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.

It is tempting to object that the contradiction has nothing in common with the assumption and therefore cannot invalidate it. However, the rules of formal logic do not demand that the contradiction has something in common with the assumption. Some attempts to change these rules were made (so-called "relevance logic", or "relevant logic"), but with little success. It is always possible to obfuscate the proof of the contradiction, making it seemingly entangled with X. We have no formal criterion able to detect any possible fictitious participation of X in the proof of the contradiction.

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wp:Strict conditional

wp:Paradoxes of material implication

wp:Relevance logic