* "only one of us can be right".

* "there is only 1 shoe".

* "only one can go".

If none are right, is that still covered under "only one". (I tend to think it's not, but the other person thinks it is.) If there turn out to be no shoes, I think I've been deceived by "only 1 shoe". Only 1 can go permits that possibility that the number who go can be zero- but the concept is a limit. So, I think it's different.

Is my interpretation similar to others?

]]>English is not my native language, so I can't claim much in the way of authority on English language parsing. But the two variants of the question on the wiki page are clearly two different problems to my mind, and apparently also to many of the people in the study:

"I have two children, at least one of them is a boy – what is the probability that the other child is a boy" got 85% "1/2" answers.

"I have two children, it is not the case that both are girls – what is the probability that both are boys" got less than 40% "1/2" answers.

That sounds like a language parsing difference to me, and claims that the two sentences are "the same" must be based on a non-universal parsing model.

I would answer 1/3 to the last question (there are four cases, one of which is excluded) and 1/2 to the first question.

To my mind the first question excludes nothing, it is equivalent to "I know the sex of one of my children, what is the probability that the other child has the same sex as the first one?", and any additonal information ("it's a boy; he's called Tommy, was born on a Tuesday, has red hair…") is merely the false trail of a trick question.

As to why I (and at least 45% of the study participants) feel that the first question describes a different situation than the second question – surely that's a linguistic question rather than a mathematical one?

]]>That's more or less what I'm saying – although I should stress that I'm not a linguist and have *zero* idea whether that's the correct use of the term "implicature" which I understand to be a somewhat technical one.

Effectively what I'm saying is (I hope) more or less what the OP is saying – that while in a great many circumstances "one of X" does indeed imply "only one of X" that certainly does not mean that it semantically *means* "only one of X" and that this is therefore the only supportable interpretation (this was roughly the point of myl's reply to one of the first comments) and it isn't (IMO) reasonable to read it as implying "only one" in this case.

*Given the length of the thread I should perhaps leave that analysis as an excercise for the reader, but it does raise some questions about in what sense and to what extent this is a real logic/probability problem rather than a language-interpretation problem.*

The question as phrased is definitely ambiguous, but based purely on anecdotal evidence, I think I would expect most people to at least be able to see the logic *behind* the 1/3 answer, and I would expect the majority of people reaching the "wrong" answer to be reaching it because of a failure to understand the mathematics, rather than as parsing the language differently.

If nothing else (and this may be my inner schoolteacher coming to the fore) working is really more important than answers here. "One out of two, because the other child is equally likely to be a boy or a girl" is not a rigorous answer, it's an answer based on intuition. Now it happens that you *can* reproduce that answer rigorously (based on the assumption that we select a child at random and make a true statement about that child's sex, rather than making a more general statement about the number of children of a particular sex that this person has) but I strongly suspect that most "incorrect" answers to this problem (like the Monty Hall problem, which has similar issues) come from getting the maths wrong or (more likely, I suspect) just not bothering to do the maths.

On a wider note, I'd point out that *all* questions contain a language interpretation element, and even a cultural element (I teach in an international school, so this is something I'm very, very aware of). For example, I once asked one of my GCSE students what the probability of drawing the Queen of Hearts from an ordinary deck of playing cards was, and she answered "one in thirty-six" because in her country decks of cards don't include the numbers 2-5. I don't think anybody would argue that the question "what is the probability of drawing the Queen of Hearts from a deck of playing cards" is not a real probability question.

]]>*Linguistic pragmatics comes from the division into syntax/semantics/pragmatics == form/meaning/usage.*

In view these, presumably main areas of linguistics, the foundational aspects of linguistics seem to be under-represented, referring to ideas begun by Chomsky. I do sense a pragmatist/ behaviorist tradition still at work despite myl's above protestations.

For more certainty of conclusions, would it than not be reasonable to add another subfield that explores language production on the basis of a logic-neurologic neuro-computational theory. Where in Broca's and Wernicke's areas (& parietal, temporal lobe functions) resides the distinction of "one" and "only one", etc, my favorite being "iff" vs. either "if" or "only if" and the now almost colloquial "and/or." This could connect linguistics with fMRI studies, which recently have been refined in several ways.

]]>This suggests a pragmatist orientation of the investigation in this thread.

Therefore, my suggestion [above] of casual English *being a cruder form* of mathematical English would be unreasonable, as based on a neural & logical theory of the human language capacity, which may be incompatible with pragmatism.

