De Morgan's Laws (Logic)/Disjunction of Negations/Formulation 1/Reverse Implication

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Theorem

$\neg \left({p \land q}\right) \vdash \neg p \lor \neg q$


Proof

By the tableau method of natural deduction:

$\neg \left({p \land q}\right) \vdash \neg p \lor \neg q$
Line Pool Formula Rule Depends upon Notes
1 1 $\neg \left({p \land q}\right)$ Premise (None)
2 2 $\neg \left ({\neg p \lor \neg q}\right)$ Assumption (None)
3 3 $\neg p$ Assumption (None)
4 3 $\neg p \lor \neg q$ Rule of Addition: $\lor \mathcal I_1$ 3
5 2, 3 $\bot$ Principle of Non-Contradiction: $\neg \mathcal E$ 4, 2
6 2 $p$ Reductio ad Absurdum 3 – 5 Assumption 3 has been discharged
7 7 $\neg q$ Assumption (None)
8 7 $\neg p \lor \neg q$ Rule of Addition: $\lor \mathcal I_2$ 7
9 2, 7 $\bot$ Principle of Non-Contradiction: $\neg \mathcal E$ 8, 2
10 2 $q$ Reductio ad Absurdum 7 – 9 Assumption 7 has been discharged
11 2 $p \land q$ Rule of Conjunction: $\land \mathcal I$ 6, 10
12 1, 2 $\bot$ Principle of Non-Contradiction: $\neg \mathcal E$ 11, 1
13 1 $\neg p \lor \neg q$ Reductio ad Absurdum 2 – 12 Assumption 2 has been discharged

$\blacksquare$

Law of the Excluded Middle

This theorem depends on the Law of the Excluded Middle, by way of Reductio ad Absurdum.

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.


Sources