George Boole

George Boole

Algebraic logic · 1815–1864 · England and Ireland

Boole helped establish an algebraic approach to logical reasoning. Modern Boolean algebra is central to digital logic, but its present notation should not be projected unchanged onto his books. The distinction is technically important: his use of addition for classes was not simply today's inclusive OR.

The historical work

Boole worked as a schoolmaster before becoming professor of mathematics at Queen's College, Cork, in 1849. Much of his advanced mathematical education was self-directed, with access to books and intellectual support; "entirely self-taught" erases those connections. His major logical works were The Mathematical Analysis of Logic in 1847 and An Investigation of the Laws of Thought in 1854. He also worked on differential equations and finite differences. Stanford Encyclopedia of Philosophy, George Boole

In the Laws of Thought, symbols can represent classes, multiplication represents their intersection, and 1-x represents complement relative to a specified universe. Addition is interpreted as aggregation of disjoint classes. The later inclusive-union convention changes that operation. The primary text and historical commentary make the distinction explicit. Boole, Laws of Thought, full text, Burris, Contributions of the Logicians, chapter on Boole

Later developments transformed this tradition. Shannon's 1937 MIT thesis applied a symbolic calculus to relay and switching circuits; its journal version appeared in 1938. This is a documented technical application, not proof that every electronic behavior is perfectly described by two-valued logic. MIT thesis record

Worked mechanism: classes and characteristic functions

Use the original teaching universe U={a,b,c,d}. Let X={a,b} and Y={b,c}. Represent membership by characteristic functions x and y taking values 0 or 1 for each element.

Element	x	y	xy	x+y	x+y−xy
a	1	0	0	1	1
b	1	1	1	2	1
c	0	1	0	1	1
d	0	0	0	0	0

The product xy selects {b}. Ordinary numerical addition x+y gives 2 at b, so it is not a characteristic function for a class. Inclusive union is represented pointwise by x+y−xy, or equivalently x+(1-x)y. The latter combines X with the part of Y outside X, making the two combined classes disjoint.

This explains why notation must be declared. In a modern Boolean-algebra text, x+y may be defined to mean OR, with 1+1=1. In ordinary arithmetic, 1+1=2. In a Boolean ring, addition may instead mean exclusive OR, with 1+1=0. These are different operations sharing a printed symbol, not contradictory arithmetic.

Idempotence does not force two elements

Every set satisfies X∩X=X. The collection of all subsets of {a,b} forms a Boolean algebra with four elements:

empty set, {a}, {b}, {a,b}

Intersection and union are its two binary operations, and complement is relative to {a,b}. All four elements satisfy intersection idempotence. Thus idempotence does not force an abstract Boolean algebra to contain exactly two elements.

The equation t²=t has only the real-number solutions 0 and 1 because real arithmetic permits factoring t(t−1)=0 and using the absence of zero divisors. That argument concerns real numbers with ordinary multiplication. It cannot be imported unchanged into an algebra of sets. A bit is a value in the two-element Boolean algebra; a Boolean algebra need not consist of a single bit.

A concrete logical deduction

Let P, Q, and R be classes in a common universe. "Every P is Q" means P∩Q=P, or pointwise p(1-q)=0. "No Q is R" means qr=0. It follows that pr=0, since p=pq gives pr=pqr=p(qr)=0.

The conclusion is "No P is R." It does not imply that any P exists. The empty class satisfies both premises. To conclude that some P is outside R, an additional existence assumption is required. Separating universal inclusion from existence prevents a common invalid syllogism.

From equations to idealized switches

In a modern ideal model, a series path conducts only if both switches conduct; parallel paths conduct if at least one does. The logical condition (a AND b) OR (a AND NOT b) simplifies to a because b OR NOT b is true. A design using the original expression and a design controlled simply by a have the same steady-state truth table.

That equivalence alone says nothing about propagation delay, transient hazards, power, or a switch stuck in an intermediate physical state. Those require a richer engineering model. Formal reasoning is strongest when its assumptions and the boundary between models are explicit.

Exercises

1. For U={1,2,3,4,5}, X={1,3,5}, and Y={3,4}, calculate intersection, union, complement of X, and symmetric difference. Give pointwise numerical formulas for union and symmetric difference using membership values x and y.
2. Construct the Boolean algebra of all subsets of {a,b}. Verify complement and idempotence for every element. Explain precisely why this refutes the claim that x²=x implies every Boolean algebra has only two elements.
3. Premises say every P is Q and no Q is R. Prove no P is R algebraically, then give a countermodel to "some P is not R." Add one premise that makes the latter conclusion valid.

Compare your reasoning with the worked solutions

1. Intersection is {3}, union is {1,3,4,5}, complement is {2,4}, and symmetric difference is {1,4,5}. Union's characteristic value is x+y−xy. Symmetric difference uses x+y−2xy, which gives zero when both membership values are one. Equivalently it is x(1-y)+(1-x)y, a sum of disjoint cases. These expressions use ordinary arithmetic on characteristic values, not an undeclared Boolean addition symbol.
2. The complements are respectively {a,b}, {b}, {a}, and the empty set for the elements empty set, {a}, {b}, and {a,b}. Intersecting any listed element with itself returns that element. There are four distinct elements despite idempotence. The invalid argument assumes the ordinary real-number zero-product rule applies to set intersection as if its elements were real numbers. The operations and their domains must be specified before algebraic inference.
3. From p=pq and qr=0, multiply the first equation by r to obtain pr=pqr=p(qr)=0. For the countermodel, choose a nonempty universe with P empty, Q empty, and R equal to the universe. Both premises hold, but no P exists. Add "some P exists"; any witness in P lies in Q and therefore outside R, establishing that some P is not R. The universal conclusion alone cannot supply a witness.