Horns of a dilemma

Dilemma \Di*lem"ma\, n. [L. dilemma, Gr. ?; di- = di`s- twice +
   ? to take. See {Lemma}.]
   1. (Logic) An argument which presents an antagonist with two
      or more alternatives, but is equally conclusive against
      him, whichever alternative he chooses.
      [1913 Webster]

   Note: The following are instances of the dilemma. A young
         rhetorician applied to an old sophist to be taught the
         art of pleading, and bargained for a certain reward to
         be paid when he should gain a cause. The master sued
         for his reward, and the scholar endeavored to elude his
         claim by a dilemma. "If I gain my cause, I shall
         withhold your pay, because the judge's award will be
         against you; if I lose it, I may withhold it, because I
         shall not yet have gained a cause." "On the contrary,"
         says the master, "if you gain your cause, you must pay
         me, because you are to pay me when you gain a cause; if
         you lose it, you must pay me, because the judge will
         award it." --Johnson.
         [1913 Webster]

   2. A state of things in which evils or obstacles present
      themselves on every side, and it is difficult to determine
      what course to pursue; a vexatious alternative or
      predicament; a difficult choice or position.
      [1913 Webster]

            A strong dilemma in a desperate case!
            To act with infamy, or quit the place. --Swift.
      [1913 Webster]

   {Horns of a dilemma}, alternatives, each of which is equally
      difficult of encountering.
      [1913 Webster]