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Traditional English grammar teaches us that a well-defined function is a function that is well defined. With the hyphen in the adjective role before the noun and without the hyphen in the role of an adverb with a past participle after the noun. (E.g., see a question about well-organized, which is following the general rule, while the current one is about mathematical jargon.)

However, continuing an argument of Mr. West, a function that is well defined is a function for which we have done a good job of giving a definition, but a function that is well-defined is an object that has been given a valid definition as a function, with every domain element given a unique image. Said that, in the second meaning (an object that has been given a valid definition as a function, with every domain element given a unique image), should the mathematicians use the hyphen in "well?defined" in the after-the-noun position?

PS. I asked the same question at math.se, but they put it on hold, claiming that it could be off-topic there. Only one person gave his/her opinion, which I liked, but I'd like to hear other arguments, too.

marked as duplicate by marcellothearcane, Edwin Ashworth, David, Davo, Mitch Sep 18 '17 at 14:09

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  • I'm trying to think of a context where "well defined" would be correct. – Hot Licks Sep 16 '17 at 19:24
  • 'Well defined' isn't used solely in math. It's used in semantics and general English. Leaving aside subject-specific usage for a moment, the 'rule' you give in your first sentence is not absolute; I follow CoBuild in hyphenating both prenominal and predicative usages. Now, how the term/s is/are used in maths is a different issue. But anyone distinguishing between a compound well-defined and a collocation well defined / defined well is duty bound to define terminology, as this isn't general practice (and off-topic on ELU). – Edwin Ashworth Sep 16 '17 at 22:47
  • Rob Arthan's answer over on Math.SE seems eminently sensible, but is not backed by supporting evidence (which I wouldn't think was that easy to find). But maths sometimes uses terms in ways that don't exactly correspond to the way they are used in everyday lives; two similar[everyday] shapes may not be similar[mathspeak]. So in your 'should we use the hyphen ...', 'we' is ill-defined. – Edwin Ashworth Sep 16 '17 at 23:00
  • Now the question is subject-specific and the normal 'rules' of English are less binding. But other rules may now be in play, sometimes more binding. This is why domain-specific questions on terminology should usually be dealt with on the dedicated websites. (Quote this on Maths.SE if you like.) But Rob Arthan gives what I consider to be the correct answer for the mathematical usage (that the fine distinction is a virtually unilateral claim/practice and should be eschewed). – Edwin Ashworth Sep 17 '17 at 15:44
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The linked argument by Mr. West, as far as I can tell, does not address the use of "well-defined" or "well defined" in predicative position, only in attributive position. It contrasts "well defined function" with "well-defined function".

While it's hard to pin down clear rules for hyphenation, it seems to me that the "some authors" who West says "insist that because 'well' is an adverb the term should not be hyphenated" are misguided, and following a misremembered or incorrectly formulated rule.

As described in Sven Yargs's answer to the following question (Should there be a hyphen in expressions such as “currently-available X”?), hyphens are generally only avoided after the adverb very, or after any adverb ending in -ly. There is no generally observed rule that I am aware of forbidding the use of a hyphen after other kinds of adverbs, such as well, quick, hard.

There are many phrases starting with "well" that are commonly hyphenated, mostly ending in participles or departicipial adjectives, like well-read and the other example West mentions, well-known. (Examples starting with other adverbs: quick-thinking and hard-working).

The hyphenation of adjective phrases in predicative position seems to be a bit more variable than that of adjective phrases in attributive position, but using a hyphen in this kind of context is considered correct by at least some stylebooks:

But well-known is also acceptable in both sentences according to The Associated Press Stylebook (AP). The rule in AP is that "when a modifier that would be hyphenated before a noun occurs instead after a form of the verb to be, the hyphen usually must be retained to avoid confusion." AP gives this example: "The man is well-known."

So, I hope the answer is now well known to you: Both well-known and well known are correct. You choose.

–"A Well-Known Problem: Well Known", by Lynn Gaertner-Johnston, Business Writing Blog