A Semantic Survey

For the discussion of math. Duh.

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The result (3H, KH, 4S) satisfies the following statement: All three cards are not hearts.

True
10
18%
False
45
82%
 
Total votes: 55

gorcee
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A Semantic Survey

Postby gorcee » Tue Oct 18, 2011 3:44 pm UTC

Situation: Three cards are drawn from a standard deck.

Statement: All three are not hearts.

Trial: 3 of Hears, King of Hearts, 4 of Spades.

Question: The trial outcome satisfies the statement. True or False?

gorcee
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Re: A Semantic Survey

Postby gorcee » Tue Oct 18, 2011 3:54 pm UTC

A bit of background (spoilered to not corrupt your response):

Spoiler:
A recent probability exam I took included the following question:

Three cards are drawn at random from a deck of 52 cards. What is the probability that all three are not hearts?

I see two ways of reading this statement:

- "All three are not hearts", with the emphasis being on "all three".
- "All three are not hearts", with the emphasis being on "not hearts".

The first admits any of the combinations: (***, H**, HH*).
The second only admits ***.

Obviously this leads to different answers.

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Qaanol
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Re: A Semantic Survey

Postby Qaanol » Tue Oct 18, 2011 6:09 pm UTC

False dichotomy. You omitted the “It’s ambiguous” option.
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gorcee
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Re: A Semantic Survey

Postby gorcee » Tue Oct 18, 2011 6:32 pm UTC

Qaanol wrote:False dichotomy. You omitted the “It’s ambiguous” option.


Fair enough. The survey isn't intended to be scientific. Mostly I want justification that the interpretation can be ambiguous (also frustrating, because the question can be written in a completely unambiguous manner).

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Re: A Semantic Survey

Postby Xanthir » Tue Oct 18, 2011 7:17 pm UTC

If you want validation that it's ambiguous, then adding an "it's ambiguous" option seems to be exactly what you should do. ^_^

Anyway, it's too ambiguous to answer well. The sentence is somewhat clumsily constructed in the first place, and English permits two valid and reasonable readings of it, which produce opposite truth values.

If I was at gunpoint and had to pick one, though, I guess I'd lean towards "All three are (not hearts)", which means the statement is false. I'd prefer to phrase the opposite intention as "Not all three are hearts".
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gorcee
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Re: A Semantic Survey

Postby gorcee » Tue Oct 18, 2011 7:28 pm UTC

Based on the logical location of the problem in the exam, it seemed as though the professor wanted us to interpret the problem as a binomial process, so the answer desired answer would have been 3-choose-0 * (p)^0(1-p)^3, where p is 1/4, or the probability of sampling a heart with replacement. I'm uncertain though, since he's out of town.

Otherwise, the answer, assuming that no card can be hearts, using sampling without replacement would be simply 39-choose-3/52-choose-3. Or, if you interpret it the other way, where all combinations are permitted so long as none of the cards is a heart, then it's 1 - 13-choose-3/52-choose-3.

Sampling with replacement wasn't explicitly specified, so I'd assume it's one of the latter cases. But there are better ways to phrase the problem:

"What is the probability of choosing 3 cards such that none of the cards are hearts?"
"What is the probability of choosing 3 cards such that no more than two of the cards are hearts?"

Both of those statements are completely unambiguous and lead the solver down the right path. I'm sure I'll argue my points back. But I shouldn't have to. I already raised concerns to the professor once that his use of language in the course is leading to confusing among the students.

And, for fuck's sake, a math professor should not be using the terms "any" and "all" interchangeably.

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Re: A Semantic Survey

Postby skullturf » Tue Oct 18, 2011 7:38 pm UTC

In most everyday contexts, I would interpret "All X are not Y" as "Not all X are Y". (All Germans are not organized, all gay men are not tidy, all mathematicians are not shy, et cetera.)

In a math context, I would hesitate more, and would not be totally sure what was meant.

In either an everyday context or a math context, I absolutely HATE it when people say "All X are not Y" to mean "Not all X are Y". But I'm very aware that this phrasing is frequently used.

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Coding
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Re: A Semantic Survey

Postby Coding » Tue Oct 18, 2011 11:44 pm UTC

I read it as "none of the cards are hearts" (or "all of the cards are non-heart cards").

I would read "not all three of the cards are hearts" as "at least one of the cards is a non-heart card."

Yesila
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Re: A Semantic Survey

Postby Yesila » Wed Oct 19, 2011 1:06 am UTC

gorcee wrote:Situation: Three cards are drawn from a standard deck.

Statement: All three are not hearts.

Trial: 3 of Hears, King of Hearts, 4 of Spades.

Question: The trial outcome satisfies the statement. True or False?


Three cards are drawn, so I read "All three" as "all of the drawn cards"
A "not heart" is either a spade, club, or diamond (or joker) so when I read the statement I read,

"All of the drawn cards are either spades, clubs, or diamonds (or jokers)" - which seems to me to (clearly) imply the case with the trial being false.

