A dilemma (Greek: δίλημμα "double proposition") is a problem offering two possibilities, neither of which is unambiguously acceptable or preferable. One in this position has been traditionally described as "being on the horns of a dilemma", neither horn being comfortable. This is sometimes more colorfully described as "Finding oneself impaled upon the horns of a dilemma", referring to the sharp points of a bull's horns, equally uncomfortable (and dangerous).
The dilemma is sometimes used as a rhetorical device, in the form "you must accept either A, or B"; here A and B would be propositions each leading to some further conclusion. Applied incorrectly, it constitutes a false dichotomy, a fallacy.
Colorful names have been given to many types of dilemmas.
Chicken or egg: which is first of two things, each of which presupposes the other
Double bind: conflicting requirements ensure that the victim will automatically be wrong.
Ethical dilemma: a choice between moral imperatives.
Extortion: the choice between paying the extortionist and suffering an unpleasant action.
Hedgehog's dilemma: the desire for intimacy going unfulfilled due to accompanying substantial mutual harm.
Fairness dilemmas: when groups are faced with making decisions about how to share their resources, rewards, or payoffs.
Hobson's choice: a choice between something and nothing; "take it or leave it".
Mesh Fencer: relates to a difficult commercial choice where neither option is cost effective but a decision has to be made.
Morton's fork: choices yield equivalent, often undesirable, results.
Prisoner's dilemma: an inability to coordinate makes cooperation difficult and defection tempting.
Samaritan's dilemma: the choice between providing charity and improving another's condition, and withholding it to prevent them from becoming dependent.
Sophie's choice: a choice between two persons or things that will result in the death or destruction of the person or thing not chosen.
Traveler's dilemma: you could make the best move possible to win the game according to its rules. But in doing so, you forfeit almost the entire value of a lost suitcase full of antiques of which you are the owner.
Zugzwang: one must move and incur harm when one would prefer to make no move (esp. in chess).
Several idioms describe dilemmas:
"Damned if you do, damned if you don't"
"Between Scylla and Charybdis"
"Lesser of two evils"
"Between a rock and a hard place", since both objects or metaphorical choices are rough.
"Between the devil and the deep blue sea"
"Out of the frying pan, into the fire"
A dilemma with more than two forks is sometimes called a trilemma (3), tetralemma (4), or more generally a polylemma.
In formal logic, the definition of a dilemma differs markedly from everyday usage. Two options are still present, but choosing between them is immaterial because they both imply the same conclusion. Symbolically expressed thus:
A
∨
B
,
A
⇒
C
,
B
⇒
C
,
⊢
C
Which can be translated informally as "one (or both) of A or B is known to be true, but they both imply C, so regardless of the truth values of A and B we can conclude C." This is a rule of inference called Disjunction elimination.
There are also constructive dilemmas and destructive dilemmas.
1. (If X, then Y) and (If W, then Z).
2. X or W.
3. Therefore, Y or Z.
1. (If X, then Y) and (If W, then Z).
2. Not Y or not Z.
3. Therefore, not X or not W.
