Rule of Material Implication/Formulation 2/Reverse Implication
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Theorem
- $\vdash \paren {\neg p \lor q} \implies \paren {p \implies q}$
Proof
By the tableau method of natural deduction:
Line | Pool | Formula | Rule | Depends upon | Notes | |
---|---|---|---|---|---|---|
1 | 1 | $\neg p \lor q$ | Assumption | (None) | ||
2 | 1 | $p \implies q$ | Sequent Introduction | 1 | Rule of Material Implication: Formulation 1 | |
3 | $\paren {\neg p \lor q} \implies \paren {p \implies q}$ | Rule of Implication: $\implies \II$ | 1 – 2 | Assumption 1 has been discharged |
$\blacksquare$
Sources
- 1964: Donald Kalish and Richard Montague: Logic: Techniques of Formal Reasoning ... (previous) ... (next): $\text{II}$: 'AND', 'OR', 'IF AND ONLY IF': $\S 5$
- 2000: Michael R.A. Huth and Mark D. Ryan: Logic in Computer Science: Modelling and reasoning about systems ... (previous) ... (next): $\S 1.2.1$: Rules for natural deduction: Exercises $1.5: \ 2 \ \text{(i)}$