Law of Identity/Formulation 1

Theorem

Every proposition entails itself:

$p \vdash p$

Proof 1

By the tableau method of natural deduction:

$p \vdash p$

Line		Pool	Formula	Rule	Depends upon	Notes
1		1	$p$	Premise	(None)

$\blacksquare$

This is the shortest tableau proof possible.

Proof by Truth Table

We apply the Method of Truth Tables (trivially) to the proposition.

$\begin{array}{|c|c|} \hline p & p \\ \hline \F & \F \\ \T & \T \\ \hline \end{array}$

$\blacksquare$

Sources

• 1964: Donald Kalish and Richard Montague: Logic: Techniques of Formal Reasoning ... (previous) ... (next): $\text{I}$: 'NOT' and 'IF': $\S 3$
• 2012: M. Ben-Ari: Mathematical Logic for Computer Science (3rd ed.) ... (previous) ... (next): $\S 2.3.3$