"Logical Atomism" by Bertrand Russell,1959

Russell's logical atomism had significant influence on the development of philosophy, especially in the first half of the 20th century. Nowhere is Russell's influence more clearly seen than in the work of his pupil Ludwig Wittgenstein. Wittgenstein's Tractatus Logico-Philosophicus appeared in 1921; in it, Wittgenstein presented in some detail a logical atomist metaphysics. (It should be noted, however, that there is significant controversy over whether, in the end, Wittgenstein himself meant to endorse this metaphysics.) In the Tractatus, the world is described as consisting of facts. The simplest facts, which Wittgenstein called “Sachverhalte”, translated either as “states of affairs” or “atomic facts”, are thought of as conglomerations of objects combined with a definite structure. The objects making up these atomic basics were described as absolutely simple. Elementary propositions are propositions whose truth depends entirely on the presence of an atomic fact, and other propositions have a determinate and unique analysis in which they can construed as built up from elementary propositions in truth-functional ways.

Partly owing to Wittgenstein's influence, partly directly, Russell's logical atomism had significant influence on the works of the logical positivist tradition, as exemplified in the works of Carnap, Waismann, Hempel and Ayer. This tradition usually disavowed metaphysical principles, but methodologically their philosophies owed much to Russell's approach. Carnap, for example, described philosophy as taking the form of providing “the logical analysis of the language of science” (Carnap 1934, 61). This originally took on the form of attempting to show that all meaningful scientific discourse could be analyzed in terms of logical combinations beginning with “protocol sentences”, or sentences directly confirmable or disconfirmable by experience. This notion of a “protocol sentence” in this tradition was originally modeled after Russellian and Wittgensteinian atomic propositions. The notion of a “logical construction” was also important for how such thinkers conceived of the nature of ordinary objects (see, e.g., Ayer 1952, chap. 3). The view that scientific language could readily and easily be analyzed directly in terms of observables gradually gave way to more holistic views, such as Quine's (see, e.g. Quine 1951), in which it is claimed that it is only a body of scientific theories that can be compared to experience, and not isolated sentences. However, even in later works growing out of this tradition, the influence of Russell can be felt.

Besides positive influence, many trends in 20th century philosophy can be best understood largely as a reaction to Russell's atomistic philosophy. Ironically, nowhere is this more true than in the later writings of Wittgenstein, especially his Philosophical Investigations (1953). Among other things, Wittgenstein there called into question whether a single, unequivocal notion of simplicity or a final state of analysis can be found (e.g., secs. 46–49, 91), and questioned the utility of an ideal language (sec. 81). Wittgenstein also called into question whether, in those cases in which analysis is possible, the results really give us what was meant at the start: “does someone who says that the broom is in the corner really mean: the broomstick is there, and so is the brush, and the broomstick is fixed in the brush?” (sec. 60). Much of the work of the so-called “ordinary language” school of philosophy centered in Oxford in the 1940s and 1950s can also been seen largely as a critical response to views of Russell (see, e.g., Austin 1962, Warnock 1951, Urmson 1956).

Nevertheless, despite the criticisms, many so-called “analytic” philosophers still believe that the notion of analysis has some role to play in philosophical methodology, though there seems to be no consensus regarding precisely what analysis consists in, and to what extent it leads reliably to metaphysically significant results. Debates regarding the nature of simple entities, their interrelations or dependencies between one another, and whether there are any such entities, are still alive and well. Russell's rejection of idealistic monism, and his arguments in favor of a pluralistic universe, have gained almost universal acceptance, with a few exceptions. Abstracting away from Russell's particular examples of proposed analyses in terms of sensible particulars, the general framework of Russell's atomistic picture of the world, which consists of a plurality of entities that have qualities and enter into relations, remains one to which many contemporary philosophers are attracted.

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Bertrand Russell described his philosophy as a kind of “logical atomism”, by which he meant to endorse both a metaphysical view and a certain methodology for doing philosophy. The metaphysical view amounts to the claim that the world consists of a plurality of independently existing things exhibiting qualities and standing in relations. According to logical atomism, all truths are ultimately dependent upon a layer of atomic facts, which consist either of a simple particular exhibiting a quality, or multiple simple particulars standing in a relation. The methodological view recommends a process of analysis, whereby one attempts to define or reconstruct more complex notions or vocabularies in terms of simpler ones. According to Russell, at least early on during his logical atomist phase, such an analysis could eventually result in a language containing only words representing simple particulars, the simple properties and relations thereof, and logical constants, which, despite this limited vocabulary, could adequately capture all truths.

