The term analytic as a kind of philosophy was first introduced to describe a new conception of philosophy appearing at the beginning of the twentieth century in the work of Bertrand Russell and G.E. Moore. These philosophers, until about 1898 had accepted a form of Hegelian idealism, as then taught in Cambridge by James Ward, G.F. Stout and McTaggart, originating in the work of T.H. Green and F.H. Bradley.
The rejection of idealism was lead by Moore, in which Russell followed for a couple of years until his assimilated of the work of Peano in 1900. Russell then worked intensively on the philosophy and foundations of mathematics until the completion of Principia Mathamatica , and came to a conception of philosophical analysis which giving a central role to the new mathematical logic.
For my present purposes it is convenient to view the course of analytic philosophy in the twentieth century as consisting of three lines of development. The first two occupied the first half of the century, and constituted the development of the conceptions of analysis due to Russell and Moore. In the first the use of mathematical logic was central and was intended to yield a new and logically rigourous philosophical method and to extend the application of the new logic into mathematics and science. In the second, common sense and ordinary language played the central role, and the aspirations of philosophers to idealised languages and special kinds of knowledge were rejected. In the second half of the century the rejection of both these conceptions of philosophical analysis became perhaps one of few unifying features of analytic philosophy.
My primary concern is with the thread of which Russell was the source, and its legacy in the twenty first century which will concern us in Part V. For the purposes of considering that legacy it is desirable to consider both its merits and defects as progressed in the twentieth century, and the criticisms which were then thought to be decisive against it. This is a period in which technical advances in logic were continous and rapid, and occurred not only in philosophy and mathematics, but also eventually in computer science, and in which there was continuous evolution in analytic methods in philosophy and for science.