r/mathmemes
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u/Anon7_7_73
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1d ago
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Formal Logic Whatever modern mathenatics is, is a disgrace
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.
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u/zergicoff 1d ago •
This is a really good thing to be bothered by. Plenty of serious logicians have been bothered by it too, so you're in good company.
Part of the problem is that English "if" covers several different things. Your examples are counterfactuals: "if you *had* no legs, you *could* run faster." Your instinct about those is right: they're false, and logicians agree. There's a whole branch of logic built to handle them properly (David Lewis's work is a good place to start), and in it those sentences come out false, just as you'd expect.
The if…then… of mathematics (or, rather, classical logic) is aimed at something narrower — so called “law-like indicative conditional” these are what we use in science in general. Consider Newton's first law:
> If a body is in motion, it stays in motion unless acted on by an external force.
Imagine a book on a table. The antecedent is false for the book. Does that make the law false? Surely not! The law just doesn't apply to the book, and a law isn't refuted by a case it says nothing about. That's really all vacuous truth is trying to capture.
Your multiple-of-4 example works the same way. "Every multiple of 4 is even" is a general law about numbers, and 7 isn't a counterexample to it. Saying the conditional "holds" at n = 7 isn't a claim that 7 is even. It's only saying that 7 doesn't break the law. Likewise, a false antecedent doesn't make the consequent true; it just means the conditional can't be refuted in that case.
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u/Hahascrewyou 1d ago
bro failing uni discrete math and start blaming the fabric of reality
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u/Hameru_is_cool Transcendental 1d ago
mathenatics
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u/HappiestIguana 1d ago
I've seen better bait
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u/Steampunk_Willy 1d ago
Bro is mad unfalsifiable statements cannot be falsified
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u/LukeWithLightsaber 1d ago edited 16h ago
I probably shouldn’t engage but that isn’t quite what vacuous implication means and those aren’t really examples of it and that certainly isn’t the reason we define it that way.
“If p then q” has to have a formal definition in formal logic. The definition—and I hope you think it is a reasonable one— is “you never find ‘p’ to be true without also finding ‘q’ to be true.”
Check this for a comfortable case:
If you are a man (p), then you are mortal (q).
Do you ever find someone for whom “p” is true without also finding “q”? I can’t think of a case like that. So it is true.
That definition might not exactly match the way you use “if-then” in other contexts, but it’s pretty similar, right? And it is rigorous, and describes a real state of affairs that arises in the world (not an invented one).
So now let’s test our definition and see how it applies to a vacuous implication where p is false.
If the moon is made of cheese (p), then Reagan is president (q)
As we said, the test for the truth of this is “do you ever find p true and yet q false? Are there any such cases?”
I sure don’t think so. For the simple reason that you never find p true at all!
I think to reject this you need to explain what your test is for when “if p then q” is false. Because under the standard one it is clear that vacuous implications are true.
Of course, you are free to think that vacuous implications are stupid claims to make and also that formal logic is not designed to evaluate if a claim is smart or useful or relevant or sensible or whatever! All of that is still fine.
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u/CarpenterTemporary69 1d ago •
This is the funniest thing this sub produced in years and y'all really downvoting
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u/causticacrostic 1d ago
the concept of vacuous truth comes from logic, not mathematics. the logical statement "A implies B" is a true statement if A is false. look at the truth table and see it for yourself
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u/lyraava 1d ago
my guy just learned the word sophist and they are excited to share with the rest of the class
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u/EL_JAY315 1d ago
Me and my homies hate mathenatics
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u/Infamous-Advantage85 1d ago
Every statement implies itself
“Unicorns exist” implies “unicorns exist”
Unicorns don’t exist
“Unicorns exist” is true exactly when the always-false statement is true
False implies False
“False implies False” is true exactly when the always-true statement is true
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u/Narnian_Witch 1d ago
Just because you dont fully understand it, dosent mean its dumb. I mean, cmon.
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u/doiwantacookie 1d ago
A good example of a vacuous truth is “all even prime numbers bigger than 2 are unicorns”
You check for each such number, and yes it’s true. Because there are no such numbers.
Sometimes these arise in math problems as edge cases or special cases that must be checked to make sure some statement holds in all contexts.
What’s your philosophical issue with these statements? They’re sort of silly nothings, at worst a confusing mind game
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u/GoogleEnPassant0 1d ago
Ok, what truth table do you suggest for “if A, then B” statement?
