Paraconsistent logic

A paraconsistent logic is a logic which gracefully deals with contradictions.

In classical logic, if $\Lambda \vdash P$ and $\Lambda \vdash \neg P$, i.e. there is some theory $\Lambda$ which allows you to show both $P$ and $\neg P$, then it is possible to prove that every formula $A$ (and so its negation $\neg A$) is true in the proof calculus; a similar model theoretic result can be derived. Classical logic, intuitionistic logic, and indeed most other logics suffer from this problem. Paraconsistent logics must not fall into this trap.

The need for paraconsistent logics comes from the nature of human reason; we appear to reason paraconsistently, and perhaps the universe itself is paraconsistent.

Some paraconsistent logics:

• Belief revision systems
• Many valued logics
• Relevant logic