1806 – 1871 | Laws of Logic and Complements
The logician who formalised the laws of negation, named mathematical induction, co-founded the London Mathematical Society, and helped make logic a branch of mathematics.
Augustus De Morgan was born on 27 June 1806 in Madurai, southern India, where his father was a colonel in the East India Company. He was blind in one eye from birth, a condition that excluded him from many activities but sharpened his intellectual focus.
The family returned to England when Augustus was seven months old. He was educated at several private schools before entering Trinity College, Cambridge in 1823 at age 16. He studied under William Whewell and George Peacock.
De Morgan graduated as fourth wrangler in 1827 but refused to take the religious test required for an MA at Cambridge, a principled stand he maintained throughout his life. He was appointed the first Professor of Mathematics at the newly founded University College London in 1828, at age 21.
27 June 1806, Madurai, India
Trinity College, Cambridge. Fourth wrangler, 1827.
Refused religious tests at Cambridge; championed secular education at UCL
Known for wit, generosity, and a vast knowledge of mathematical history. Prolific correspondent (over 10,000 letters survive).
De Morgan spent most of his career at University College London, serving as Professor of Mathematics from 1828 to 1831 and again from 1836 to 1866. He resigned twice on principle: first over the dismissal of a colleague, and finally over the refusal to appoint a Unitarian to a chair.
He was an extraordinary teacher who influenced a generation of British mathematicians, including James Joseph Sylvester and William Stanley Jevons. His textbooks on algebra, differential calculus, and logic were widely used.
In 1866, he co-founded the London Mathematical Society and served as its first president. This institution became one of the most important mathematical societies in the world.
Appointed first Professor of Mathematics at University College London, age 21
Published "Essay on Probabilities" and began work on formal logic
Published "Formal Logic" on the same day Boole published "Mathematical Analysis of Logic"
Co-founded the London Mathematical Society; served as first president
De Morgan worked during a period when logic was being transformed from a branch of philosophy into a branch of mathematics.
For over two thousand years, logic meant Aristotle's syllogistic logic. De Morgan and his contemporaries sought to extend logic beyond the categorical syllogism, making it algebraic and quantitative.
The 1820s-1860s saw the reform of British universities: new institutions like UCL (1826), the end of religious tests, and the modernisation of the Cambridge Mathematical Tripos. De Morgan was at the centre of these reforms.
British algebraists (Peacock, Gregory, Boole) were freeing algebra from arithmetic, treating it as the study of formal operations. De Morgan applied this approach to logic, treating logical operations as algebraic ones.
"The two great branches of exact thinking are mathematics and logic, and De Morgan did more than any man of his time to show the connection between them."
— Alexander Macfarlane, Ten British Mathematicians of the Nineteenth CenturyDe Morgan's laws state that negation distributes over conjunction and disjunction by swapping them:
NOT (A AND B) = (NOT A) OR (NOT B)
NOT (A OR B) = (NOT A) AND (NOT B)
In set-theoretic notation:
(A ∪ B)' = A' ∩ B'
(A ∩ B)' = A' ∪ B'
These dualities are fundamental to Boolean algebra, digital circuit design, set theory, and programming. Every if statement you simplify uses De Morgan.
De Morgan's laws are axioms of Boolean algebra. In any Boolean algebra (sets, propositions, digital logic), the complement operation exchanges join and meet:
(a ∨ b)' = a' ∧ b'(a ∧ b)' = a' ∨ b'
This duality extends to arbitrary operations: every theorem in Boolean algebra has a dual obtained by swapping ∧ and ∨, and swapping 0 and 1.
De Morgan's laws are essential for circuit simplification. A NAND gate implements NOT(A AND B) = (NOT A) OR (NOT B). Since NAND gates alone can implement any Boolean function, De Morgan's duality shows that all logic can be built from a single gate type.
Every programmer uses De Morgan's laws when simplifying conditionals:
!(isActive && isValid)
!isActive || !isValid
De Morgan's laws extend to:
De Morgan's 1847 book Formal Logic extended Aristotelian syllogistic in several ways:
De Morgan's relation algebra was ahead of its time. He showed that many inferences depend on the properties of relations (e.g., "if A is a parent of B and B is a parent of C, then A is a grandparent of C") and cannot be captured by traditional syllogistic.
This work influenced Charles Sanders Peirce, who developed it into a full algebra of relations, and ultimately fed into modern predicate logic and relational databases.
De Morgan is credited with coining the term "mathematical induction" in 1838, distinguishing the mathematical method of proof from the philosophical concept of inductive reasoning.
The method itself was known earlier (used by Pascal, Fermat, and others), but De Morgan gave it its name, formalised it explicitly, and advocated for its central role in mathematical proof.
He articulated the principle clearly: if a property holds for n = 1, and if its truth for n = k implies its truth for n = k + 1, then it holds for all natural numbers.
Lattices with an involution satisfying De Morgan's laws but not necessarily complementation. These arise in many-valued logic and fuzzy logic. A De Morgan algebra is weaker than a Boolean algebra.
Published posthumously in 1872, this witty collection examines mathematical cranks, circle-squarers, and pseudoscience. It remains a delightful read and a model of mathematical debunking.
De Morgan was a meticulous historian of mathematics. His biographical essays and historical notes in the Penny Cyclopaedia (over 850 articles) remain valuable references.
In 1865, De Morgan and two of his students, Arthur Cowper Ranyard and George Campbell De Morgan (his own son), founded the London Mathematical Society (LMS).
