# False Statement implies Every Statement/Formulation 2

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## Theorem

- $\vdash \neg p \implies \left({p \implies q}\right)$

## Proof 1

By the tableau method of natural deduction:

Line | Pool | Formula | Rule | Depends upon | Notes | |
---|---|---|---|---|---|---|

1 | 1 | $\neg p$ | Assumption | (None) | ||

2 | 2 | $p$ | Assumption | (None) | ||

3 | 1, 2 | $\bot$ | Principle of Non-Contradiction: $\neg \mathcal E$ | 2, 1 | ||

4 | 1, 2 | $q$ | Rule of Explosion: $\bot \mathcal E$ | 3 | ||

5 | 1 | $p \implies q$ | Rule of Implication: $\implies \mathcal I$ | 2 – 4 | Assumption 2 has been discharged | |

6 | $\neg p \implies \left({p \implies q}\right)$ | Rule of Implication: $\implies \mathcal I$ | 1 – 5 | Assumption 1 has been discharged |

$\blacksquare$

## Proof 2

By the tableau method of natural deduction:

Line | Pool | Formula | Rule | Depends upon | Notes | |
---|---|---|---|---|---|---|

1 | 1 | $\neg p$ | Assumption | (None) | ||

2 | 1 | $p \implies q$ | Sequent Introduction | 1 | False Statement implies Every Statement: Formulation 1 | |

3 | $\neg p \implies \left({p \implies q}\right)$ | Rule of Implication: $\implies \mathcal I$ | 1 – 2 | Assumption 1 has been discharged |

$\blacksquare$

## Sources

- 1946: Alfred Tarski:
*Introduction to Logic and to the Methodology of Deductive Sciences*(2nd ed.) ... (previous) ... (next): $\S \text{II}.13$: Symbolism of sentential calculus - 1964: Donald Kalish and Richard Montague:
*Logic: Techniques of Formal Reasoning*... (previous) ... (next): $\text{I}$: 'NOT' and 'IF': $\S 5$: Theorem $\text{T18}$ - 1973: Irving M. Copi:
*Symbolic Logic*(4th ed.) ... (previous) ... (next): $2.4$: Statement Forms