On this page
Every logic gate in your physics chapter and every if-statement in a Python program runs on an algebra with only two values. It was written down by a shoemaker's son from Lincoln who never took a degree. This post covers who George Boole was, what his 1854 book set out to do, a worked example in his own notation, and the mistakes students make when they mix his algebra with today's.
Who was George Boole?
His MacTutor biography gives his birth in Lincoln on 2 November 1815 and his death at Ballintemple, County Cork, on 8 December 1864. His father made shoes and loved science. When the family business failed, George became an assistant teacher at 16 to support his parents and siblings, and he opened his own school at 19.
He taught himself higher mathematics, and later felt he had nearly wasted five years trying to learn calculus without a skilled teacher. In 1844 the Royal Society gave him its Royal Medal for a paper on a general method in analysis. In 1849 he became the first professor of mathematics at Queen's College, Cork, and in 1857 he was elected a Fellow of the Royal Society.
What was the Laws of Thought trying to do?
The book's full title is An Investigation of the Laws of Thought, on Which Are Founded the Mathematical Theories of Logic and Probabilities. Its first sentence sets the goal: "The design of the following treatise is to investigate the fundamental laws of those operations of the mind by which reasoning is performed; to give expression to them in the symbolical language of a Calculus, and upon this foundation to establish the science of Logic and construct its method".
Letters stand for classes of things. If x means "men" and y means "good things", xy means things that are both. 1 − x means everything that is not x.
Why does x² = x matter?
Boole argued that if x and y have exactly the same meaning, then xy = x, so xx = x, which he wrote as x² = x. He then pointed out that among ordinary numbers only 0 and 1 obey this law, since 0² = 0 and 1² = 1. So he compared logic with an algebra whose symbols take only the values 0 and 1.
From x² = x he derived x − x² = 0, so x(1 − x) = 0. In words: the class of things that are both x and not x is empty. Boole showed that this is the principle of contradiction, which he quotes Aristotle calling the most certain of all principles.
Our post on Claude Shannon tells that part of the story.
Worked example: "or" in Boole's notation
Boole gives two ways to write "either x's or y's". When the classes cannot overlap, it is x(1 − y) + y(1 − x). When they can, it is x + y(1 − x). Put in every pair of values 0 and 1:
x = 0, y = 0: x + y(1 − x) = 0, and x(1 − y) + y(1 − x) = 0.
x = 1, y = 0: 1 + 0 = 1, and 1 + 0 = 1.
x = 0, y = 1: 0 + 1 × 1 = 1, and 0 + 1 = 1.
x = 1, y = 1: 1 + 1 × 0 = 1, and 0 + 0 = 0.
So x + y(1 − x) behaves like today's OR gate and x(1 − y) + y(1 − x) like XOR. Multiplying out, x + y(1 − x) = x + y − xy. With NOT as 1 − x and AND as xy, you can check one of De Morgan's laws by algebra: 1 − (x + y − xy) = 1 − x − y + xy = (1 − x)(1 − y). Not (x or y) equals (not x) and (not y).
What mistakes do students make?
Treating Boole's + as today's +. In modern Boolean algebra 1 + 1 = 1. Boole used + to join classes with no common members, which is why his inclusive "or" needs the y(1 − x) term.
Solving x² = x. It is a law every symbol obeys, not an equation to solve.
Mixing up OR and XOR. They differ only in the row x = 1, y = 1. Check that row first in any truth-table question.
Dropping the brackets in NOT. 1 − (x + y − xy) changes every sign inside.
Where does this appear in your syllabus?
The JEE (Main) 2026 syllabus lists "Logic gates (OR. AND. NOT. NAND and NOR)" in the physics unit on electronic devices; our post on semiconductors and logic gates covers that chapter. CBSE Class 11 computer science students meet the same algebra in our guide to Boolean logic. Check the syllabus for your exam year.
If truth tables still feel like guesswork, the Fermi Concept Library covers every concept in physics, chemistry, biology, mathematics and computing, so you can review AND, OR and NOT before practising gate problems.
What can a student take from Boole's way of working?
Get guidance early. Boole regretted the years he spent learning calculus alone.
Let notation carry ideas. Writing xx as x² led him straight to the 0 and 1 algebra.
Check an analogy before trusting it. He stressed that xy = yx holds in logic for reasons quite distinct from those in arithmetic, even though the two laws look the same.
Next step: build the truth table for x(1 − y) and describe in one sentence which things it picks out.
