## Ask Language Log: One = only one?

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Keith Ellis got into an argument with a friend about the meaning of the number one, and asked us for help:

In a discussion I had today with someone about the probability puzzle of "one of my two children is a boy, what is the probability that my other child is a girl?" we got hung up on her (very strong) inference of "only one of my two children is a boy" from "one of my children is a boy". […]

She insisted that if one "takes the statement literally" that the statement necessarily has this ["one is" == "only one is"] meaning.

Keith's friend has some distinguished company on the losing side of this argument (which, as Keith notes, can sometimes "get a little argumentative"). Here's J.S. Mill in *An Examination of Sir William Hamilton’s Philosophy*, 1865:

This is not the only case in which Sir W. Hamilton insists upon wrapping up two different assertions in one form of words, and demands that they shall be considered one assertion. He strenuously contends that the form "Some A is B" […] ought in logical propriety to be used and understood in the sense of "some and *some only*." No shadow of justification is shown for thus deviating from the practice of all writers on logic, and of all who think and speak with any approach to precision, and adopting into logic a mere *sous-entendu* of common conversation in its most imprecise form. If I say to any one, "I saw some of your children to-day," he might be justified in inferring that I did not see them all, not because the words mean it, but because, if I had seen them all, it is most likely that I should have said so : though even this cannot be presumed unless it is presupposed that I must have known whether the children I saw were all or not. But to carry this colloquial mode of interpreting a statement into Logic, is something novel.

The general consensus among philosophers of language, logicians, linguists, and others who study such things is that John Stuart Mill and Keith are right, while Sir William Hamilton and Keith's friend are wrong.

The "colloquial" moves from "some" to "not all", or from "one" to "only one", are specific examples of a general phenomenon that has come to be known as "scalar implicature". Thus Ira Noveck and Anne Reboul, "Experimental Pragmatics: A Gricean turn in the study of language", T*rends in Cognitive Sciences*, 12(11): 425-431 2008:

One class of proposals has been influenced by an account developed by Laurence Horn […] For those cases like Some, he argues that the derivation of generalized implicatures relies on pre-existing linguistic scales consisting in a set of expressions ranked by order of informativeness, e.g. <some, all>, where the former is less informative than the latter. When a speaker uses a term that is low in order of informativeness (e.g. some in Mill’s example), the speaker can be taken to implicate that the proposition that would have been expressed by the stronger term in the scale (All) is false. This can be generalized to a host of scales.

Keith wants more:

What is/are the common usage(s) of statements in the form of *one S is P*? Are the distinct contexts in which people consistently intend *only one S is P*? Distinct contexts in which they rarely or never intend *only one S is P*? Are generalizations available or even possible about this usage in this regard?

Luckily for him, Noveck and Reboul survey experimental work on roughly these same questions in the paper cited above. They observe that

Generally, […] the current data show that semantic meanings of weak terms are the ones that are readily accessible to children and adults while the narrowed meanings are associated with extra effort, leading to the intriguing developmental trends and the reaction-time slowdowns. This argues against default accounts that would expect narrowed meanings to be primary and to occur automatically and would expect semantic meanings to be the result of cancellations.

In the (standard) terminology that they're using, the "semantic meanings" are things like "one = at least one", and the "narrowed meanings" are things like "one = only one". This answers Keith's second question:

For whatever perverse reasons, I am practically *starving* for an/the appropriate technical term to describe the *only one S is P* usage of *one S is P*. It feels like it's on the tip of my tongue…exclusive? Exclusionary?

The only term I know for the "one = only one" interpretation is *narrowed* (because the available part of the pragmatic scale is reduced). It would be plausible to for some derivative of Latin *excludo* ("to shut out, exclude; to cut off, remove, separate from any thing") to be applied to this phenomenon, but I'm not aware of any general practice of this kind. Perhaps a reader can help out.

[For some additional illustrations — literally — see "Scalar Implicature in the funny papers", 9/13/2007.]

## Peter S. said,

July 22, 2011 @ 6:53 am

If I said "two of my children are boys," I think you would have much stronger grounds to conclude that the rest are girls.

## Jesse Sheidlower said,

July 22, 2011 @ 7:12 am

Factoid: Larry Horn, cited in one of the quotations above, is credited by the OED with the coinage of "scalar implicature"; the first example OED gives is from Horn's 1972 Ph.D. dissertation.

## David L said,

July 22, 2011 @ 7:52 am

Surely you're not suggesting that the strictly logical (i.e. as interpreted by a Professional Logician) meaning of "one of my two children is a boy" is the only permissible meaning? I'm with Keith's friend on this — if someone said to me "one of my two children is a boy," I would interpret the unspoken other half of the sentence to be "and the other one isn't." (I don't agree with Keith's friend that this interpretation is a matter of "taking the statement literally," however).

Of course, this would be an odd way for anyone to say "I have two children, a boy and a girl," but then Professional Logicians use language in an idiosyncratic way.

[(myl) Everyone from Mill onward has distinguished between what someone uses a phrase to convey, in context, and what the same phrase "means" in and of itself.

A: Is X a good writer?

B: His handwriting is quite legible.

Sensible people don't conclude from this that legible handwriting implies poor writing skills; but sensible people also recognize that a plausible interpretation in context of B's response may be that X is a poor writer.

There is plenty of room for disagreement about the boundaries of the phenomena — commonplace conversational implicatures may become conventional connotations and even denotations of words and phrases — and even about the basic mechanisms involved (see for example "post-Gricean" ideas like Relevance Theory).

But I'm not clear what you have against "Professional Logicians", or what you think that they're getting wrong here. ]

## J.W. Brewer said,

July 22, 2011 @ 8:08 am

Further to David L, I think the trouble with Keith's friend is that he is assuming that the question complied with usual Gricean default maxims because the questioner is trying to be maximally or appropriately informative in context. In fact, people who like to ask other people questions of this form should perhaps instead be assumed to be acting in bad faith, or at least anti-socially and anti-cooperatively. So a different, more suspicious set of maxims should apply. The wiki article http://en.wikipedia.org/wiki/Boy_or_Girl_paradox notes that the original formulations of the question (attr. M. Gardner) were less ambiguous, but one of them (despite using the clearer "at least one") was still fatally ambiguous. The version Keith's friend is dealing with gratuitously piles a further ambiguity upon that ambiguity. It no longer remotely resembles a logic/probability question. It is purely a trick question in which the questioner hopes (for reasons primatologists will understand) to demonstrate some geeky social superiority to you because you did not guess the right formal-logic notation he was hiding behind ambiguous use of natural language.

The wiki article also notes that the question in any form presupposes three assumptions about human reproduction, at least two of which are materially false.

## Oskar said,

July 22, 2011 @ 8:20 am

This little problem is a classic in probability theory demonstrating that you should always use rigorous arguments and never rely on intuition when solving problems (most people instinctively say that the probability is 1/2, when it is in fact 2/3).

About the linguistic point, I once heard a fantastic variant of this problem that not only solves the puzzle of the meaning of "one", but is even more mind-bending than the original version. It goes like this: "I have two children. One of them is a boy born on a Tuesday. What's the probability that I have two boys?". In this formulation of the problem, it's really quite obvious that "one" means "one or more".

The answer, by the way, is very different.

## David L said,

July 22, 2011 @ 8:29 am

@myl: In your post you said that "Sir W Hamilton and Keith's friend are wrong," which I took to mean that you think they are interpreting the statement incorrectly, in some absolute sense.

I don't have anything against Professional Logicians except when, as J.W. Brewer says, they expect non-professionals to interpret a sentence exactly as they do.

## James said,

July 22, 2011 @ 8:42 am

David L., the claim is that "at least one" is the

semantic value, while "only one" is animplicature. Thus,This is consistent with the claim. Interpreting the unspoken other half of the sentence is quite plausibly a way of teasing out implicature.

## GeorgeW said,

July 22, 2011 @ 9:11 am

Peter S.: "If I said "two of my children are boys," I think you would have much stronger grounds to conclude that the rest are girls."

I think one could reasonable infer two things: (1) You have more than two children, and (2) the others are girls.

