Logic

Logic changed more in a single short book published in 1879 than it had in the roughly two thousand years since Aristotle first organized valid argument into syllogistic figures. Gottlob Frege's Begriffsschrift, literally concept script, introduced a formal notation capable of expressing quantification, statements about all or some members of a domain, and multiple, nested quantifiers within a single sentence, a capacity Aristotelian syllogistic had never possessed and that ordinary language handles only awkwardly and often ambiguously. Frege's own two dimensional notation, built from vertical and horizontal strokes, proved too unwieldy to survive in that exact form, but the underlying idea, that logical structure can and should be represented with a precision language itself does not supply, became the field's new working assumption.

The notation's descendants reshaped the rest of mathematics and philosophy built on top of it. Bertrand Russell and Alfred North Whitehead's Principia Mathematica, published between 1910 and 1913, adopted quantificational logic in something close to its modern symbolic form to attempt a full derivation of mathematics from logical axioms, and the predicate calculus every introductory logic course now teaches, with its universal and existential quantifiers ranging over variables, traces its lineage directly to Frege's 1879 innovation rather than to any refinement of Aristotle's original syllogism. Logic before Frege asked which arguments among a fixed, limited set of sentence forms were valid; logic after him could express and evaluate an open ended range of structures no syllogism had ever been built to state.