Constructive Dilemma/Formulation 2
Theorem
| \(\ds \paren {p \implies q} \land \paren {r \implies s}\) | \(\) | \(\ds \) | ||||||||||||
| \(\ds p \lor r\) | \(\) | \(\ds \) | ||||||||||||
| \(\ds \vdash \ \ \) | \(\ds q \lor s\) | \(\) | \(\ds \) |
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 | $p \lor r$ | Premise | (None) | ||
| 3 | 1, 2 | $\paren {p \lor r} \land \paren {p \implies q} \land \paren {r \implies s}$ | Rule of Conjunction: $\land \II$ | 2, 1 | Associativity is implicit | |
| 4 | $\paren {\paren {p \lor r} \land \paren {p \implies q} \land \paren {r \implies s} } \implies \paren {q \lor s}$ | Theorem Introduction | (None) | Constructive Dilemma: Formulation 3 | ||
| 5 | 1, 2 | $q \lor s$ | Modus Ponendo Ponens: $\implies \mathcal E$ | 4, 3 |
$\blacksquare$
Sources
- 1973: Irving M. Copi: Symbolic Logic (4th ed.) ... (previous) ... (next): $2$ Arguments Containing Compound Statements: $2.3$: Argument Forms and Truth Tables: Exercise $\text{I} \ \mathbf k.$
- 1973: Irving M. Copi: Symbolic Logic (4th ed.) ... (previous) ... (next): $3$: The Method of Deduction: $3.1$: Formal Proof of Validity: Rules of Inference: $5.$
- 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: $3$