Begriffsschrift, a Formula Language, Modeled upon That of Arithmetic, for Pure Thought
The notation that gave logic the quantifier — so 'all' and 'some' could at last be calculated.
Before 1879, logic could not properly say 'every' and 'some' in the same breath. One small book fixed that — and quietly founded modern logic.
Giving logic a grammar for 'all' and 'some'
For two thousand years, formal logic was Aristotle's logic of syllogisms — fine for 'all men are mortal,' but helpless before a sentence like 'every number has a larger number,' which nests one 'every' inside one 'some.' Frege rebuilt logic from the ground up. He stopped reading a sentence as a subject joined to a predicate and started reading it as a function fed with arguments, then added a precise symbol for generality. With that, 'all,' 'some,' and the relations between things could finally be written down exactly and reasoned about by rule.
A quiet professor in Jena
Gottlob Frege was a mathematics lecturer at the University of Jena when, in 1879, he published a slim 88-page book with an odd title — Begriffsschrift, 'concept-script' — printed in a sprawling two-dimensional notation of his own design. He was reaching for an old dream of Leibniz: a written language so exact that reasoning could be done almost mechanically. Almost nobody understood it. The leading reviewer dismissed it as a worse version of work George Boole had already done. Frege's genius went largely unrecognised for decades, until Bertrand Russell and others saw what he had built.
Why it matters
Frege's concept-script is the foundation of all modern logic, and through it of computer science. The idea that you can write statements with total precision and then check each step of an argument by fixed rules is exactly what lets a computer verify a proof, run a database query, or follow a program's logic. He also pinned down the 'if … then' of logic — true unless the 'if' part holds and the 'then' part fails — the same rule a circuit or a line of code obeys.
From sentences to blueprints
Think of the difference between describing a machine in flowing prose and drawing it as a blueprint. Prose is rich but ambiguous; a blueprint pins down every part and connection, so anyone can build from it without guessing. Frege turned the loose prose of logic into a blueprint. His quantifier is the blueprint's way of saying 'this holds for every part,' and his conditional is a precise wire: a signal goes out unless the input is on and the output is off.
Where it sits
Frege stands between George Boole, who in 1854 turned logic into a kind of algebra, and the great foundations of the twentieth century. His predicate logic became the language in which David Hilbert posed his 1900 problems, in which Kurt Gödel proved (1931) that no such system can prove all the truths of arithmetic, and in which Alan Turing (1936) defined what a computation is. Every one of those landmarks, also in this Library, is written in the grammar Frege invented.