THE LOGIC NOTES

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

  1. The simplest contradiction consists of an atomic formula and its negation, so p and NOTp.
  2. ALLx(Fx IMP Gx) and NOTALLx(Fx IMP Gx) together constitute a contradiction.
  3. The set {p OR q,  NOTp AND NOTq} is inconsistent, but it is not a contradiction in the narrow sense specified here.

Links