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The expression I could not care less originally meant 'it would be impossible for me to care less than I do because I do not care at all'. It was originally a British saying and came to the US in the 1950s. It is senseless to transform it into the now-common I could care less. If you could care less, that means you care at least a little. The original is quite sarcastic and the other form is clearly nonsense. The inverted form I could care less was coined in the US and is found only here, recorded in print by 1966.

The question is, something caused the negative to vanish even while the original form of the expression was still very much in vogue and available for comparison - so what was it? There are other American English expressions that have a similar sarcastic inversion of an apparent sense, such as Tell me about it!, which usually means 'Don't tell me about it, because I know all about it already'. The Yiddish I should be so lucky!, in which the real sense is often 'I have no hope of being so lucky', has a similar stress pattern with the same sarcastic inversion of meaning as does I could care less.

Source: Which is correct: ‘I couldn't care less’ or ‘I could care less’?(Dictionary.com)

As further proof, I offer this excerpt taken from the American sitcom show Cheers, which ran from 1982 to 1993. In season 4; episode 6, “I will gladly pay you Tuesday” (1984), the bar owner, Sam Malone says:

SAM: Well truth is, I've had a lot of trouble of loaning people money in the past. Y'know hurt a lot of feelings. Lost a couple of buddies once. So, I've decided from now on if anybody asks me, I'm just gonna giv'em the money and never expect to see it again.

DIANE: You'll see it again. Oh, you'll see it again. Diane Chambers always honours her debts.

SAM: Hey, I could care less, really. You walk out that door, I'm not even going to think about that money again.

[…]

CARLA: By the way, Sam, has Diane paid back that loan yet?

SAM: No, Carla, she hasn't. But I could care less.

(Carla is complimenting Diane on her new cashmere sweater)

CARLA: Must have cost you an arm and a leg, huh?

DIANE: No, no, it was marked down 2 percent at Haute and Bothered last week.

CARLA: Well, we girls deserve to splurge on ourselves once in a while—don't you think, Sam?

SAM: I could care less
_{script: Cheers}

Aficionados of the show will confirm that the character, Diane Chambers, was always very careful with her grammar, and choice of words. She would not have missed the opportunity to have corrected Sam's ‘I could care less’. This suggests that the (American) idiom was well established by the 1980s and its meaning was clearly understood.

UPDATE:

On Google Books, the earliest recorded American usage of I could care less is from 1962 in the National Association of Marketing Officials

Foreign buyers, in general, could care less where these products come from within the USA, as long as location of source does not specifically influence quality, appearance or price.

But there is evidence to suggest that the inverted version was commonly heard before the 60s in From the Sourdough Crock 1960 - California, a reader complains

Dear Uncle Flabby:

I get sick and tired of hearing people say, “I could care less,” which doesn't mean what they mean to mean at all. If they would only stop and think about it, they would know that what they are trying to say is, “I couldn't care less,” which means “I don't care at all,” Don't you agree?

(Partial) Source: http://blog.dictionary.com/could-care-less/

TL;DR: May have turned up because of:

1. young punks not bothering to learn the right usage, or


2. Yiddish humor, or


3. it doesn't matter - people got the point, and the more conservative ones got annoyed by the usage (which I think was a bonus to the implied meaning, and accelerated its use)

Discussions of of 'could care less' in reference works

People interested in English usage have been noticing and commenting on the formal illogic of the wording "could care less" for at least the past fifty years. From Wilson Follett & Jacques Barzun, Modern America Usage (1966):

could care less, couldn't care less. To the generations of world-weary young, the avowal of indifference I couldn't care less is ever new. To others it is trite and glib, but at least its logic is clear: one cares so little about something or other that caring less is beyond one's powers. Speakers who mean to convey this but change couldn't to could are not jaded but deaf to LOGIC. To say My boss could care less (about something) implies unmistakably that the boss takes an interest. All that remains in doubt is the degree of that interest, which may be the highest possible.

