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:

$\vdash \paren {\paren {p \implies q} \implies p} \implies p$

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 $12 \ (11)$