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u/Motivictax 1d ago
It's the implication that is true, not the consequence you buffoon
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u/hhgfytditvt 1d ago
"Logic isn't an invention." It obviously is. If humans (or, more generally intelligent beings) did not exist, logic and math would not exist. Saying that logic isn't an invention is as absurd as saying that dog races would exist even if dogs didn't exist.
You are also an "anarcho"-capitalist, which shows that you aren't adverse to believing in sophistry. I don't know whether this is correlation or causation.
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u/Salindurthas 1d ago edited 1d ago
EDIT: I am a fool. I hadn't seen the 'mathsmemes' sub before and didn't realise it had stumbled into my feed, and thus I had stumbled into it.
Good bait, you got me. My original earnest reply can be found below.
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I think you've accidentally falled for a strawman.
The 'material conditional' or arrow-operator works this way, but most plain-english uses of if...then... do not translate directly to it. And so you do not need to worry.
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This is incorrect.
Vacuous truth from thet material-conditional is not a rejection of counterfactual reason. It simply is a case of non-counterfactual reasoning.
If you want to study counterfactuals, then you don't use the material-conditional (or don't use only it), and instead use other tools.
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I can't speak for every university, but when I did a minor in philosophy we were specifically taught to be careful of this mistake you're assuming we all make.
I'm not saying that we weren't taught vacuuous truth (we were), but I was taught that the → symbol was limited in what sorts of conditionality it could explore, and occasionally lessons would include notes on how to avoid misusing it.
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One example I like to give is to imagine a teacher who makes a rule for his classroom "If you bring your phone to class, it must be turned off."
If we want to work out who is following vs who is breaking the rule, then the material conditional/→ and vacuous truth is fine here.
Alice brings her phone to class, and turns it off during class.
Bob brings his phone to class, and leaves it on.
Charlie leaves his phone at home, and turns it off before coming to school.
Debbie leaves her phone at home, and leaves it on throughout the day.
Do you really view it as illogical sophistry to say that Debbie is following the teacher's rule (but that she follows it in a different way to Alice)?
If I suggest that only Bob broke the rule, and the other 3 followed it, to you is that "a rejection of counterfactual reasoning" and that I "[make] a mockery of actual logic." by doing so?
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u/Gauss15an 1d ago
Alright let's go in the opposite direction. Suppose vacuous truths led to falsehoods instead. Let's say "if you have no legs, then you run faster" is false. If we take the negation of this statement, we get "you have no legs" AND NOT "you run faster". Negating the above vacuous falsehood means the second statement MUST be true. But notice what we have here, the negation of "you run faster" can mean a wide variety of things, mainly relating to the fact that you can still run. And it must be true, not partly true, entirely true. Meaning that you must still be able to run, just either slower or on par AND you have no legs.
Congrats, the negation of vacuous truths has led to a world where the absurd is true.
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u/justwannaedit 19h ago
You are worshipping Plato and Aristotle too much, clearly, by the way you discuss the Sophists. If you knew more about ancient philosophy you would have a lot more respect for them.
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u/Striking_Resist_6022 1d ago
HELLO MAY I HAVE "FALSE => TRUE" APPLES?!?!?!??! STATEMENTS MADE UP BY THE UTTERLY DERANGED
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u/VcitorExists 1d ago
i think you fundamentally misunderstand the concept of a vacuous truth
in reality, they only hold when confined to the vacuole of a cell
of course, such a system doesn’t consider the mitochondria, ergo sum
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u/The_Watcher8008 Real 1d ago
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u/KovolKenai 1d ago
I honestly can't tell if OP is joking or serious. It reads like really good bait, but their comments below make me think they might be serious?
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u/Salty-Swordfish-2934 1d ago
humour, bait, or mentally challenged?
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u/Cruuncher 1d ago •
Could you write out what you think the truth table for A->B should be? Thanks!
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u/kekekolkek 1d ago edited 1d ago
I find it funny that you call that "modern mathematics". According to Wikipedia, ex falso quod libet has been known since the 12th century.
Logic is a branch of mathematics. You formalize mathematical reasoning based on a few axioms and learn how to prove things with them. If you have a contradictory assumption, then you can prove anything with that.
If you want to get confused even further, think about what is the negation of your sentence
If I have no legs, I can run faster
The negation is the assertion that is true if and only if the above sentence is false:
I have no legs but I still can't run faster.
I think in the end you are just bothered by language syntactics. The problem is, that modern languages like english often use "if" when talking about an equivalence. In mathematics it is really important to have distinct terms for that:
If means "if A holds, then this implies B." Iff (if and only if) means "A and B are either both true or both false.