## slobone said,

July 22, 2011 @ 9:17 am

@Oskar, pardon my dumbidity, but why is it 2/3 rather than 1/3?

## J.W. Brewer said,

July 22, 2011 @ 9:33 am

Oskar: interestingly enough I have seen the "Tuesday" variant where the poser of the question insisted (in, frankly, kind of an asshole way) that contrary to your interpretation it was unambiguously obvious that "one and only one" rather than "one or more" of the kids was a boy-born-on-Tuesday, i.e. that the other child was necessarily either a girl or a boy-born-on-a-day-other-than-Tuesday. Now, from a Gricean perspective, if it's not "one and only one," the Tuesday detail is presumably a red herring. But that brings us back to the question of whether it is sensible in this context to assume that the questioner is the sort of nice Griceanly-cooperative person who would not include a red herring. That is sort of an extra-linguistic or at least extra-semantic question.

And of course any implicit assumption that children are equally likely (in the contemporary U.S.) to be born on any of the seven days of the week is materially false — a nontrivial number of births are the result of pre-scheduled c-sections or induced labor, and ceteris paribus such events are less likely to be prescheduled for weekends or holidays.

## Eric P Smith said,

July 22, 2011 @ 9:52 am

The degree of implicature depends crucially on the context. “I know one person who may be able to help us” implicates very weakly, if at all, that I know only one such person. By contrast, “My father has one arm” very strongly implicates that he has not two, and no-one would seek to make it watertight by saying “My father has one and only one arm,” except possibly in the context of a logic puzzle.

Sometimes, regrettably, mathematicians expect non-professionals to interpret a sentence exactly as they do. But more often, in my experience, they are acutely aware of possible ambiguities, and that is exactly why they develop watertight terminology. A mathematician will never say “This equation has one solution”, precisely because it is ambiguous. She will say “This equation has one and only one solution”, or, using mathematical jargon, “This equation has a unique solution”. She will not even say “This equation has only one solution” for fear that she may be taken to mean “at most one solution".

If the word ‘one’ is being used as a quantity rather than a count, I don’t think there is any ambiguity. By any token, “This cost me one dollar” is false if in fact it cost me two dollars.

## J.W. Brewer said,

July 22, 2011 @ 10:15 am

I should probably modify the "trick question" rhetoric of my first comment in light of the oft-useful maxim that one should not attribute to malice what can be adequately explained by stupidity. It is certainly possible that in some instances the posers of these questions are so confident of the meaning they intend to convey they do not adequately focus on whether the actual words they are using will unambiguously convey that meaning. OTOH, my impressionistic sense is that the sort of people who like logic puzzles are not infrequently also the sort of people who like puns and similar wordplay, and/or are ignorant prescriptivists who think that their phrasing has One Right Meaning and the fact that they know many fellow-users of the English languge may interpret it otherwise just means that Those People Are All Wrong.

## Brett said,

July 22, 2011 @ 10:26 am

The mention of someone with at least one arm reminded me of my favorite puzzles that I learned from Martin Gardner. Paraphrased, it's: "How likely is it that the next U.S. Open tennis champion will have more than the average human number of arms?" Of course, it is overwhelmingly likely, since that average is slightly less than 2.

## Faldone said,

July 22, 2011 @ 10:32 am

We have to remember the context here. This is a logic problem. If "one of my two children is a boy" is taken to mean "one and only one of my two children is a boy" the problem is trivial and therefore cannot be the correct interpretation.

@slobone. The possibilities are the older child is the given boy. The younger child can then be, with equal possibility, a boy or a girl. Or the younger child is the given boy. Again the older child can be a boy or a girl. But the case of the younger and the older child being a boy is not two separate possibilities. Thus we have boy-boy, boy-girl, and girl-boy. Girl-girl is not a possibility and the probability of the other child being a girl in the original problem is 2/3.

## Simon Spero said,

July 22, 2011 @ 10:41 am

It's surprising that there isn't an obviously correct word for "only one S is P usage of one S is P", unless you count

#$thereExistExactlyas a word.The concept is used quite heavily in Description Logics with "Q" in their name (see http://www.win.tue.nl/~aserebre/ks/Lit/dlhb-appendix.pdf *, which refers to the more general case as "Qualiﬁed exact restriction").

—–

* Baader, F. (2003). The Description Logic Handbook Theory, Implementation, and Applications. Cambridge, UK: Cambridge University Press.

## Richard Hershberger said,

July 22, 2011 @ 10:49 am

I think the key to understanding this sort of logic puzzle is that they don't use standard English. Standard English carries all the implicature others has discussed. Logic puzzle English does not. Standard English is the default. If someone asks a question, it is reasonable to assume it uses standard English unless context makes it clear that it is in logic puzzle English. Anyone posing such a question without sufficient context, and using this as a zinger to count coup is somewhere on the spectrum ranging from cluelessness at one end to assholery at the other.

The reverse phenomenon is also pretty common: someone posing a puzzle but botching the logic puzzle English. Many statements of the Monty Hall problem include an unstated assumption that Monty always opens a door and offers the contestant the opportunity to switch choices. The "correct" answer depends that Monty have no personal volition, but rather go through this ritual every time. If the statement of the problem leaves this ambiguous, there is no solution.

## Dennis Paul Himes said,

July 22, 2011 @ 10:51 am

@Brett: That depends on whether by "average" you mean "mean" or "median". The median number of arms is surely two.

## Oskar said,

July 22, 2011 @ 11:04 am

@slobone: Usually the question is "What's the probability that i have two boys?", and then it's 1/3, but the way it was phrased in the blog post was "what is the probability that my other child is a girl?". Hence, 2/3.

## Keith M Ellis said,

July 22, 2011 @ 11:19 am

Thanks for posting this, Mark! Your response has been very helpful and much appreciated by me.

I'm more than a little taken aback by the, er,

vigorof J. W. Brewer's assertions of the correctness of his assumption of bad-faith in how I phrased the puzzle. It's also somewhat disheartening because my friend, while not claiming that I, personally, chose my wording in bad-faith when I mentioned this to her, did assert that she's encountered the non-narrowed usage of "one is" before and has concluded that it's deliberately misleading. (She didn't assume bad-faith on my part specifically because the context in which we were discussing this was such that we were both familiar with the puzzles we'd already mentioned and there was no expectation by either of us that I was posing this as a challenge to her. Quite the opposite, really.)I can accurately attest (though you'll have to take my word for it) to my own state of mind when I described the problem and that, in truth, I gave zero explicit thought to whether I should use "one" or "at least one" and that there was no intent to mislead whatsoever. I didn't even look this up and/or otherwise quote someone else; the wording I chose was extemporaneous and intuitive.

Which J. W. Brewer and my friend my find difficult to believe…until you take into account my deep and long familiarity with categorical and other logic, and philosophy in general, and also math statements of this and similar natures. It would never occur to me

in the context of technical languageto ever use or parse "one is" or "some are" as anything other than theirsemantic meanings.Mark elided an interesting and, I think, relevant bit about usage: my friend said that she'd seen just yesterday a Sesame Street skit

This speaks to the one thing in the research that Mark cites that surprised me: that the logician's usages are more accessible for children and that, apparently (if I read this correctly), they have to learn to infer the implied, narrowed usages.

On the one hand, this is surprising because my friend's assertion that the narrowed usage is the more "literal" exemplifies the notion that the logician's usages are arguably "unintuitive" and "unnatural" so, assuming this, one would guess that children would start with the narrowed usage and learn the logician's usages for only those rare and technical contexts in which it applies.

On the other hand, this arguably

isn'tsurprising because, contra my friend's assertion, the narrowed usage isn't "literal", it's an inference from implicature. With this in mind, it makes sense to think that a child would first parse such statements quite "literally"—saying one ball is red doesn't in any way say that the other balls aren't red. Only, in time, do children learn to parse usages that rely upon implicature. And, of course they take time and effort to do so—it requires so much more context and work.Which then goes to explain the apparent paradox of the first view: most usages rely upon scads of implied contextual meaning to correctly parse and we learn to do so…and find it very difficult to

notdo so when the occasion arises that a bare, "literal" parsing is most appropriate. So the logician's usage, insofar as it's arguably "unintuitive" and "unnatural", is thus simply because it's an example of learning tounlearna long-acquired generalization for certain specific, exceptional cases.I think this is an important and relevant point. This is something that comes up often in discussions of the Monty Hall Problem, about which I've devoted a small website.