From Theodore Bernstein, Miss Thistlebottom's Hobgoblins (1971):

COULD CARE LESS The standard idiom, or perhaps smart retort, is, of course, "I couldn't care less." But many conversationalists these days either because their hearing is defective or because they are in an inordinate hurry, or merely because they think it’s cute, shorten it to "I could care less." Here are the words of an editor, writing to a journalism publication: "Graduates since the U. of W. journalism school went to theory and research could care less about achieving objectivity of any degree in newswriting." At the moment the senseless abbreviated form has not really taken hold, but at the next moment it may, who knows? If it does, it will join "more than I can help" [cross reference omitted] as a piece of apparently reverse English.

It is striking that those who say or write could care less leave logic intact in the comparable expression I couldn't agree more. Perhaps the greater mental emphasis that falls on agree in the second expression keeps their minds on the sense.

From William Labov, Language in the Inner City: Studies in the Black English Vernacular (1972):

Speakers of English are not unaccustomed to sudden inversions of positive and negative. I couldn't care less seems to mean the same as I could care less. In Maine and surrounding eastern New England, young people say So don't I to mean the same thing as their parents' So do I. Even without negative concord, we have learned that not every negative means what it might.

Yet another perspective comes from Patricia O'Conner & Stewart Kellerman, Origins of the Specious: Myths and Misconceptions of the English Language (2009):

The psychologist Steven Pinker has an interesting take on "I could care less," which he calls "an alleged atrocity" and a favorite target of language pundits. As he points out in his book The Language Instinct [1994], the melodies and stresses in intonation between "I couldn't care less" an "I could care less" are completely different and convey a youthful sarcasm: "By making an assertion that is manifestly false or accompanied by ostentatiously mannered intonation, one deliberately implies its opposite."

Early occurrences of 'X couldn’t [or could not] care less'

I was quite surprised at how little time—roughly 20 years—separates the earliest instances of "X couldn't [or could not] care less" and "X could care less" in the relevant sense. Here is the Ngram chart for "could not care less" (red line) vs. "could care less" (blue line) for the years 1900–2005:

I omitted "couldn't care less" from the chart because Ngram doesn’t graph contractions properly (owing to confusion over the embedded apostrophe), but the earliest match for that phrase is 1945.

As the Ngram suggests, instances of "X could not [or couldn't] care less" are quite rare before about 1950. According to Max Cryer, Who Said That First? The Curious Origins of Common Words and Phrases (2010), the first recorded usage is from 1945:

The phrase surfaced publicly in 1945 when BBC war correspondent Stewart MacPherson commented on a commando operation in which the British gave the impression they were going on a picnic: "They just couldn't care less."

Cryer also notes, as the first published instance of "couldn't care less," a memoir by Anthony Phelps titled I Couldn't Care Less (1945). A Google Books search finds this book, too, and the next-earliest match is for The Spectator (July 11, 1947) [combined snippets], which appears to include two instances of "could not care less," although the second may be from a slightly later issue of the same periodical:

The speed of the action is held up over Troy and Rory's reunion, and, truth to tell, we could not care less. In point of fact, Troy's clinical observation of the degenerate dependents of the old stage baronet, as almost Volpone-like they manoeuvre for position, is more entertaining than Rory’s detection.

and

Afterwards I enquired what had happened that we should have had the speech [by Churchill] relaid. It appeared that either English folk had approached the manageress and had also asked that the speech should be laid on. But this was not the explanation. For the manageress had been adamant. Mr. Churchill might be speaking, but she "could not care less." The matter had been settled, not by the English, but by the Irish. For they, too, wanted to hear Mr. Churchill.

In Britain, the expression "could not care less" took off in 1948, with instances appearing in Parliamentary Debate records, in a history of the Second World War, in a book about opera, in a collection of essays, and in such periodicals as The Aeroplane, British Poultry Field and World Poultry Digest, Country Life, Journal of the Institution of Civil Engineers,
The Journalist, The National [and English] Review, Overseas Education, and The Spectator (again).

