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.

---

wp:Strict conditional

wp:Paradoxes of material implication

wp:Relevance logic