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): $\S 2.3.3$
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