In the United States, the explosive year for "couldn't [or could not] care less" was 1950. From Henry Green, Nothing:

"What's the gal done then Jane?"

"Nothing so far as I know, nothing at all. I couldn't care less. But just because John is one of my oldest friends I don't see why I should like his daughter even if, as you remember perfectly well, at one time I loved her mother, oh so dearly!"

From William Purcell, Five Minutes to Twelve (1950):

Significantly, "couldn't care less" has become the motto of an age in which there is more needing to be cared about urgently than for a very long time.

From "Ruth Prepares for Her Baby," in Life magazine (April 3, 1950):

Ruth Khama looked at her before answering the question and her blue-green eyes were very serious. "I couldn't care less," she said slowly and with precision. "I really don't care. Except if it is a boy, I would want him to be dark. And if it’s a girl, I suppose I’d like her to be light."

From "It Won’t Do!," an unsigned editorial, in Life magazine (April 17, 1950):

John Strachey in his own right is a gifted political essayist. He says that he never joined the Communist party, and we couldn't care less whether he did or didn't. During the 1930s he worked hard for Communism and he was known both in Europe and in the U.S. as one of its most brilliant exponents.

From Richard Williams, "Politician Without a Future," in Life magazine (May 1, 1950):

But Lee could not care less about pleasing everybody: "I'm fed up with most politicians anyway, and that goes for Republicans as well as Democrats. . . . Why, there are members of my own party who'd have this state in worse shape than it was when I took office, if I'd let them."

From "Taxes and Politics," an unsigned editorial, in Life magazine (November 27, 1950):

Now it would be stupid to say that the people are up in arms against excess profits taxes. We have a strong suspicion that "the people" couldn't care less about this particular issue. But the election results do suggest that people care about the performance of the Truman administration, and it could be that the President's occasional similarity to the mule is not as popular as it, or something, was in 1948.

Early occurrences of 'X could care less'

The earliest matches for "could care less" that my Google Books searches turned up were from 1962. From Ann Pugh, Heidi (1962):

HEIDI. [Running to Dete] Look, Aunt Dete, yellow buttercups —

[But DETE, who could care less, does not even turn around. No one except a worried PETER notices OLD FRANZ, who enters from off Right of ramp and is coming down , smoking a pipe and carrying a load of firewood. ...]

From a letter to the editor by Eric Hepborn of Chicago, Illinois, in Ebony (November 1962):

To begin with, your title should not have been "How to Meet Negroes," but rather "How to Meet the Negro Bourgeoisie" ... or approximately ten percent of the so-called Negro populace. Only the bourgeoisie are overwhelmingly obsessed to meet, rub shoulders and party with white society. The large mass of "Negroes" (whatever that is) could care less.

From Lee Kalcheim, And the Boy Who Came to Leave” (first produced 1965, first published 1966), in Playwrights for Tomorrow, volume 2 (1966):

PETER. Oh, screw . . tell me . . tell me. I'm dying . . I really give a shit . . I could care less . . (Paul laughs) So tell me . . go ahead, screwhead . .

PAUL. You hit it, friend . . You know the magic answer . .

PETER. OK . . OK . . funny man . . tell me . .

PAUL You know yourself . .

PETER Yeah?

PAUL. You could care less . .

From "Music Soaring in a Darkened World," in Life magazine (July 29, 1966):

"I could care less than a damn about seeing," Ray [Charles] says. "Maybe I'm better off this way. Seeing a person means nothing. You can't tell by looking at somebody what he's like. I keep hearing people say, 'I saw that guy and hated him on sight.' Or, 'I just didn’t like his looks.' Well, I don't feel that way. I can't. I judge a person only by how he reacts to me, how he treats me. I give you a clean sheet of paper and it's up to you to mess it up. Seeing's not all that important, as best as I can remember. People say, ‘What would you do if you could see?’ Well, I don't know, I'd still be a musician."

