Modus Ponendo Ponens/Variant 3
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Theorem
- $\vdash \paren {\paren {p \implies q} \land p} \implies q$
Proof 1
By the tableau method of natural deduction:
Line | Pool | Formula | Rule | Depends upon | Notes | |
---|---|---|---|---|---|---|
1 | 1 | $\paren {p \implies q} \land p$ | Assumption | (None) | ||
2 | 1 | $p \implies q$ | Rule of Simplification: $\land \EE_1$ | 1 | ||
3 | 1 | $p$ | Rule of Simplification: $\land \EE_2$ | 1 | ||
4 | 1 | $q$ | Modus Ponendo Ponens: $\implies \mathcal E$ | 2, 3 | ||
5 | $\paren {\paren {p \implies q} \land p} \implies q$ | Rule of Implication: $\implies \II$ | 1 – 4 | Assumption 1 has been discharged |
$\blacksquare$
Proof by Truth Table
We apply the Method of Truth Tables.
As can be seen by inspection, the truth values under the main connective are true for all boolean interpretations.
$\begin{array}{|ccccc|c|c|} \hline ((p & \implies & q) & \land & p) & \implies & q \\ \hline \F & \T & \F & \F & \F & \T & \F \\ \F & \T & \T & \F & \F & \T & \T \\ \T & \F & \F & \F & \T & \T & \F \\ \T & \T & \T & \T & \T & \T & \T \\ \hline \end{array}$
$\blacksquare$
Sources
- 1959: A.H. Basson and D.J. O'Connor: Introduction to Symbolic Logic (3rd ed.) ... (previous) ... (next): $\S 3.6$: Reference Formulae: $RF \, 20$