The issues of how to parse questions in such specialized contexts as test-taking has come up before at LL. It often seems to me that many (but certainly not all!) of the complaints about ambiguity of questions in standardized tests and similar (such as logic/probability puzzles like this) are much weaker than they seem because parsing these questions

as test questionsprovides a whole bunch of contextual information about how they should be, or necessarily must be, understood.## Eric P Smith said,

July 22, 2011 @ 11:39 am

To make this topical: An official police statement in the UK today says, “There has been one or several powerful explosions in the government district in Oslo.”

But not two and only two?

## Keith M Ellis said,

July 22, 2011 @ 11:40 am

@Richard Hershberger, you wrote:

On my MHP site, I phrase the problem as so:

…and the footnote reads:

…and my footnote to my footnote:

All of which is to both mostly agree with you but slightly disagree with you. I think it's necessary, and necessary for good-faith, to include the stipulation that Monty must necessarily open a losing door that is not the door you chose after you've made your choice.

But, on the other hand, most or all of the alternatives produce either unsolvable or uninteresting problems. For this reason, and per what I wrote in my previous comment, I'm not completely sympathetic to such complaints about the MHP or, indeed, in general. One of the biggest reasons such careful stipulations are omitted is because often they're only necessary if you're trying to parse the problem statement in a void of context. Which, sure, to be rigorous in formulating a problem statement, we should try to approach. But, as we know from the context of LL, the notion that it's possible to parse sentences entirely outside of external context is itself nonsensical.

## jan said,

July 22, 2011 @ 11:50 am

Here's a related one.

The customer's purchases total $19.01. Not wanting $.99 in change, the customer tells the cashier, "I have one cent," and hands over a nickel. Or, "I have a penny" and hands over a nickel.

## wally said,

July 22, 2011 @ 11:57 am

@slobone

I think that as the wikipedia article referenced above shows, the problem at the top of the post is ambiguous, but the most reasonable interpretation of the question as stated is that the answer is 1/2.

The article also states "The paradox has frequently stimulated a great deal of controversy. Many people argued strongly for both sides with a great deal of confidence, sometimes showing disdain for those who took the opposing view." So I'll duck now.

## Chandra said,

July 22, 2011 @ 12:03 pm

I can only see Keith Ellis's friend being "wrong" in the context of such a probability puzzle or discussion. In any ordinary context she would be quite right, probably even on legal grounds (if, for example, someone were to employ similar phrasing to cover up a deceit, and then try to use the hyper-logical definition as a defense).

## Rodger C said,

July 22, 2011 @ 12:13 pm

@Dennis Paul Himes: Don't you mean modal?

## Adrian said,

July 22, 2011 @ 12:22 pm

@Keith: your friend would've been right if she hadn't misused the word "literally". Perhaps the word "idiomatically" would be a suitable replacement.

@Eric: "several" *literally* means more than one.

## Keith M Ellis said,

July 22, 2011 @ 12:36 pm

I don't think that's correct; it's certainly not been demonstrated.

{(myl) Bronston v. United States, decided by the U.S. Supreme Court in 1973, let Mr. Bronston off the hook for a different kind of alleged-perjury-by-implicature, but the decision also includes this footnote:

So if you're under oath, watch out. (See here for some additional details and discussion, including the question of whether truth is like pie.)]

Common, non-technical usages are what I was interested in when I wrote to Mark; the ambiguities in the context of the specific problem itself are not of much interest to me for reasons similar to why I find alternative parsings of the ambiguous portions of the MHP statements to be relatively uninteresting.

I haven't attempted to read and digest the Noveck and Reboul paper, but Mark says that it "survey[s] experimental work on roughly these same questions". (

Thesebeing my questions about whether any generalizations can be made about when speakers prefer the semantic or implicative usages.)I could be mistaken, but I didn't get the impression from what Mark wrote that we can assume that the semantic usage only appears in technical usage by Logicians and their peers while, in common language, the implicative usage is universal. Eric P. Smith asserts that in common language this varies by context and provides two examples. At the very least, the implication of the examples Smith chose is that the usage will vary relative to how unusual/common "only one" or "all" applies in the real world to the objects being discussed. Relatively few people have less than two arms whereas, in contrast, in relatively few cases would a person be able to come up with one and only one person that could possibly help them.

It seems to me to be true that, as in the case of the Logician's usage, the choice to use "one" purely widely and not narrowly is itself implicatively determined.

## Kevin said,

July 22, 2011 @ 1:01 pm

One = only one?

I guess that's one way to look at it.

## Brett said,

July 22, 2011 @ 1:04 pm

@Dennis Paul Himes, Rodger C: Clearly, both the median and modal numbers of arms are 2. However, I (and I know I am not alone in this) would only use "average" to refer to the mean. As I recall, this agrees with Gardner's usage in Aha! Insight (from which the puzzle came), In contrast, Darrell Huff discusses how obfuscating which meaning is meant is a skillful method of How to Lie With Statistics.

I think the use of "average" only for "mean" may be a product of the fact that when the "median" is actually used (presumably because it, unlike the mean, is minimally effected by the long tail may thus be more illustrative of typical values), it tends to be described specifically as the "median," to avoid confusion with the mean.

## Coby Lubliner said,

July 22, 2011 @ 1:15 pm

A peeve: I have a hard time seeing the need for the verb

implicate(whose ordinary meaning is `involve in an incriminating way,' e.g. in a scandal) when it seems to differ only slightly, if at all, fromimply; and even less so for the nounimplicature, which implies (implicates?) the existence ofimplicaturain Latin, since-tureis not, as far as I know, a productive suffix in English: words likenature, quadrature, sutureetc. come from the corresponding Latin ones. Or does someone nate, quadrate or sute?## Jerry Friedman said,

July 22, 2011 @ 1:20 pm

Out of 40 Google hits (counting duplicates) I found on

one of my children is a boy, most were about this "probability" puzzle. There were five (not counting a duplicate or one withonly one…) that were written by people talking about their children in natural language. Three of those people said they have exactly one son, and two of them said so somewhere else not long before they wrote the phrase; in context, I feel sure both of them had exactly one son at the time of writing.I looked at the Noveck and Reboul paper, and they don't mention this particular construction. By the way, I'm not convinced that the meaning found by little children, or by adults who don't take or aren't given time to think, is the meaning the phrase has in itself. (Due to another implicature in ordinary language,

don't take time to thinkdoesn't includearen't given time to think, at least for me.)At least one thing is clear to me: if you want to pose puzzles that are really about probability rather than semantics, you need to say

exactly oneorat least oneorthe older oneor something unambiguous like those.@Dennis Paul Himes: I don't remember ever hearing

averageused to meanmedian, and the OED doesn't mention that possibility. If you have citations and share them here, Jesse Sheidlower might be interested (or might not), and he's already shown up in this thread.## Ray Dillinger said,

July 22, 2011 @ 1:24 pm

Logic problems like this aren't using conversational English. They are using mathematics, which are *expressed*, somewhat perversely, with English words and sentence structure.

If you want to get the "correct" answer you must first translate to the language of Mathematics, which is usually expressed in its own purely ideographic writing system where each symbol has exactly one meaning. You then solve the problem and translate again back to English.

These translations are lossy. In the same way a Finnish speaker's word for maternal-grandmother loses part of its meaning when translated into English as "grandmother", an English phrase like "one of my children is" loses part of its meaning when translated into a THERE_EXISTS proposition in Mathematics.

## Jerry Friedman said,

July 22, 2011 @ 1:31 pm

I wrote, "By the way, I'm not convinced that the meaning found by little children, or by adults who don't take or aren't given time to think, is the meaning the phrase has in itself."

Le me rephrase that. I'm not clear on what is supposed to be special about that meaning. In particular, I don't see what relevance it has to the understanding of puzzles and whether Keith M Ellis or his friend is right.

I should also say that I realize a lot of what I posted is merely agreeing with Eric P Smith.

