A limited number of options (usually two) is given, while in reality there are more options. A false dilemma is an illegitimate use of the "or" operator. Putting issues or opinions into "black or white" terms is a common instance of this fallacy. "The fallacy of false dilemma is committed when an argument limits consideration of positions on an issue to two mutally exclusive ones, thereby setting up an apparent dilemma, when there are other positions that could be offered." (Bowell & Kemp, Critical Thinking, p. 164)

Examples

1. Either you're for me or against me.
2. America: love it or leave it.
3. "There is a tough choice facing the Government and the nation: either we cut taxes and increase everyone's spending power, thereby providing much needed stimulation to the economy, or we increase spending on health and education. It is impossible to do both; and without a tax cut the economy will remain weak. So increased spending on health and education will have to wait." (Bowell & Kemp, Critical Thinking, p. 160.)
4. "Either our fellow citizens are good or the're bad. If they're good, laws to deter crime aren't needed. But if they're bad, laws to deter crime won't succeed. So laws to deter crime either are not needed or won't succeed." (Kahane & Cavender, Logic and Contemporary Rhetoric, p. 58.)
5. Arguably, the famous Euthyphro dilemma: either the good is good because God commands it, or God commands it because it is good. If one believes that the good or moral is equivalent to whatever God commands, the first option leads to the charge that the good is thereby arbitrary, or on the second option, God is unnecessary for the good. Is there a third option?
6. Did you walk to school or take a lunch? (This was a favorite of mine in middle school for befuddling classmates.)
7. "A second false dilemma has threatened to dominate the argument on national defense... The choice, of course, is between the desperate alternatives, either universal atomic death or complete surrender to Communism. The Catholic mind, schooled in the traditional doctrine of war and peace, rejects the dangerous fallacy involved in this casting up of desperate alternatives. Hidden beneath the fallacy is an abdication of the moral reason and a craven submission to some manner of technological or historical determinism." (Miller, War in the Twentieth Century, p.259)

Proof

Two Approaches

Grasping the Horns of the Dilemma

Challenge one or both of the alternatives given and show it to be a false option.  

Splitting the Horns of the Dilemma 

Identify the options given and show (with an example) that there is an additional option.

A clue to look for when detecting false dilemmas is whether the second option is the inverse of the second, that is, whether the options are symmetrical, A and not A. Notice, for example, that in the Euthyphro dilemma the second option is not quite the inverse of the first, which would be, "either the good is good because God commands it, or the good is good not because God commands it". Or, with respect to America: "love it or don't love it". This isn't a fail safe clue that a false dilemma has been proposed, but more often than not, one has. A color must be either red, or not red; but it needn't be either red or blue. While it is always the case of a proposition that A or not A is true, it is rare that there are only two asymmetrical options.

"In effect, Burgess presents his opponents with a dilemma: Are you advancing a linguistic thesis? Or are you advocating a scientific revolution? Either way, your proposition is implausible... Notice that no reasons are given for thinking that the alternatives considered above are the only possible ones. Why must Burgess's opponents take one or the other of these horns? Well, what other possibilities are there? How else can one rationally reject Burgess's conclusion?" (Chihara, Constructibility and Mathematical Existence, p. 184.)

Structure

False Dilemma

Either P or Q.
If P then R.
If Q then S.
Therefore, either R or S.

Either-Or

Either P or Q.
Not P.
Therefore, Q.