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Unit 2: A Free Man’s Worship by Bertrand Russell

some important sense reducible to logic), his refining of the predicate calculus introduced by Notes
Gottlob Frege (which still forms the basis of most contemporary logic), his defense of neutral
monism (the view that the world consists of just one type of substance that is neither exclusively
mental nor exclusively physical), and his theories of definite descriptions and logical atomism.
Along with G.E. Moore, Russell is generally recognized as one of the founders of modern
analytic philosophy. Along with Kurt Gödel, he is regularly credited with being one of the
most important logicians of the twentieth century.
Over the course of his long career, Russell made significant contributions, not just to logic and
philosophy, but to a broad range of subjects including education, history, political theory and
religious studies. In addition, many of his writings on a variety of topics in both sciences and
humanities have influenced generations of general readers.
After a life marked by controversy—including dismissals from both Trinity College, Cambridge,
and City College, New York—Russell was awarded the Order of Merit in 1949 and the Nobel
Prize for Literature in 1950. Noted for his many spirited anti-war and anti-nuclear protests,
Russell remained a prominent public figure until his death at the age of 98.

2.2 Russell’s Work in Logic

Russell’s main 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, his impressively
general theory of logical relations, his formalization of the reals, 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 response to the paradox came with the development of his theory of types between
1903 and 1908. It was clear to Russell that some form of restriction needed to be placed on the
original comprehension (or abstraction) axiom of naive set theory, the axiom that formalizes
the intuition that any coherent condition or property 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,
together with 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.” Sentences about these objects will then always be higher in
the hierarchy than the objects themselves.
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 three-volume
work he co-authored with Alfred North Whitehead, viz. 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 came under attack: the simple theory for being too

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