# Proof by Contradiction/Variant 3/Formulation 2

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## 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 | $\left({p \implies \neg p}\right) \implies \neg p$ | Rule of Implication: $\implies \mathcal I$ | 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 | $\left({p \lor p}\right) \implies p$ | Axiom $A1$ | ||||

2 | $\left({\neg p \lor \neg p}\right) \implies \neg p$ | Rule $RST \, 1$ | 1 | $\neg p \, / \, p$ | ||

3 | $\left({p \implies \neg p}\right) \implies \neg p$ | Rule $RST \, 2 \, (2)$ | 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: $D \, 1$ - 1964: Donald Kalish and Richard Montague:
*Logic: Techniques of Formal Reasoning*... (previous) ... (next): $\text{I}$: 'NOT' and 'IF': $\S 5$: Theorem $\text{T20}$