Principle of Non-Contradiction/Proof Rule

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.

As a proof rule it is expressed in the form:

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

It can be written:

$\ds {\phi \quad \neg \phi \over \bot} \neg_e$

Tableau Form

Let $\phi$ be a well-formed formula in a tableau proof.

The Principle of Non-Contradiction is invoked for $\phi$ and $\neg \phi$ in the following manner:

			Pool:					The pooled assumptions of $\phi$
								The pooled assumptions of $\neg \phi$
			Formula:					$\bot$
			Description:					Principle of Non-Contradiction
			Depends on:					The line containing the instance of $\phi$
								The line containing the instance of $\neg \phi$
			Abbreviation:					$\operatorname {PNC}$ or $\neg \EE$

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

• This is a rule of inference of the following proof systems:
  • Definition:Natural Deduction

Technical Note

When invoking the Principle of Non-Contradiction in a tableau proof, use the {{NonContradiction}} template:

{{NonContradiction|line|pool|first|second}}

or:

{{NonContradiction|line|pool|first|second|comment}}

where:

line is the number of the line on the tableau proof where the Principle of Non-Contradiction is to be invoked

pool is the pool of assumptions (comma-separated list)

first is the first of the two lines of the tableau proof upon which this line directly depends

second is the second of the two lines of the tableau proof upon which this line directly depends

comment is the (optional) comment that is to be displayed in the Notes column.

Sources

• 2000: Michael R.A. Huth and Mark D. Ryan: Logic in Computer Science: Modelling and reasoning about systems ... (previous) ... (next): $\S 1.2.1$: Rules for natural deduction