De Morgan's Laws¶

In propositional logic and Boolean algebra, De Morgan's laws, also known as De Morgan's theorem, are a pair of transformation rules that are both valid rules of inference.

Version: v1 · 2026-09-28 · History
Domain-specific #: 8872
Domain group: Formal Sciences
Origin domain: Mathematics
Subdomains: Mathematical Logic, Boolean Algebra → Mathematics

Core Idea¶

De Morgan's Laws is treated here as the recurring mathematics, logic, and statistics identity summarized by this source-grounded definition: In propositional logic and Boolean algebra, De Morgan's laws, also known as De Morgan's theorem, are a pair of transformation rules that are both valid rules of inference.

In propositional logic and Boolean algebra, De Morgan's laws, also known as De Morgan's theorem, are a pair of transformation rules that are both valid rules of inference. They are named after Augustus De Morgan, a 19^th-century British mathematician. The rules allow the expression of conjunctions and disjunctions purely in terms of each other via negation.

The rules can be expressed in English as. The negation of "A and B" is the same as "not A or not B". The negation of "A or B" is the same as "not A and not B".

For De Morgan's Laws, the abstraction is narrower than the article's general subject matter: a positive case must preserve In propositional logic and Boolean algebra, De Morgan's laws, also known as De Morgan's theorem, are a pair of transformation rules that are both valid rules of inference. Retaining only the name, a familiar example, or a downstream effect is insufficient. The specialist roles and tests remain anchored in mathematics, logic, and statistics, which is why this identity is domain-specific rather than prime.

How would you explain it like I'm…

Flip-the-Not Rules

If it's NOT true that 'you have a cat AND a dog,' then you're missing at least one: no cat, or no dog. And if it's NOT true that 'you have a cat OR a dog,' then you have no cat AND no dog. De Morgan's laws are these two rules for flipping 'not' through 'and' and 'or.'

Flipping And and Or With Not

De Morgan's Laws are two rules in logic about the words 'and', 'or', and 'not'. The first says: 'not (A and B)' means the same as 'not A, or not B.' The second says: 'not (A or B)' means the same as 'not A, and not B.' So when you put 'not' in front of a grouped statement, the 'and' flips to 'or', or the 'or' flips to 'and', and each part gets its own 'not.' They're named after Augustus De Morgan, a British mathematician from the 1800s.

Negation Duality Laws

De Morgan's laws are a pair of transformation rules in propositional logic and Boolean algebra, and both are valid rules of inference. They state that the negation of a conjunction equals the disjunction of the negations, ¬(A ∧ B) ≡ ¬A ∨ ¬B, and the negation of a disjunction equals the conjunction of the negations, ¬(A ∨ B) ≡ ¬A ∧ ¬B. In words: "not (A and B)" is "not A or not B," and "not (A or B)" is "not A and not B." This means "and" and "or" can each be expressed in terms of the other using negation. A common mistake they help avoid is thinking "not (A and B)" means "not A and not B." They are named after the 19th-century British mathematician Augustus De Morgan.

De Morgan's laws, also called De Morgan's theorem, are a pair of transformation rules in propositional logic and Boolean algebra, each of which is a valid rule of inference. They are named after Augustus De Morgan, a 19th-century British mathematician. The first law states that the negation of a conjunction is equivalent to the disjunction of the negations: ¬(A ∧ B) ≡ ¬A ∨ ¬B. The second states that the negation of a disjunction is equivalent to the conjunction of the negations: ¬(A ∨ B) ≡ ¬A ∧ ¬B. Together they show that conjunction and disjunction can each be expressed purely in terms of the other plus negation, a duality that is used when simplifying Boolean expressions or logic circuits. Because they are valid rules of inference, either side may be replaced by the other in any derivation. The identity of the concept is this specific pair of equivalences, not negation or Boolean algebra in general.

Structural Signature¶

Sig role-phrases:

Defining carrier — The corpus of documents containing "cats" or "dogs" can be represented by four documents.
Constitutive relation — This transformation demonstrates how negation is distributed across a conjunction by negating each component and switching the logical operator.
Operating condition — De Morgan's formulation was influenced by the algebraization of logic undertaken by George Boole, which later cemented De Morgan's claim to the find.
Recognition evidence — Nevertheless, a similar observation was made by Aristotle, and was known to Greek and Medieval logicians.
Admissible variation — For example, in the 14^th century, William of Ockham wrote down the words that would result by reading the laws out.
Characteristic consequence — The proof that \overline{A\cap B} = \overline{A} \cup \overline{B} is completed in 2 steps by proving both \overline{A\cap B} \subseteq \overline{A} \cup \overline{B} and \overline{A} \cup \overline{B} \subseteq \overline{A\cap B} .
Failure boundary — In its application to the alethic modalities of possibility and necessity, Aristotle observed this case, and in the case of normal modal logic, the relationship of these modal operators to the quantification can be understood by setting up models using Kripke semantics.

