The Strange Algebra Where AND becomes Times

Imagine you are sorting a bag of mixed LEGO bricks. You decide to pick out all the red bricks. Then, from those red bricks, you pick out all the ones that are also 2x4 blocks. You’ve just done something strangely similar to what happens when you multiply numbers. Red times 2x4 equals red 2x4.

Now imagine you reach into a bag and just grab. You pick out all the red bricks. Then you put them back, and you pick out all the 2x4 bricks. You end up with two piles: a red pile and a 2x4 pile. If you were forced to write this as an addition problem, you’d write “red + 2x4.” But here’s a problem: what if some bricks are both red and 2x4? Then you’ve counted them twice. In ordinary math, 1 red brick plus 0 other red bricks equals 1 red brick. But in this strange sorting world, a red brick that is also 2x4 is not two bricks; it’s one brick. Plain addition just doesn’t work.

This was the exact puzzle a 19th-century mathematician named George Boole faced. He wanted to build an algebra of ideas—a way to use numbers and equations to do logical thinking. He wanted to prove that logic could be turned into math.

The Genius Who Learned Everything on His Own

George Boole was born in 1815 in Lincoln, England, into a poor family. His father was a shoemaker who loved science but wasn’t good at making money. Boole had no fancy schooling. Instead, he taught himself French, German, Italian, and advanced mathematics by reading borrowed books. By age 16, he had to work as a teacher to support his family. He opened his own small school at 19 and ran it for 15 years while studying math at night.

Most people think Boole was just a logician. But he was already a famous mathematician before he wrote a single word about logic. In 1844, he won the Royal Society’s gold medal—the first time they gave it to a mathematician—for his work on differential equations, which are equations used to describe things like motion or growth. Only later did he turn his attention to logic.

In 1847, a public fight broke out between two philosophers arguing about who invented a certain idea about how to express sentences with “all” and “some.” This fight pushed Boole to finally write down the system he had been thinking about since childhood. Within a few months, he wrote a short book called The Mathematical Analysis of Logic. His sister later said that when the idea finally clicked, “he was literally like a man dazzled with excess of light.”

Seven years later, he published a bigger, more polished book: The Laws of Thought. This is where his real system took shape.

The Basic Trick: Classes Become Numbers

Boole’s central idea was this: let the number 1 stand for the universe of everything you’re talking about. If you’re sorting LEGO bricks, 1 means all LEGO bricks. If you’re talking about living creatures, 1 means all creatures. If you’re talking about true statements, 1 means the set of all possible cases where things are true.

Now let a letter like x stand for a class (a group) of things. So “x” might mean “all red LEGO bricks.” Then what does “1 - x” mean? It means everything in the universe that is not red. If you have “x” and you think of it as a number, then “x” can only be either 0 or 1. A brick either is red (1) or it’s not red (0).

Boole said that the operation “times” (multiplication, like xy) means and. So “x times y” means “things that are both x and y.” If x means red and y means 2x4 blocks, then xy means red 2x4 blocks.

This is where the math gets interesting. In ordinary numbers, 1 times 1 is 1. And 1 times 0 is 0. That works fine. But here’s the strange rule: x times x equals x. If you take all red bricks and pick out the red bricks again, you just get red bricks. So x² = x. This is called the idempotent law (idem means “same,” potent means “powerful”—same-power). Ordinary numbers don’t do this (2² = 4, not 2). But in Boole’s algebra, it’s the whole point.

Why Addition Broke

Addition was where Boole got stuck. If x means red bricks and y means 2x4 bricks, what does x + y mean? In ordinary math, it’s simple. But in logic, if you add the red bricks to the 2x4 bricks, you’ve counted the red 2x4 bricks twice. So “x + y” only makes sense if x and y have nothing in common. If they overlap, the expression is “uninterpretable”—it doesn’t mean anything.

This made Boole’s algebra weird. He could only add things that were completely separate. If you wanted to say “things that are red or 2x4” (including things that are both), he had to write something clunky like “x + (1-x)y,” which means “red bricks plus non-red bricks that are 2x4.” That’s a sum of two non-overlapping groups, so it’s safe.

A younger logician named William Stanley Jevons later argued that Boole should just make a new rule: x + x = x. If you add red bricks to red bricks, you still just have red bricks. This would let him use addition as “or.” But Boole refused. He saw—correctly, given his system—that if x + x = x, then ordinary algebra would let you subtract x from both sides and conclude x = 0, which would mean every class is empty. That would destroy everything. So Boole stuck with his limited addition.

The Great Theorems

Even with these weird rules, Boole developed powerful tools.

The Expansion Theorem says that any logical expression can be broken down into pieces that show exactly what happens in every possible case. For example, if you have two variables x and y, you can write any expression as a combination of four basic building blocks: xy (both x and y), x(1-y) (x but not y), (1-x)y (y but not x), and (1-x)(1-y) (neither). The coefficients (the numbers multiplying these blocks) tell you what happens in each case. This is very close to what we now call a truth table.

The Elimination Theorem let Boole remove a middle term from two premises. In ordinary logic, you might say “All humans are mortal” and “Socrates is human” to conclude “Socrates is mortal.” The middle term is “human.” Boole’s method could eliminate “human” algebraically and produce the right conclusion.