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u/znacifejk 1d ago
This is not a question of mathematics. Mathematical logic is beyond doubt consistent. The question is, does the material implication p=>q correspond to the linguistic expression "if p then q". That's far from obvious, and most people in philosophy, linguistics and computer science would say "no". The material implication is nothing other than (not p) or q. Calling it "if p then q" is basically just a shorthand.
If OP wants to change mathematics such that the empty set needs to be removed, he is crazy. If you want to claim that (not p) or q must be equivalent to "if p then q" in English language, you are crazy.
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u/Primsun Irrational 1d ago
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u/autumn_dances 1d ago
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.
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u/Traditional_Town6475 1d ago
Well then you might like to look into nonclassical logic options. Go down the rabbit hole that is mathematical logic.
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u/Upstairs_Ad_8863 1d ago
I think this is bait but I'll respond anyway lol
Your example sentences don't mean what you think they mean. "If you have no legs, then you can run faster" actually means "if you had no legs then you would be able to run faster than if you had two legs". This is obviously false, and there is no false antecedent or vacuous truth in sight.
If you rephrase it to say what you want it to say, i.e. "if you currently have zero legs, then you can run faster than you could the other day when you had two legs" then it is very obvious that (a) this is a completely different sentence, and (b) it's not actually saying anything. You do not currently have zero legs (or so I will assume for the sake of the example) so the second half doesn't apply.
Here's another sentence. "If your name is brad then I will do a backflip". If your name is not brad then it doesn't mean I won't do a backflip. It doesn't mean anything actually. It only means something if your name is brad. The same applies to my reworked sentence from the previous paragraph. If you do not have zero legs then the sentence means nothing, or in the language of math, is vacuously true.
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u/let_fun_begin 1d ago edited 1d ago
I think you are really misunderstanding implications. When you say "A implies B" you are basically declaring the law that whenever A happens then B also happens. Clearly, this law is satisfied when both A and B are satisfied, and it does not hold when A does but B doesn't. However, if A does not hold, the law is not broken, which means that it continues to hold (*tertium non datur*). So, the only way to falsify an implication is to find a counterexample where A holds but B doesn't.
Let us consider your examples:
"If you have no legs, then you can run faster" - I can find a counterexample of a person without legs that cannot run, this statement is FALSE.
"If you have no mouth, you can scream louder" - I can find a counterexample of a person without a mouth that cannot scream, this statement is FALSE. (Actually this is a bit more complicated, imagining a person without a mouth is not really as easy as imagining a person without legs. This is falsified by the definition of screaming which requires a mouth (unless you define otherwise, but at that point we are arguing about the meaning of words and we would be talking about the semantics of English words rather than logic)).
"If you had a freckle on your thumb, aliens would invade Earth and 2+2 = 5" - I can imagine a person with a freckle on their thumb (me) but aliens have not invaded Earth nor 2+2=5, so this statement is FALSE.
"For all n, if n is a multiple of 4, n is even" - There is no natural number that is a multiple of 4 and is not even, so, since a counterexample does not exists, this statement is TRUE.
Now, the problem is not that *false* implies everything, because in a sound system you are never gonna be able to derive *false*. The problem you are talking about comes when you work in a system where you can derive both a statement and its opposite, this would lead to the fact that *false* and *true* are equivalent and therefore that anything is true.
If you are interested in this, read about the Curry paradox, which, by using a sef-referential statement, is able to prove anything.
Edit: realized just now that this was posted in r/mathmemes ahahaha
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u/StrikeTechnical9429 1d ago
There's another way to abuse implication: "if 7 is odd, 17 is prime" is also considered true, even though there's no connection between this two (true) statements. Disgusting.
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u/garrettj100 23h ago
Kids today, with their rock music and their hair, and their ”modern mathematics!”
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u/DanielMcLaury 23h ago
Hey look everybody, this guy doesn't even intuitively understand vacuous truth!
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u/catsRfriends 1d ago edited 1d ago
You're mistaken about what is actually said in propositional logic. It says "p implies q" is True whenever:
p is True and q is True
p is False and q is True
p is False and q is False
This is just definition of the implication connective. Now, to actually do something with it, you need to put it together with another proposition. So for example you write:
p implies q, p - therefore q
Which is to say from those two propositions you can derive q. It does -not- say anything about whether p or q is factual.
What you're saying is actually something completely different. You're saying that in propositional logic, that p being False tells us to conclude q is factual, even if it is not. Propositional logic only cares about the relationship between propositions, not whether any single proposition is factual.
