De Morgan's law
English
Alternative forms
- DM (initialism)
Etymology
Named after the British mathematician and logician Augustus De Morgan (1806–1871), who first formulated the laws in formal propositional logic.
Pronunciation
- Lua error in Module:parameters at line 331: Parameter 1 should be a valid language or etymology language code; the value "US" is not valid. See WT:LOL and WT:LOL/E. enPR: dēmôrʹgĭnz.lô', IPA(key): /dɨˈmɔɹɡɪnzˌlɔ/
Noun
De Morgan's law (plural De Morgan's laws)
- (mathematics, logic) Either of two laws in formal logic which state that:
- 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).
- 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:
- 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
- The negation of a disjunction is the conjunction of the negations; expressed in propositional logic as: ¬ (𝑝 ∨ 𝑞) ⇔ (¬ 𝑝) ∧ (¬ 𝑞)
- The negation of a conjunction is the disjunction of the negations; expressed in propositional logic as: ¬ (𝑝 ∧ 𝑞) ⇔ (¬ 𝑝) ∨ (¬ 𝑞)
- (mathematics) Either of two laws in set theory which state that:
- The complement of a union is the 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′
- (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
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law of set theory
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Further reading
De Morgan’s laws on Wikipedia.Wikipedia