De Morgan's law

English Alternative forms en DM initialism Etymology Named after the British mathematician and logician Augustus De Morgan (1806–1871), who first formulated the laws in formal en propositional logic. Pronunciation dēmôrʹgĭnz.lô' US, en /dɨˈmɔɹɡɪnzˌlɔ/ 1 Noun De Morgan's law en mathematics logic Either of two laws in formal en logic which state that: The en negation of a en conjunction is the en disjunction of the negations; expressed in propositional logic as: ¬ (𝑝 ∧ 𝑞) ⇔ (¬ 𝑝) ∨ (¬ 𝑞) en 2004 August J. L. Schellenberg The Atheist’s Free Will Offence 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). The negation of a disjunction is the conjunction of the negations; expressed in propositional logic as: ¬ (𝑝 ∨ 𝑞) ⇔ (¬ 𝑝) ∧ (¬ 𝑞) en mathematics Either of two laws in en set theory which state that: The en complement of a en union is the en intersection of the complements; as expressed by: (𝐴 ∪ 𝐵)′ = 𝐴′ ∩ 𝐵′ The complement of an intersection is the union of the complements; as expressed by: (𝐴 ∩ 𝐵)′ = 𝐴′ ∪ 𝐵′ en mathematics loosely Any of various laws similar to De Morgan’s laws for set theory and logic; for example: ¬∀𝑥 𝑃(𝑥) ⇔ ∃𝑥 ¬𝑃(𝑥) Translations law of formal logic Czech: cs De Morganův zákon m Dutch: nl wet van De Morgan f Finnish: fi De Morganin laki German: de De Morgen'sches Gesetz n Hebrew: he Polish: prawo De Morgana n law of set theory Czech: cs De Morganův zákon m Dutch: nl wet van De Morgan f Finnish: fi De Morganin laki German: de De Morgen'sches Gesetz n Hebrew: he Polish: prawo De Morgana n Further reading De Morgan’s laws en eponyms