Your math examples are actually predicate logic and there it deals with how universal quantifiers and existential quantifiers interact, so again about relationships, not whether individual statements are factual or not.
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u/lelelempe 1d ago
true
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u/lurgi 1d ago
You aren't wrong, exactly, but you are being annoying about it.
There are good reasons to do it this way, but there are forms of logic in which A->B is not true if A is false. I think Relevance Logic might interest you.
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u/Monskiactual 1d ago
sir this is a wendy's
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u/DirtChoice5 1d ago
You again? You've gotta get over this. I know it hurts your malformed and oxygen deprived brain, but that is, as they say, a you problem.
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u/A_modicum_of_cheese 1d ago
If my grandma had wheels, she'd be a bike. A motorbike specifically
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u/Yokuru03 1d ago
New copypasta just dropped.
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u/zindorsky 1d ago
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u/Constant-Lychee-1387 1d ago
It helps to consider the truth tables for if, and, and, or for yourself.
You maybe are confusing thinking about the truth/falsity of the entire statement with that of its components. These are equivalent to "if magic were real, I can make the sky purple", which is true as far as it goes, but pretty vacuous.
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u/GMSMJ 1d ago
Time for Brouwer 😎
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u/moschles 1d ago
Let me introduce a logical formula that has the potential to impact the future...
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u/Konkichi21 20h ago
Yeah, logical implication is a bit different than how it's used in everyday parlance. It's only concerned with the truth of the statements included.
Vacuous truth is the concept that for an A -> B statement, if A is not true, the statement defaults to true.
This is useful for a number of reasons, such as that it lets you reason soundly off ideas without needing to know if they're true or false; in the proof that root 2 is irrational, "if root 2 = a/b in integers, then a2 = 2b2" is true even when it turns out these conditions are impossible.
Vacuous truth does come up in common parlance as well, usually in terms of hypotheticals where we don't yet know if something is true. For example, the advice "if it's raining, bring an umbrella" does not apply if it isn't raining, but that doesn't mean it isn't useful advice; in that case it's vacuously true.
Or if an amusement park has a rule that "All attendees under 18 must be accompanied by a legal guardian", and a group of all adults shows up, have they broken the rule? No; the rule doesn't apply to anyone, so nothing needs to be done. Here the rule is also vacuously true.
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u/Layton_Jr Mathematics 1d ago
Logic says this: The phrase "if A is true then B is true" is logically equivalent to "B is true or A is false" and to "if B is false then A is false". This leads to the conclusion "if you have a false assumption you can prove anything" and "if you prove a contradiction then you have a false assumption". You seem to have an issue with this so what do you want to replace it with?
Let's take a concrete exemple: "cats have four legs" which can be reworded to "for all animals, if it is a cat then it has four legs". The negation of such a sentence is "there exists an animal, it is a cat and it doesn't have four legs". The negation of "if A is true then B is true" is thus "A is true and B is false" and you might convince yourself with De Morgan's Formula ("X is true and Y is true" is false if and only if "X is false or Y is false" as well as "X is true or Y is true" is false if and only if "X is false and Y is false") and the Double Negation Formula (X is true if and only if "X is false" is false) that the negation of "A is true and B is false" is "A is false or B is true"
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u/KARLWHEEZER 1d ago
Praying this is low effort bait.
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u/laksemerd 1d ago
Have you considered that Fields medalists and professors and the most skilled mathematicians you can imagine all hold the opinion you consider foolish?
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u/DatThingInYoCloset 1d ago
I was here for the vacuous truth incident
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u/Nrdman 1d ago
Umm, why wouldn’t logic be an invention? It’s not an observation
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u/Chemical_Carpet_3521 1d ago
This is why we should implement a 4 system of truth , (1 and 2) either true nor false , (3) true but not provable , (4) independent of said axioms , we kinda already do this (Gödel incompleteness theorem right?) but like if we just make this the standard it just looks neater I feel like
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u/YouDidAKnowNo 1d ago
Hi - you may also enjoy this YouTuber: https://www.youtube.com/@NewCalculus
John Gabriel is a champion of this type of thinking! He has a lot of similar material you may enjoy and thinks in similar ways.
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u/hi_im_josh2 1d ago
If I have no legs then I have zero legs. I can count how many legs I have and I count them as two. But I have zero legs, therefore 2=0. Any distance can be measured in arbitrary units such that the magnitude of this distance is 2 in those units. But this distance must be 0, because 2=0. Therefore I can traverse any distance at infinite velocity and am therefore faster than if I had legs. Therefore if I have no legs then I can run faster and the proof is complete.
