Source Summary: Boole 1854
Full title: An Investigation of the Laws of Thought, on which are Founded the Mathematical Theories of Logic and Probabilities
Authors: George Boole
Affiliation: Queen’s College, Cork
Venue: Walton and Maberly, London (1854). Project Gutenberg eBook #15114.
PDF: papers/boole-1854-an-investigation-of-the-laws.pdf
Contributions
- Boolean algebra — a formal algebraic system for logical operations, using symbols and equations to represent logical reasoning.
- Laws of thought as mathematics — proved that classical logical inference (Aristotelian syllogisms) is a special case of algebraic manipulation with 0 and 1.
- Probability theory extension — extended the logical calculus to probability, showing that uncertain inference follows the same algebraic structure.
- Symbolic method — demonstrated that complex logical arguments can be reduced to solving equations, automating the verification of logical validity.
Methodology
The system
Boole assigns symbols (x, y, z, …) to classes (sets of objects sharing a property):
- x = the class of “men”
- y = the class of “mortal things”
- z = the class of “Athenians”
Operations:
- xy = intersection (things that are both x and y)
- x + y = union (exclusive, in Boole’s original — later generalized)
- 1 = the universe of all things
- 0 = the empty class
- 1 − x = the complement of x (things that are not x)
The fundamental law
x² = x
(or equivalently: x(1−x) = 0)
This holds because any class intersected with itself is unchanged — something is either in the class or not; there are no fractional memberships. This law is what distinguishes Boolean algebra from ordinary algebra.
Logical inference as equation solving
A syllogism like “All men are mortal; Socrates is a man; therefore Socrates is mortal” becomes:
y = vx (all x are y, for some v)
z ≤ x (Socrates is in x)
∴ z ≤ y (Socrates is in y)
Boole showed how to translate any valid logical form into an equation, solve it algebraically, and read off the conclusion — mechanically, without intuition.
Probability extension
In the final sections, Boole develops a probability calculus based on the same algebraic foundation. If P(x) = probability of event x, then the laws of combination follow the same rules as Boolean operations on classes. This was an early attempt at a unified theory of uncertain reasoning.
Structure of the book
- Part I (Chapters 1–13): The logic of classes — the core Boolean algebra
- Part II (Chapters 14–21): Application to probability
- Chapter 0: Introduction and statement of the program
- Chapter 13: Relation of the system to Aristotelian logic (shows syllogisms are special cases)
Historical context
Boole was building on Leibniz’s dream of a calculus ratiocinator — a mechanical method for reasoning. He was also responding to contemporary debates about the nature of logic (was it empirical or purely formal?). His answer: logic is a branch of mathematics, specifically algebra restricted to the values 0 and 1.
This was controversial. Logic had been the province of philosophy since Aristotle; Boole claimed it was mathematics.
Limitations
- Boole’s original system used exclusive OR (x + y means x or y but not both), which is awkward; later work by Jevons and others moved to inclusive OR
- The probability section contains technical errors in the treatment of dependent events
- No connection was made to circuit theory (that would come 84 years later via Shannon)
Key References
- Leibniz — calculus ratiocinator concept (predecessor)
- Whately 1826 — Elements of Logic (standard logic text Boole was responding to)
- De Morgan 1847 — Formal Logic (contemporary work, complementary)
- Shannon 1938 — Applied Boole’s algebra to switching circuits