# De Morgan's law

## English

### Alternative forms

• DM (initialism)

### Etymology

Named after its eponym, the British mathematician and logician Augustus De Morgan (1806–1871), who first formulated the laws in formal propositional logic.

### Pronunciation

• (US) enPR: dēmôrʹgĭnz.lô', IPA(key): /dɨˈmɔɹɡɪnzˌlɔ/

### Noun

De Morgan's law (plural De Morgan's laws)

1. (mathematics, logic) Either of two laws in formal logic which state that:
1. The negation of a conjunction is the disjunction of the negations; expressed in propositional logic as: ¬ (𝑝 ∧ 𝑞) ⇔ (¬ 𝑝) ∨ (¬ 𝑞)
• 2004 August, J. L. Schellenberg, “The Atheist’s Free Will Offence” in the International Journal for Philosophy of Religion, volume 56, № 1, pages 11–12
Let ‘F’ stand for the state of affairs that consists in finite persons possessing and exercising free will. Let ‘p’ stand for ‘God exists’; ‘q’ for ‘F obtains’; ‘r’ for ‘F poses a serious risk of evil’; and ‘s’ for ‘There is no option available to God that counters F.’ With this in place, the argument may be formalized as follows:
(1) [(p & q) & r] → s Premiss
(2) ~s        Premiss
(3) ~[(p & q) & r]   1, 2 MT
(4) ~(p & q) v ~r    3 DM
(5) r         Premiss
(6) ~(p & q)      4, 5 DS
(7) ~p v ~q      6 DM
(3) follows from the conjunction of (1) and (2) by modus tollens; De Morgan’s law applied to (3) yields (4); (4) and (5) together lead to (6) by disjunctive syllogism; and another application of De Morgan’s law takes us from (6) to the final conclusion, according to which either God exists or there is free will (but not both).
2. The negation of a disjunction is the conjunction of the negations; expressed in propositional logic as: ¬ (𝑝 ∨ 𝑞) ⇔ (¬ 𝑝) ∧ (¬ 𝑞)
2. (mathematics) Either of two laws in set theory which state that:
1. The complement of a union is the intersection of the complements; as expressed by: (𝐴 ∪ 𝐵)′ = 𝐴′ ∩ 𝐵′
2. The complement of an intersection is the union of the complements; as expressed by: (𝐴 ∩ 𝐵)′ = 𝐴′ ∪ 𝐵′
3. () Any of various laws similar to De Morgan’s laws for set theory and logic; for example: ¬∀𝑥 𝑃(𝑥) ⇔ ∃𝑥 ¬𝑃(𝑥)