>In a "letter to the editor" of my local newspaper, the
>business manager of Plumbers and Pipe Fitters Local 99 wrote:
>        As I speak to our retirees they are concerned
>        with the proposed changes to our Medicare in
>        this country. They could give a damn about
>        Whitewater.        ^^^^^^^^^^^^^^^^^
>"Could care less" (as a variation on "couldn't care less") has
>often been discussed here. But this is the first time I've
>noticed a similar, apparently ironic, use of "could" in other
>expressions. I suppose it might be a simple transference of
>form, by analogy with "could care less."
>I have the odd feeling that this has been going on all the
>time, and I just haven't picked up on it. I could give a damn
>about the "correctness" of the expression, but if others are
>able to cite similar, apparently formulaic, usages of "could",
>I'd be interested to hear about them.
Like could care less, give a damn is a Negative Polarity Item, that is, a phrase that is ordinarily used only within the scope of semantic negation of some kind (not, never, only, rarely, few, etc.). Hence the perceived strangeness of They could give a damn, which has no overt negative, but means the same thing as the same phrase with a negative. I.e, the business manager was saying that his members couldn't give a damn.

Give a damn is a member of the open Minimal Direct Object class of NPI's, like lift a finger, drink a drop, do a thing, eat a bite, etc. The implication of all of them is that, if one can't even Verb a Minimal Direct Object, why, then, one couldn't Verb any Direct Object at all. Thus it's an idiomatic intensification of a negative. But it does usually require a negative to intensify.

However, there apparently is such a thing as negation by association. Like what happened to French pas from ne...pas, which is now usable as a negative in its own right, from long association in the discontinuous morpheme with the overt negative ne, give a damn and could care less have, in American usage at least, come to have their own quasi-independent negative force.

Give a damn has been used independently of negatives for at least 25 years in America. I published a paper (J. Lawler, Ample Negatives, in Papers from the Tenth Regional Meeting, Chicago Linguistic Society (CLS 10)) in 1974 that remarked on this topic, among other negative phenomena.

Your question about the use of could is interesting; the usual NPI idiom is I don't give a damn, and it seems to me that it would have been much stranger for the business manager to have said They give a damn about Whitewater instead of They could give a damn, at least in the intended sense. So the use of could somehow contributes to the acceptability of the locution. My guess is that the conditional nature of the modal auxiliary provides enough negative force to allow give a damn to work. Consider the alternatives (in the intended sense of They don't give a damn):

   *They give a damn about Whitewater.
   *They can give a damn about Whitewater.
   *They may give a damn about Whitewater.
   *They might give a damn about Whitewater.
    They could give a damn about Whitewater.
All other modals fail, as does the generic present, but could works. It might be, as you suggest, analogized from could(n't) care less, which seems to require a modal, and prefers could:
    *They care less
    ?They can care less
    *They don't care less
    ?They can't care less
It wouldn't be the first time analogy has created a new idiom.
  - John Lawler       Linguistics Department and Residential College     University of Michigan

    "Language is the most  massive  and  inclusive  art  we know,  a           - Edward Sapir
      mountainous and anonymous work of unconscious generations."       Language (1921)

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