Rules in propositional logic and set theory stating that the negation of a conjunction equals the disjunction of the negations, and the negation of a disjunction equals the conjunction of the negations (¬(A∧B) ⇔ ¬A∨¬B and ¬(A∨B) ⇔ ¬A∧¬B), with corresponding set complement analogues.

Principle

Negation distributes over conjunction and disjunction by exchanging their roles; in Boolean algebra this is an instance of duality under complementation.

Demonstration

Truth-table verification shows ¬(A∧B) and (¬A∨¬B) share the same truth values for all A,B. In sets, (A ∩ B)^c = A^c ∪ B^c and (A ∪ B)^c = A^c ∩ B^c relative to a fixed universal set.

Misapplication

Misplacing the scope of negation in natural-language statements (e.g., 'not both' vs 'neither') or applying classical De Morgan equivalences in logics without classical negation (intuitionistic logic does not validate both equivalences without further double-negation reasoning).

Consequence

Enable mechanical rewriting of logical formulas and set expressions, simplification of conditions, and derivation of dual statements; essential in digital logic design and proof transformations.

Reversal

The inverted statements are the originals: applying negation to the right-hand forms returns the left-hand forms (demonstrates the two-way equivalence); more conceptually, swapping ∧ and ∨ while negating yields the logical dual.

Boundary

Valid in classical propositional and predicate logic and Boolean algebras; care required in nonclassical logics (intuitionistic, paraconsistent) where equivalences may fail or require qualification. In set theory complements require a specified universal set.

Semantic Tension

Tension between formal symbolic manipulation (where De Morgan is an algebraic identity) and natural-language negation (where pragmatic scope and implied quantifiers can change meaning); also tension when moving to constructive logics.

Synthesis

De Morgan's laws are duality rules that convert negated conjunctions into disjunctions of negations and vice versa, providing a syntactic and semantic bridge for transforming logical and set expressions while cautioning about scope and logic framework.