# Dilemma

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.

## Related termsEdit

Several idioms describe dilemmas:

A dilemma with more than two forks is sometimes called a trilemma (3), tetralemma (4), or more generally a polylemma.

## Use in logicEdit

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:

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

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.

### Constructive dilemmasEdit

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

### Destructive dilemmasEdit

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