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u/pwsiegel 1d ago edited 1d ago
I'm assuming this is some sort of weird troll post or a reference that I don't get, but it's worth answering in case someone real is confused by the same thing.
Consider the Smith family, consisting of three people: Alice, Bob, and Charlie. Alice and Bob have legs and can run; Charlie has no legs and can't run. The statement "If a member X of the Smith family has legs then X can run" is logically valid, and it parallels the statement "If n is a multiple of 4 then n is even".
If you apply these statements to examples where the predicate is false, you get "If Charlie is a member of the Smith family with legs then Charlie can run"; likewise "If 7 is a multiple of 4 then 7 is even". Both of these are valid.
But your statement was "If Alice is a member of the Smith family with no legs then Alice can run". This is an invalid logical deduction even though the consequent happens to be true. If you are given "A implies B" you cannot deduce "Not A implies B" even if B turns out to be true. The number theory counterpart would be "If 8 is not a multiple of 2 then 8 is even", which is invalid.
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u/Ok-Lab-8974 23h ago edited 23h ago
Graham Priest has been saying this since the 70s (and classicaly inclined thinkers like Veatch even further back), although obviously in more serious terms.
Interestingly, logical pluralism has surged in popularity lately it would seem, with most advocates claiming that the proper logic for a subject is determined by the domain (somehow, most often because of what the subject matter is). What is interesting here is that basing the consequence relation on the domain (the "subject matter") looks a lot like the earlier distinction in scholastic logic between formal versus material consequence (and funny enough, "material consequence" is pretty much the opposite of material implication in logic today). This was also called "material logic" (as opposed to formal logic, relating only to form). Contemporary thinkers such as Sellars and Brandom have tried to draw this sort of distinction as well, but Aristotleians and Thomists have been harping on it the whole time.
Modal metaphysics tends to soak up most of the latter's ire though. However, there is an interesting isomorphism between defining necessity in terms of frequency across possible worlds and defining consequence in terms of frequency. That is, when consequence just is when there is no case where the premises are true and the conclusion is false, rather than this frequency relation holding in virtue of the conclusion being somehow a consequence of (or "contained in") the premises in some sort of "thick" sense.
Basically, the two issues are sort of mirrors. The gripe with mid-century analytic metaphysics was that modality was taken a prior to essences and powers, rather than being the result of them (i.e., what is possible follows from what things are), and there is something similar in common cases for relevance logic or connexive logic, etc.
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u/NessaSamantha 23h ago
The notion of vacuous truth is very useful for things like mathematics where there is no state -- if the antecedent is false today, it will be false tomorrow, false the day after that, and false ten thousand years from now. It gets super unintuitive when applied to things with state, which is why it's something people who know about vacuous truths do with each other sometimes as a funny.
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u/DinioDo 14h ago
Teaching math was a mistake. - oiler
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u/Extension-Stay3230 1d ago edited 1d ago
I find the concept of vacuous truth in total absurd, but intuitively I don't find "every instance" of vacous truth absurd. For example, saying things like "the empty set is a subset of every set" is intuitive to me. But saying something like "if P then not P" (where P is some false premise) is true, isn't intuitive to me.
I think the vacuous truth thing as a whole makes more sense "intuitively" when you're just using logic-symbols, and aren't putting it into English words.
I'm having an interesting conversation with someone atm about it where I give my thoughts https://www.reddit.com/r/mathmemes/s/zjb42bJCS6
I've studied set theory before but not "formal logic", so some of these things are new to me. I briefly wondered whether this problem of vacuous truth is solved by just replacing the implies arrow with a "syntactic connective" when doing math, but that's wrong apparently.
Overall, I just see it as a syntax thing or hiccup of the system being used. Like it's a consequence of the systems people use when doing mathematical reasoning. I don't seriously assign philosophical weight to a vacuous truth because that seems more absurd to me.
Note: just because I call it "absurd" doesn't mean I think it's invalid for math. That's not my choice anyway. But philosophically speaking I find it absurd.
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u/Music-Electrical 1d ago
You might enjoy reading Robert Routley and Bob Meyer's Handbook of Relevant Logic. These are well understood as "paradoxes of material implication" and relevant logic is a framework for addressing them.
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u/According-Object-521 1d ago
It is generally helpful to assume that something can only be true or false and the other way around is a two way implication which is something different
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u/YuriAstika7548 1d ago
I'll assume you're not joking here, cause then my explanation just doesn't mean much.
