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.

For a mathematician, an inconsistent theory is completely useless. Some philosophers disagree:

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

But a mathematician insists: an inconsistent theory is completely useless, since all statements (in the given language) are theorems! The reason is, proof by contradiction. No matter which statement X is in question, we always can prove X as follows:

• Assume that X is false;
• ... (put the contradiction here);
• the assumption leads to a contradiction, thus X is true.

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wp:Strict conditional

wp:Paradoxes of material implication

wp:Relevance logic