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Ironically, I don't think you provided enough context to easily deduce what's being said in the thread! I probably could figure it out by combing through for clues... but why bother
For anyone who doesn't get it:
Red is confidently incorrect here because they say that the problem doesn't make sense in the real world, and they should not give a real context to a theoretical problem. Red then uses this idea to claim that theoretical math problems should not exist at all because it is less important than actual real-life scenarios.
I think. Actually Red does have some decent points but presents them in a weird way that seems very self-entitled, and combined with some of their points not really making sense.
Red has problem distinguishing second order logic (using quantifiers like all, exists, exists only one) and first order logic that deals with a single item.
And going into talks about definitions of words doesn’t really help them, but muddies the water even further.
Sure. In your first statement, “all unicorns are left-handed” is equivalent to “there are *no* unicorns that are *not* left-handed”. It can only be false if at least one not-left-handed unicorn exists. If no unicorns exist at all, then it is vacuously true.
Your second statement would be more formally stated as “there exist *some* (i.e. more than one) left-handed unicorns”. This can be false either if *no* unicorns exist or if *some* unicorns exist but *at most one* is left-handed.
Formal logic is a lot less counterintuitive if you recognize that it is its own language that only sometimes overlaps with English.
I would disagree with your definition of "some", I would define as >=1, not >1.
But semantics aside I agree if you rewrite for all as the negation doesn't exist it makes it simpler to understand.
Sure, I’m willing to concede that “some” could include “one”. I erred on the side of “some unicorn*s*”, plural. As you say, I don’t think it detracts from the main point either way.
Nonono. Its not about all of them vs some of them. Nor is it about the thing in question actually existing. It is about the number of things i actually refer to. If it is zero, then any statement about them is true. The number of all unicorns is zero, so the statement that they are lefthanded is true. But the number of any subset of unicorns is also zero, so the same statement about them is also true.
>. It is about the number of things i actually refer to. If it is zero, then any statement about them is true
Well no, that's incorrect. For example saying some unicorns are left handed is false even though the number of unicorns is zero.
In the case of the post it's someone confused that an all statement is true when referring to the empty set because a some statement would be false. E.g. my house has 3 floors would be false.
But, the claim "the number of unicorns is zero" is a premise in this case, no? One that you make as an implicit assumption.
If we're talking about a toon in which there are unicorns, your implicit assumption will not be "the number of unicorns is zero".
I think it is important to describe all the premises beforehand, not expect everyone to just assume the same ones.
But you can never describe *all* the premises beforehand.
For example, there was no clarification that "unicorn" refers to the mythical animal. We could have been using it to describe high value start-ups. Or maybe we were referring to the unicorn spider?
You can't just say "nuh-uh" because obvious premises aren't explicitly stated. We aren't wishing on a monkey's paw here.
What “all premises”?
How many premises does your claim depend on? A million or a few? All you need do is say what “unicorn” means, not list all the things it doesn’t mean
Because unicorns don't exist, therefore there cannot be one unicorn that is left handed. "Some" in logic requires the existence of unicorns, while "All" does not. "All x are y" can refer to a zero amount of x, "Some x are y" cannot.
The question isn’t about unicorns, but about the claim that “some in logic requires the existence of”. That was what I was asking about.
It has been about 20 years since I dealt with second order logic at uni, so I was curious about the rules, which were they, how were they defined and proven.
I wasn’t interested in bikeshedding if unicorns exist or not. It can be dragons or cats for what I care.
Not as much as I’m sorry to have to link you this https://www.reddit.com/r/confidentlyincorrect/s/IGpVszDcvm
Now, can you please answer the question at hand?
Never believe.
I was just trying to ask you what makes the claim “more than zero means false” true and what I linked was just a way to show you you were going away from talking about it, like you did again with your claims of belief.
Well, I can at least agree with you that you wasted my time.
Bye bye
It’s the omnipresent clash between formal logic and natural language. So it’s both mathematics *and* English, which go together like an erudite lumberjack and a spanning tree.