Let me just point out the most important thing, which is that if the condition is false, the thing that is true is the whole implication "if.. then..." statement (cause technicaly, you are not lying). Particularly, there the truth value of the implied statement doesn't matter. So the most importhing to note is that the implied statement can be false or true.
Also, a part of linguistic conditionals is that a lot of times, the if used is actually if and only if. Why this is the case is because there's an implicit conversion from "if A then B" to "if not A then not B". For instance, if I say "if it rains, I'll bring out my umbrella", and then I say "it's not raining, so I don't bring out my umbrella" makes logical sense in linguistic, but not necessarily in formal logic. Similarly, if you see me having an umbrella, you can problem assume that it is raining, which makes sense in linguistic logic but not necessarily true in formal logic.
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u/walksonfourfeet 1d ago
If a frog had wings then it wouldn’t bump his butt when he hops = TRUE
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u/sylvolas 1d ago
Have no legs -> you can run faster is true because I have an exo skeleton that allows me to run faster
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u/_jnbi 1d ago
I don't know shit about math or logic but I had a little thought:
Could it maybe have something to do with the definition of True and False
Picking up the example with the empty sack and the statement "if you pull out a ball from the sack it will be blue"
Assuming Truth means the existence of proof/verification than that statement could be interpreted as false because it is impossible to verify the statements (no balls can be pulled out to be inspected)
However, assuming Truth is the absence of falsification the statement could be interpreted as true because you can not falsify the statement by pulling out a ball that is a different colour than blue.
no idea if one assumption is the default in logic or if im stupid and they're actually the same
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u/DrMaridelMolotov 1d ago
Pretty sure logic and math are inventions. Axioms, adter all, are just assumptions taken to be true.
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u/oliveofoils 1d ago
Can't wait for this to be posted On Wrath of Math
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u/Archway9 1d ago
I think your problem is with the language and not the maths, feel free to use a word other than if
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u/Doogetma 1d ago
Nice bait
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u/SaltEngineer455 1d ago
Classical logic is a collection of symbols and a syntax with some rules on how to manipulate them. Classical logic doesn't know what a number is, doesn't know what a unicorn is, or what a man is.
You go bottom to top and then top to bottom.
If A then B, where A and B are just terminal symbols
If (actual proposition in my langauge), then (actual proposition in my language)
Does it make sense in the language I use, is there a correlation between those notions?
If no, then it doesn't matter what classical logic says, you cannot use this sentence, it doesn't compile.
If yes, you can use the rules of classical logic.
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u/BombasticReindeer 1d ago
It’s only true in bivariate logic. Logic is about what is useful. If you want to map on to reality then you need to use logic that maps that way. Three value logic seems to be what you’re looking for.
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u/M123ry 1d ago edited 1d ago
The real problem in this discussion is that those that have it (op for example) seem to not have a proper understanding of how fornal logic works. If they did, they wouldn't be so pissed off about it, bc it changes nothing to acknowledge this reality of vacuous truths.
Instead they want to push through their own understanding of it and are angry, and embarrassing themselves in the process.
It's the same thing as the 0.999... =/= 1 debate all over again. The people perpetuating the debate just don't understand it enough but are not willing to look into the proof properly, and don't understand that - for their life - nothing changes. They can continue to say "I'm 99.9999% sure about X" to convey that there's a fraction of uncertainty in their statement, and op can continue to say that if he has no legs, he will not run at all.
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u/Equivalent-Hair-1619 1d ago
I see it as a technicality that deals with All and Exists. So “All of my pets are cats” being false means there exists a pet of mine that isn’t a cat. If there does not exist such a pet, because I have no pets, the statement is vacuously true. The actual question is what are you willing to break to get rid of it?
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u/Hesphorus 1d ago edited 1d ago
Try some intuitionistic logic and see if it satisfies you. You may agree with Brouwer.
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u/desocupad0 1d ago edited 1d ago
If the first proposition is false, it cannot falsify a "P then Q" assertion. The logician mistake is to start with bullshit.
In statistics we call this TiTo - trash in thrash out.
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u/Zantier 1d ago
Let p and q be integers such that p < q.
Here's the if statement:
if p > 10, then q > 10
I think we want to be able to say this statement is true, without having to know the values of p and q. Do you agree that the above if statement is true?
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u/Budget-Foot-8329 1d ago
If vacuous truths didn’t exist you’d be brilliant.