To get the other interpretation I would expect to see something like "(the drawn cards) are not (all) three hearts."

I was always bad at counting though and all but failed the probability portion of my "probability and statistics" class back in 19(cough)(cough) because I always tended to read these statements "wrong."

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Re: A Semantic Survey

Postby yurell » Wed Oct 19, 2011 1:23 am UTC

I took it to mean "All three cards are (not hearts)", and figured if they wanted the other meaning they'd have said "Not (all three cards are hearts)".

That said, it does appear a little ambiguous.
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Re: A Semantic Survey

Postby phlip » Wed Oct 19, 2011 4:05 am UTC

I am aware that there exist some people who use the "All X are not Y" to mean "¬(∀ : X → Y)" instead of "∀ : X → ¬Y", but then I am also aware that to do so is dumb, so I voted "false".

To disagree with the OP, I don't think the emphasis matters. "All three are not hearts", with the emphasis on "All three" still doesn't admit a hand of two hearts and a spade... only one of them is not a heart, not all three.

The only time I would read "all X are not Y" as meaning "not all X are Y" is in a context where the speaker was specifically contrasting it as being the opposite of "all X are Y". And even then I'd still think "but wait, that's not the opposite of... ooooohhhh, they mean it in the stupid way".

If you really mean "not all X are Y" then say "not all X are Y". Don't just shove the word "not" into the sentence at random and just hope the result is comprehensible.

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Re: A Semantic Survey

Postby Tirian » Wed Oct 19, 2011 5:05 am UTC

I agree with phlip. If there is ambiguity in casual English, then there is none mathematically. ∀x:~Hx and ~∀x:Hx are profoundly different logical statements, and there is no mistaking that "all (three) cards are not hearts" is the former.

I hope that generous partial credit is given to the students who give the correct answer to the question that wasn't asked if this is the first time they've been exposed to the concept, but take this opportunity to understand that mathematics is a precisely worded business.

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Re: A Semantic Survey

Postby Jyrki » Wed Oct 19, 2011 5:39 am UTC

I would phrase it either as "Not all three are hearts" (meaning the negation of "all three are hearts") or "All three are non-hearts" (where the admittedly contrived word "non-heart" means a card from a suit other than hearts). Don't know, if that would be any clearer way to put it, but may be the hyphenated version makes it clear that the negation applies to "hearts". I voted for the latter interpretation anyway.

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Re: A Semantic Survey

Postby gorcee » Wed Oct 19, 2011 2:29 pm UTC

I understand the mathematical background. However, there is no precedent for formal mathematical semantics in the class. Furthermore, there is plenty of precedent in the textbook problems and homework where the phrases were worded in completely unambiguous ways. Even still furthermore, the professor uses the words "any" and "all" interchangeably.

Another way of possibly looking at it is by taking the converse of the categorization (hearts) and phrasing the statement as a question: "Here are some cards (3H, KH, 4S). Are all three hearts?" Two possible grammatically-sound responses exist: "No, not all three are hearts," and "No, all three are not hearts." Crafting those sentences into formal mathematical logic makes them quite distinct; however, both grammatical forms have been used in class and in the textbook.

Thus, barring the a priori expectation of formalism, I argue that students who responded in either manner should be awarded full credit, because either answer could be given in a form consistent with the prior presentations of the material in the class.

Edit:

Here's a great example of the precedent set for lack of mathematical formalism in this class. I sent the professor an email taking issue with a web-based homework problem that appeared to be giving the wrong answer, and I wanted clarification on his definition of mutually exclusive events. (I know what mutually exclusive events are, but I wanted to answer questions in a manner consistent with the class).

His response (emphasis mine):

"Are events A, B and D mutually exclusive?"

is asking whether A and D, A and B, B and D... are mutually exclusive.

Usually we say X and Y are mutually exclusive if the intersection of X and Y is zero. For more than two events E_1, ... E_n we say that they are mutually exclusive if any distinct pair E_i and E_j are mutually exclusive. This is how it was used in class on Tuesday.


(In this example, AB = 0, but AD != 0 and BD != 0.)

So if I used "any distinct pair", then ABD would be mutually exclusive, because AB is mutually exclusive. But that's not the case, because "any distinct pair" is not the same bloody thing as "all distinct pairs".

Wikipedia phrases it just fine: "events E1, E2, ..., En are said to be mutually exclusive if the occurrence of any one of them automatically implies the non-occurrence of the remaining n − 1 events" (http://en.wikipedia.org/wiki/Mutually_exclusive_events).

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Re: A Semantic Survey

Postby Qaanol » Wed Oct 19, 2011 6:22 pm UTC

gorcee wrote:
"Are events A, B and D mutually exclusive?"

is asking whether A and D, A and B, B and D... are mutually exclusive.

Usually we say X and Y are mutually exclusive if the intersection of X and Y is zero. For more than two events E_1, ... E_n we say that they are mutually exclusive if any distinct pair E_i and E_j are mutually exclusive. This is how it was used in class on Tuesday.


