Contradiction Glossary
Definition
A contradiction is a pair of formulae one of which is the negation of the other, so any two formulae of the form A and NOTA.
Comments
Since a contradiction cannot be true, one of the basic rules of the natural deduction calculus is that the absurd constant ⊥ is an immediate consequence of any contradiction.
In classical logic, a contradiction entails every formula whatsoever. In paraconsistent logics, this is not the case.
In the tableau calculus, a contradiction in a branch closes that branch. Contradictions in all branches close the tableau.
Examples
- The simplest contradiction consists of an atomic formula and its negation, so p and NOTp.
- ALLx(Fx IMP Gx) and NOTALLx(Fx IMP Gx) together constitute a contradiction.
- The set {p OR q, NOTp AND NOTq} is inconsistent, but it is not a contradiction in the narrow sense specified here.