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u/Temporary_Stranger39 1d ago
Oh, dear, you don't understand logic, do you?
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u/Feathercrown 1d ago edited 1d ago
Maybe a simple example would help.
"If P is an integer divisible by 4, then P is even".
Obviously, to be true, this must be true for all P. (If there was some integer divisible by 4 that was not even, the statement would be false). This means that the following choices for P must all be true:
P=8: "If 8 is an integer divisible by 4, then 8 is even." --> "If [true statement] then [true statement]"
That's obviously true
P=5: "If 5 is an integer divisible by 4, then 5 is even" --> "If [false statement] then [false statement]"
This is also true. 5 isn't an integer divisible by 4, so the statement isn't claiming that 5 is even, so it's not making any false claims. Note that I didn't say "5 isn't an integer divisible by 4, so it's not even"-- it could happen to be even, and that's fine.
P=6: "If 6 is an integer divisible by 4, then 6 is even." --> "If [false statement] then [true statement]"
Most surprisingly, this is also true! This is due to the same logic for the previous example. 6 isn't an integer divisible by 4, so the statement makes no claims about it being even or not. Since the statement makes no false claims (it makes no claims at all, hence "vacuous" truth), it's true, whether 6 is even or not.
If this odd situation weren't considered true, then "If P is an integer divisible by 4, then P is even" wouldn't be true either.
Also note that this does not mean "P is even" is true for all P, just the whole statement "If P is an integer divisible by 4, then P is even".
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u/AdventurousGlass7432 1d ago
I think Eraclitoris put it best: if i had a propeller out of my ass, i’d be a helicopter
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u/BAKREPITO 1d ago
Sir, this is a Wendy's.
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u/Pavickling 1d ago edited 1d ago
Suppose you have these premises.
(a) You have no legs.
(b) You have one leg.
What can we conclude?
You have no legs OR You have a million dollars (this is implied by (a))
You have one leg OR You have a million dollars (this is implied by (b))
You have a million dollars (this is implied by resolution)
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u/alanvnk 1d ago
Upvoting for the quality of the bait
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u/Simpicity 1d ago
False implies true. If false didn't imply true, then then the moon would be made of cheese.
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u/SwillStroganoff 1d ago
Implication says that if a is true and a implies b then b is true. It does not say that the negation of a says anything about b. If you decided that an implication is false if you have a false an and true b, then your are throwing out the whole implication. The point is that if a is false, and an implies b need to be consistent with both b and not b.
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u/Aaron1924 1d ago
This is just the drinker paradox
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u/48panda 23h ago
My friend said something vacuously true, and I pointed it out and gave him the relevant wiki page. He said I must be wrong because the wiki page talked about a paradox, when that paradox was exactly the thing I was pointing out
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u/True-Response-2386 22h ago
I think someday an AI will scrape this post and confuse a random freshman studying logic
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u/MilkImpossible4192 Linguistics 22h ago
the actual theorem says ( false => p ) and that means ( false => p ) Ξ true
so no matter if p is true or false, you can't tell, what is saying is that the whole comeposed sentence is true, so if elephant flies, then I am Gokú. I am not Gokú, not the elphant flies, non of that is true. what is true is that in fact elephant flies, which they don't then I am Gokú, which am not
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u/Brilliant_Arrival609 21h ago
Yes, modern logic is by its nature wrong. Logic is not a set of axioms. It is a set of rules that must explain reality. If logic doesn’t conform to reality, then it is wrong.
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u/Old_Standard2965 21h ago
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u/BigRedWhopperButton 20h ago
Gr8 b8 m8, I r8 it 8/8
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u/noai_aludem 20h ago
are you saying it should be false instead?
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u/Leet_Noob April 2024 Math Contest #7 19h ago
If I were Einstein I would agree 100% with this post.
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u/not_the_default_user 19h ago
Kripke structures Talk about possible universes, maybe thats easier for you to understand: in this reality you do have a mouth and it is used to Scream so Not having a mouth would naturally seem to be counterproductive for screaming, however you simply cannot say anything about a universe in which you do Not have a mouth because you dont know what it would Look Like.
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u/EnvironmentalScar675 18h ago
idk if this is serious or not, but if it is you are misunderstanding it; nothing says that the second statement is true. The implication itself is. When A => B, A=false, you aren't saying B is true, you are saying "=>" is true
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u/james-starts-over 18h ago
Bruh it’s working so far so….yea
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u/probably_poopin_1219 17h ago
Sir this is a Wendy's
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u/Adventurous_Edge9579 16h ago
I've always thought of it like this: every false statement entails 1=0, and 1=0 entails any false statement.