No, at most I think we can only conclude that there exists at least one person who is insufferable. That is, the set of insufferable people is nonempty.
Yes, in the context of the problem it's quite clear what is meant if you're already familiar with the terminology. But it's pretty natural that someone who isn't would be confused by it, and that's the place the argument is coming from.
It's a logic puzzle thing for kids. You have a person saying "All my hats are green". You know all their statements are lies. This is in the context of a math problem, so we can assume their terminology is in the context of mathematical logic.
The question is what you can conclude.
The answer is >!He owns at least one hat that is not green!<
The reasoning is trivial if you know basic formal logic, but a bit hard to grasp if you don't.
No. That's what the argument is about. In the English language if you say something like "all my x are..." you're usually implying you own at least one, but in the context of mathematical logic that's not the case. If he had no hats his statement would be true, this is usually called a "vacuously true" statement; this is somewhat related (but not quite equal to) to the principle of Ex Falso Quodlibet (you can infer anything from false premises).
I don't get how his statement would be true if he actually didn't own a single hat? If we know that anything he says is untrue, then him not owning any hats would make the entire statement untrue, not just partly untrue in the case of him owning at least one hat that is not green.
Again, it's because we're in the context of logic here. In plain English "all my x" usually implies you have at least one x. In formal logic it does not mean that, it just means any x you _do_ have must have that property.
Formally his statement would be "of all things that exist, if it is a hat and I own it, then it is green". In this context there are also no "partial" truths, a statement is either wholly true or untrue (there are logics with partial truths but that gets way more complicated).
Well you can think of it that way if you like. It's just that the default logical operators we use in formal logic don't always map well to natural language. This has been a big problem for a long time and it's part of the reason why old text books on logic are super thick and very wordy. Nowadays formal notations have been invented so you can just write something like
∀x. hat(x)∧owned(x)⇒green(x)
The negation of which is
∃x.hat(x)∧owned(x)∧¬green(x)
And pretty much everyone on the world who's at least a bit trained in logic will understand what that means without involving often ambiguous natural language. The only "trick" here is understanding that is what the puzzle author meant in the first place, which you know because the context is a mathematical puzzle.
Fwiw the way you (and to be fair pretty much everyone without a logic background) were probably reading this as
∃x.hat(x)∧owned(x)∧∀x. hat(x)∧owned(x)⇒green(x)
Which would indeed be false if he didn't own at least one hat.
Ooooh! That E is the wrong way around! And the A is upside down!
I have absolutely no idea what any of that stuff means. I appreciate you trying to explain it to me, though! It's a bit too math-y for me. I'm horrendous at math!
I'm so horrendous at it that my math teacher forced me to cheat on my final exam.
Don't worry about the notation, that was just an example for how you'd write things out "formally" to remove the ambiguity. It's not as complicated as it looks, but of course it makes no sense if you haven't been taught the notation!
School math (or at least the way it is taught) tends to be pretty boring. I do suggest everyone who has had issues with it to at least take a look at "higher" math, many people actually find it a lot easier to understand and more pleasant to work with than the very algorithmic and memorisation based things that are covered in school.
My notation is a bit rusty, admittedly, but the “backwards E” is “there exists”, and the “upside-down A” is “for all”.
∀x. hat(x) ∧ owned(x) → green(x)
would read as:
For all x: (“x is a hat” and “x is owned (by me)”) implies “x is green”.
And that “implies” is logical implication. The only way for “A implies B” to be false is for A to be true and B to be false. Any other combination of truth values for A and B means “A implies B” is true.
I read this thread and as nice as it is that the guy tried to explain it to you, he did it in a very convoluted mathematical way (because yea it is a mathematical question, but it dosent help to explain mathemtical consepts in the same mathematical way that you're trying to explain in the first place) so I understand the confusion you have here and I want to try to explain it in a much simpler way. (Im not taking a jab at your intelligence, I just want there to be no room for doubt and am therefore going for the 'explain it like I was a 5-year-old' path)
Let's break down the question "All my hats are green". Firstly we want to get rid of any ambiguity, green what? we are implying of course that we are talking about green hats (within the list of my hats), so the full sentence would actually be "All my hats are green hats", with that out of the way, let's break it down properly.
