Axiom:Principle of Non-Contradiction

Context

The principle of non-contradiction (PNC) is one of the axioms of natural deduction.

Axiom

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

$\vdash \neg \left({p \land \neg p}\right)$

Otherwise known as:

• Principium contradictionis, Latin for principle of contradiction;
• Law of Contradiction.

Alternatively:

$p \land \neg p \dashv \vdash \bot$

By proof by contradiction it can be seen that the two formulations are equivalent.

It can be written:

$\displaystyle {{p \land \neg p} \over \bot} \textrm{PNC}$

where the symbol $\bot$ (called bottom) signifies contradiction.

It is also equivalent to the rule of not-elimination.

It is one of the cornerstones of Aristotelian logic, along with the law of the excluded middle.

Truth Table Demonstration

We apply the Method of Truth Tables to the proposition $\neg \left({p \land \neg p}\right)$.

As can be seen by inspection, the truth value of the main connective, that is $\neg$, is $T$ for each model of $p$.

$\begin{array}{|ccccc|} \hline \neg & (p & \land & \neg & p)\\ \hline T & F & F & T & F \\ T & T & F & F & T \\ \hline \end{array}$

$\blacksquare$

Sources

• Donald Kalish and Richard Montague: Logic: Techniques of Formal Reasoning (1964): $\text{II}: \S 3$: Theorem $\text{T36}$
• Alan G. Hamilton: Logic for Mathematicians (1978): $\S 1.2$: Example $1.6 \ \text{(b)}$