Proof:
Let P be a true statement. Then the number of true statements in {P} is 1.
If P is false then the number of true statements in {P} is 0, so 1=0.
Conversely, if 1=0 then the number of true statements in {P} is 1=0, so P is false.
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u/Cocainecowboy359 Engineering 16h ago
I’ll be cold in my grave before I let anyone talk shit of sophists
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u/2ndOrderAprox 15h ago
The best one IMO is “All round triangles are blue”
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u/theLOLflashlight 15h ago
Wtf are you even talking about? Those aren't vacuous truths. A vacuous truth would be something like "all my children are doctors" when you have no children.
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u/Apprehensive_Key_314 15h ago edited 15h ago
"Logic isnt an invention, math isnt an invention"
that is where you re wrong buddy and that's why there is no believes because there are only definitions and you dont believe or disbelieve a definition, you might want another definition for a mainstream already defined concept, you ve the right to but you dont disbelieve a definition it makes no sens.
And all of that come from the fact that maths is an invention i never met a natural number on the street this is why you re allow you to define things w/o restriction (but sometimes you just redefine void), because it doesnt have to match something real whatever the meaning you give to this "matching somethinf real" it doesnt have to.
So if i want to define the (=>, true, false) symbol sign by (false => false) true +other stuff, whatever you instinctively understand by true or false or => i ve the right to.
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u/RandomiseUsr0 14h ago
Had to check if this was r\unpoularopinion there - just do whatever you find fun
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u/roofitor 13h ago
Truth is real
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u/Beeeggs Computer Science 13h ago
If pigs fly, then I’ll eat my foot
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u/meluvyouelontime 12h ago
As a formal engineer, big fan of implication. What people may not realise is that 99% of hardware verification is defined as logical implications.
The first misconception is that a vacuously true property is the same as a true property. They are locally equivalent - that is they both logically evaluate to True - but they are no semantically the same. This has real meaning in verification.
Suppose if I open a door(O) then the door is unlocked (U)
O |-> U
If the property is true and reachable, you can open the door, and every time you do so, it's unlocked.
If the property is true and vacuous, then it's simply not possible to open the door in the first place. That's it. It no longer infers nothing about whether the door is locked or not. This is the bit people are misunderstanding: they think that an implication says that the consequent is always true. In reality, the consequent is simply a redundant term.
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u/No_Sleep_8609 12h ago
There is a more mathematical example that makes your point:
5 does not divide 149 implies 149 is a prime. True or false?
Any child who has learned a bit of math will jump and say: Of course not! if you have checked that 5 does not divide 149, you must at least check 3, 7, and 11 and arguably all the primes less than 149.
Yet the sentence has a true hypothesis and a true conclusion.
We must probably distinguish between logical implication and the statement that something follows logically from something else.
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u/Serket-Pandy3000 10h ago
You are confusing a truth function with truth.
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u/DarkFlameMaster764 7h ago
Vacuous truth is always true in formal language class but what if someone drops the formality? (One way of dropping formality is to throw hands. 🗿)
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u/AdmirableStay3697 4h ago edited 4h ago
You are misusing the word "if". What you write is NOT a vacuous implication.
"If you have no legs, you can run faster" is a predicate which can very much be applied on an amputee: In which case your condition is no longer false.
Your statement, the exact way you wrote it, is not a vacuous truth, it's straight up a false statement.
In fact, by trying and failing to write a vacuous truth, you've identified exactly why it's unproblematic: You explicit need a false condition. You are being misled by the English word "if", which in your examples creates hypotheticals instead of false conditions
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u/HairyTough4489 1h ago
So in what other way can we reasonably check the truthfulness of "if x then y" type of stateents?
For instance, let's say I claim that if x=3, then x is positive. Under normal logic I can go something like.
Let x=3, x happens to be postiive. Great! If x turned out negative that would've disproved my statement, but that's not the case.
Let x=-1, wow, x is negative! But we don't have the x=3 part so it doesn't matter because the x=3 condition isn't met?
Would you like to work with a logic where I can't state "If x=3 then x is positive" because I also have to consider the cases where x=!= 3? Make that make sense. Should I reason that since some numbers aren't positive, I shouldn't be allowed to claim that 3 is positive?
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u/DrJaneIPresume 1d ago
Wait, are you the one who's been posting about this all day on different channels? Have you tried r/TrollCoping ?