Frege founded modern mathematical logic in his 1879 "Begriffsschrift," inventing quantificational predicate logic, and pursued logicism, the project of deriving arithmetic from logic alone, in "The Foundations of Arithmetic" (1884). In "On Sense and Reference" (1892) he distinguished a sign's sense (its mode of presentation) from its reference (the object it picks out) to explain why true identity statements like "the morning star is the evening star" can be informative rather than trivial.
Intellectual setting
Frege worked in relative institutional isolation at Jena, and his logical notation and logicist program went largely unrecognized until Bertrand Russell, upon discovering the paradox that bears his name in Frege's system, brought Frege's work to the attention of the wider philosophical community.
Key ideas
Sense and reference
Frege distinguished the sense of an expression, the particular way it presents its object, from its reference, the object itself, explaining how two expressions can refer to the same thing while differing informatively in meaning.
Logicism and predicate logic
Frege invented modern quantificational logic in the "Begriffsschrift" and used it to pursue logicism, the attempt to derive the truths of arithmetic from purely logical axioms and definitions, a project undermined when Russell discovered a contradiction in its underlying set-theoretic assumptions.
Legacy
Frege is regarded as the founder of modern logic and analytic philosophy of language; his sense-reference distinction remains foundational to philosophy of language, and his logical notation underlies contemporary formal logic despite Russell's paradox forcing a revision of his original logicist program.