From Kansas Agriculture: Annual Report (1966[?]):

First of all, as farm broadcasters, to tell the farm story to the urban audience, we feel we have to capture that urban audience. We get letters which express a disinterest in farm programming. They could care less whether hog or cattle prices are up or down, or about livestock pests.

From an interview with Gene Willis, in Studs Terkel, Division Street: America (1967)

But I don't bother anybody and I don't like anybody to bother me. If I were walkin' down the street and they were robbin' a bank, the guy would walk out and say, "Hi, Gene, how are you?" And if I knew him or didn't know him, I'd say, "Hi, how are you?" And as long as he didn't step on my foot while I was walkin' by the bank. Let him do what he wants to do. It’s none of my goddam business. I could care less. (Indignantly.) Nothin' bothers me more than people that are tryin' all the time to say, "Oh, did you see what he did over there?" It's none of your business, he's not bothering you. If nobody hurt me, let them do what they want to do. I'm glad they're gettin' away with it. If they can get away with it, all the more power to 'em, that's their business. If I were a police officer, then of course that's my problem. I'm pleasant, get along, courteous.

From Simeon Booker "Adam Clayton Powell: The Man Behind the Controversy," in Ebony (March 1967):

But Adam could care less. He defied the rules. He was a loner, too independent to be a team member on any venture from civil rights to socializing. For his own protection, he was crafty and elusive and little people loved him because he didn't give a damn.

From "Requiem for a Heavyweight," in Billboard (October 28, 1967):

Since [Murray the K] Kaufman left the station, the programming has been stagnant and insipid. This was not a slow development; it happened suddenly . . . ostensibly the moment Kaufman left the air. Obviously, there's no one qualified to pick "tomorrow's music . . . progressive rock." No one knows what's happening musically. Worse, the station and its management could care less about progressive rock. From a progressive rock album station, the image of WOR-FM has changed to just another rocker. From aiming its programming at young adults and the people who've grown up with rock ‘n’ roll since 1955, the station is going for teeny boppers.

An earlier sense of 'couldn’t care less'

Authorities who have pointed to Phelps’s book as the first instance of "couldn't care less" have ignored or overlooked two earlier instances of the expression “couldn't care less” that use it in a different way—not to mean "I already don't care at all," but to mean "I care with a steadfastness that nothing can lessen." From "In the Brera," in The Church Standard (April 27, 1901):

”Yes. I wronged him shamefully. . . . It was seven years ago, and I haven't seen him since. I had almost forgotten—but I always meant to tell you, Gertrude. And then—it didn’t seem worth while—but I must now. Will it—will it make you care less for me darling?”

"Oh! You are frightening me—for nothing—I know you are,” she said, with a little nervous laugh. I couldn't care less—now; and you are a goose to say such a thing! But—tell me, at once—it wasn't about—about—a woman—was it?"

And from John Randall, With Soul on Fire (1919):

"It isn't possible that she,—" He took the words out of my mouth before I could frame them.

"That she is beginning to care less? You don't know her if you think that. She's the kind that couldn't care less; it's always more and more, with women like her. Her love is a mighty passion for self-giving; it grows and grows, until it becomes capable of the supreme renunciation, if need be, for the sake of the happiness of the one loved. ... If they cared less, if their love wasn’t so great, I shouldn't feel as anxious as I do."

Conclusions

From the various search results compiled here, it appears that "could not care less" achieved mass appeal in British English publications beginning in 1948, that "couldn't care less" followed suit in U.S. English publications beginning in 1950, and that "could care less" achieved some level of popular-culture breakthrough (in the United States) in 1966–1967.

