Every processor, from the smallest microcontroller to the most powerful server, is built from logic gates: AND, OR, NOT and their combinations. The fact that logical reasoning can be turned into an electrical circuit is not obvious. It is the result of two steps separated by almost a century: George Boole's algebra in the mid-nineteenth century, and Claude Shannon's thesis in 1937.
George Boole and the algebra of logic
George Boole was a largely self-taught English mathematician who became a professor at Queen's College, Cork, in Ireland. In 1847 he published The Mathematical Analysis of Logic and in 1854 his fuller work, An Investigation of the Laws of Thought.
The underlying idea
Since Aristotle, logic had been studied as a verbal discipline made of syllogisms. Boole proposed treating it as an algebra. Propositions become variables that can take only two values, which we can write as 1 (true) and 0 (false). The words "and", "or", "not" become operations with precise rules.
- AND: x · y equals 1 only if both variables equal 1.
- OR: x + y equals 1 if at least one of them equals 1.
- NOT: the complement of x equals 1 if x equals 0, and vice versa.
A curious property of this algebra is that x · x = x: stating the same thing twice adds nothing. In ordinary algebra this holds only for 0 and 1, and that is precisely what links Boole's logic to a system with just two values.
Truth tables
The clearest way to describe these operations is a table listing every combination:
| x | y | x AND y | x OR y | NOT x |
|---|---|---|---|---|
| 0 | 0 | 0 | 0 | 1 |
| 0 | 1 | 0 | 1 | 1 |
| 1 | 0 | 0 | 1 | 0 |
| 1 | 1 | 1 | 1 | 0 |
In the same years Augustus De Morgan formulated the laws that bear his name: negating an AND is equivalent to taking the OR of the negations, and vice versa. Anyone who writes code uses them every day, often without knowing it, when rewriting a condition like not (a and b) as (not a) or (not b).
For decades Boole's algebra remained a topic in logic and the philosophy of mathematics, with no obvious practical application.
Claude Shannon and relays
In 1937 Claude Shannon, a young student at MIT, completed his master's thesis entitled A Symbolic Analysis of Relay and Switching Circuits, published the following year. Shannon had worked on Vannevar Bush's differential analyzer, an analogue computer controlled by very complicated relay circuits, and he knew Boole's algebra from his mathematics studies.
The insight
Shannon noticed that a switch has two states, open or closed, exactly like a Boolean variable. And that the arrangement of switches corresponds to the operations:
- Two switches in series: current flows only if both are closed. That is an AND.
- Two switches in parallel: current flows if at least one is closed. That is an OR.
- A normally closed contact, which opens when the relay is energised, implements a NOT.
A powerful consequence followed: any switching circuit can be described by a Boolean expression, and that expression can be simplified using the rules of the algebra. A circuit with many relays could be replaced by an equivalent one with fewer components, with mathematical proof that the behaviour stayed identical.
A practical example
Think of a staircase light controlled by two switches, one at the bottom and one at the top. The light must change state every time either switch is flipped. The function is an exclusive OR (XOR): the light is on when the two switches are in different positions.
In Boolean form: L = (A AND NOT B) OR (NOT A AND B).
The ordinary two-way switch wiring is the physical implementation of this expression. Shannon provided the method for moving systematically from a specification like this to the circuit, and from the circuit to its simplest version.
From the thesis to computers
Shannon's thesis turned circuit design from an empirical craft into a mathematical discipline. Within a few years the same ideas were applied to the relays of the first electromechanical calculators, then to vacuum tubes, then to transistors and integrated circuits. The technology changed, but the logic remained Boole's.
Shannon did not stop there. In 1948 he published a paper on the mathematical theory of communication that founded information theory and popularised the term "bit". But the 1937 thesis alone would be enough to secure him a place in the history of computing.
Why it still matters
Every time a programmer writes a condition in an if, they are using Boole's algebra. Every time a compiler simplifies a logical expression, it applies the same laws. Every time a hardware designer synthesises a circuit from a description, they follow the path Shannon laid out. It is one of the rare cases in which an abstract theory, born with no practical purpose, became the foundation of an entire industry.
In short
- George Boole, with his works of 1847 and 1854, turned logic into an algebra with two values and operations such as AND, OR and NOT.
- In 1937 Claude Shannon showed that circuits of switches and relays follow Boole's algebra exactly.
- Switches in series implement an AND, in parallel an OR: any circuit can be described and simplified with Boolean rules.
- This correspondence underlies all digital circuits, from relays to microprocessors.
