What is the origin of the word badass? Why a "bad" ass/"bad" + "ass"? What is an ass that is bad and how can an ass that is bad describe a tough person?
The earliest bad-ass
According the Random House Historical Dictionary of American Slang (1994), the word bad-ass is attested first as an adjective (1955), then as a noun (1956), and then as a verb (1974–1977). Here are the main entry and the three earliest occurrences identified for the adjective form:
bad-ass or bad-assed adj. bad (in any common sense, esp. in slang senses).—usu. considered vulgar. Also semi-adv.
1955 in J. Blake, Joint 110. Wanted to be a hard-nosed badass type. 1958 Stack a Lee 2: There was this bad ass Indian they call Geronimo....Over in the corner sit old bad ass Stack. 1959 Abrahams, Deep Down 138: Before I throw open my bad-ass cashmere and pull my bad-ass gun....I had that old bad-ass Benny Long in my thirty-eight sights.
The same dictionary cites a 1971 article in Playboy magazine quoting the liner notes of Bo Diddley's most recent LP, which describe him as "the most outrageous bad-assed guitar man alive."
RHHDAS's entry for the noun form of the word starts this way:
bad-ass n. 1. A dangerous, browbeating individual; bully.—usu. considered vulgar.
1956 A[merican] S[peech] XXXI 191: A marine who postures toughness is sarcastically labeled a badass. 1961 Peacock, Valhalla 345 [ref. to Korean War]: They's a lot of bad asses at South Camp, ain't they? 1970 Ponicsan, Last Detail 7: Bad-Ass...in Navy parlance means a very tough customer.
And for the verb:
bad-ass v. to bully; to behave like a bully.—usu. considered vulgar.
1974–77 Heinemann Close Quarters 96: "Listen," he said again, trying to bad-ass me. "When I tell you to slow down , that's exactly what I mean."
It's interesting that both the earliest adjective citation and the earliest noun citation emphasize the sarcasm implicit in the usage. To get a sense of where the construction came from, however, we need to look at the entire -ass family as it existed in the 1950s and 1960s.
Members of the mid-20th-century -ass family
Today various slang terms have -ass tacked onto their rear ends—and a surprisingly large number of them arose by the end of the 1950s. Here is a rundown of some of the more prominent members of this family, with first cited occurrence in the Random House Dictionary of Historical American Slang or in Chapman & Kipfer, Dictionary of American Slang, third edition (1995), and the first fairly reliable relevant Google Books match.
bad ass Earliest cited occurrence in RHHDAS: 1955 (adj.), 1956 (n.). Earliest relevant Google Books match: John Oliver Killens, Youngblood (1954) [combined snippets]:
Joe looked down into Mr. Pete's fat face. Where had he heard those words before? Bad ass [racial slur]. Smart ass [racial slur] — He tried to remember. It was as if everything that happened had been done before — Bad ass [racial slur]. Feeling his entire body grow hot with anger. Control yourself, Joe. Don't get in no trouble with white folks.
Earliest relevant Google Books match for bad-assed: George Garrett, Which Ones Are the Enemy? (1961) [combined snippets]:
"Man. I like this grappa," he'd say. "When I get going and get high on grappa I'm nothing but big and bad-assed and mean and, buddy, you better look out. You better make way for me." That may be the way he saw himself. I would say he was the mildest, quietest kind of drunk.
big-ass Earliest cited occurrence in RHHDAS: 1945 (adj.), 1963–1964 (n.). Earliest relevant Google Books match: Thomas Williams, Town Burning (1959) [combined snippets]:
Them old biddies this side of Bank Street—now I don't mean your maw, Johnny, she ain't as bad as the rest — but them big society big-ass biddies with their Women's Club and all. You know what I mean? Sometimes I wish I'd of stayed in the Army; they want a man to kiss their ass like that.
Earliest relevant Google Books match for big-assed: Irwin Shaw, The Young Lions (1948) [combined snippets]:
"You-all go on ahead now and take off like a big-assed bird if they spot you."
Noah and Burnecker looked at each other. They wanted to say something to Rickett, standing scornfully at the window, the BAR loose in his big hands, but they didn't know what to say.
candy-ass Earliest cited occurrence in RHHDAS: 1952 [actually 1953, referring to 1942 usage] (adj.). 1966 (n.). Earliest relevant Google Books match: Leon Uris, Battle Cry (1953) [snippet]:
Huxley had the time he had bid for; he had the conditions he wanted and he spared no rods on us. And then we began to pass outfits along the route, crapped out and exhausted. “Candy-ass Marines,” our boys would shout as we flashed past them.
crazy-ass Not cited in RHHDAS or DAS: Earliest relevant Google Books match: Thomas Phillips, Kangaroo Hollow (1954) [combined snippets]:
He wanted to get back to the line and said so, in the middle of a cafe session while pretty girls walked by.
"For what, you crazy-ass fool?" Bamey demanded.
dumb-ass Earliest cited occurrence in RHHDAS: 1957 (adj.), 1958 (n.). Earliest relevant Google Books match:John Oliver Killens, And Then We Heard the Thunder (1962) [snippet]:
"...or some kind of hifalutin shit like that. You too educated for us dumb-ass colored soldiers."
half-ass Earliest cited occurrence in RHHDAS: 1863 (adj.), 1929–1933 (adv.), 1929–1933 (n). Earliest relevant Google Books match: Joseph Freeman, An American Testament (1936) [combined snippets]:
Robert and I liked this editorial, but Louis furiously disagreed. No socialist, he said, ought to like anything about J. P. Morgan as a social figure. There was nothing especially attractive about his belief that his money belonged to him. Every parasite believed that; he had to kid himself as well as the public. If the Masses editor were a real revolutionary socialist instead of a half-ass intellectual he would not applaud Morgan's noble attitude toward his money; he would point out where that money came from from the sweat and blood of the workers.
