Principle of Non-Contradiction

Theorem

The Principle of Non-Contradiction is a valid argument in types of logic dealing with negation $\neg$ and contradiction $\bot$.

This includes classical propositional logic and predicate logic, and in particular natural deduction.

Proof Rule

If we can conclude both $\phi$ and $\neg \phi$, we may infer a contradiction.

Sequent Form

$p, \neg p \vdash \bot$

Explanation

The Principle of Non-Contradiction can be expressed in natural language as follows:

A statement can not be both true and not true at the same time.

This means: if we have managed to deduce that a statement is both true and false, then the sequence of deductions show that the pool of assumptions upon which the sequent rests contains assumptions which are mutually contradictory.

Thus it provides a means of eliminating a logical not from a sequent.

Also known as

The Principle of Non-Contradiction is otherwise known as:

• Principium Contradictionis, Latin for principle of contradiction
• Rule of Not-Elimination
• Law of Contradiction
• Law of Non-Contradiction

Also see

• Definition:Logical Not
• Definition:Contradiction
• Law of Excluded Middle

Sources

• 1988: Alan G. Hamilton: Logic for Mathematicians (2nd ed.) ... (previous) ... (next): $\S 1$: Informal statement calculus: $\S 1.2$: Truth functions and truth tables: Example $1.6 \ \text{(b)}$
• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): contradiction
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): contradiction