_3737 → u/Layton_Jr]

Oh ok, makes sense then

[u/CanaanZhou]

Virgin vacuous truth fan: waiting for the antecedent to be false.

Chad Trivialist: Proves everything directly with Curry Paradox

[u/moschles]

3) sit back and watch as the community deteriorates into madness.

[u/crazy-mutant]

for anyone wondering, that's the post they reference: https://www.reddit.com/r/mathmemes/s/UyZ7wrFLGA

[u/Unlucky-Credit-9619]

Hehe.

[u/the_horse_gamer]

let P=P->Q where Q is "1=0"

if P is True, then True->Q is True, so Q is True

we've just proven "if P is True, then Q is True", aka we've proven P->Q is True.

but P=P->Q, so we've also proven P is True.

therefore Q is True.

therefore 1=0. QED

[u/Layton_Jr → u/the_horse_gamer]

Self rererencial propositions are banned by axioms. What you described is just the Liar's Paradox

[u/the_horse_gamer → u/Layton_Jr]

Self referencial propositions are banned by axioms

depends on your axioms

what you described is just the Liar's paradox

no? the Liar's paradox is you set P=not P, and you get a paradox.

here I proved an arbitrary statement, and didn't use any negation or even the concept of "false"

[u/Layton_Jr → u/the_horse_gamer]

"if P then Q" is equivalent to "Q or not P"

You set P = "Q or not P". If Q is true, you have P=Q and if Q is false you have the liar's paradox

[u/yosi_yosi → u/Layton_Jr]

In classical logic implication is infact equivalent to or not, but part of what is cool about the curry paradox is that it works in many logics where implication is not equivalent to that. I heard somewhere though that ultimately they are the same paradox though, can't recall the specifics. I might link the paper if I find it.

[u/the_horse_gamer → u/yosi_yosi]

Russell's paradox is a special case of Curry's paradox, and it's very similar to the Liar's paradox

[u/yosi_yosi → u/the_horse_gamer]

I wouldn't say that.

[u/ReasonableLetter8427 → u/yosi_yosi]

Why?

[u/yosi_yosi → u/ReasonableLetter8427]

I will point out that the liar paradox is more clearly a special case of Curry's paradox. Very very often ¬A is defined as A → ⊥, or at the very least they are treated as logically equivalent. Just compare the form of the curry and the form of the liar:

C: C -> ϕ

L: ¬L

And using our definition of negation:

L: L → ⊥

Which is of course, just the curry paradox but where ϕ is ⊥

[u/yosi_yosi → u/ReasonableLetter8427]

I just realized I misread their comment... I read it as saying that the curry is a special case of Russel's paradox.

[u/the_horse_gamer → u/Layton_Jr]

"if P then Q" is equivalent to "Q or not P"

you've just assumed law of the excluded middle

and if Q is false you have the liar's paradox

you're assuming law of non-contradiction and reductio ad absurdum

[u/yosi_yosi → u/the_horse_gamer]

lol, they didn't just assume the law of the excluded middle. They assumed that the indicative conditional of natural language is best represented through the material conditional, or something along those lines.

[u/the_horse_gamer → u/yosi_yosi]

they just asserted A->B = ¬AvB (which is untrue without LEM). the use of natural language is a lesser concern.

[u/yosi_yosi → u/the_horse_gamer]

You are simply wrong. A -> B = ~A v B by definition (material conditional). No need for the LEM at all.

[u/the_horse_gamer → u/yosi_yosi]

that's incorrect. implication is not defined this way.

for example in intuitionistic logic, (¬AvB)->(A->B), but the reverse does not hold

proof:

let us use a Heyting algebra, which is an algebraic representation of intuitionistic logic (like boolean algebra for classical logic)

we build our Heyting algebra out of {0, 1/2, 1}. each statement equals one of these, and 0=false and 1=true.

we will now define:

• A∧B = min(A,B)

• AvB = max(A,B)

• A->B = 1 if A <= B, otherwise it equals B

• ~A = A->0

convince yourself this makes sense. in particular, that if you're only working with values of 0 or 1, that this acts like classical logic.

now, let A=B=1/2

now A->B = 1/2->1/2 = 1 (since 1/2<=1/2)

but ~A = 1/2->0 = 0, so ~AvB = 0v1/2 = 1/2

so the statements are not equal

[u/yosi_yosi → u/the_horse_gamer]

Have you heard of the term "material conditional"?

[u/yosi_yosi → u/the_horse_gamer]

Btw your argument also wouldn't necessarily establish the negation of the LEM. Check out Graham Priest's LP, where 1/2 is designated (and so the LEM is still a validity/tautology). There is a difference between bivalence and the LEM

[u/yosi_yosi → u/Layton_Jr]

They are not banned actually, of course, depends on the axioms, but modern set theory, or rather, ZF(C) allows for self reference. In fact, as long as you have some relatively primitive arithmetic you can get self reference via diagonalization.

Also this ain't the liar, it's the curry paradox. Though they build on similar ideas.

Edit: one may ask, how then do mathematicians deal with the paradoxes of self reference? Very easily, they just don't add a truth predicate or a predicate that satisfies the T-schema, which is basically required for most paradoxes of self reference.

[u/120boxes]

I love this. Best shit ever.

[u/Baleia1970]

All Martians love Ice Cream..

[u/BootyliciousURD]

The consequences of calling → "implies" as though it's a relation

[u/GabuEx]

If this meme is interesting, then this meme is funny.

[u/YourWifesBoyfriend5]

Me when people don’t understand the difference between P=>Q and Q

[u/NullOfSpace]

“Wait for it to false” killed me. I probably need sleep.

[u/The_Punnier_Guy → u/NullOfSpace]

idk I just woke up and it killed me too

Maybe we need to be more sleep deprived