What It Is Not¶

Not the whole field of mathematics, logic, and statistics. The node requires the specific identity stated by In propositional logic and Boolean algebra, De Morgan's laws, also known as De Morgan's theorem, are a pair of transformation rules that are both valid rules of inference.
Not an over-broad reading. For a refined version of the failing law concerning existential statements, see the lesser limited principle of omniscience {\mathrm {LLPO}} , which however is different from {\mathrm {WLPO}} .
Not an over-broad reading. Evaluating Search B, the search "(NOT cats)" will hit on documents that do not contain "cats", which is Documents 2 and 4.
Not an over-broad reading. Let's consider the statement: "It is not the case that a number is both even and positive".
Not automatically Lindenbaum–Tarski algebra. Retrieval proximity does not establish equivalence; the two identities must be compared by carrier, operation, and failure boundary.

Scope of Application¶

De Morgan's Laws applies literally inside mathematics, logic, and statistics wherever the source-defined carrier and relation can be established. Its documented habitats include:

In computer engineering. De Morgan's laws are widely used in computer engineering and digital logic for the purpose of simplifying circuit designs.
Generalising De Morgan duality. The existence of negation normal forms drives many applications, for example in digital circuit design, where it is used to manipulate the types of logic gates, and in formal logic, where it is needed to find the conjunctive normal form and disjunctive normal form of a formula.
Worked Example. Here is a more concrete example to illustrate how De Morgan's laws operate in practice.
Negation of a disjunction. In the case of its application to a disjunction, consider the following claim: "it is false that either of A or B is true", which is written as.
Negation of a conjunction. The application of De Morgan's theorem to conjunction is very similar to its application to a disjunction both in form and rationale.
But, using De Morgan's laws,. In its application to the alethic modalities of possibility and necessity, Aristotle observed this case, and in the case of normal modal logic, the relationship of these modal operators to the quantification can be understood by setting up models using Kripke semantics.

Outside mathematics, logic, and statistics, the name should be retained only when these same operational conditions survive; otherwise the comparison belongs to the broader parent Pattern or should be marked as analogy.

Clarity¶

A clear use of De Morgan's Laws names the carrier, the operative relation, and the conditions under which the source treats the identity as present. The minimal definition is In propositional logic and Boolean algebra, De Morgan's laws, also known as De Morgan's theorem, are a pair of transformation rules that are both valid rules of inference. The strongest recognition evidence in the frozen account is: Nevertheless, a similar observation was made by Aristotle, and was known to Greek and Medieval logicians. A report should distinguish that evidence from a proxy, consequence, or common implementation. It should also state the qualification For a refined version of the failing law concerning existential statements, see the lesser limited principle of omniscience {\mathrm {LLPO}} , which however is different from {\mathrm {WLPO}} . so that a reader can reproduce the classification rather than infer it from topical resemblance.

Manages Complexity¶

De Morgan's Laws compresses multiple mathematics, logic, and statistics details into a stable diagnostic relation. The source shows both the central mechanism—this transformation demonstrates how negation is distributed across a conjunction by negating each component and switching the logical operator.—and the practical consequence—the proof that \overline{A\cap B} = \overline{A} \cup \overline{B} is completed in 2 steps by proving both \overline{A\cap B} \subseteq \overline{A} \cup \overline{B} and \overline{A} \cup \overline{B} \subseteq \overline{A\cap B} . This compression makes cases comparable while leaving parameters, conventions, exceptions, and evidential quality explicit. It is lossy by design: local history and implementation details may be omitted only when they do not alter the defining relation.

Abstract Reasoning¶

Type the carrier. Identify the mathematics, logic, and statistics entities to which the claim applies.
State the relation. Use the source-grounded identity: In propositional logic and Boolean algebra, De Morgan's laws, also known as De Morgan's theorem, are a pair of transformation rules that are both valid rules of inference.
Check operation and conditions. De Morgan's formulation was influenced by the algebraization of logic undertaken by George Boole, which later cemented De Morgan's claim to the find.
Demand recognition evidence. Nevertheless, a similar observation was made by Aristotle, and was known to Greek and Medieval logicians.
Test variation. Change an implementation or setting while preserving for example, in the 14^th century, William of Ockham wrote down the words that would result by reading the laws out.
Run the collapse test. Remove the defining operation; if the label still seems equally apt, only a topic or correlate was retained.
Reduce cautiously. When the specialist conditions cannot be carried, route the residual comparison to Pattern.

Knowledge Transfer¶

Within the home domain. Knowledge about De Morgan's Laws transfers literally when a new case preserves the same carrier type, relation, and recognition test. De Morgan's laws are widely used in computer engineering and digital logic for the purpose of simplifying circuit designs. The existence of negation normal forms drives many applications, for example in digital circuit design, where it is used to manipulate the types of logic gates, and in formal logic, where it is needed to find the conjunctive normal form and disjunctive normal form of a formula.

