# Non-Equivalence as Disjunction of Conjunctions/Formulation 2

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## Theorem

- $\vdash \paren {\neg \paren {p \iff q} } \iff \paren {\paren {\neg p \land q} \lor \paren {p \land \neg q} }$

## Proof

By the tableau method of natural deduction:

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

1 | 1 | $\neg \paren {p \iff q}$ | Assumption | (None) | ||

2 | 1 | $\paren {\neg p \land q} \lor \paren {p \land \neg q}$ | Sequent Introduction | 1 | Non-Equivalence as Disjunction of Conjunctions: Formulation 1 | |

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

4 | 4 | $\paren {\neg p \land q} \lor \paren {p \land \neg q}$ | Assumption | (None) | ||

5 | 4 | $\neg \paren {p \iff q}$ | Sequent Introduction | 4 | Non-Equivalence as Disjunction of Conjunctions: Formulation 1 | |

6 | $\paren {\paren {\neg p \land q} \lor \paren {p \land \neg q} } \implies \paren {\neg \paren {p \iff q} }$ | Rule of Implication: $\implies \mathcal I$ | 4 – 5 | Assumption 4 has been discharged | ||

7 | $\paren {\neg \paren {p \iff q} } \iff \paren {\paren {\neg p \land q} \lor \paren {p \land \neg q} }$ | Biconditional Introduction: $\iff \mathcal I$ | 3, 6 |

$\blacksquare$

## Sources

- 1964: Donald Kalish and Richard Montague:
*Logic: Techniques of Formal Reasoning*... (previous) ... (next): $\text{II}$: 'AND', 'OR', 'IF AND ONLY IF': $\S 5$: Theorem $\text{T87}$