# Dilemma

A dilemma (Greek δί-λημμα "double proposition") is a problem with at least two solutions or possibilities. None of the solutions are practically acceptable; a person in this position has been traditionally described as being impaled on the horns of a dilemma, neither horn being comfortable.

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 in this way, it may be a fallacy, a false dichotomy.

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:

${\displaystyle A\vee B,A\Rightarrow C,B\Rightarrow C\vdash C}$

This 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."

Horned dilemmas can present more than two choices. The number of choices of Horned dilemmas can be used in their alternative names, such as two-pronged (two-horned) or dilemma proper, or three-pronged (three-horned) or trilemma, and so on.

Constructive dilemmas--

1. (If X, then Y) and (If W, then Z).
2. X or W.
3. Therefore, Y or Z.

Destructive dilemmas--

1. (If X, then Y) and (If W, then Z).
2. Not Y or not Z.
3. Therefore, not X or not W.

## Responses to a Dilemma

In Zen and the Art of Motorcycle Maintenance, Robert Pirsig outlines possible responses to a dilemma. The classical responses are to either choose one of the two horns and refute the other or alternatively to refute both horns by showing that there are additional choices. Pirsig then mentions three illogical or rhetorical responses. One can "throw sand in the bull's eyes" by, for example, questioning the competence of the questioner. One can "sing the bull to sleep" by, for example, stating that the answer to the question is beyond one's own humble powers and asking the questioner for help. Finally one can "refuse to enter the arena" by, for example, stating that the question is unanswerable.

## Other pages

The English Wiktionary has a dictionary definition (meanings of a word) for: dilemma