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
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