# Law of Excluded Middle for Two Variables

## 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:

$\vdash ((p \land q) \lor (\lnot p \land q)) \lor ( (p \land \lnot q) \lor (\lnot p \land \lnot q))$
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 axioms of logic 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.