"Logic and Knowledge" by Bertrand Russell,1956,

Russell's contributions to logic and the foundations of mathematics include his discovery of Russell's paradox, his defense of logicism (the view that mathematics is, in some significant sense, reducible to formal logic), his development of the theory of types, and his refining of the first-order predicate calculus.

Russell discovered the paradox that bears his name in 1901, while working on his Principles of Mathematics (1903). The paradox arises in connection with the set of all sets that are not members of themselves. Such a set, if it exists, will be a member of itself if and only if it is not a member of itself. The paradox is significant since, using classical logic, all sentences are entailed by a contradiction. Russell's discovery thus prompted a large amount of work in logic, set theory, and the philosophy and foundations of mathematics.

Russell's own response to the paradox came with the development of his theory of types in 1903. It was clear to Russell that some restrictions needed to be placed upon the original comprehension (or abstraction) axiom of naive set theory, the axiom that formalizes the intuition that any coherent condition may be used to determine a set (or class). Russell's basic idea was that reference to sets such as the set of all sets that are not members of themselves could be avoided by arranging all sentences into a hierarchy, beginning with sentences about individuals at the lowest level, sentences about sets of individuals at the next lowest level, sentences about sets of sets of individuals at the next lowest level, and so on. Using a vicious circle principle similar to that adopted by the mathematician Henri Poincaré, and his own so-called "no class" theory of classes, Russell was able to explain why the unrestricted comprehension axiom fails: propositional functions, such as the function "x is a set," may not be applied to themselves since self-application would involve a vicious circle. On Russell's view, all objects for which a given condition (or predicate) holds must be at the same level or of the same "type."

Although first introduced in 1903, the theory of types was further developed by Russell in his 1908 article "Mathematical Logic as Based on the Theory of Types" and in the monumental work he co-authored with Alfred North Whitehead, Principia Mathematica (1910, 1912, 1913). Thus the theory admits of two versions, the "simple theory" of 1903 and the "ramified theory" of 1908. Both versions of the theory later came under attack for being both too weak and too strong. For some, the theory was too weak since it failed to resolve all of the known paradoxes. For others, it was too strong since it disallowed many mathematical definitions which, although consistent, violated the vicious circle principle. Russell's response was to introduce the axiom of reducibility, an axiom that lessened the vicious circle principle's scope of application, but which many people claimed was too ad hoc to be justified philosophically.

Of equal significance during this period was Russell's defense of logicism, the theory that mathematics was in some important sense reducible to logic. First defended in his 1901 article "Recent Work on the Principles of Mathematics," and then later in greater detail in his Principles of Mathematics and in Principia Mathematica, Russell's logicism consisted of two main theses. The first was that all mathematical truths can be translated into logical truths or, in other words, that the vocabulary of mathematics constitutes a proper subset of that of logic. The second was that all mathematical proofs can be recast as logical proofs or, in other words, that the theorems of mathematics constitute a proper subset of those of logic.

Like Gottlob Frege, Russell's basic idea for defending logicism was that numbers may be identified with classes of classes and that number-theoretic statements may be explained in terms of quantifiers and identity. Thus the number 1 would be identified with the class of all unit classes, the number 2 with the class of all two-membered classes, and so on. Statements such as "There are two books" would be recast as statements such as "There is a book, x, and there is a book, y, and x is not identical to y." It followed that number-theoretic operations could be explained in terms of set-theoretic operations such as intersection, union, and difference. In Principia Mathematica, Whitehead and Russell were able to provide many detailed derivations of major theorems in set theory, finite and transfinite arithmetic, and elementary measure theory. A fourth volume was planned but never completed.

Russell's most important writings relating to these topics include not only Principles of Mathematics (1903), "Mathematical Logic as Based on the Theory of Types" (1908), and Principia Mathematica (1910, 1912, 1913), but also his An Essay on the Foundations of Geometry (1897), and Introduction to Mathematical Philosophy (1919).