Beyond the home domain. No canonical parent is asserted for De Morgan's Laws. An outside case receives the specialist name only when the same typed roles and rejection conditions can be filled literally; otherwise the comparison remains an analogy pending later graph densification.

Examples¶

Canonical¶

For example, from knowing it not to be the case that both Alice and Bob showed up to their date, it does not follow who did not show up. This case is canonical because it supplies a concrete carrier and lets the defining relation be checked rather than merely named.

Mapped back: carrier → the entities in the documented case; operation → In propositional logic and Boolean algebra, De Morgan's laws, also known as De Morgan's theorem, are a pair of transformation rules that are both valid rules of inference; recognition evidence → Nevertheless, a similar observation was made by Aristotle, and was known to Greek and Medieval logicians

Applied / In Practice¶

Let's consider the statement: "It is not the case that a number is both even and positive". The applied case shows how the identity is used under a second setting or qualification while keeping the same operative relation.

Mapped back: changed setting → Worked Example; invariant → In propositional logic and Boolean algebra, De Morgan's laws, also known as De Morgan's theorem, are a pair of transformation rules that are both valid rules of inference; boundary → the case exits the class when for a refined version of the failing law concerning existential statements, see the lesser limited principle of omniscience {\mathrm {LLPO}} , which however is different from {\mathrm {WLPO}}

Structural Tensions¶

T1 — Stable identity versus admissible variation. For a refined version of the failing law concerning existential statements, see the lesser limited principle of omniscience {\mathrm {LLPO}} , which however is different from {\mathrm {WLPO}} . The tension matters because emphasizing only one side either dissolves the identity or overstates what the evidence and domain conventions warrant.

Diagnostic: Which changes preserve the defining relation, and which replace it?

T2 — Recognition versus proxy. Evaluating Search B, the search "(NOT cats)" will hit on documents that do not contain "cats", which is Documents 2 and 4. The tension matters because emphasizing only one side either dissolves the identity or overstates what the evidence and domain conventions warrant.

Diagnostic: Does the cited evidence establish the identity or only a correlated sign?

T3 — Definition versus implementation. Let's consider the statement: "It is not the case that a number is both even and positive". The tension matters because emphasizing only one side either dissolves the identity or overstates what the evidence and domain conventions warrant.

Diagnostic: Is the observed implementation constitutive, optional, or merely common?

T4 — Scope versus overextension. Using De Morgan's laws, this statement can be rewritten to read: "The number is either not even or it is not positive". The tension matters because emphasizing only one side either dissolves the identity or overstates what the evidence and domain conventions warrant.

Diagnostic: Can every claimed application fill the same typed roles without metaphor?

T5 — Transfer versus domain accent. The corpus of documents containing "cats" or "dogs" can be represented by four documents. The tension matters because emphasizing only one side either dissolves the identity or overstates what the evidence and domain conventions warrant.

Diagnostic: Does the receiving case instantiate De Morgan's Laws literally, co-instantiate Pattern, or only resemble it?

T6 — Autonomy versus reduction. This transformation demonstrates how negation is distributed across a conjunction by negating each component and switching the logical operator. The tension matters because emphasizing only one side either dissolves the identity or overstates what the evidence and domain conventions warrant.

Diagnostic: What does De Morgan's Laws distinguish that the broader parent Pattern leaves together?

Structural–Framed Character¶

De Morgan's Laws is structural-leaning. Its structural side is the repeatable organization summarized by In propositional logic and Boolean algebra, De Morgan's laws, also known as De Morgan's theorem, are a pair of transformation rules that are both valid rules of inference. Its framed side is the mathematics, logic, and statistics vocabulary that fixes the carrier, evidence, exceptions, and admissible transformations.

Evaluative weight: the identity can be stated descriptively even when applications carry practical stakes. Human-practice dependence: the source-grounded carrier determines whether the relation exists independently or is constituted by a practice. Institutional origin: disciplinary conventions stabilize the name and test. Vocabulary portability: De Morgan's formulation was influenced by the algebraization of logic undertaken by George Boole, which later cemented De Morgan's claim to the find. Import versus recognition: literal transfer requires the same mechanism; shape alone is analogy.

Its portable skeleton is Pattern. Its character: a recurring specialist identity whose thin organization can be abstracted, while its operational meaning remains domain-bound.