De Morgan served as its first president from 1866 to 1867. In his inaugural address, he articulated a vision of a society open to all who pursued mathematics seriously, regardless of institutional affiliation or social standing.
The LMS became a model for national mathematical societies worldwide. Today it is one of the oldest and most respected mathematical societies, publishing several leading journals and awarding prestigious prizes including the De Morgan Medal.
The founding of the LMS reflected De Morgan's lifelong commitment to making mathematics accessible and professional, moving it beyond the gentleman-amateur tradition of British mathematics.
The highest honour awarded by the LMS, given every three years for outstanding contributions to mathematics. Recipients include Bertrand Russell, G. H. Hardy, Michael Atiyah, and Andrew Wiles.
The society today has over 3,000 members and publishes the Bulletin, Journal, and Proceedings of the London Mathematical Society, among other publications.
De Morgan's vision of an inclusive, professional mathematical community helped transform British mathematics from a Cambridge-centred, amateur pursuit into a modern professional discipline.
De Morgan approached logic as an algebraist: abstract the formal structure, study its laws, and let the symbols do the thinking.
Study logical reasoning as practised
Introduce notation for operations
Find algebraic identities
Extend to new domains
De Morgan was above all a teacher. He wrote textbooks designed to build understanding, not merely to display technique. His "Elements of Algebra" (1835) was revolutionary in its explanatory approach, anticipating modern pedagogical methods by a century.
De Morgan believed that understanding the history of a mathematical idea was essential to understanding the idea itself. He brought historical perspective to all his work, tracing concepts from their origins to their modern forms.
De Morgan maintained extensive correspondences, notably with Boole (they developed Boolean logic in parallel) and Hamilton (over 1,000 letters survive). He tutored Ada Lovelace in mathematics.
Not to be confused with the Irish mathematician William Rowan Hamilton (De Morgan's friend), Sir William Hamilton was a Scottish philosopher who claimed priority over De Morgan in extending Aristotelian logic.
Hamilton (the philosopher) accused De Morgan of plagiarism in 1846, claiming that De Morgan had stolen his idea of "quantification of the predicate" — extending syllogistic logic by quantifying both subject and predicate.
The dispute was bitter and public, playing out in the pages of journals and the Athenaeum. De Morgan's supporters (including Boole) sided with him, while Hamilton's philosophical allies backed the philosopher.
Modern historians generally agree that both men arrived at their ideas independently, and that De Morgan's version was mathematically superior. Hamilton's approach was philosophically motivated but technically flawed.
De Morgan resigned from UCL twice on principle: first in 1831 when the university dismissed a colleague without just cause, and finally in 1866 when it refused to appoint a Unitarian professor. His integrity was absolute.
De Morgan refused to sit for a Cambridge MA because it required subscribing to the Thirty-Nine Articles of the Church of England. He was not anti-religious, but fiercely opposed to religious coercion in education.
De Morgan's laws are axioms of Boolean algebra. Every digital computer operates on Boolean logic, making De Morgan's identities among the most frequently applied mathematical results in human history.
De Morgan's work on relations paved the way for Peirce's relation algebra and, through it, for modern predicate logic. His notion of the universe of discourse is standard in logic and model theory.
De Morgan algebras (lattices with a De Morgan involution) are studied in their own right. They arise in many-valued logic, fuzzy sets, and rough set theory.
De Morgan's laws are fundamental to compiler optimisation, circuit minimisation (NAND/NOR logic), and program verification. Every software engineer learns them.
De Morgan's relation algebra influenced Codd's relational model. SQL query optimisation uses De Morgan-like transformations to simplify WHERE clauses with NOT, AND, and OR.
The London Mathematical Society, which De Morgan founded, remains a pillar of the international mathematical community. The model of a national professional mathematical society spread worldwide.
De Morgan's laws are used in every step of logic circuit design. NAND and NOR gates are "universal" gates precisely because De Morgan's laws allow AND/OR conversion. VLSI chip design relies on these transformations millions of times per chip.
Compiler optimisers apply De Morgan's laws to simplify Boolean expressions in code. Program verification tools (model checkers, theorem provers) use them to normalise logical formulas. Every conditional expression benefits.
Hardware and software verification requires manipulating complex logical formulas. De Morgan's laws are essential for converting between conjunctive and disjunctive normal forms, which are the standard representations used by SAT solvers.
Search engines use Boolean logic to combine queries. De Morgan's laws help optimise NOT queries: "NOT (A OR B)" becomes "NOT A AND NOT B", enabling efficient index intersection rather than expensive complement operations.
Sophia Elizabeth De Morgan (1882). Written by his wife, this affectionate biography provides intimate details of De Morgan's character, teaching, and intellectual life.
Augustus De Morgan (1872, posthumous). A witty collection of essays on mathematical cranks and paradoxes. Entertaining and still relevant as a guide to recognising pseudomathematics.
George Boole (1847). Published the same day as De Morgan's Formal Logic. Reading both together reveals how two independent minds converged on algebraic logic.
Adrian Rice (2011). A detailed study of the British contribution to mathematical logic, with extensive treatment of De Morgan's role in the development of formal logic and relation algebra.
George Boole (1854). The fuller development of Boolean algebra, building on the foundations that Boole and De Morgan laid in 1847. Essential context for understanding De Morgan's contribution.
Augustus De Morgan (1847). The original work. Dense but rewarding. The section on relations anticipates modern relational logic by decades.
"The moving power of mathematical invention is not reasoning but imagination."
— Augustus De MorganAugustus De Morgan · 1806 – 1871