## Keith M Ellis said,

July 22, 2011 @ 1:38 pm

I'm a little startled at regulars of LL confidently asserting a universal particular common-language English usage on the basis of their own common-language usage and intuition.

## Keith M Ellis said,

July 22, 2011 @ 1:55 pm

And, I'd like to reiterate that I didn't query Mark in order to determine which of I or my friend was right. I knew and know that my position was correct within the limited context of mathematical language. I don't think either of us asserted that a particular usage was universal in casual English—certainly I didn't. Rather, I asked Mark if he knew how usage varied in casual English and if any generalizations could be made about it.

It's pretty obvious that

one S is Pis just a special case of the larger set ofsome S are Pstatements. And that therefore the cited paper is relevant.## Dennis Paul Himes said,

July 22, 2011 @ 2:28 pm

I don't have any citations on hand as to the use of "average" to sometimes mean "median". That was based on 1) my experience as a native speaker of English, and 2) the fact that the tennis puzzle isn't much of a puzzle unless there's a good possibility that one it's posed to will interpret the word "average" in the puzzle to mean "median" (or perhaps "mode"). The aha factor of the puzzle, of course, assumes that this is an incorrect but common usage and that the mistaken person will recognize that fact once it's pointed out.

My experience is that while the word "average" is usually used to mean "mean", it's ambiguous enough that I try to remember to say either "mean" or "median" in those situations where it makes a difference. One situation where "median" is relatively common (although I don't have anything to back up this claim, I admit) is when the possible values are small nonnegative integers, such as number of arms.

BTW, the Merriam-Webster online dictionary's first definition for "average" is "a single value (as a mean, mode, or median) that summarizes or represents the general significance of a set of unequal values", but then in the "synonym discussion" it seems to assume that "average" is "mean", and should be distinguished from "median".

## Jonathan Mayhew said,

July 22, 2011 @ 2:34 pm

If you imagine this conversation:

A. One of my parents was born in Arizona.

B. Where was the other one born?

A. Arizona.

You would conclude that A. is not being very Gricean. Logic brain-teaser, though, are not very Gricean.

## Bloix said,

July 22, 2011 @ 3:01 pm

I don't think this is a question of implicature or of logic at all. "One" does not mean "at least one" or "one or more than one." It means "one."

If I tell the IRS, "I made $100,000 last year," and I made $200,000, I'm a liar.

It's true that by describing what we know, it is understood that we are not describing what we don't know. So, if I tell my wife about a guy I've just met who told me he has two kids, and I say, "one is a boy," that doesn't mean that the other is not a boy – it might mean that I don't know.

But if I'm talking about my own two kids, then it is a certainty that I know the gender of both. So if I say, "one of my two kids is a boy," I'm saying that precisely one is a boy. If in fact two are boys, then I've told a lie.

## James said,

July 22, 2011 @ 3:45 pm

Cory L.:

There is, in fact, a very important difference between 'implicate', in this technical context, and 'imply' (and a parallel difference between 'implication' and 'implicature'). If you assert a proposition p, what is implied is whatever follows from p, whereas what is implicated is what follows from the fact that you asserted p in the context you did. The difference is important especially when Gricean rules come into play.

## Keith M Ellis said,

July 22, 2011 @ 3:48 pm

I keep seeing universal assertions about usage based upon nothing more than personal usage and intuition. Is this the day when LL regulars are possessed by the spirits of prescriptivist peevers? WTF?

## Adrian said,

July 22, 2011 @ 4:02 pm

@Keith: You're being unfair. As far as I can see, the comments are not based on "personal usage and intuition" but on experience. In other words, we aren't being peevers, we're being descriptivists.

## Keith M Ellis said,

July 22, 2011 @ 4:27 pm

@Adrian, descriptivism

isn'tnormative generalizing to all usages from personal experience.## Hermann Burchard said,

July 22, 2011 @ 4:35 pm

It is fun to transate the "one & only one" logic into "if & only if" which is so very common in maths. In English we have the happy abbreviation "iff" [do recall seeing an ugly "dund" in German]:

If the universe of my children is known to have cardinality exactly two, then in plain English

"Exactly one of my children is a boy" can be expressed logically by saying "my first child is a girl iff my second child is a boy."

Give the universe of my children is {a, b}, but not a=b, then the above is, of course, logically equivalent with "my second child is a girl iff my first child is a boy." If you have any doubts, recall that by contraposition we have:

[If a is a girl then b is a boy] iff [If b is a girl than a is a boy].

Formal proof: Let m(x) stand for "x is a boy." We may avoid quantifiers and use "v" for the inclusive "or," "A=>B" for "if A then B," "~" for not (sorry), and "A==B" for "A and B are logically or semantically equivalent." Then

at least one of my children is a boy == m(a)vm(b) == ~m(a)=>m(b)

at most one of my children is a boy == ~m(a)v~m(b) == m(a)=>~m(b)

Then we find effortlessly:

~m(a) m(b) == exactly one of my children is a boy.

BTW, my maths students picked up on "only if" vs. "if" after a short discussion with examples, but probably required that introduction to this somewhat specialized precision, which clearly is absent in colloquial speech (not the proper linguistic term). Some took a logic course offered in the phil department.

Please report any logical flaws above (likely because of senior brainsmanship).

## Hermann Burchard said,

July 22, 2011 @ 4:41 pm

Typo correction,

above please read in the third paragraph from the bottom:Then we find effortlessly:

~m(a) iff m(b) == exactly one of my children is a boy.

## Eric P Smith said,

July 22, 2011 @ 4:43 pm

@Brett, @Jerry: In mathematicians’ usage, ‘average’ has the meaning of ‘a measure of central tendency’. Thus the arithmetic mean, the median, the mode, trimmed means, etc are all ‘averages’. There are (at least) three measures of central tendency that go by the name 'mean': the arithmetic mean, the geometric mean, and the harmonic mean.

I think that in common usage, a person using the word ‘average’ as a noun and being precise is likely to be referring to the arithmetic mean’ My only peeve in this context is not against Joe Public: it is against Microsoft Excel for pandering to Joe Public by giving their function that returns the arithmetic mean the name ‘AVERAGE’.

There is of course a much looser usage of ‘average’ as an adjective with the meaning ‘not quite up to standard’, as in an ‘average performance’.

## Dan Hemmens said,

July 22, 2011 @ 4:49 pm

Tangential linguistic note about the 1/2 or 1/3 (or 2/3 depending on what you're asking about) interpretation of the puzzle.

It strikes me that the two ways of looking at the actual *mathematical* puzzle highlight a *third* distinct meaning that "one" could have in this sentence.

The two meanings discussed in the OP are "at least one" and "exactly one" but even if we ignore the "exactly one" interpretation there is still an ambiguity between "at least one" and "one identified individual."

If we take "one" to mean "at least one" then the odds of both of Xs children being boys are one in three, by the working outlined above.

On the other hand, if we take "one" to mean "a specific one" then the probability remains 1/2, because the sex of one child does not depend on the sex of the other.

I can only speak for myself, but in the sentence described in the OP it's the third "a specific one" meaning that feels most natural. I wouldn't take "one of them is a boy" as precluding the other from being a boy as well, but nor would I take it as making a more general probabilistic statement that "at least one" is a boy.

## Dan Hemmens said,

July 22, 2011 @ 5:12 pm

A couple more points, apologies for double posting:

@Bloix:

Don't your examples rather undermine your point? If "one of his children is a boy" could mean either "I have certain knowledge that one of his children is a boy, and am uncertain about the sex of the other" or "I am certain that one and only one of his children is a boy" why can it not, in the context of a probability exercise where it is clear that some information is being deliberately concealed from the reader, mean "you are informed that one of his children is a boy, and are given no information about the other"?

@Eric P. Smith

I'm not sure, I think the average member of the public uses "average" to mean either arithmetic mean, or mode, or median largely interchangeably (and to be fair, they often *are* very similar to each other), and I suspect that this is true even if the average member of the public is attempting to be precise in their usage.

Take, for example, the old "Bushism" about how awful it was that fifty percent of High School students were below average – this was considered funny because "everybody knows" that fifty percent of people are always below average because that's how an average works, but this is true only if "average" means "median".

