De Morgan's law

plural De Morgan's laws · alternative DM

1. Either of two laws in formal logic which state that:
   1. human sciencesThe negation of a conjunction is the disjunction of the negations; expressed in propositional logic as: ¬ (𝑝 ∧ 𝑞) ⇔ (¬ 𝑝) ∨ (¬ 𝑞)

      1 more example

      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).

      2004 August, J. L. Schellenberg, “The Atheist’s Free Will Offence”, in International Journal for Philosophy of Religion, volume 56, № 1, pages 11-12

   2. human sciencesThe negation of a disjunction is the conjunction of the negations; expressed in propositional logic as: ¬ (𝑝 ∨ 𝑞) ⇔ (¬ 𝑝) ∧ (¬ 𝑞)
2. Either of two laws in set theory which state that:
   1. mathematicsThe complement of a union is the intersection of the complements; as expressed by: (𝐴 ∪ 𝐵)′ = 𝐴′ ∩ 𝐵′
   2. mathematicsThe complement of an intersection is the union of the complements; as expressed by: (𝐴 ∩ 𝐵)′ = 𝐴′ ∪ 𝐵′
3. broadlymathematicsAny of various laws similar to De Morgan’s laws for set theory and logic; for example: ¬∀𝑥 𝑃(𝑥) ⇔ ∃𝑥 ¬𝑃(𝑥)