Earliest relevant Google Books match for half-assed: Travis Ingham, Young Gentlemen, Rise (1935) [combined snippets]:
"Of course I see," he said, taking out a pipe and filling it. "I saw it yesterday when I had to rescue you from that half-assed bastard from Dubuque or wherever it is that he lays sewer pipe. I knew it the night before when you refused to go to Number Thirty-two with the crowd and see an exhibition."
jive-ass Earliest cited occurrence in RHHDAS: 1967 (adj.). Earliest relevant Google Books match (one of many in 1969, actually): Esquire, volume 72 (1969) [combined snippets]:
Suddenly a rangy Negro, a high-school student, is out of the aisle, yelling at Jenkins : "How can you sit there and tell these jive-ass hypocritical lies? You got this, like, jive-ass educational bullshit, and it's, like, one big jive-ass...." This word jive-ass starts getting to the kid. It starts wrapping around him like a turban, like a flamenco sash, like an epiphyte, like a flag, like a python, an octopus. "... I mean, like, you talk jive-ass, man, then you do jive-ass, and this hypocritical jive-ass ain't relevant to the Third World peoples, it's just jive-ass, and you sit there with both hands on the jive-ass hypocritical jive-assing jive-ass!"
"Right on! Right on!"
lazy-ass Not cited in RHHDAS or DAS. Earliest relevant Google Books match: Glenn Ross, The Last Campaign (1962) [combined snippets]:
"Sonovabitches! Capture a perfeckly good piece of equipment, and that lazy ass Doglane throws it away! I wouldn't even a rolled the thing down the hill, if I'd a thought he was gonna get in on the deal."
smart-ass Earliest cited occurrence in DAS: "By 1960." Earliest relevant Google Books match: Leon Wilson, Sinners Come Away (1949) [combined snippets]:
Pres stepped on the man's hat as he went to the front of the cell. He kicked it back. "One of these smart-ass Northern boys," he said. "We'll slow him down. We'll put a string on him by God and give him to Pop for a pet."
Earliest relevant Google Books match for smart-assed: Ed McBain, The Mugger (1956) [combined snippets]:
The little girl goes with them to Reno—Dad conveniently has business there at the same time Mom must establish residence—and through an unvarying progression of mincing postures and bright-eyed, smirking little-girl facial expressions, convinces Mom and Dad to stay together eternally and live in connubial bliss with their mincing, bright-eyed, smirking little smart-assed daughter.
Are the asses biblical donkeys or mammalian posteriors?
The upsurge in -ass compounds in the 1950s follows some crucial precedents. First, both half-ass (by 1936) and half-assed (by 1935) were established forms; and second, so were big-ass (1945, according to RHHDAS) and big-assed (1948). Another form already in place was smart-ass (1949).
There are strong reasons to suppose that half-ass arose originally in reference to the mule (half-ass and half-horse), but it's harder to argue that everyone in 1950 was on board with this etymology. After all, if "mule" were the idea in all writers' and speakers' minds, why would any of them express the adjective as half-assed instead of as half-ass (or maybe half-asinine)?
In my view, the more plausible explanation is that when the flood of -asses inundated the language in the 1950s and 1960s, English speakers were already divided on the question of what the -ass referred to in any particular case. As a result, we often see both -ass and -assed forms—and even within the -ass segment of speakers and writers, the association with hooved beasts of burden is far from unanimous.
I wrote an answer here on how the meanings of bad and other similar words have changed over time (and actually it was a dupe, and the previous questions' answers may be just as good). The comment on my answer is spot on—that words have often tended to just reverse their meaning.
If you imagine, at one point, the meaning of the word badass to be the opposite of what it is now—someone actually bad (negatively), it makes much more sense. Ass has often been used to pejoratively refer to someone stupid, or to insult people, and appending bad sounds just about right.
I'd say badass, at face value, sounds like it is meant in a negative way, to mean someone who is belligerent or reckless (a bad ass/bad donkey?), so people started using it to mean the opposite. Similar to how other words that imply danger have changed over time to mean something positive (because danger is sometimes seen as cool), badass may have at some point changed to mean someone who is dangerous or reckless in a cool way.
note: I am not saying that the word badass was actually meant in a bad way at some point—just suggesting that thinking about it that way might help it make sense.
According to Etymonline.com:
"tough guy," 1950s U.S. slang, from bad + ass (n.2).
slang for "backside," first attested 1860 in nautical slang, in popular use from 1930; chiefly U.S.; from dialectal variant pronunciation of arse (q.v.). The loss of -r- before -s- attested in several other words (such as burst/bust, curse/cuss, horse/hoss, barse/bass). *Indirect evidence of the change from arse to ass can be traced to 1785 (in euphemistic avoidance of ass "donkey" by polite speakers) and perhaps to Shakespeare, if Nick Bottom transformed into a donkey in "A Midsummer Night's Dream" (1594) is the word-play some think it is. Meaning "woman regarded as a sexual object" is from 1942. Colloquial (one's) ass "one's self, one's person" attested by 1958.