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.)
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2 Paradoxes associated with contradiction 3 Proof by contradiction 4 Contradictions and common sense psychology |
(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.)
The answer becomes clear when we spell out the steps of the argument. For example, take the theistic version:
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.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:
But atheists have no less reason to celebrate then theists, for this argument is also valid:
It certainly seems a bit odd that one could 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 curious step here is step 6. But 6 can be demonstrated to be true based on the truth table for material implication. Here's how: material implication is defined such that any conditional with a false antecedent is true. Since every instance of "x is odd and x is not odd" is a contradiction, its instances are always false; and since they are false, any conditional that has one as an antecedent is always true. "5 is odd and 5 is not odd" is one such instance; therefore, step 6 is one such conditional. But this is no less true of "If 5 is odd and 5 is not odd, then God does not exist"; we could, therefore, construct the atheistic version of this argument with no greater difficulty. Thus, once the contradiction is supposed, everything and nothing follows.
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))