## Ian Preston said,

July 22, 2011 @ 5:18 pm

I agree with those who understand

averageas an unspecific term for any measure of central tendency, covering not only the median and arithmetic mean but also the geometric mean, harmonic mean, and so on. Contrary to what Jerry Friedman says, it seems to me that the OED does admit the possibility of it denoting the median:It's not very difficult to Google up examples of

averagebeing used explicitly formedian:## Ellen K. said,

July 22, 2011 @ 7:14 pm

I'm thinking a lot of the people who commented after it didn't read the first reply (fairly early in the replies) from Eric P Smith. He gave a good example of where "one" does not imply "only one". “I know one person who may be able to help us.” Rather, the implicature is "one that I have in mind and am talking about".

## Rodger C said,

July 22, 2011 @ 8:17 pm

This is all reminding me of Jonathan Miller's

Beyond the Fringeskit where he played Bertrand Russell recalling his first meeting with G. E. Moore. I quote from memory: "'Moore, do you have any apples in that bag?' 'No.' 'Do you havesomeapples?' 'No.' 'Do you haveapples?' 'Yes.' Then he offered me one, and we have been fast friends ever since."## bloix said,

July 22, 2011 @ 9:11 pm

Dan Hemmens –

In this case, the puzzle takes the form of a story problem in which a father is talking about his children. We are entitled, I think, to assume that the father has the information usually known to fathers and that he speak standard English. Given those assumptions, we are entitled to conclude that "one of my children is a boy" means just that – one, not two, not zero, but one. The meaning, "one of the children is a boy and I'm not sure about the other one" is not permitted when a father is speaking. Otherwise, why do this as a story problem?

## Hermann Burchard said,

July 22, 2011 @ 9:36 pm

My stat prof colleague told me that stat guys don't say average but expected value, same as mean but with less ambiguity.

BTW, "average" is from Arab العوار, al awar (I believe a form al avarya may exist) "damage" from maritime law concerning insurance of ship's cargo. In English Wikipedia this occurs as "General Average."

This goes back to Phoenician, I think. The ancient mariners knew to apply probability theory and insure according to average damage. French Avarie, Italian Avaria, German Havarie.

## Steve Morrison said,

July 22, 2011 @ 9:43 pm

Gardner reduced the gimmick to its essentials in another riddle: "I have two U.S. coins whose total value is 30¢. One of them is not a nickel. What are the coins?" Of course the answer is "A quarter and a nickel".

## Jerry Friedman said,

July 22, 2011 @ 9:46 pm

I wonder whether the formulation of the puzzle has an effect on the difficulty many of us have in understanding

oneasat least oneorone or more. If we simply made that change—"at least one of my two children is a boy, what is the probability that my other child is a girl?" the question wouldn't make sense to me. Other than one or both of them? For me, if the question hasat least one, it needs to be reworded. Maybe "At least one of my two children is a boy. What is the probability that I have a daughter?" Thus I think any wording withthe otherstrengthens the implicature, already strong for many people as seen in the comments and in my Google search, thatonehere means "exactly one".(I'd give the combination of

at least onewiththe otheran asterisk, except that the Wikipedia article quotes "Mr. Smith says: 'I have two children and at least one of them is a boy.' Given this information, what is the probability that the other child is a boy?" from an article by Fox and Levav behind a paywall. Preview messing up—I hope the link posts all right.)On the subject of Noveck and Reboul, even though philosophically the question of "some or all" versus "some but not all" is analogous to that of "one or more" versus "exactly one", I doubt it follows that the patterns of non-technical usage must be the same. Finally, I agree with Eric P Smith that the context makes a good deal of difference, and I'd expect

I know one person who may be able to help usto be understood as "exactly one" less thanOne of my two children is a boy. For me,One person who can help us is Doehas even less of an implicature of "exactly one".And I still don't see how the hearer's processing time is relevant to your question about when speakers intend which meaning, or to other questions of yours.

@Ian Preston: I'd argue with you about the OED entry, but the point is academic; your Google results clearly show that some people do use

averageto meanmedian.@Dennis Paul Himes: I needn't have doubted you on the meaning of

average, but on a side note, I'd say that something like the number of a person's legs is not a good situation to use the median. If there's one person in the world with 3 legs and one with 4, the median is 2, but two surgical operations could change that to 1. (I mean my numbers to be exact, so I didn't saytwo or more surgical operations.)## Rebecca said,

July 22, 2011 @ 9:53 pm

I agree with Ellen K. Regarding Eric P. Smith's example. "one" can easily be used to just bring focus to a particular entity without even a weak implication that it is unique. I can start a story with "one of my friends just got back from vacation"… without any flavor of their being only one such friend.

But there are other instances where I'm not pointing to a particular person, and, like the father example, I know full well that the "one" is not unique, yet the natural reading is still one=at least one, for example:

"One of my kids can help you with that.". If anything, this seems biased towards. Dreading where one=any one

Of course, like some other examples, this is getting away from "one of S is P" into more complex predicates, so it may not be relevant to the original question.

It seems to me, it really is all about context, and it's more complex than just what people can be supposed to know.

## Jerry Friedman said,

July 22, 2011 @ 11:29 pm

Supplementing my previous post with data: a search at COCA found 17 hits for "at least one of [wildcard, up to 9 words] other", with none in which

othercontrasted withat least one. For "one of [wildcard, up to 9 words] other", there were 3208 hits, and 31 out of the 85 I looked at had the relevant meaning.However, I did find two examples at Google with an ease that surprised me. The first is in some strange dialect, but appears to be about guessing what cards the author's Texas Hold 'Em opponents had.

"I had them on Ax and Kx respectively, at least one suited, the other with maybe an 8-10 in reserve."

The second is clearer to me.

"Basically, in pairs of identical twins with at least one homosexual, in 68% of cases the other identical twin was NOT homosexual! In comparison, in control groups consisting of non-identical twins and non-twin sibling pairs, in 79% of cases with at least one homosexual, the other sibling was NOT homosexual."

Nevertheless, I think the COCA results show that

at least onecontrasts withthe othervery rarely, suggesting thatthe othermight indeed implicate (?) thatonemeansexactly one.## Jerry Friedman said,

July 22, 2011 @ 11:49 pm

Sorry, maybe I should compare paraphrases to paraphrases. COCA had no hits on "exactly one of [wildcard, up to 9 words] other", but 9 of the 20 hits on "only one of [wildcard, up to 9 words] other" had the relevant meaning.

For equal time for paraphrases of the semantic meaning, none of the 19 COCA hits on "one or more of [wildcard, up to 9 words] other" had the relevant meaning.

## alexw said,

July 23, 2011 @ 8:07 am

I might have missed it in all the comments, but I have come across the argument that "one" and other numerals are different in this regard and only takes the narrow reading, perhaps because "a" does seem to mean "at least one" in most contexts. Kratzer's paper on intermediate scope of specific indefinites encodes this difference into the conventional meaning of "one".

I don't totally buy it–Jan's "one penny" example is felicitous to me–but I do think "one" has a much stronger tendency to take the narrow reading than other numerals and determiners.

## Rebecca said,

July 23, 2011 @ 10:09 am

Cursed autocorrect. In the third paragraph of my commet, "towards. Dreading" should read "towards a reading".

## J.W. Brewer said,

July 23, 2011 @ 10:37 am

I wonder if part of Mr. Ellis' problem is that he's engaging in implicature of his own. He apparently had not previously considered implicature based on people's experience or intuitions about how actual human beings typically talk about their own actual children. Rather, he is working with some rival assumption like "pick the meaning that will yield an interesting and nontrivial logic/probability problem." But that only works if you already know the respective "answers" that would correctly flow from different understandings of the question. If you already know what the question is "supposed" to mean based on that sort of sense, you may have a blind spot as to what the actual words being used will in context be taken to mean by someone who does not already know the answer to the question.

