Law of Excluded Middle for Two Variables
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Theorem
- $\vdash (p \land q) \lor (\lnot p \land q) \lor (p \land \lnot q) \lor (\lnot p \land \lnot q)$
Proof
By the tableau method of natural deduction:
| Line | Pool | Formula | Rule | Depends upon | Notes | |
|---|---|---|---|---|---|---|
| 1 | $p \lor \lnot p$ | Law of Excluded Middle | (None) | |||
| 2 | $q \lor \lnot q$ | Law of Excluded Middle | (None) | |||
| 3 | $(p \lor \lnot p) \land (q \lor \lnot q)$ | Rule of Conjunction: $\land \II$ | 1, 2 | |||
| 4 | $((p \lor \lnot p) \land q) \lor ((p \lor \lnot p) \land \lnot q)$ | Sequent Introduction | 3 | Conjunction Distributes over Disjunction | ||
| 5 | 5 | $(p \lor \lnot p) \land q$ | Assumption | (None) | ||
| 6 | 5 | $(p \land q) \lor (\lnot p \land q)$ | Sequent Introduction | 5 | Conjunction Distributes over Disjunction | |
| 7 | $(p \lor \lnot p) \land q \implies (p \land q) \lor (\lnot p \land q)$ | Rule of Implication: $\implies \II$ | 5 – 6 | Assumption 5 has been discharged | ||
| 8 | 8 | $(p \lor \lnot p) \land \lnot q$ | Assumption | (None) | ||
| 9 | 8 | $(p \land \lnot q) \lor (\lnot p \land \lnot q)$ | Sequent Introduction | 8 | Conjunction Distributes over Disjunction | |
| 10 | $(p \lor \lnot p) \land \lnot q \implies (p \land \lnot q) \lor (\lnot p \land \lnot q)$ | Rule of Implication: $\implies \II$ | 8 – 9 | Assumption 8 has been discharged | ||
| 11 | $((p \lor \lnot p) \land q) \lor ((p \lor \lnot p) \land \lnot q) \implies ((p \land q) \lor (\lnot p \land q)) \lor ( (p \land \lnot q) \lor (\lnot p \land \lnot q))$ | Sequent Introduction | 7,10 | Constructive Dilemma | ||
| 12 | $((p \land q) \lor (\lnot p \land q)) \lor ( (p \land \lnot q) \lor (\lnot p \land \lnot q))$ | Modus Ponendo Ponens: $\implies \mathcal E$ | 11, 4 |
$\blacksquare$
Law of the Excluded Middle
This theorem depends on the Law of the Excluded Middle.
This is one of the logical axioms that was determined by Aristotle, and forms part of the backbone of classical (Aristotelian) logic.
However, the intuitionist school rejects the Law of the Excluded Middle as a valid logical axiom.
This in turn invalidates this theorem from an intuitionistic perspective.