Peirce's Law/Formulation 2
Theorem
- $\vdash \paren {\paren {p \implies q} \implies p} \implies p$
Proof 1
By the tableau method of natural deduction:
| Line | Pool | Formula | Rule | Depends upon | Notes | |
|---|---|---|---|---|---|---|
| 1 | 1 | $\paren {p \implies q} \implies p$ | Assumption | (None) | ||
| 2 | 1 | $p$ | Sequent Introduction | 1 | Peirce's Law: Formulation 1: $\paren {p \implies q} \implies p \vdash p$ | |
| 3 | $\paren {\paren {p \implies q} \implies p} \implies p$ | Rule of Implication: $\implies \II$ | 1 – 2 | Assumption 1 has been discharged |
$\blacksquare$
Proof by Truth Table
We apply the Method of Truth Tables to the proposition.
As can be seen by inspection, the truth values under the main connective are $\T$ for all boolean interpretations.
$\begin{array}{|ccccc|c|c|}\hline ((p & \implies & q) & \implies & p) & \implies & p \\ \hline \F & \T & \F & \F & \F & \T & \F \\ \F & \T & \T & \F & \F & \T & \F \\ \T & \F & \F & \T & \T & \T & \T \\ \T & \T & \T & \T & \T & \T & \T \\ \hline \end{array}$
$\blacksquare$
Source of Name
This entry was named for Charles Sanders Peirce.
Sources
- 1965: E.J. Lemmon: Beginning Logic ... (previous) ... (next): Chapter $2$: The Propositional Calculus $2$: $2$: Theorems and Derived Rules: Exercise $5 \ \text{(c)}$
- 1964: Donald Kalish and Richard Montague: Logic: Techniques of Formal Reasoning ... (previous) ... (next): $\text{I}$: 'NOT' and 'IF': $\S 3$: Exercises, Group $\text{I}: \ 14$
- 1964: Donald Kalish and Richard Montague: Logic: Techniques of Formal Reasoning ... (previous) ... (next): $\text{I}$: 'NOT' and 'IF': $\S 5$: Theorem $\text{T23}$
- 1996: H. Jerome Keisler and Joel Robbin: Mathematical Logic and Computability ... (previous) ... (next): $\S 1.14$: Exercise $12 \ (11)$