Law of Identity/Formulation 2
Theorem
Every proposition entails itself:
- $\vdash p \implies p$
Proof 1
By the tableau method of natural deduction:
| 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:
| 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
- 1910: Alfred North Whitehead and Bertrand Russell: Principia Mathematica: Volume $\text { 1 }$ ... (previous) ... (next): Chapter $\text{I}$: Preliminary Explanations of Ideas and Notations
- 1959: A.H. Basson and D.J. O'Connor: Introduction to Symbolic Logic (3rd ed.) ... (previous) ... (next): $\S 3.6$: Reference Formulae: $RF \, 3$
- 1975: T.S. Blyth: Set Theory and Abstract Algebra ... (previous) ... (next): $\S 1$. Sets; inclusion; intersection; union; complementation; number systems
- 1980: D.J. O'Connor and Betty Powell: Elementary Logic ... (previous) ... (next): $\S \text{I}: 3$: Logical Constants $(2)$
- 1988: Alan G. Hamilton: Logic for Mathematicians (2nd ed.) ... (previous) ... (next): $\S 1$: Informal statement calculus: $\S 1.3$: Rules for manipulation and substitution