User:Boris Tsirelson/Sandbox1

Consistent or inconsistent

If a theory states that 2+2=5, it is a paradox but not yet a contradiction. By "paradox" people may mean

• a contradiction;
• an apparent contradiction;
• something counterintuitive;
• something surprising;
• something ironic;

etc. In contrast, a contradiction (in a mathematical theory) is, by definition, a pair of theorems (of the given theory) such that one is the negation of the other. Thus, two theorems

$2+2=4,$

$2+2=5$

are still not a contradiction. Two theorems

$2+2=4,$

$2+2\neq 4$

are a contradiction.

If a contradiction exists in a given theory, this theory is called inconsistent. Otherwise, if no contradiction exist (rather than merely not found for now), the theory is called consistent.

An inconsistent theory is completely useless. Some philosophers disagree:

Superstitious dread and veneration by mathematicians in face of a contradiction (Ludwig Wittgenstein)

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wp:Strict conditional

wp:Paradoxes of material implication

wp:Relevance logic