Proof by Contradiction/Variant 3/Formulation 2
Theorem
- $\vdash \paren {p \implies \neg p} \implies \neg p$
Proof 1
By the tableau method of natural deduction:
| Line | Pool | Formula | Rule | Depends upon | Notes | |
|---|---|---|---|---|---|---|
| 1 | 1 | $p \implies \neg p$ | Assumption | (None) | ||
| 2 | 1 | $\neg p$ | Sequent Introduction | 1 | Proof by Contradiction: Variant 3: Formulation 1 | |
| 3 | $\paren {p \implies \neg p} \implies \neg p$ | Rule of Implication: $\implies \II$ | 1 – 2 | Assumption 1 has been discharged |
$\blacksquare$
Proof 2
This proof is derived in the context of the following proof system: Instance 2 of the Hilbert-style systems.
By the tableau method:
| Line | Pool | Formula | Rule | Depends upon | Notes | |
|---|---|---|---|---|---|---|
| 1 | $\paren {p \lor p} \implies p$ | Axiom $\text A 1$ | ||||
| 2 | $\paren {\neg p \lor \neg p} \implies \neg p$ | Rule $\text {RST} 1$ | 1 | $\neg p \, / \, p$ | ||
| 3 | $\paren {p \implies \neg p} \implies \neg p$ | Rule $\text {RST} 2 \, (2)$ | 2 |
$\blacksquare$
Sources
- 1959: A.H. Basson and D.J. O'Connor: Introduction to Symbolic Logic (3rd ed.) ... (previous) ... (next): $\S 4.7$: The Derivation of Formulae: $\text D 1$
- 1964: Donald Kalish and Richard Montague: Logic: Techniques of Formal Reasoning ... (previous) ... (next): $\text{I}$: 'NOT' and 'IF': $\S 5$: Theorem $\text T20$