An important precursor of the Analytic Philosophy tradition was the Logicism developed during the late 19th Century by Gottlob Frege. Logicism sought to show that some, or even all, of mathematics was reducible to Logic, and Frege's work revolutionized modern mathematical Logic. In the early 20th Century, the British logicians Bertrand Russell and Alfred North Whitehead continued to champion his ideas (even after Russell had pointed out a paradox exposing an inconsistency in Frege's work, which caused him, Frege, to abandon his own theory). Russell and Whitehead's monumental and ground-breaking book, "Principia Mathematica" was a particularly important milestone. Their work, in turn, though, fell prey to Kurt Gödel's infamous Incompleteness Theorems of 1931, which mathematically proved the inherent limitations of all but the most trivial formal systems.
Both Russell and Whitehead went on to develop other philosophies. Russell's work was mainly in the area of Philosophy of Language, including his theory of Logical Atomism and his contributions to Ordinary Language Philosophy. Whitehead developed a metaphysical approach known as Process Philosophy, which posited ever-changing subjective forms to complement Plato's eternal forms. Their Logicism, though, along with Comte's Positivism, was a great influence on the development of the important 20th Century movement of Logical Positivism.
The Logical Positivists campaigned for a systematic reduction of all human knowledge down to logical and scientific foundations, and claimed that a statement can be meaningful only if it is either purely formal (essentially, mathematics and logic) or capable of empirical verification. The school grew from the discussions of the so-called "Vienna Circle" in the early 20th Century (including Mauritz Schlick, Otto Neurath, Hans Hahn and Rudolf Carnap). In the 1930s, A. J. Ayer was largely responsible for the spread of Logical Positivism to Britain, even as its influence was already waning in Europe.
The "Tractatus" of the young Ludwig Wittgenstein, published in 1921, was a text of great importance for Logical Positivism. Wittgenstein is one of the 20th Century's most important philosophers. A central part of the philosophy of the "Tractatus" was the picture theory of meaning, which asserted that thoughts, as expressed in language, "picture" the facts of the world, and that the structure of language is also determined by the structure of reality. At the time he was convinced that the publication of the "Tractatus" had solved all the problems of philosophy: They exist only because language is used in the wrong way. Wittgenstein later abandoned his early work. He began to see language not as a model of reality, but as a system that can generate meaning intrinsically. He looked at language as a kind of game which generates its own rules and which we use in different ways. This approach led to the development of "Ordinary Language Philosophy."
Ordinary Language Philosophy shifted the emphasis from the ideal or formal language of Logical Positivism to everyday language and its actual use, and it saw traditional philosophical problems as rooted in misunderstandings caused by the sloppy use of words in a language. Some have seen Ordinary Language Philosophy as a complete break with, or reaction against, Analytic Philosophy, while others have seen it as just an extension or another stage of it. Either way, it became a dominant philosophic school between the 1930s and 1970s, under the guidance of philosophers such as W. V. O. Quine, Gilbert Ryle, Donald Davidson, etc.
Quine's work stressed the difficulty of providing a sound empirical basis where language, convention, meaning, etc, are concerned, and also broadened the principle of Semantic Holism to the extreme position that a sentence (or even an individual word) has meaning only in the context of a whole language. Ryle is perhaps best known for his dismissal of Descartes' body-mind Dualism as the "ghost in the machine", but he also developed the theory of Philosophical Behaviorism (the view that descriptions of human behavior need never refer to anything but the physical operations of human bodies) which became the standard view among Ordinary Language philosophers for several decades.
Another important philosopher in the Analytic Philosophy of the early 20th century was G. E. Moore, a contemporary of Russell at Cambridge University (then the most important center of philosophy in the world). His 1903 "Principia Ethica" has become one of the standard texts of modern Ethics and Meta-Ethics, and inspired the movement away from Ethical Naturalism (the belief that there exist moral properties, which we can know empirically, and that can be reduced to entirely non-ethical or natural properties, such as needs, wants or pleasures) and towards Ethical Non-Naturalism (the belief that there are no such moral properties). He pointed out that the term "good", for instance, is in fact indefinable because it lacks natural properties in the way that the terms "blue", or "smooth," have them. He also defended what he called "common sense" realism (as opposed to Idealism or Skepticism) on the grounds that common sense claims about our knowledge of the world are just as plausible as those other metaphysical premises.