# Principle of Non-Contradiction/Sequent Form/Formulation 2/Proof 1

## Theorem

- $\vdash \neg \paren {p \land \neg p}$

## Proof

By the tableau method of natural deduction:

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

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

2 | 1 | $p$ | Rule of Simplification: $\land \mathcal E_1$ | 1 | ||

3 | 1 | $\neg p$ | Rule of Simplification: $\land \mathcal E_2$ | 1 | ||

4 | 1 | $\bot$ | Principle of Non-Contradiction: $\neg \mathcal E$ | 2, 3 | ||

5 | $\neg \left({p \land \neg p}\right)$ | Proof by Contradiction: $\neg \mathcal I$ | 1 – 4 | Assumption 1 has been discharged |

$\blacksquare$

## Sources

- 1964: Donald Kalish and Richard Montague:
*Logic: Techniques of Formal Reasoning*... (previous) ... (next): $\text{II}$: 'AND', 'OR', 'IF AND ONLY IF': $\S 3$: Theorem $\text{T36}$