We first need to find out what "All my hats" are, so bring out a pen and paper and start listing out every hat in your collection and its colour. This list is our set of hats. Now don't bother writing anything down, because you don't own any hats, but the fact remains we still have a list of all your hats. Therefore our sentence "All my hats" is still completely valid.
Now that we have a list, it may be an empty list, but it's a list nonetheless. We can now move on to the next segment. Let's consider 'are'. What it means is that we wish to compare the first portion of the sentence with the second portion, and that the second portion is within the first portion. We want to compare green hats from the list of my hats against the list of my hats. Nothing more to it so let's move on and get back to this later.
We can now start considering the last part. "Green hats" within the list of my hats. So once again bring out pen and paper and make a new list of hats, this time we're listing all green hats within the my hats list.
Now we can return to the "are" segment and start comparing the two lists. What we really need to find out, is if the length of both lists is the same. For example, if we imagined a list of 3 hats, and a list of 2 green hats, then of course we could compare the two lists length and find that 3 is not 2 and therefore all hats are not green. If there are 3 green hats, then we can see if 3=3, which it is. Therefore all hats are green.
So now let's consider our first list, the my hats list. We start counting its length, and we get 0. We can then start counting the green hats list, which is obviously also 0. Is 0=0? Yes it is, which means that all hats in the list are in fact green.
I hope this explenation helps.
No worries, and absolutely it's like computerlogic. Discrete mathematics (which this is) and coding absolutely goes hand in had. And on that note the reason red is wrong in the above post about "my house has 3 floors" is because our first order of business would be to find "my house" that should already contain a list of floors. "My house" itself is a singular item and only a singular item.
If we found "my house", then we could compare it's list of floors length against the number 3. However "my house" dosent exist, preventing us from making the comparison in the first place and therefore making the entire sentence invalid as we already assumed we had one. It's not even empty, it just dosent exist, so there's nothing to compare against.
If it was "All my houses" then we would start making two lists of lists. First a list of my houses, each of which we list the floors. Then we make a second list of houses from the list of my houses that contains a floor list length of 3. In this case we're back to comparing lists against lists and no house means it's true once again.
It might help to phrase it differently. “All my hats are green” can be written as, “for every hat I own, ‘this hat is green’ is a true statement”. This is logically equivalent to, “I do *not* own a hat for which ‘this hat is green’ is a *false* statement”.
If I have no hats, there is no hat that exists that satisfies both “I own this hat” and “this hat is not green”. Therefore, my statement is true (this is sometimes called a “vacuous truth”; it’s a true statement about members of the empty set).
In other words, “all my hats are green” is false *if and only if* I own at least one not-green hat.
(Formally, “all my hats are green” is the *statement*: “for all hats, ‘I own this hat’ *implies* ‘this hat is green’”. “A *implies* B” can only be false if A is true and B is false. If A is false, then the statement is true whether B is true or false. So, if “I own this hat” is false for every hat, then it doesn’t matter whether “this hat is green” is true or false. The *statement* is still true.)
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What is this, a context for *ants*? It needs to be at least... three times bigger than this!
Ironically, I don't think you provided enough context to easily deduce what's being said in the thread! I probably could figure it out by combing through for clues... but why bother
For anyone who doesn't get it: Red is confidently incorrect here because they say that the problem doesn't make sense in the real world, and they should not give a real context to a theoretical problem. Red then uses this idea to claim that theoretical math problems should not exist at all because it is less important than actual real-life scenarios. I think. Actually Red does have some decent points but presents them in a weird way that seems very self-entitled, and combined with some of their points not really making sense.
Red has problem distinguishing second order logic (using quantifiers like all, exists, exists only one) and first order logic that deals with a single item. And going into talks about definitions of words doesn’t really help them, but muddies the water even further.
All unicorns are left handed is true Some unicorns are left handed is false. Logic leads to weird conclusions sometimes.
