Law of Identity/Formulation 1
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Theorem
Every proposition entails itself:
- $p \vdash p$
Proof 1
By the tableau method of natural deduction:
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): $\S 2.3.3$