I am struck by the fact that apparently the number of people giving the falsely-claimed-to-be-correct* answer (1/3 or 2/3, depending on formulation) varies systematically with the phrasing of the problem. While some of the people doing work in this area seem to think this shows something about cognition, or the ability of hapless lay people to think clearly about probability if not properly primed, I think it may be more of a linguistics issue. The variation in response means, I think, that the implicit minor premise that the different formulations of the problem all "mean" exactly the same thing is inaccurate, for at least one useful sense of the word "mean." If enough native speakers are "misunderstanding" the question you have posed to them, you should be open to the possibility that it's not their fault, it's your fault, because the question apparently does not mean (or at least does not unambiguously mean) what you intended it to mean.

*Falsely claimed to be correct because the odds that a particular birth will be a boy versus a girl are not 50/50, as has been known for a very long time by anyone with even a modest acquaintance with the data. (This is without even getting into the further complication that the sex of a particular child is unlikely in practice to be entirely independent of the sex of prior children born in the same household, and the sex of the first two children may have an impact on whether the household "stops at two" rather than having a third child — although of course some forumulations of the problem would include men who have children by multiple women w/o having necessarily bothered to marry either of them). I find this feature of the problem to make it as irksome as, say, a "logic question" that falsely presupposes the earth's orbit around the sun to be circular, although it has been known at least since Kepler's day that that is not the case. This may be an idiosyncratic reaction on my part. There's obviously a fuzzy and subjective line between what's a permissible simplifying assumption and what's Just Flat Untrue.

## wally said,

July 23, 2011 @ 12:38 pm

@bloix "Otherwise, why do this as a story problem?"

1) With a story problem, you can introduce ambiguity, and then crow about how stupid the people are who interpret it differently from you. (See Parade magazine)

2) The logic problem is given as a stylized and simplified version of what could be a reasonable statement. "Oh what a week this has been. Both of my kids have been sick. I even took my 18 year old son to the emergency room"* What is the probability that both my kids are boys? This has extraneous info, but is a natural way to get to the same place logically.

This has been a delightful thread, by the way.

* True story at my house.

## Brett said,

July 23, 2011 @ 2:24 pm

@Eric P Smith: I do have a Ph.D. in mathematics, so I am familiar with mathematical usage. I think that any of the arithmetic, geometric, or harmonic means (or other one of an infinite class of such structures) would be "an average," but only the arithmetic mean is "the average." (However, I'm not sure how many mathematicians would accept the median even as "an average.")

This kind of distinction is a pretty common on in English. A character may be "a hero" in a story without being "the hero," etc. I'm now curious what linguistic research has to say about this question.

## Keith M Ellis said,

July 23, 2011 @ 2:38 pm

Well, yes, I did engage in implicature which is

whyI didn't much consider how "actual human beings typically talk about their own actual children". (As opposed to how virtual human beings atypically talk about other people's virtual children.)I expected that because from its form this was manifestly a toy problem in probability, reinforced by the context that

she and I were discussing toy problems in probability, I expected that this would strongly inform how she approached the problem. Also, her personality and academic history gave me good reason to believe that she'd prior experience with this logician's usage of "one". It never occurred to me to worry about a possible misunderstanding.In contrast, if I had been describing this problem to someone I knew had little or no experience with the logician's usage and with these sorts of problems, I would have made a deliberate effort to phrase it differently so as to avoid the confusion. For example, "We know that someone has two children but for whatever reason we only know the gender of one of them and that it's a boy." (I think these sorts of problems are intrinsically interesting and I have zero interest in them as a means to make other people look bad and I'm still quite offended at your implication to the contrary.)

My sense is that through no fault of his own, Mark's choices in what to quote from my email and what to elide, and what to emphasize in his response here, has given readers something of the wrong idea about my conversation with my friend, my interest and intent in writing LL about it, and what I was specifically asking.

I wrote to Mark:

…which he then uoted. My questions were about how such statements are used in common language and whether there's any patterns to the usage, and what is the terminology involved. I

didn'twrite looking for an authority to take my side.And I

certainlywasn't looking for an opportunity for someone to project onto me their accumulated issues with people they believe pose puzzles for the sake of making others look foolish; and subsequently to hear (repeatedly) how such problem-writers and tellers in their oversimplifications are thetruefools in this interaction. If I'd wanted that sort of thing, I'd go to a Mensa meeting or hang out with maladjusted adolescents. But I repeat myself.## Jerry Friedman said,

July 23, 2011 @ 4:03 pm

Speaking of assumptions about the probabilities of boys and girls and children with intersex phenotypes, I've been wondering how Chinese people, from the cities where the one-child policy applies, would answer this question. I suspect they'd bring different presuppositions about the sex of the children. Or am I wrong?

## Hermann Burchard said,

July 23, 2011 @ 4:52 pm

@Rebecca:

Auto-correct? Is that what deleted one "iff" in my above comment, forcing me to self-correct at July 22, 2011 @ 4:41 pm?## Dan Hemmens said,

July 23, 2011 @ 5:50 pm

*Falsely claimed to be correct because the odds that a particular birth will be a boy versus a girl are not 50/50, as has been known for a very long time by anyone with even a modest acquaintance with the data.That doesn't mean that the answer is not correct, only that it's not correct to an arbitrarily high level of precision.

## Ellen K. said,

July 23, 2011 @ 6:28 pm

@Dan Hemmings: That ratio, and the corresponding percentage, assumes two digit precision. It assumes 50% of births are male, 50% female, to the nearest percent. A quick investigation on the internet, and a little math, shows that's not the case.

## Rebecca said,

July 23, 2011 @ 6:44 pm

@Hermann Burchard: The autocorrect I was cursing was on my iPad. I don't know the situation on your input device. I've not seen evidence that the blog is doing any correcting, but I imagine that there might be characters that have a formatting role and could mess things up if we slip them in, thinking of them just as formal symbols. I don't know if that's the case with your comment.

## hector said,

July 23, 2011 @ 7:32 pm

Keith Ellis refers to himself as using "mathematical language," and it seems to me the problem starts right there. I question whether the phrase "the language of mathematics" is anything but a metaphor, given that mathematical notation does everything it can to (a) remove ambiguity and implication, and (b) abstract itself from context.

Real languages, on the other hand, are rife with implication and rely on context. Which is why spoken-word logic problems are so problematic, being open to interpretation.

This is why a lot of people (like me) find "toy problems in probability" so irritating and tiresome. Woe betide you if you disagree with the questioner's translation of logic into the living language. It's not because we're "maladjusted adolescents;" it's because we find the whole exercise dubious and inherently quarrelsome.

## Dan Hemmens said,

July 23, 2011 @ 7:38 pm

That ratio, and the corresponding percentage, assumes two digit precision. It assumes 50% of births are male, 50% female, to the nearest percentI'm pretty sure it doesn't.

If we're being technical here, the precision of "50%" is ambiguous – there is no indication whatsoever that it is intended to be read to the nearest percentage – it could just as easily be to the nearest 10% or for that matter the nearest 50%, we have no way of knowing because it isn't specified. The fact that there are two digits in the number "50" makes no difference because (a) the second digit is a zero and (b) the number of significant figures is only a rough indicator of precision in any case.

On a point of general usage, I'm also really not sure there's any mileage in the claim that "50/50" (or any other round percentage) implies precision to the nearest 1% in any situation where it is normally used.

## Richard Hershberger said,

July 23, 2011 @ 8:29 pm

@Keith M Ellis:

Regarding the wording of the Monty Hall problem, you are demanding that the person being posed the problem work out every possible interpretation of a vaguely worded question, analyze the implications of each interpretation, and determine based on unstated assumptions which one of these is considered interesting to the person posing the problem. That seems rather a lot to ask. The last step is particularly onerous. How am I to know what you consider interesting? Perhaps you are interested in whether or not I spot the ambiguities, and your correct answer is that the problem is unsolveable as stated.

There is, through historical accident, a culturally traditional interpretation of the problem, with a solution deriving from this interpretation. I know this solution because I am familiar with discussions of the problem. But were I to encounter it de novo, I could merely guess what the poser considered correct.