Structural Core vs. Domain Accent¶

What is skeletal. In propositional logic and Boolean algebra, De Morgan's laws, also known as De Morgan's theorem, are a pair of transformation rules that are both valid rules of inference. The stable skeleton is the typed relation expressed in that definition and the entry's recognition and collapse tests. The source identifies these operative conditions: The corpus of documents containing "cats" or "dogs" can be represented by four documents. This transformation demonstrates how negation is distributed across a conjunction by negating each component and switching the logical operator. It further constrains recognition and variation through: De Morgan's formulation was influenced by the algebraization of logic undertaken by George Boole, which later cemented De Morgan's claim to the find. Nevertheless, a similar observation was made by Aristotle, and was known to Greek and Medieval logicians.

What is domain-bound. mathematics, logic, and statistics supplies the operative entities, technical vocabulary, warrants, and exceptions that make De Morgan's Laws literal. Its documented scope includes the condition that De Morgan's laws are widely used in computer engineering and digital logic for the purpose of simplifying circuit designs. Another bounded application condition is that The existence of negation normal forms drives many applications, for example in digital circuit design, where it is used to manipulate the types of logic gates, and in formal logic, where it is needed to find the conjunctive normal form and disjunctive normal form of a formula. These are not decorative examples; they determine which carrier and evidence can fill the abstraction's roles.

Why no parent is asserted. Removing those specialist details does not currently yield one live catalog node that is a necessary genus for every instance. The entry is therefore approved as unparented rather than attached by topical resemblance. Its collapse evidence remains specific—For example, in the 14^th century, William of Ockham wrote down the words that would result by reading the laws out.—and future graph densification may discover a defensible relation only if it preserves that boundary.

Approved unparented node. No current live node supplies a defensible necessary genus or structural prerequisite for De Morgan's Laws. The reviewed identity is: In propositional logic and Boolean algebra, De Morgan's laws, also known as De Morgan's theorem, are a pair of transformation rules that are both valid rules of inference. The accelerated suggestion was declined because topical or lexical similarity does not establish hierarchy; the node is admitted without a parent pending later graph densification.
Related reasoning operations. Evidence, representation, comparison, classification, transformation, or evaluation may participate in particular cases, but participation does not make any one of them a necessary parent of every instance.

Neighborhood in Abstraction Space¶

De Morgan's Laws sits in a moderately populated region (44^th percentile for distinctiveness): it has near-neighbors but no dense thicket of look-alikes.

Family — Formal Logic & Semantic Systems (18 abstractions)

Nearest neighbors

Computed from structural-signature embeddings · 2026-10-08

Not to Be Confused With¶

Pattern. The parent omits the specialist differentia. Tell: Can the case establish In propositional logic and Boolean algebra, De Morgan's laws, also known as De Morgan's theorem, are a pair of transformation rules that are both valid rules of inference?
Lindenbaum–Tarski algebra. The quotient algebra of formulas or sentences of a logical theory by provable equivalence, with logical connectives inducing well-defined algebraic operations on equivalence classes. Tell: Which entry's carrier, operation, and failure condition are satisfied?
Disjunctive normal form. A Boolean formula represented as a disjunction of conjunctions of literals—an OR of AND terms. Tell: Which entry's carrier, operation, and failure condition are satisfied?
Boolean algebra. An algebraic structure with conjunction, disjunction and complementation satisfying laws that model two-valued logic and set operations. Tell: Which entry's carrier, operation, and failure condition are satisfied?
A measurement, proxy, or consequence. Those may provide evidence without being the identity. Tell: Would De Morgan's Laws remain present if the detector or downstream effect changed?
A metaphorical analogue. A similar shape outside mathematics, logic, and statistics lacks the specialist mechanism. Tell: Do the native roles transfer literally, or only the parent Pattern?

References¶

Frozen Wikipedia discovery revision: https://en.wikipedia.org/wiki/De_Morgan%27s_laws (revision 1367345874).
Preserved source candidate: https://www.taylorfrancis.com/books/mono/10.4324/9781315510897/introduction-logic-irving-copi-carl-cohen-kenneth-mcmahon
Preserved source candidate: http://hyperphysics.phy-astr.gsu.edu/hbase/Electronic/DeMorgan.html
Preserved source candidate: https://books.google.com/books?id=NdAjEDP5mDsC&pg=PA81
Preserved source candidate: https://mathworld.wolfram.com/deMorgansLaws.html
Preserved source candidate: http://www.mtsu.edu/~phys2020/Lectures/L19-L25/L3/DeMorgan/body_demorgan.html
Preserved source candidate: https://web.archive.org/web/20080323122125/http://www.mtsu.edu/~phys2020/Lectures/L19-L25/L3/DeMorgan/body_demorgan.html
Preserved source candidate: https://link.springer.com/10.1007/978-3-031-59797-8_3
Preserved source candidate: https://www.cambridge.org/core/product/identifier/9781107280991/type/book

The frozen Wikipedia revision is discovery provenance. The retained source set was reviewed for identity, formal or operational relation, and scope. The encyclopedia's structural synthesis is bounded to those claims; a thin authority surface is recorded as a nonblocking source-strengthening repair rather than concealed.