AUGUSTUS DEMORGAN

(1806 -1871)

Augustus DeMorgan was born in 1806 in Madras Province, India, where his father was employed by the East India Company. He received his early education in English private schools, which he hated. He had lost the use of one eye in infancy which made him shy and solitary and it exposed him to schoolboy pranks. One such prank was for a boy to sneak up on DeMorgan on his blind side, and holding a sharp-pointed penknife to his cheek, to suddenly call his name. When DeMorgan turned around he would be poked by the point of the knife on his face. DeMorgan did not allow other children to bully him and proved this by chasing down and beating up the boy who did this.

DeMorgan attended Trinity College in Cambridge. He was recognized as far superior in mathematical ability to any other person there, but his refusal to commit to studying resulted in his finishing only fourth in his class. He referred to himself as a "Christian unattached" which was not acceptable by the university for proceeding into the M.A. degree program so he was ineligible forfellowship.

He then decided to try for the Bar, but a short time after entering Lincoln's Inn he learned that he might have a chance to teach mathematics at the newly formed University of London. DeMorgan had strong support from the leading Cambridge mathematicians which helped him be appointed as the first professor of mathematics to the University of London (later called University College) in 1828. He taught here for thirty years.

DeMorgan published first rate elementary texts on arithmetic, algebra, trigonometry and calculus, and important treatises on the theory of probability and formal logic. Many of his papers dealt with the possibility of establishing a logical calculus and the fundamental problem of expressing thought by means of symbols. DeMorgan pointed out that every science that has thrived, has thrived upon its own symbols, and that logic had not developed any symbols. This was what he set out to remedy. He realized the close relationship which existed between logic and pure mathematics, and believed these disciplines should be treated jointly and not separately. His studies in logic were of the highest value both in illuminating new areas and in encouraging others to follow in his footsteps.

DeMorgan's reputation comes mostly from his famous writings, such as, Trigonometry and Double Algebra, Formal Logic and memoirs in Cambridge Philosophical Transactions. He also tutored private pupils, performed consulting services as an actuary and published an enormous amount of articles for biographical dictionaries, historical series, composite works and encyclopedias. His main fields were astronomy, mathematics and biography.

The awakening of logic began with DeMorgan and George Boole. Much of DeMorgan's work was an enormous extension of Boole's work. Although DeMorgan did not truly discover the following laws of logic or set theory, he is given credit for formally stating them as they are shown, which is why they are named after him.

DEMORGAN'S LAWS OF LOGIC:

The following statements are considered logically equivalent, which can be seen from the truth tables shown below:

(a.) not (p or q) = (not p and not q)

symbolically: (p V q)' = (p' & q')

TRUTH TABLES:
p q not(p or q)
T T F
T F F
F T F
F F T
not p not q (not p and not q)
F F F
F T F
T F F
T T T

(b.) not (p and q) = (not p or not q)

symbolically: (p & q)' = (p' V q')

TRUTH TABLES:
p q not(p and q)
T T F
T F T
F T T
F F T
not p not q (not p or not q)
F F F
F T T
T F T
T T T

These statements show that negating an OR makes it an AND and negating an AND makes it an OR.

DEMORGAN'S LAWS FOR SETS:

Shown below are the corresponding statements for sets:

(a.) (A & B)' = A' v B'

(b.) (A v B)'= A' & B'

Cheryl Haefner

[Ed. note] DeMorgan's Laws can be proved easily and today, might almost seem trivial. But taken in the context that he was deriving a manipulative algebra for logic and set theory (referred to currently as the propositional calculus and set calculus respectively), his laws of negation substitution become two of the most frequently applied theorems in modern proof theory.

Robert H. Orr