Destructive Dilemma/Formulation 2
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Theorem
\(\ds \paren {p \implies q} \land \paren {r \implies s}\) | \(\) | \(\ds \) | ||||||||||||
\(\ds \neg q \lor \neg s\) | \(\) | \(\ds \) | ||||||||||||
\(\ds \vdash \ \ \) | \(\ds \neg p \lor \neg r\) | \(\) | \(\ds \) |
Proof
By the tableau method of natural deduction:
Line | Pool | Formula | Rule | Depends upon | Notes | |
---|---|---|---|---|---|---|
1 | 1 | $\paren {p \implies q} \land \paren {r \implies s}$ | Premise | (None) | ||
2 | 2 | $\neg q \lor \neg s$ | Premise | (None) | ||
3 | 1 | $p \implies q$ | Rule of Simplification: $\land \EE_1$ | 1 | ||
4 | 1 | $r \implies s$ | Rule of Simplification: $\land \EE_2$ | 1 | ||
5 | 5 | $\neg q$ | Assumption | (None) | ||
6 | 1, 5 | $\neg p$ | Modus Tollendo Tollens (MTT) | 3, 5 | ||
7 | 1, 5 | $\neg p \lor \neg r$ | Rule of Addition: $\lor \II_1$ | 6 | ||
8 | 8 | $\neg s$ | Assumption | (None) | ||
9 | 1, 8 | $\neg r$ | Modus Tollendo Tollens (MTT) | 4, 8 | ||
10 | 1, 8 | $\neg p \lor \neg r$ | Rule of Addition: $\lor \II_2$ | 9 | ||
11 | 1, 2 | $\neg p \lor \neg r$ | Proof by Cases: $\text{PBC}$ | 2, 5 – 7, 8 – 10 | Assumptions 5 and 8 have been discharged |
$\blacksquare$
Sources
- 1973: Irving M. Copi: Symbolic Logic (4th ed.) ... (previous) ... (next): $2$ Arguments Containing Compound Statements: $2.3$: Argument Forms and Truth Tables: Exercise $\text{II} \ \mathbf {18}.$
- 1973: Irving M. Copi: Symbolic Logic (4th ed.) ... (previous) ... (next): $3$: The Method of Deduction: $3.1$: Formal Proof of Validity: Rules of Inference: $6.$
- 1980: D.J. O'Connor and Betty Powell: Elementary Logic ... (previous) ... (next): $\S \text{II}$: The Logic of Statements $(2): \ 1$: Decision procedures and proofs: $4$