## J.W. Brewer said,

July 23, 2011 @ 9:31 pm

Presumably to a logician or mathematician 50% *really means* 0.500000 etc., with zeroes out past the horizon. The contention that it conveys "close enough to 50% for present purposes" or perhaps "as close to 50% as the measuring instrument we are using can accurately determine" is invoking implicature. By "being technical," Dan Hemmens is leaving logic behind for the viewpoint of a mere engineer or observational scientist. And to be fair, the standard answers floating around the how-many-boys problem apparently tend to be expressed not in percentages but as 1/2 (the official "wrong" answer) and, depending on phrasing, either 1/3 or 2/3 (the official "right" answer(s)). I have no idea how to gauge what degree of precision (as expressed in significant figures or otherwise) is typically intended or understood to be conveyed by these answers, except to note that lots of standard probability problems involve sequences of coin flips and I believe those problems assume a substantially fairer coin than one which (to perhaps rather vulgarly analogize the human uterus to a coin and boys to heads), regularly comes up heads somewhere between 51% and 52% of the time over sequences of millions of flips. But as I said, the imprecision and/or falsity of this aspect of the problem may well be something that bothers me more than others, and was a structural feature of the problem long before Mr. Ellis had this particular dispute about the meaning of "one" in his phrasing of the problem.

However, this suggests an interesting way of trying to separate out the effects of child-specific implicature: if Mr. Ellis' original question had instead been phrased as "one of the two coins I just flipped came up heads; what is the probability that the other one came up tails" would the subsequent discussion have gone differently? I think it well might have — that the "one and only one" reading feels substantially weaker, because of various salient ways in which people typically think and talk about their own children differently than they think and talk about coins they have flipped. But that's just my unsubstantiated intuition.

## Keith M Ellis said,

July 23, 2011 @ 10:13 pm

This is a red herring, though I'll give you the benefit of the doubt and assume that it's in good faith.

When I wrote "the language of mathematics" I was not referring to "English transliterations of specific mathematical statements", which would probably be rightly considered something of a metaphor. I was referring to "the technical nomenclature" of mathematics (as well as philosophy) where this usage of "one" is normative, just like "passive voice" has a particular meaning within the technical context of linguistics. You may not like that specialists have nomenclature that doesn't quite fit with your non-specialized usage, but that doesn't mean that the specialists are "wrong" to use it.

It does mean that when talking to non-specialists these specialists (and those fluent in their technical dialect) should be aware that the non-specialists may misunderstand some of their usages and therefore adjust accordingly. But, indeed, I never argued otherwise and, indeed, I do exactly that. I had assumed, with good reason, that my friend was also familiar this technical usage, just like you would assume that other regular LL readers will know what you mean when you refer to the "passive voice".

Furthermore, this usage of "one" and "some" in math and philosophy is precisely as a result of (b). As discussed at length in Mark's post and in the resulting comments, the narrow reading of these two words relies upon context

notits absence. Philosophers and mathematicians embraced these particular technical usages after some careful analysis indicated that they are the most "literal", context-independent usages.I think it's the insistence that it's dubious and the subsequent impulse to quarrel about it to be reminiscent of maladjusted adolescents.

I've maintained that Monty Hall Problem page for almost sixteen years now. When it was young, and was the only resource on the MHP on the web (and when it was among about four), it got quite a bit of traffic. I've sixteen years of experience discussing the MHP with correspondents and others, including newspaper reporters and professors and students. Given that large number, I can say with extremely rare and possibly unique experience and authority that the percentage of people who have trouble with the MHP as a result of the ambiguities of how it's written in English is very close to zero percent. It is a non-issue.

It plays a disproportionate role in certain forums such as the Wikipedia Talk page for the MHP. Some people, when told they were wrong about something, and can't directly contest it (as is often the case with math problems and related), will then retreat to "oh, I answered a different problem than what you asked and it's your fault that I misunderstood you"

The rest of us, when told that our answer of 50-50 to the MHP was wrong, either simply assert that no, that's the right answer (sometimes repeatedly, at length, eventually concluded with a "but I don't really care about this anyway") or they accept the correction and then earnestly attempt to understand the problem and its solution. It's only a certain kind of perverse personality that argues about the problem and the solution while simultaneously contending that such problems are tiresome and dubious.

Well, no, that's not what

I"demand", obviously since I very carefully worded my presentation of the MHP on my page to avoid forcing the reader to such an exercise. I even explain at length in footnotes how the lack of such careful language causes some readers to interpret the problem other than it was intended. I question whether it is really reasonable for readers to interpret inmostof these ways. But I certainly don't demand it and, as is demonstrated by my page, I don't expect it because right at the outset I make the effort to avoid it.## Bathrobe said,

July 23, 2011 @ 10:18 pm

"One of my two children is a boy. And so is the other one."

A logician might find no argument with this kind of utterance. But in most situations it will be regarded as deliberately humorous.

## Keith M Ellis said,

July 23, 2011 @ 10:32 pm

You're awfully committed to arguing the validity of a point-of-view that you yourself had already allowed, as you do again here, as possibly being idiosyncratic. Yet you've written paragraphs and paragraphs explaining how views other than your own are unreasonable. Do you have an angel sitting on each shoulder? (Hint: go with the one with the halo. And, really, whether your view is idiosyncratic or not is empirically available.)

## etv13 said,

July 24, 2011 @ 6:11 am

"I know one person who can help us" versus "I know a person who can help us" — it seems to me that the choice of "one" versus "a" is meaningful. Which is to say, if a person said, "I know a person who can help us," I would not necessarily assume "only one," but if somebody said "I know one person who can help us," I would. "I know one person" just feels unnatural to me, as opposed to "I know a person," unless the intetion is to say there's only one person who can help.

## Ellen K. said,

July 24, 2011 @ 7:56 am

@etv13. For me, the "only one" reading requires a extra amount of stress on the word one and thus is not a possible reading in well written and edited prose. Though, then again, that's a statement that's unlikely to appear in edited writing. :) I do think someone might say (or write) those words when they have one particular friend in mind without it meant to be a comment on any of their other friends.

## Richard Hershberger said,

July 24, 2011 @ 8:31 am

@ Keith M. Ellis:

"Well, no, that's not what I "demand"…" Fair enough. I was sloppy in my wording. What you actually wrote was "I'm not completely sympathetic to such complaints about the MHP or, indeed, in general." To which I should have responded, to the extent that your sympathy is incomplete, this is unreasonable. Either the puzzle is written in natural English, with all the implicature natural English carries, or it is written in logic puzzle English, where no implicit assumptions are allowed. If the person posing the problem insists on the right to mix the two, then "solving" the problem is reduced to correctly guessing when to interpret the language and natural English and when as logic puzzle English. This is about as pointless and uninteresting an exercise as one could ask for.

## Hermann Burchard said,

July 24, 2011 @ 8:49 am

@Bathrobe: Imagine the following situation, an acquaintance "A" of you "Y" is starting a club for raising boys in a way she thinks beneficial to them. She informs you:

A: To become a member of my club you need to be parent of at least one boy.

Y: One of my two children is a boy. And so is the other one.

A: Good, you can join.

Not deliberately humorous. A bit stilted perhaps.

## Hermann Burchard said,

July 24, 2011 @ 8:55 am

@Rebecca: Thanks. Probably, my careless self omitted "iff" and then failed to proofread that line.

I had hoped your comment offered a viable "out."

## J.W. Brewer said,

July 24, 2011 @ 9:01 am

Let me make two more observations going back not to the general structure of the problem but the specific phrasing used. I wonder if the issue is not so much the meaning of "one" even in a specific subject-matter context but the meaning of the construction "one of my two X's is Y." When it's put that narrowly, I have trouble finding values for X and Y where there's not a very strong implicature that my other X is not-Y. So, to take some real-life examples, "One of my two cars is a Honda" (when they are both Hondas) or "One of my [understood to be two] parents went to Z University" (when they both did) both feel indefensible. Unfortunately, poking around COCA/COHA I only found two or three instances that even loosely fit this pattern, so not enough data to be sure. Can anyone suggest a counterexample, i.e., a sentence of this pattern where it seems plausible that the speaker's other X might well also be Y? The "my" seems crucial to the introspective result; if, analogously to Gardner's apparent original formulation of the how-many-boys problem, I said "one of Mr. Smith's two cars is a Honda" the implicature that Mr. Smith's other car was not-a-Honda feels less strong, because there are more plausible scenarios in which I might know that Mr. Smith has two cars but not know or be unable to recall the make of both of them.

