# False Statement implies Every Statement/Formulation 2

## Theorem

$\vdash \neg p \implies \paren {p \implies q}$

## Proof 1

By the tableau method of natural deduction:

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

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

$\blacksquare$