As for what triggered the emergence of these various formulations, it's difficult to say. The wording "couldn't care less" had appeared in print (with a different meaning) 44 years before the two 1945 occurrences noted above. And phrases of the type "nobody [or no one] could care less" have appeared in print going back to the early 1800s. The only unmistakable trends appear to be that "couldn't care less" began to catch on toward the end of the Second World War, evidently in Britain, and that Life magazine is responsible for an inordinate number of U.S. instances of "couldn't care less" from the year 1950.

Given that only a dozen years separate the 1950 mushrooming of "couldn't care less" in the United States and the first instances of "could care less" there, both phrases may have been sufficiently new that some people didn't have to unlearn the older, more formally correct idiom in favor of the newer, stranger one; they could simply have misheard it from the get-go.

Update (December 11, 2018): Early Elephind instances of 'could care less'

Multiple instances of "could care less" in the sense of "could not care less" appear in a newspaper article from 1952, quoting a letter from an "AWL" husband to his estranged wife, in "English Instructress Tells Why Her Marriage Came to a Full Stop," in the Perth [Western Australia] Mirror (June 28, 1952). The letter, actually written in February 1949, uses one or another version on the expression seven times:

"Please don't try to answer this, or ring . . . I will work out my own destination. Maybe it will be too late, but I don't care.

". . . . Say what you like, talk how you like with all your clearcut logic (it sickens me), but I know that I did love you with all the passion and love that is possible of a man (if you can call me a man in your idea) and now I could care less.

". . . If I've been wrong to you I will pay the rest of my life for taking you from a professional job and ending you in the mire. If I find my love for you so big, I'll stop at nothing to get you back. But at the present time I could care less.

...

". . . . I did not treat you as a chattel. I don't care how you take it, I could care less.

"I've had a few drinks tonight as I guess you will analyse from my writing, but as I can care less, I'm writing how I feel. I will post this irrespective of my feeling in the morning.

...

"Would I be able to plan for home and grounds, etc., without worry or wondering whether it's up to Zoe's standards?

"I could care less.

"A man's a man for all that.

"I don't care. how .or why I write at the present time—I'm writing how I feel and I could car less.

"Goodnight Zoe and goodbye if you wish it —I could care less."

Although the letter writer's serial usage of "could care less" seems unlikely to have influenced subsequent occurrences of the expression, it does show that at least one person was using it informally from very nearly the beginning of recorded use of "couldn't care less."

An Elephind database search turns up four matches for "could care less" in the relevant sense between September 8, 1960, and November 14, 1960. The sources are Molly Oharra, "Music Makers," in the Coronado [California] Eagle and Journal (September 8, 1960):

People who ordinarily could care less about a symphony orchestra have been known to see him, if only out of curiosity.

Jack Stevens, "Stevens Report: Clock Watchers, What Time Is It Really Now?" in the Coronado [California] Eagle and Journal (October 13, 1960):

And, as far as the condition of light in the early mornings, I could care less, as I never acknowledge the presence of a new day until long after breakfast anyway.

Ann Landers, "Ann Landers... Answers Your Problems," in the Madera [California] Tribune (October 20, 1960):

Dear Ann: My girl friend and I are having an argument. You know that common expression: "I couldn’t care less." Well she says it's "I COULD care less." Please tell us in your column which is right.—TOOTH AND TOENAIL

and in a letter to the editor by Myrtle Mancebo to the Santa Cruz [California] Sentinel (November 14, 1960):

Too many people in this day and age could care less about anything that does not penetrate their immediate existence.

The fact that all of these instances appear in California newspapers is striking, but I should note that the California Digital Newspaper Collection is one of the few included in Elephind searches that makes local newspapers from the 1950s and later accessible to online searches. If other parts of the United States were similarly represented in search results, we might very well find instances of "could care less" there as well. Still, of the four articles cited from 1960, only the Ann Landers column involves a nationally syndicated source; the other three were locally written in the municipalities of Coronado and Santa Cruz, California. The instance from 1960 that Mari-Lou A cites in her answer is likewise from California.