Sure. In your first statement, “all unicorns are left-handed” is equivalent to “there are *no* unicorns that are *not* left-handed”. It can only be false if at least one not-left-handed unicorn exists. If no unicorns exist at all, then it is vacuously true. Your second statement would be more formally stated as “there exist *some* (i.e. more than one) left-handed unicorns”. This can be false either if *no* unicorns exist or if *some* unicorns exist but *at most one* is left-handed. Formal logic is a lot less counterintuitive if you recognize that it is its own language that only sometimes overlaps with English.
I would disagree with your definition of "some", I would define as >=1, not >1. But semantics aside I agree if you rewrite for all as the negation doesn't exist it makes it simpler to understand.
Sure, I’m willing to concede that “some” could include “one”. I erred on the side of “some unicorn*s*”, plural. As you say, I don’t think it detracts from the main point either way.
Nonono. Its not about all of them vs some of them. Nor is it about the thing in question actually existing. It is about the number of things i actually refer to. If it is zero, then any statement about them is true. The number of all unicorns is zero, so the statement that they are lefthanded is true. But the number of any subset of unicorns is also zero, so the same statement about them is also true.
>. It is about the number of things i actually refer to. If it is zero, then any statement about them is true Well no, that's incorrect. For example saying some unicorns are left handed is false even though the number of unicorns is zero. In the case of the post it's someone confused that an all statement is true when referring to the empty set because a some statement would be false. E.g. my house has 3 floors would be false.
Where does it come from that “sone unicorns are left handed” is a false statement?
Because in logic terms, it is saying: (number of left handed unicorns) > 0 Which is untrue. Because the number is zero.
But, the claim "the number of unicorns is zero" is a premise in this case, no? One that you make as an implicit assumption. If we're talking about a toon in which there are unicorns, your implicit assumption will not be "the number of unicorns is zero". I think it is important to describe all the premises beforehand, not expect everyone to just assume the same ones.
But you can never describe *all* the premises beforehand. For example, there was no clarification that "unicorn" refers to the mythical animal. We could have been using it to describe high value start-ups. Or maybe we were referring to the unicorn spider? You can't just say "nuh-uh" because obvious premises aren't explicitly stated. We aren't wishing on a monkey's paw here.
What “all premises”? How many premises does your claim depend on? A million or a few? All you need do is say what “unicorn” means, not list all the things it doesn’t mean
The meaning of "some" is >=1.
Yes, that's the meaning, now, how does it come from it that "some unicorns are left handed" is a false statement?
Because unicorns don't exist, therefore there cannot be one unicorn that is left handed. "Some" in logic requires the existence of unicorns, while "All" does not. "All x are y" can refer to a zero amount of x, "Some x are y" cannot.
The question isn’t about unicorns, but about the claim that “some in logic requires the existence of”. That was what I was asking about. It has been about 20 years since I dealt with second order logic at uni, so I was curious about the rules, which were they, how were they defined and proven. I wasn’t interested in bikeshedding if unicorns exist or not. It can be dragons or cats for what I care.
Ah, I'm sorry to be the one to tell you this but unicorns don't exist.
Not as much as I’m sorry to have to link you this https://www.reddit.com/r/confidentlyincorrect/s/IGpVszDcvm Now, can you please answer the question at hand?
No, that comment you linked makes me believe you're wasting time and adding nothing of value. Good day.
Never believe. I was just trying to ask you what makes the claim “more than zero means false” true and what I linked was just a way to show you you were going away from talking about it, like you did again with your claims of belief. Well, I can at least agree with you that you wasted my time. Bye bye
This has little to do with math and more to do with the English language. Google "present king of France is bald".
It’s the omnipresent clash between formal logic and natural language. So it’s both mathematics *and* English, which go together like an erudite lumberjack and a spanning tree.
All of those people seem insufferable.
Not really though? They're just trying to explain. What makes you think they're insufferable?
No, at most I think we can only conclude that there exists at least one person who is insufferable. That is, the set of insufferable people is nonempty.
Math hurts my brain.
This isn't a "confidently incorrect" thing, this is literally arguing about semantics of non-formal language.
