English Dictionary

De Morgan's law

IPA（発音記号）

/dɨˈmɔɹɡɪnzˌlɔ/

名詞

広義

The negation of a conjunction is the disjunction of the negations; expressed in propositional logic as: ¬ (𝑝 ∧ 𝑞) ⇔ (¬ 𝑝) ∨ (¬ 𝑞)

日本語の意味

命題論理におけるド・モルガンの法則。すなわち、論理積（AND）の否定は、それぞれの命題の否定を論理和（OR）で結んだものに等しいという法則。

De Morgan's law explains why the negation of a conjunction is equivalent to the disjunction of the negations.

連言の否定が各命題の否定の選言であるという法則は、(A と B) の否定が (A の否定) または (B の否定) と同値である理由を説明する。

意味（1）

The negation of a conjunction is the disjunction of the negations; expressed in propositional logic as: ¬ (𝑝 ∧ 𝑞) ⇔ (¬ 𝑝) ∨ (¬ 𝑞)

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).

意味（2）

The negation of a disjunction is the conjunction of the negations; expressed in propositional logic as: ¬ (𝑝 ∨ 𝑞) ⇔ (¬ 𝑝) ∧ (¬ 𝑞)

意味（3）

The complement of a union is the intersection of the complements; as expressed by: (𝐴 ∪ 𝐵)′ = 𝐴′ ∩ 𝐵′

意味（4）

The complement of an intersection is the union of the complements; as expressed by: (𝐴 ∩ 𝐵)′ = 𝐴′ ∪ 𝐵′

意味（5）

(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)

De Morgan's laws （ plural ）