(In this example, AB = 0, but AD != 0 and BD != 0.)

So if I used "any distinct pair", then ABD would be mutually exclusive, because AB is mutually exclusive. But that's not the case, because "any distinct pair" is not the same bloody thing as "all distinct pairs".

Erm, in standard mathematical writing, “any” and “all” have essentially the same meaning in reference to elements of a set. The usual way to parse “If any x satisfies P” is as “If each x satisfies P” or “If P is satisfied regardless of x”. This usage generally can be thought of as “For any arbitrarily chosen x, no matter which one it happens to be”.

It may seem rather strange, but when a mathematician wants to specify that something need only be true for a single element, the standard phrasing is “If there exists an x that satisfies P”. Of course, an experienced mathematician will generally avoid “any” in logical statements, and stick is “for all” and “there exists”.
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antonfire
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Re: A Semantic Survey

Postby antonfire » Wed Oct 19, 2011 7:56 pm UTC

gorcee wrote:
"Are events A, B and D mutually exclusive?"

is asking whether A and D, A and B, B and D... are mutually exclusive.

Usually we say X and Y are mutually exclusive if the intersection of X and Y is zero. For more than two events E_1, ... E_n we say that they are mutually exclusive if any distinct pair E_i and E_j are mutually exclusive. This is how it was used in class on Tuesday.


(In this example, AB = 0, but AD != 0 and BD != 0.)

So if I used "any distinct pair", then ABD would be mutually exclusive, because AB is mutually exclusive. But that's not the case, because "any distinct pair" is not the same bloody thing as "all distinct pairs".
For what it's worth, the professor's way of phrasing things is so common in mathematics that it took me a few moments to see the ambiguity. Now is a good time to get used to it.

On the other hand, if the sentence had been "If any distinct pair Ei and Ej are mutually exclusive, we say that E1, ..., En are mutually exclusive", then I'd lean strongly in the other direction. Ah, English.

Anyway, if it really bugs you, you should be talking to your professor about it, not to the xkcd forums.
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gorcee
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Re: A Semantic Survey

Postby gorcee » Wed Oct 19, 2011 8:08 pm UTC

Qaanol wrote:Erm, in standard mathematical writing, “any” and “all” have essentially the same meaning in reference to elements of a set. The usual way to parse “If any x satisfies P” is as “If each x satisfies P” or “If P is satisfied regardless of x”. This usage generally can be thought of as “For any arbitrarily chosen x, no matter which one it happens to be”.

It may seem rather strange, but when a mathematician wants to specify that something need only be true for a single element, the standard phrasing is “If there exists an x that satisfies P”. Of course, an experienced mathematician will generally avoid “any” in logical statements, and stick is “for all” and “there exists”.


Short version: There is a variation in usage between "for any" and "if any".

I see where you're coming from, and I had written something up, but I decided to dig through some of my books for examples.

From a book grabbed off my desk (Computer Methods for Ordinary Differential Equations and Differential-Algebraic Equations, SIAM Press): "The linear multistep method is 0-stable iff all roots xi_i of the characteristic polynomial rho(xi) satisfy..."

The same page: "Also, for any consistent method the polynomial rho has the root 1."

In the former case, we're talking about a condition/property applied to every element of a well-defined set; namely, the roots of rho. In the latter case, we're talking about a more general category of things (methods) being applied to something that has some property.

Now, from another book (Mathematics for Computer Science, MIT Press):

"The proof is by induction. Let P(n) be the proposition that if any one square of a 2^n x 2^n courtyard must be left blank, then there exists a tiling of the remainder."
"But only terminating states are those with z = 0, so if any terminating state (x,y,0) is reachable, then y=yx^0=d as required."

So, in this case, "for any" is certainly synonymous with "for all blah in Blah", but "if any blah" means "if there is at least one blah".

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Re: A Semantic Survey

Postby gorcee » Wed Oct 19, 2011 8:24 pm UTC

antonfire wrote:Anyway, if it really bugs you, you should be talking to your professor about it, not to the xkcd forums.


I plan to, but I wanted to get a consensus to see if I'm crazy or not. I was curious if this a domain-specific grammar, and that my experience as an engineer/applied mathematician may differ from that of someone in algebra/analysis. FWIW, an informal survey of my co-workers (all engineers/computer scientists, masters level and above) the card example to permit (3H, KH, 4S). The more pure-math oriented folks on this forum seem to lean the opposite way. It certainly wouldn't be the first distinction in nomenclature between the fields (i vs. j, anyone?) but it is interesting, upon deeper inspection, to see how the fields, which carry the same level of formalism (despite what objections pure math folks might have to say), differentiate the way they word statements.

Just in diving through my books here, applied texts seem to focus on the negative effect of a single, "polluting" condition ("an LTI system is unstable if any eigenvalue has positive real part"), whereas pure texts seem to put them emphasis on the constructive effect of all properties satisfying some condition ("an LTI system is stable iff all eigenvalues have negative real part").


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