De Morgan's Laws (Logic)/Disjunction/Formulation 1/Forward Implication
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Theorem
- $p \lor q \vdash \neg \paren {\neg p \land \neg q}$
Proof
By the tableau method of natural deduction:
Line | Pool | Formula | Rule | Depends upon | Notes | |
---|---|---|---|---|---|---|
1 | 1 | $p \lor q$ | Premise | (None) | ||
2 | 2 | $\neg p \land \neg q$ | Assumption | (None) | ||
3 | 2 | $\neg p$ | Rule of Simplification: $\land \EE_1$ | 2 | ||
4 | 2 | $\neg q$ | Rule of Simplification: $\land \EE_2$ | 2 | ||
5 | 5 | $p$ | Assumption | (None) | ||
6 | 2, 5 | $\bot$ | Principle of Non-Contradiction: $\neg \EE$ | 5, 3 | ||
7 | 7 | $q$ | Assumption | (None) | ||
8 | 2, 7 | $\bot$ | Principle of Non-Contradiction: $\neg \EE$ | 7, 4 | ||
9 | 1, 2 | $\bot$ | Proof by Cases: $\text{PBC}$ | 1, 5 – 6, 7 – 8 | Assumptions 5 and 7 have been discharged | |
10 | 1 | $\neg \paren {\neg p \land \neg q}$ | Proof by Contradiction: $\neg \II$ | 2 – 9 | Assumption 2 has been discharged |
$\blacksquare$
Sources
- 1965: E.J. Lemmon: Beginning Logic ... (previous) ... (next): Chapter $1$: The Propositional Calculus $1$: $5$ Further Proofs: Résumé of Rules: Theorem $36 \ \text{(a)}$