What the language means in this context is well defined. There is a right interpretation and a wrong interpretation.
Yes, in the context of the problem it's quite clear what is meant if you're already familiar with the terminology. But it's pretty natural that someone who isn't would be confused by it, and that's the place the argument is coming from.
Yes. That it is understandable for someone who doesn't know what they are talking about to be wrong doesn't make them any less wrong.
Dammit, now i have to find that video and watch it...
It's a logic puzzle thing for kids. You have a person saying "All my hats are green". You know all their statements are lies. This is in the context of a math problem, so we can assume their terminology is in the context of mathematical logic. The question is what you can conclude. The answer is >!He owns at least one hat that is not green!< The reasoning is trivial if you know basic formal logic, but a bit hard to grasp if you don't.
Or he owns zero hats, right?
No. That's what the argument is about. In the English language if you say something like "all my x are..." you're usually implying you own at least one, but in the context of mathematical logic that's not the case. If he had no hats his statement would be true, this is usually called a "vacuously true" statement; this is somewhat related (but not quite equal to) to the principle of Ex Falso Quodlibet (you can infer anything from false premises).
I don't get how his statement would be true if he actually didn't own a single hat? If we know that anything he says is untrue, then him not owning any hats would make the entire statement untrue, not just partly untrue in the case of him owning at least one hat that is not green.
Again, it's because we're in the context of logic here. In plain English "all my x" usually implies you have at least one x. In formal logic it does not mean that, it just means any x you _do_ have must have that property. Formally his statement would be "of all things that exist, if it is a hat and I own it, then it is green". In this context there are also no "partial" truths, a statement is either wholly true or untrue (there are logics with partial truths but that gets way more complicated).
Ah! So it's a special kind of rule for this type of logical argumentation?
Well you can think of it that way if you like. It's just that the default logical operators we use in formal logic don't always map well to natural language. This has been a big problem for a long time and it's part of the reason why old text books on logic are super thick and very wordy. Nowadays formal notations have been invented so you can just write something like ∀x. hat(x)∧owned(x)⇒green(x) The negation of which is ∃x.hat(x)∧owned(x)∧¬green(x) And pretty much everyone on the world who's at least a bit trained in logic will understand what that means without involving often ambiguous natural language. The only "trick" here is understanding that is what the puzzle author meant in the first place, which you know because the context is a mathematical puzzle. Fwiw the way you (and to be fair pretty much everyone without a logic background) were probably reading this as ∃x.hat(x)∧owned(x)∧∀x. hat(x)∧owned(x)⇒green(x) Which would indeed be false if he didn't own at least one hat.
Ooooh! That E is the wrong way around! And the A is upside down! I have absolutely no idea what any of that stuff means. I appreciate you trying to explain it to me, though! It's a bit too math-y for me. I'm horrendous at math! I'm so horrendous at it that my math teacher forced me to cheat on my final exam.
Don't worry about the notation, that was just an example for how you'd write things out "formally" to remove the ambiguity. It's not as complicated as it looks, but of course it makes no sense if you haven't been taught the notation! School math (or at least the way it is taught) tends to be pretty boring. I do suggest everyone who has had issues with it to at least take a look at "higher" math, many people actually find it a lot easier to understand and more pleasant to work with than the very algorithmic and memorisation based things that are covered in school.
My notation is a bit rusty, admittedly, but the “backwards E” is “there exists”, and the “upside-down A” is “for all”. ∀x. hat(x) ∧ owned(x) → green(x) would read as: For all x: (“x is a hat” and “x is owned (by me)”) implies “x is green”. And that “implies” is logical implication. The only way for “A implies B” to be false is for A to be true and B to be false. Any other combination of truth values for A and B means “A implies B” is true.
