Law of Identity/Formulation 2/Proof 1

Theorem

Every proposition entails itself:

$\vdash p \implies p$

Proof

By the tableau method of natural deduction:

$\vdash p \implies p$

Line		Pool	Formula	Rule	Depends upon	Notes
1		1	$p$	Premise	(None)
2			$p \implies p$	Rule of Implication: $\implies \II$	1 – 1	Assumption 1 has been discharged

$\blacksquare$

This is the second shortest tableau proof possible.

Sources

• 1964: Donald Kalish and Richard Montague: Logic: Techniques of Formal Reasoning ... (previous) ... (next): $\text{I}$: 'NOT' and 'IF': $\S 5$: Theorem $\text{T1}$
• 1965: E.J. Lemmon: Beginning Logic ... (previous) ... (next): Chapter $2$: The Propositional Calculus $2$: $2$: Theorems and Derived Rules: Theorem $38$
• 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