Contradiction

A contradiction is the simultaneous acceptance and denial of a proposition or statement. "It was the best of times and the worst of times" is a poetic example: the contradiction is being used to enhance the poetic nature of the piece.

According to J.Sarfati, a contradiction is defined as the conjunction of the affirmation and denial of a premise, in the same time, place, and sense (i.e. p and not-p, or in symbolic form, p.~p).

In a proof, if two contradictory inferences can be drawn from the premise, this indicates either that a premise is false or that the argument is invalid.

It is often held that from a contradiction anything can be inferred. This can be demonstrated:

1. Take any contradiction, “A” and “Not A”, and take it to be true.
2. Take anything that you want to prove, “P”
3. Now because “A and Not A” is true, “A” is true, by virtue of the simplification rule.
4. “A or P” is true just so long as both Not A and P are false.
5. So, As “A” is true, “A or P” must be true.
6. If “A or P” is true, then one of “A” or “P” must be true.
7. As “Not A” is true then “A” is false which means that the remaining term in “A or P” must be true.
8. “P”, absolutely anything you like, is true.