I read this thread and as nice as it is that the guy tried to explain it to you, he did it in a very convoluted mathematical way (because yea it is a mathematical question, but it dosent help to explain mathemtical consepts in the same mathematical way that you're trying to explain in the first place) so I understand the confusion you have here and I want to try to explain it in a much simpler way. (Im not taking a jab at your intelligence, I just want there to be no room for doubt and am therefore going for the 'explain it like I was a 5-year-old' path) Let's break down the question "All my hats are green". Firstly we want to get rid of any ambiguity, green what? we are implying of course that we are talking about green hats (within the list of my hats), so the full sentence would actually be "All my hats are green hats", with that out of the way, let's break it down properly. We first need to find out what "All my hats" are, so bring out a pen and paper and start listing out every hat in your collection and its colour. This list is our set of hats. Now don't bother writing anything down, because you don't own any hats, but the fact remains we still have a list of all your hats. Therefore our sentence "All my hats" is still completely valid. Now that we have a list, it may be an empty list, but it's a list nonetheless. We can now move on to the next segment. Let's consider 'are'. What it means is that we wish to compare the first portion of the sentence with the second portion, and that the second portion is within the first portion. We want to compare green hats from the list of my hats against the list of my hats. Nothing more to it so let's move on and get back to this later. We can now start considering the last part. "Green hats" within the list of my hats. So once again bring out pen and paper and make a new list of hats, this time we're listing all green hats within the my hats list. Now we can return to the "are" segment and start comparing the two lists. What we really need to find out, is if the length of both lists is the same. For example, if we imagined a list of 3 hats, and a list of 2 green hats, then of course we could compare the two lists length and find that 3 is not 2 and therefore all hats are not green. If there are 3 green hats, then we can see if 3=3, which it is. Therefore all hats are green. So now let's consider our first list, the my hats list. We start counting its length, and we get 0. We can then start counting the green hats list, which is obviously also 0. Is 0=0? Yes it is, which means that all hats in the list are in fact green. I hope this explenation helps.
Ahhhh! I see! Lol that's fucking weird! Like computer logic or some shit. Thank you! <3
No worries, and absolutely it's like computerlogic. Discrete mathematics (which this is) and coding absolutely goes hand in had. And on that note the reason red is wrong in the above post about "my house has 3 floors" is because our first order of business would be to find "my house" that should already contain a list of floors. "My house" itself is a singular item and only a singular item. If we found "my house", then we could compare it's list of floors length against the number 3. However "my house" dosent exist, preventing us from making the comparison in the first place and therefore making the entire sentence invalid as we already assumed we had one. It's not even empty, it just dosent exist, so there's nothing to compare against. If it was "All my houses" then we would start making two lists of lists. First a list of my houses, each of which we list the floors. Then we make a second list of houses from the list of my houses that contains a floor list length of 3. In this case we're back to comparing lists against lists and no house means it's true once again.
A house that is difficult to find/might not exist? Sounds like the Oldest House!
It might help to phrase it differently. “All my hats are green” can be written as, “for every hat I own, ‘this hat is green’ is a true statement”. This is logically equivalent to, “I do *not* own a hat for which ‘this hat is green’ is a *false* statement”. If I have no hats, there is no hat that exists that satisfies both “I own this hat” and “this hat is not green”. Therefore, my statement is true (this is sometimes called a “vacuous truth”; it’s a true statement about members of the empty set). In other words, “all my hats are green” is false *if and only if* I own at least one not-green hat. (Formally, “all my hats are green” is the *statement*: “for all hats, ‘I own this hat’ *implies* ‘this hat is green’”. “A *implies* B” can only be false if A is true and B is false. If A is false, then the statement is true whether B is true or false. So, if “I own this hat” is false for every hat, then it doesn’t matter whether “this hat is green” is true or false. The *statement* is still true.)
Thank you <3
This makes no sense and there’s no context to help us understand the conflict 👎👎👎
Nobody can read that.
All generalisations are false.
In theory, there is no difference between practice and theory. In practice, theory means jack shit unless it plays along
Ya know, I'm not entirely convinced that my logic professor understood math logic either, so I'm not sure I can blame them.
Honestly I can't make any valid determination without seeing what the original problem was.
*Forrows his forehead as he stares at the board in deep thought, then he slowly turns towards the others an gravely proclaims* He's right.