Law of Identity/Formulation 2

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Theorem

Every proposition entails itself:

$\vdash p \implies p$


Proof 1

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

$\blacksquare$


Proof 2

Using a tableau proof for instance 1 of a Hilbert proof system:

$p \implies p$
Line Pool Formula Rule Depends upon Notes
1 $\paren {p \implies \paren {\paren {p \implies p} \implies p} } \implies \paren {\paren {p \implies \paren {p \implies p} } \implies \paren {p \implies p} }$ Axiom 2 $\mathbf A = p, \mathbf B = p \implies p, \mathbf C = p$
2 $p \implies \paren {\paren {p \implies p} \implies p}$ Axiom 1 $\mathbf A = p, \mathbf B = p \implies p$
3 $\paren {p \implies \paren {p \implies p} } \implies \paren {p \implies p}$ Modus Ponendo Ponens: $\implies \mathcal E$ 1, 2
4 $p \implies \paren {p \implies p}$ Axiom 1 $\mathbf A = p, \mathbf B = p$
5 $p \implies p$ Modus Ponendo Ponens: $\implies \mathcal E$ 3, 4

$\blacksquare$


Proof by Truth Table

We apply the Method of Truth Tables to the proposition.

As can be seen by inspection, the truth value under the main connective is $\T$ throughout.

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

$\blacksquare$


Also see

Some sources, for example 1980: D.J. O'Connor and Betty Powell: Elementary Logic, use the statement:

$\vdash p \implies p$

to be the defining property of a tautology.


Sources

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