[(myl) I'm afraid that you're confused by terminology.

Linguistic pragmatics comes from the division into syntax/semantics/pragmatics == form/meaning/usage.

Philosophical pragmatism is based on a particular view of the inter-relationship of theory and practice.

You could study linguistic usage without being a philosophical pragmatist, or caring about those philosophical issues one way or the other.

And you could be a philosophical pragmatist without having any interest in or ideas about the use of language.

There's also an ordinary-language use of pragmatist, in the sense of someone who "concentrates on practicalities and facts rather than theory and ideals". This has no systematic connection with either of the other meanings.]

]]>Interestingly enough, I think after some reflection that the common "wrong" but nonetheless supposedly "intuitive" answer of 1/2 (adopting those simplifying assumptions) may itself, at least for some wordings on the question, based on interpreting the word "one" with the wrong implicature — an implicature which is certainly not required but nonetheless apparently just plausible enough to lead some material number of native speakers astray. Given the length of the thread I should perhaps leave that analysis as an excercise for the reader, but it does raise some questions about in what sense and to what extent this is a real logic/probability problem rather than a language-interpretation problem. If you specify the simplifying assumptions and keep rewording the question to block various superficially-plausible-but-wrong implicatures that experience shows people have gotten tripped up by, how high can you drive the rate of correct answers? (This is not meant to sound petty or carping – the wiki article claims that it's been empirically found that different phrasings can yield substantially different percentages, which I find interesting.) On the other hand, maybe a particular plausible-but-wrong implicature is itself an artifact of a failure to bring to the problem some important conceptual piece of probability theory, and those who understand that piece will not be tempted to see that implicature in the wording.

]]>"I know a man with one leg, called George"

"Really? What's the other one called?"

K.

]]>I'm pretty sure you're *not* understanding first order approximation right.

The actual secondary sex ratio of humans is somewhere in the region of 105:100. The exercise simplifies this to 1:1 for ease of calculation.

Using the "real" values in the calculation gives an answer of 21/61 for the probability that both children are boys. True, this is slighly larger than the "false" answer of 1/3, but it is *far* closer to the "false" answer than to the *intuitive* answer of 1/2. Since the whole purpose of the exercise is to highlight a feature of condutional probability and not to actually predict, to the nearest percentage, the probability of the child having a particular sex (which, given that the child is fictional, is a fruitless endeavour at the best of times) quibbling over a difference of less than 2% seems perverse.

]]>About your placing in opposition (1) "mathematical [English] language" *vs*. (2) "casual English [language]".

It is tempting to posit (2) as being a cruder form of (1), due to imperfect logic/ mathematical formation in most if not all human minds (== logical brain functions). In a perfectly formed mind the two types of language ought to coincide.

]]>Umm … I'm pretty sure those aren't the only two options.

Surely it is also reasonable to suggest that the puzzle is written in logic-puzzle-English, where the implicit assumptions are those common to logic (or more specifically, probability) puzzles: The question will be solvable, it will have a non-trivial answer, the answer will not rely on wordplay or pedantry (which rules out answers such as "her other child is a girl because one means only one" or something like "she has no children because she is a fictional construct"), you will be expected to make simplifying assumptions, and so on.

Now there certainly *is* a school of irritating quasi-logic puzzle which involves puzzling out the deliberately obscurantist language of the puzzle (like the "words ending in gry" puzzle) but this is not a question of that type.

]]>Umm … you may certainly presume that if you wish. I'm not sure what your evidence is. My personal experience is that pure mathematicians barely ever work with actual numbers, while applied mathematicians are perfectly happy with approximations. I'd also point out that the original construction of this problem – and the initial solution which you insist on calling "false" was put together *by a mathematician*

*By "being technical," Dan Hemmens is leaving logic behind for the viewpoint of a mere engineer or observational scientist.*

Well … yes. Because your complaint was based on observational science.

You can take it one of two ways.

Firstly, you can look at this problem from the point of view of an observational scientist, in which case it is *perfectly reasonable* to treat the probabilities of children being boys or girls as 50/50 in order to get a good first order approximation.

Alternatively, you can look at this problem as a logician or a mathematician, in which case the real world data is *completely unimportant*, because what matters is the underlying mathematics.

I'd also note that from the point of view of a "mere observational scientist" the solution to the question is actually: "since the child has already been born, their sex is determined and is not, therefore, a question that requires a probabilistic solution. If the child is a boy, he is a boy, if the child is a girl, she is a girl. This can be easily ascertained by experiment."

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