The Solution Theorem let him solve an equation for one variable in terms of others, sometimes introducing “arbitrary parameters” (v, w, etc.) that represented “some.” If you said “All X is Y,” he’d write “x = vy,” where v is a mysterious variable meaning “some.” You could then eliminate v and get “x = xy,” which is cleaner but loses information.

The Rule of 0 and 1

The deepest and simplest idea in Boole’s system is what modern scholars call the Rule of 0 and 1. Boole said you could test whether a logical argument is valid by only checking what happens when every variable is either 0 or 1. If the equations work out for all possible combinations of 0 and 1, then the argument is valid.

Think about this: instead of checking every possible red brick or every possible creature, you just check four tiny cases (0 or 1 for each variable). If the algebra works in those four cases, it works for all real-world situations. This is because the idempotent law x² = x forces every variable to behave like it can only be “completely present” or “completely absent.” There’s no “half red.”

This rule is the ancestor of the truth tables you might see in computer science classes.

What Boole Missed

Boole’s system was revolutionary, but it had problems. His book was famously confusing. One reviewer later said that anyone reading it “will receive an unpleasant surprise when he discovers how ill constructed his theory actually was and how confused his explanations of it.”

The biggest issue: his algebra only worked perfectly for “universal” propositions—statements like “All X is Y.” For “particular” propositions like “Some X is Y,” he struggled. He used the trick with the parameter v (for “some”), but this introduced arbitrary symbols that made the system messy. He actually waited until the last chapter of his big book to even discuss how to handle “some” statements, and his treatment there was so compressed that almost nobody could follow it.

Boole also missed the chance to invent truth tables, even though he had all the pieces. He never quite saw that his own expansion theorem was already doing what truth tables do.

Why It Matters Today

Boole died in 1864 at age 49, from pneumonia caught after walking through a rainstorm to give a lecture. He was too stubborn to change out of wet clothes.

His algebra was later simplified by others (including Jevons and an American philosopher named Charles Sanders Peirce) into what we now call Boolean algebra. This is the logic that runs computers. Every search engine, every video game, every smartphone uses circuits built on the idea that you can do “and,” “or,” and “not” using 0 and 1. When your computer adds numbers, it’s actually doing a kind of logic called binary arithmetic that Boole’s ideas made possible.

But Boole himself would probably object. He didn’t think he was building a practical tool. He thought he was uncovering the mathematical laws of thought itself. In his second book, he tried to apply his algebra to philosophical arguments about God’s existence, concluding that you couldn’t prove God’s existence with pure logic alone. This part of his work never caught on.

So here’s the strange truth: Boole created a system that was partly wrong, partly confused, and partly incomplete—and it also happened to be one of the most important inventions in the history of human thought. He showed that logic could become algebra, that reasoning could become calculation. And nobody has fully agreed on exactly how his system works, even 170 years later.

Appendices

Key Terms

Term	What it does in Boole’s system
Universe (1)	The set of everything you’re talking about; the starting point.
Class (x, y, z)	A group of things, like “red bricks” or “mortal beings.”
Multiplication (xy)	And: things that belong to both classes.
Addition (x + y)	Or, but only works if the classes have nothing in common.
1 - x	Not: everything in the universe that isn’t in class x.
0	The empty class: nothing.
Idempotent law (x² = x)	The rule that makes this whole system work; says a class combined with itself is just the same class.
Constituents	The basic building blocks of a logical expression; each corresponds to one possible combination of presence/absence.
Expansion Theorem	Breaks any logical expression into its constituent pieces with coefficients.
Elimination	Removing a middle term (like “human”) from two premises to get a direct connection.
Rule of 0 and 1	A method: check if an argument works when every variable is only 0 or 1; if it does, it works for everything.

Key People

George Boole (1815–1864): A self-taught mathematician from a poor family who won a gold medal for his math and later invented a revolutionary algebra for logic.
William Stanley Jevons (1835–1882): A younger logician who argued with Boole about whether x + x should equal x, and who later simplified Boole’s system into something closer to modern Boolean algebra.
Charles Sanders Peirce (1839–1914): An American philosopher and logician who gave the name “Boolean algebra” to the simplified version of Boole’s system, though Boole himself never used that name.

Things to Think About

Boole’s system only makes sense if every variable can only be 0 or 1. But real life has gray areas—things that are partly true or partly present. Could you handle that in an algebra? What would it mean to be 0.3 red?
Boole thought he was discovering the laws of how the human mind actually thinks. Do you think we think in binary? When you consider a possibility, do you say “yes” or “no,” or do you sometimes hold both options open?
The Rule of 0 and 1 says you only need to check four cases to test an argument with two variables. But with 100 variables, you’d have 2¹⁰⁰ cases—more than the number of atoms in the universe. Is this really a shortcut?
Boole got the idea for his algebra from working with differential equations (equations about change). That’s a strange source for a system about static logic. What does it tell you about how creative people find ideas?

Where This Shows Up

Computer circuits: Every “AND,” “OR,” and “NOT” gate in a computer chip does Boole’s algebra with electricity.
Search engines: When you type “cats AND dogs,” the search engine is doing Boole’s multiplication (but with millions of web pages instead of variables).
Programming: Conditional statements (“if this AND that, then do something”) are Boole’s logic.
Philosophy: The question of whether logic is discovered (like math) or invented (like chess pieces) is still debated, and Boole’s work sits at the center of it.