The second observation is that perhaps there's something akin to a garden-path problem here. If "one of my X's is Y" has been taken to convey that my other X is not-Y, the immediately following clause "what's the probability that my other X is not-Y" will seem close to nonsensical — dude, I thought you just told me that your other X is not-Y, so wouldn't it be 100%? That sense of bafflement ideally should lead the listener to go back and reanalyze the first half of the sentence to look for an alternate interpretation that makes the second half non-nonsensical, but sometimes (perhaps especially in conversation rather than reading?) the listener simply gets stuck and can't regroup and reanalyze what's gone before.

## J.W. Brewer said,

July 24, 2011 @ 9:11 am

In light of Hermann Burchard's example, I should probably ask for a counterexample where the "one of my two X's is Y" is not immediately followed by information about the other X.

## Rebecca said,

July 24, 2011 @ 12:40 pm

@J.W. Brewer: I don't know if this is what you're looking for, but if "Y" is something transient enough that the speaker might not always know its applicability to all X, then it's not hard to concoct examples that fit:

(hearing the refrigerator door open from another room, I say to someone on the phone): Well, one of my (two) kids is in the kitchen now, so I don't know if there will be any brownies left to take to the picnic.

I can easily imagine saying this while holding no strong opinion about whether there's one or two or more kids in the kitchen. I heard one, so I went with the weaker statement.

It really seems to me to be the case that the "one and only one" implicatures are not tied very closely (if at all) to the linguistic structures.

## J.W. Brewer said,

July 24, 2011 @ 1:47 pm

Rebecca, I think that's an excellent counterexample. I had been thinking of some similar ones while driving to church earlier this morning. For some Y's, "one of my two X's is Y" may plausibly carry the implicature "and i don't know whether my other X is Y or not-Y." I suspect that in conversation there might be some intonational clues that would help the listener choose between that implicature and the "and my other X is not-Y" implicature. (Of course, the sex of ones own children is a Y where the I-don't-know implicature seems far-fetched.)

I still think, however, that it's very hard to come up with X's and Y's where, with no further information immediately given, "and my other X is also Y" is a minimally plausible implicature, although one can imagine non-strained sentences where further information is given, e.g. in the form "one of my two X's is Y, but the other is Z" where Z is a subset or intensification of Y and thus necessarily implies Y. Perhaps more to the point, in most discourse situations the implicature "and although I know perfectly well whether my other X is Y or not-Y, I'm just not going to tell you" would not be expected or understood, because (I would contend) it violates various Gricean maxims. But Mr. Ellis is entirely right that if the listener knows that she is being given a "toy problem in probability," it is not a normal discourse context, and perhaps the intentional omission of information that one would normally expect a Griceanly-cooperative speaker to supply (either expressly or by implicature) is affirmatively necessary in the specific discourse context, because otherwise the problem just won't work as a problem.

But knowing that the nature of the discourse means the usual Gricean maxims will not fully apply (or will not apply in their usual form) may not always provide sufficient cues as to what specifically different assumptions to bring to the interpretative table. This is, I would suggest, why stating the problem with an explicit "one or more [or "at least one"] of my two children is a boy," along the lines of the Gardner formulation, is helpful. It (when shifted from "Mr. Smith's children" into first person) is a sufficiently weird thing for a father to say about his own children (in most discourse contexts) that it not only preempts the particular confusion that occurred here but more generally helps get the listener into the proper frame of mind for the particular unusual form of discourse.

## Rebecca said,

July 24, 2011 @ 2:20 pm

@J.W. Brewer –

I tend to agree about the minimal plausibility of "and my other X is also Y" without further info. But if we don't restrict the form of the shared property to "is Y", it's not too hard to find examples, like the one I gave upthread:

One of my (two) kids can help you with that.

For me, the implication that I'm referring to exactly one is extremely weak, the more likely reading is that I'm suggesting that any of the kids could help. Obviously, there's some presumed context, that somebody mentioned a difficult or unpleasant task.

Maybe there's something about modals and their implicatures that makes this example irrelevant to the discussion, but it still feels to me like it's all a matter a context, not a conventional implicature. Kind of like rounding in math. In any practical situation, whether you round up or down depends on the task at hand, not on rounding rules one learned in school. That's the feel I get.

## Dan Hemmens said,

July 24, 2011 @ 4:09 pm

Presumably to a logician or mathematician 50% *really means* 0.500000 etc., with zeroes out past the horizon.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."

## Dan Hemmens said,

July 24, 2011 @ 4:39 pm

Either the puzzle is written in natural English, with all the implicature natural English carries, or it is written in logic puzzle English, where no implicit assumptions are allowed.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.

## Ellen K. said,

July 24, 2011 @ 8:13 pm

If I'm understanding first order approximation right, then, no, it's not, at the present, a good first order approximation. There's already plenty of information that the ratio is different. We're beyond first order approximation.

## Hermann Burchard said,

July 25, 2011 @ 1:07 am

@Keith M Ellis:

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.

## Dan H said,

July 25, 2011 @ 5:42 am

If I'm understanding first order approximation right, then, no, it's not, at the present, a good first order approximation. There's already plenty of information that the ratio is different. We're beyond first order approximation.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.

## Keith said,

July 25, 2011 @ 9:29 am

Eric P Smith, writing "My father has one arm"…, reminded me of a joke that relies on this mechanism.

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

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

K.

## J. W. Brewer said,

July 25, 2011 @ 2:34 pm

Dan H., I think what you're saying is that the context/genre of the question means that a non-perverse listener/reader should know to interpret it with an implicature like "this question seeks the answer you would reach if you set aside whatever real-world knowledge you may happen to have about demographics and human reproduction that bear on the probability of sons v. daughters in fathers-with-two-and-only-two-children situations and instead use certain simplifying assumptions that are sufficiently obvious in context that they don't need to be specified expressly." Is that it? Maybe there's a less clunky way to formulate that implicature, and maybe the context really does make it a very strong implicature, but it's not an implicature-free reading of the question. Maybe there are no implicature-free readings of the question (not that I'm accusing anyone of having made the specific claim that there are).

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.

## Hermann Burchard said,

July 25, 2011 @ 4:00 pm

wiki:

Implicature is a technical term in the pragmatics subfield of linguistics […]This suggests a pragmatist orientation of the investigation in this thread.

Therefore, my suggestion [above] of casual English

being a cruder formof 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.]

## Hermann Burchard said,

July 25, 2011 @ 5:53 pm

@myl: Apparently, I really was confused by terminology, the latter now much clearer in my mind thanks to your explanation:

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.

## Dan H said,

July 26, 2011 @ 3:32 am

Is that it? Maybe there's a less clunky way to formulate that implicature, and maybe the context really does make it a very strong implicature, but it's not an implicature-free reading of the question.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.

## j-g-faustus said,

August 10, 2011 @ 1:06 am

@DanH

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.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?

## Paul Kay said,

August 31, 2011 @ 4:48 pm

If the puzzle was really posed in the words, "one of my two children is a boy, what is the probability that my other child is a girl?" as stated originally, it seems to me that English grammar only admits the 'only one' interpretation because "

my otherchild" does not compute on the 'at least one' reading. This hypothesis predicts that the following attempt at a sentence fails, which for me it does: "At least one of my two children is a boy, what is the probability that my other child is a girl?"My other X(s), likethe other X(s)presupposes a fixed subset s of a set X and denotes the complement of s in X. Mark is (as usual) right that the relevant scholarly consensus is that under normal circumstances numerals are only upper bounded by contextual reasoning (technically "implicature"). But the insights of Mill, Grice and Horn don't seem relevant to this case because the internal semantics of the sentence rules out the broad interpretation of the numeral. The folks who suppose the answer to the puzzle is different from .5 are, I think, erroneously equating the original question with, "One of my two children is a boy, what is probability that I have a daughter?" That is not the same question as "One of my two children is a boy, what is the probability that my other child is a girl?”## lucia said,

September 1, 2011 @ 5:25 pm

I have a question based on a blog claim. Can "only one" be interpreted to include none? I've got three examples, because it seems to me that in some cases, I take the meaning to suggest there is exactly 1 and sometimes not: Examples:

* "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?