Principle of explosion
The principle of explosion is a logical principle which states that if we accept that a certain statement is true at the same time it is not true, then any other statement, no matter how false, can be proven. In other words, from a false premise, any imaginable conclusion can be reached. It is used to illustrate the importance of having a set of axioms which are self-consistent.
Suppose we accept both of the statements "red is a color" and "red is not a color" as true.
Since "red is a color" is true, at least one of the two statements "red is a color" and "Obama is a competent president" is true, and so the single statement "red is a color, or Obama is a competent president, or both" is true. Since we know "red is not a color" is true, the only clause in "red is a color, or Obama is a competent president" which can be true it the latter, ie, "Obama is a competent president."
From the false assumption "red both is and isn't a color," we have reached a false conclusion.