Broadly speaking, a **contradiction** is when two or more statements, ideas, or actions are seen as incompatible. One must, it seems, reject at least one of the ideas outright.

In logic, contradiction is defined much more specifically, usually as the simultaneous assertion of a statement and its negation, also known as its denial. (See: the Law of non-contradiction.) This, of course, assumes that "negation" has a non-problematic definition. (Coming sometime: article on negation.)

Table of contents |

2 Paradoxes associated with contradiction 3 Proof by contradiction 4 Contradictions and common sense psychology |

## Colloquial use

In everyday speech, "contradiction" may be used in a much less rigorous way than in formal logic. For example, there is nothing*logically*contradictory involved in a man condemning the members of his church for not giving the church enough financial support even though he never puts anything in the collection plate when it goes around. In ordinary language we would be quite inclined to say that his actions contradict his words, but the immediate connection of this usage to the logical usage is unclear. Hypocrisy is certainly lamentable but it's hard to say that it's

*logically incoherent*--our hypothetical church-goer, after all, is not clearly asserting

*anything*by refusing to put money in the collection plate, let alone the logical negation of what he asserted.

(One way to understand the colloquial usage might be to shift grounds from *logical* contradiction to what some philosophers describe as a *performative* contradiction. A hypocrite is not *saying* anything that contradicts the general principles that he asserts to be true; but his actions, in some sense, presuppose that those principles are false. Similarly, "I cannot assert anything." is a sentence that no-one can truly utter. This is not because of a logical contradiction in the sentence--it is, for example, true of the brain-dead. But there *is* a performative contradiction involved in the *act* of saying it; for to say it presupposes that you *can* assert something.)

## Paradoxes associated with contradiction

Contradiction is associated with several notorious paradoxes. One of these is that in first-order predicate calculus*any proposition*can be derived from a contradiction. Thus, for example, the following argument is

*strictly valid:*

- 5 is even.
- 5 is odd.
- Therefore, God exists.

*also*valid:

- 5 is even.
- 5 is odd.
- Therefore, God does not exist.

*validly*infer

*either*"God exists" or "God does not exist" from the seemingly completely unrelated propositions "5 is even" and "5 is odd." And if that seems odd, how much odder must it seem that you can validly infer

*both!*How did that happen?

The answer becomes clear when we spell out the steps of the argument. For example, take the theistic version:

- 5 is even. (premise)
- 5 is odd. (premise)
- If 5 is even then 5 is not odd. (def. of even)
- 5 is not odd. (M.P. 1, 3)
- 5 is odd and 5 is not odd. (Conj. 2, 4)
- If 5 is odd and 5 is not odd, then God exists. (Def. M.I.)
- Therefore, God exists. (M.P. 5, 6)

Because you can validly infer anything from contradictory premises, contradictions are said to be logically explosive in first-order logic. Of course, while this result may seem curious, it poses no burly metaphysical problems. For while both arguments are strictly *valid,* neither of them could possibly be *sound*--since the two premises contradict one another, there is no way that they could *both* be true.

## Proof by contradiction

In deductive logic (and thus, also, in mathematics), a contradiction is usually taken as a sign that something has gone wrong, that you need to retrace the steps of your reasoning and "check your premises." This has been used to great effect in mathematics through the method of proof by contradiction (also known as**indirect proof**): since a contradiction can

*never*be true, it can thus never be the conclusion of a valid argument with all true premises. To construct a proof by contradiction, then, you construct a valid proof from a set of premises to a conclusion that is a logical contradiction. Since the conclusion is false, and the argument is valid, the only possibility is that one or more of the premises are false. This method is used in many key mathematical proofs, such as Euclid's proof that there is no greatest prime, and Cantor's diagonal proof that there are uncountably many real numbers between 0 and 1.

(ironic: you can't know that correct, formal reasoning will lead to consistent conclusions. (specify in what sense this is true))

## Contradictions and common sense psychology

Common sense suggests that people do hold many contradictory beliefs, and this is confirmed by psychologists. (?)Being non-contradictory seems to be central to people's conceptions of what "reason" is, and what it means to be "reasonable".

What to make of it: 1) be very, very careful, so as to not be contradictory. 2) embrace contradiction as part of human nature. 3) ?