# True Statement is implied by Every Statement/Formulation 2/Proof 1

Jump to navigation
Jump to search

## Theorem

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

## Proof

By the tableau method of natural deduction:

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

1 | 1 | $q$ | Assumption | (None) | ||

2 | 1 | $p \implies q$ | Sequent Introduction | 1 | True Statement is implied by Every Statement: Formulation 1 | |

3 | $q \implies \paren {p \implies q}$ | Rule of Implication: $\implies \mathcal I$ | 1 – 2 | Assumption 1 has been discharged |

$\blacksquare$

## Sources

- 1964: Donald Kalish and Richard Montague:
*Logic: Techniques of Formal Reasoning*... (previous) ... (next): $\text{I}$: 'NOT' and 'IF': $\S 5$: Theorem $\text{T2}$