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To answer this question, it is important to immediately make a clear reservation about what can be called analytical philosophy. Two significant options for this issue::
1) The entire tradition that goes back to the works of Frege, Russell, Moore, and Wittgenstein; 2) The specific philosophical program of the first half of the twentieth century.
In the first case, it is difficult to identify the main ideas, because now ” analytical philosophy “is such a wide clearing that it is more appropriate to call it simply” modern philosophy”, often with the addition of”Anglo-American”. There are simply no “general” ideas here, and disputes are being waged on all fronts. Sometimes a specific writing style of “analytical philosophers” is distinguished, implying a desire for clarity of expression and reliance on logical argumentation. However, these features are neither necessary nor specific to modern “analytical” philosophy.
In the second case, “analytical philosophy” refers to the ideas of specific philosophers and the associated philosophical program. Usually, G. Frege, J. Moore, B. Russell and L. Wittgenstein are considered the fathers of analytical philosophy. Meanwhile, from a historical and philosophical point of view, it is difficult to find some idea that would be unambiguously shared by all the listed philosophers. So “anti-metaphysics” was clearly not characteristic of Frege, Moore, or Russell, and Wittgenstein is an ambiguous figure here. Perhaps the most significant idea that I think best unites the first analytic philosophers (although Frege's key figure for the entire tradition is, of course, somewhat different here) was formulated by Russell in his book on Leibniz: “The truth that all philosophy must begin with the analysis of propositions is so obvious that it may be difficult even to demand proof for it.”
However, the concept of “analysis” is already different for different analytical philosophers. Thus, Moore understood” analysis ” as conceptual analysis, i.e., the division of propositions (a proposition is, very roughly, a sentence) into separate concepts, concepts, and then the analysis of the concepts themselves, i.e., their division into other concepts. For Russell, analysis was primarily a logical analysis, the discovery of the logical structure that is subject to the propositions of everyday language. This is one step away from the idea shared by Russell and early Wittgenstein that the world is structured similarly to language, and that to adequately describe the world, one must distil some “pure” language with a logical form. However, the fundamentals of the method and structure of this ideal language were not developed by Russell or Wittgenstein, but by Gottlob Frege. Frege, in turn, reacted to the so-called “crisis in the foundations of mathematics “(in the center of the question of the possibility of “deducing” mathematics from certain foundations) and the psychologism that was gaining popularity at that time, which assumed all forms of our thinking, including logic and mathematics, to be derived from the human mental structure, and not universal.
The program of analytical philosophy is often associated with the activities of the so-called Vienna Circle, whose members, mostly scientists or philosophers who were scientists, sought to shield empirical science from various attacks from traditional and contemporary metaphysics using methods of formal logic and radical criticism of the previous metaphysical tradition. It was supposed to do this by reducing all scientific knowledge to protocol sentences that capture direct empirical experience. The language of protocol sentences in this case is the same ideal language that Frege, Russell, and Wittgenstein worked on.
An important milestone in the decline of the theoretical program of analytical philosophy was the critique of the division into analytical and synthetic judgments by W. W. O. Quine.
It is best to read primary sources on this topic, i.e.
Фреге (Begriffsschrift, eine der arithmetischen nachgebildete Formelsprache des reinen Denkens),
Moore (Moore J. E. The Nature of Moral Philosophy [and other works] / Preface by A. F. Gryaznov and L. V. Konovalova; Translated from English, comp. and note by L. V. Konovalova, Moscow: Respublika Publ., 1999, 351 p.)
Russell (Russell B. The philosophy of logical atomism. – Tomsk: Aquarius, 1999)
Wittgenstein (Logical and philosophical treatise).
However, these works may seem very complex, so as an introduction, you can pay attention to:
An excellent collection of articles on the history of analytical philosophy has recently been published: Michael Beaney (ed.) The Oxford Handbook of the History of Analytical Philosophy. Oxford: Oxford University Press, 2013.
A classic work in Russian is the book by J. R. R. Tolkien. Passmore's One Hundred Years of Philosophy. J. Passmore One hundred years of philosophy. Translated into Russian by I. V. Borisov, L. B. Makeev, A. M. Rutkevich, 1998. Moscow: “Progress-Tradition”, 1998. – 496 p.