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A dilemma (Greek: δίλημμα "double proposition") is a problem offering two possibilities, neither of which is unambiguously acceptable or preferable. The possibilities are termed the horns of the dilemma, a clichéd usage, but distinguishing the dilemma from other kinds of predicament as a matter of usage.
The dilemma is sometimes used as a rhetorical device. Its isolation as textbook material has been attributed to Hermogenes of Tarsus in his work On Invention. C. S. Peirce gave a definition of dilemmatic argument as any argument relying on excluded middle.
In the form "you must accept either A, or B"; here A and B would be propositions each leading to some further conclusion, and applied incorrectly, it constitutes a false dichotomy, that is, a fallacy. Traditional usage distinguished the dilemma as a "horned syllogism" from the sophism that attracted the Latin name cornutus. The original use of the word horns in English has been attributed to Nicholas Udall in his 1548 book Paraphrases, translating from the Latin term cornuta interrogatio.
In propositional logic, dilemma is applied to a group of rules of inference, which are in themselves valid rather than fallacious. They each have three premises, and include the constructive dilemma and destructive dilemma. Such arguments can be refuted by showing that the disjunctive premise — the "horns of the dilemma" — does not in fact hold, because it presents a false dichotomy. You are asked to accept "A or B", but counter by showing that is not all. Successfully undermining that premise is called "escaping through the horns of the dilemma".
Dilemmatic reasoning has been attributed to Melissus of Samos, a Presocratic philosopher whose works survive in fragmentary form, making the origins of the technique in philosophy imponderable. It was established with Diodorus Cronus (died c. 284 BCE). The paradoxes of Zeno of Elea were reported by Aristotle in dilemma form, but that may have been to conform with what Plato said about Zeno's style.
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