De Morgann.
/də ˈmɔːɡ(ə)n/, U.S. /də ˈmɔrɡ(ə)n/ Logic and Mathematics.
I. Compounds.
1. attributive and in the genitive, esp. in De Morgan's law, De Morgan's theorem. Designating two laws of the propositional calculus, (a) that the negation of a conjunction is logically equivalent to the alternation of the negations of the conjoined expressions, and (b) that the negation of an alternation is logically equivalent to the conjunction of the negations of the alternated expressions; (also) designating the analogous truths in the algebra of classes.The laws may be stated symbolically as ∼(p.q) ≡ ∼p ∨ ∼q, and ∼(p ∨ q) ≡ ∼p.∼q, which may be rendered as: (a) if p and q are not jointly true, then p is false or q is false; (b) if neither p nor q is true, then p is false and q is false.
ΚΠ
1912 Mind 21 530 De Morgan's Theorem holds only for extensional disjunction.
1932 C. I. Lewis & C. H. Langford Symb. Logic ii. 33 These always follow from their correlates by some use of De Morgan's Theorem.
1950 W. V. Quine Methods of Logic (1952) §10. 53 De Morgan's laws are useful in enabling us to avoid negating conjunctions and alternations.
2002 J.-Y. Béziau in W. A. Carnielli et al. Paraconsistency xxiv. 480 From the fact that we obviously have ⊢ a → a, we get ⊢ ¬a ∨ a, by application of the De Morgan law for implication.
II. Simple uses.
2. = De Morgan's theorem at sense 1.
ΚΠ
1965 P. Caws Philos. of Sci. xlii. 325 Which by De Morgan is seen to be equivalent.
2009 T. Gover Pract. Study of Argument (ed. 7) 404 Answers to Selected Exercises... 9. ~(A ⋅ B) by De Morgan.
This entry has been updated (OED Third Edition, March 2014; most recently modified version published online March 2022).
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n.1912