Analytic philosophy is one of two major branches of philosophy defined by its emphasis on formal logic, philosophy of language and scientism. Prevalent in the Anglo-American world from the early 20th century to the present day, analytic philosophy is the dominant philosophical discourse in academia and many mathematicians and computer scientists find themselves intrigued by it, due to the close relationship it often has with their areas of study.

Analytic philosophy is one of two major branches of philosophy defined by an emphasis on formal logic, philosophy of language and scientism. Prevalent in the Anglo-American world from the early 20th century to the present day, analytic philosophy is currently the dominant philosophical discourse in academia and many mathematicians and computer scientists find themselves intrigued by it, due to the close relationship it often has with their areas of study.

Analytic philosophy began in the United Kingdom with Bertrand Russell and G. E. Moore rejecting the then prevalent idealism, originally inspired by Immanuel Kant and G. W. F. Hegel. These British idealists were in turn reacting to their predecessors, the British empiricists. Though empiricism was not at the heart of Russell's idea of philosophy, it would later be revived by the logical positivists as part of analytic philosophy's development.

Russell, inspired by and collaborating with Gottlob Frege and Ludwig Wittgenstein, sought to reduce the world to atomic facts in the form of propositions, amenable to logical calculus, upon which mathematics was also to be founded. This idea of reducing the world to its simplest constituent facts formed logical atomism, the first major movement of analytic philosophy. Ludwig Wittgenstein would later reject this philosophy, discussing its failings in his Tractatus Logico-Philosophicus by establishing the limitations of language and of philosophical discourse, as well as criticising Frege and Russell's attempts to derive mathematics from set theory without contradictions, citing limitations in logic and language that would later earn Kurt Gödel eternal fame when he conclusively proved the existence and severity of these limitations.

With the logical atomist project indefinitely on hold, a new movement emerged, inspired by Wittgenstein's Tractatus: the logical positivists. These philosophers, primarily based in Vienna and including the likes of Moritz Schlick and Rudolf Carnap, wanted not only to reduce the world to factual propositions, but also sought to violently eliminate all metaphysical statements from philosophical discourse as being 'nonsensical'. Obviously the inspiration they drew from Wittgenstein's work was fundamentally a misunderstanding of Wittgenstein's intent, and the logical positivists arguably took Wittgenstein's closing statement 'whereof one cannot speak, thereof one must be silent' to justify the idea of requiring that all propositions be subject to a verification principle before being deemed acceptable or intelligible.

Of course, the verification principle could not verify its own significance, and likewise the logical positivists came under criticism for twisting the facts to suit their anti-metaphysical or anti-idealist agendas, and with Wittgenstein himself not only refusing to support their movement, but eventually actively criticising their movement with his magnum opus Philosophical Investigations, their movement would fizzle out. Nevertheless, as logical atomism before it, logical positivism inspired the next generation of analytic philosophy and further emphasised the philosophy of language, which of course Wittgenstein reinforced and shaped with his Investigations, shifting the paradigm from language-by-ostension to language-through-use.

From Wittgenstein's later work analytic philosophy focused in language-use and epistemology, exploring and developing linguistics and questioning classical definitions of knowledge. W. V. O. Quine published a significant paper rejecting the division of analytic and synthetic propositions, and Wittgenstein's posthumously published On Certainty highlighted the extreme specificity and technicality involved when using the term 'knowledge' under the tripartite definition or similar definitions, and how certainty is much more of an ordinary variant of knowledge in certain forms of life. Analytic philosophers also busied themselves with objecting to continental philosophy in the form of structuralism and post-structuralism and what analytic philosophers deemed to be outdated ways of understanding linguistics; this is perhaps best represented by the clash between speech-act proponent John Searle and Jacques Derrida—the father of deconstruction.

Despite its loss of favour after Wittgenstein's later philosophy, logical positivism influenced psychology with the verification principle, and the behaviourist movement emerged directly from that influence. Behaviourism in turn influenced functionalism and other psychological schools of thought, and thus psychology and philosophy became very closely linked in the English-speaking world, just as psychoanalysis walked hand-in-hand with philosophy during the structuralist and post-structuralist phases in continental philosophy. Analytic philosophy can therefore be cited as a major influence for cognitive science and behavioural psychology, and logical positivism, though it failed philosophically, directly influenced scientific study with its method.

Another important aspect of analytic philosophy is the philosophy of science itself. Analytic philosophy has always been supportive of science, and several key analytic philosophers, like Karl Popper, are identified as important characters in the refinement of the scientific method. Popper himself posited the notion of falsification, stipulating that, rather than trying to prove their hypotheses (which could appear to be correct even if there is a coincidence involved), scientists should instead seek to disprove their theories, thereby eliminating the possibility that their theory is correct coincidentally. Analytic philosophy was influential in refining the approaches to theoretical sciences and clarifying the importance of making predictions with a theory, rather than just constructing a model to fit the hitherto observed facts.

Analytic philosophy persists to the present day; among Anglo-American countries, it is still the most prevalent school of thought, though as the advances have become more incremental and petty, more thinkers are turning to continental philosophers for different approaches and perspectives.