In this common fallacy, the arguer reasons as though only two alternatives exist, when, in fact, additional reasonable alternatives are being ignored.
The false dichotomy can take one of two forms:
Simple Constructive Dilemma:
Either a or b.
If a then c.
If b then c.
Which could be true for the trivial examples such as the following:
Either pink or green.
If pink then color.
If green then color.
Which would not be true for the following:
Either God or evolution.
If God then God.
If evolution then God.
Disjunctive syllogisms revolve around "not":
Either d or e.
A fallacious disjunctive syllogism is found in the typical creationist fallacy:
Either God or Darwinian-evolution.
A logical error occurs when contrary propositions (at most one will be true, but both may be false) are confused with contradictory propositions (exactly one will be true).
Let's assume that the color in question is red.
Contrary propositions: either black or white
Contradictory propositions: either black or not-black
Only the contradictory proposition (red is not-black) is true.
A disjunction with contradictory disjuncts (black, not not-black) is logically true (not-black is not black) and follows the law of the excluded middle.
By contrast, a disjunction is only logically contingent when its disjuncts are contraries (black or white). That is the disjunction is true only when the the alternatives truly are either black or white. Or, expressed differently, the truth of the disjunction is contingent upon one of the alternatives' actually happening to be true—either white is true, or black is true.