Submitted to Dr. Murdith Mc Lean, Professor, St. John's College

University of Manitoba

November, 5th, 1987.

by George S. Sagi.




1. My purpose presenting this essay is to examine Russell’s working method, and to understand what he meant by 'logical analysis1 A further aim is to show Russell's almost fanatical belief in the purity and absoluteness of mathematics. Thirdly, to demonstrate, how his mathematical pre-disposition, his frame of mind, affected his logical-analytical work.

2. The sources I have relied upon were the lectures given by Dr. Mc Lean, and Russell's original publications, in about ninety percent of this paper

3. The results expected by showing Russell's work and his personal biases which are not motivated by my critical intention. But it is for the purpose of showing his method of presentation, which is interwoven by a mixture of logical and psychological passages. For example, he never described in advance, in his referenced volumes, nor did he give a clear definition of what he meant by ‘logical analysis’. Therefore to understand him better, it is instructive to look at both the content and form of his presentation.

4. My method of presentation. Russell's development of thought is best illustrated in historical order of the publications of his works. I am highlighting in 'boldface' those terms he used whenever he switched his line of thought to a new concept or term. This is a great help for me to summarize at the end how he progressed from one idea to another. Also, this method may assists me to see the sum total of his great achievements.




1.  The dominant philosophy during Russell's university years was Bradley’s Analysis, Idealism and monistic views contrast with pluralism. “After a brief excursion into absolute idealism, which lasted until 1898, Russell formulated a dualistic theory of reality: a dualism of mind and matter—and of universals and particulars that despite of modifications, especially toward neutral monism, he retained throughout his work.” (Ref 7)

2.  Young Russell grew up in a turbulent period of history. It was a period of rapid industrialization, accompanied by industrial unrest, the birth of organized labour movements, the First World War, and the birth of the first state, which was built upon the philosophical doctrine of Communism. Great progress was made by science and technology in the same period; the births of quantum theory, the discovery of Radioactivity, Einstein’s Theory of Relativity are just a few scientific milestones. In the from of technology, rapid progress was made from the steam engine to the combustibles, electrical generation systems and hundreds of new devices, the birth of automobiles, aviation, radio communication, etc. marked his early years. Those rapid changes had left their effects on Russell’s keen interest. His first book was published in 1896, titled German Social Democracy, when Russell was 24 years old. Everything interested him, driven by his flair for discovery; he investigated and wrote about many subjects, in more than sixty books, during his long life-span of almost 98 years. He won the Nobel Prize for literature in 1950, which is one of the proofs of his outstanding and diverse talents, in addition to mathematics and philosophy. It took him about ten years to publish Principia Mathematica (3 vols. 1910-13) together with his former professor and friend Dr. Whitehead. A pioneering work in mathematical logic.

3.  I think, Russell became almost possessed by mathematics early in his life. Instead of using my own description of his infatuation here is his own;

“Mathematics, rightly viewed, possesses not only truth, but supreme beauty, a beauty cold and austere, like that of sculpture, without appeal to any part of out weaker nature without the gorgeous trappings of paintings or music, yet sublime pure and capable of a stern perfection such as only the greatest art can show. The true spirit of delight, the exaltation, the sense of being more than man which is the touchstone of the highest excellence, is to be found in mathematics as surely as in poetry. What is best in mathematics deserves not merely to be learnt as a task, but to be assimilated as a part of daily thought, and brought again and again before the mind with ever-renewed encouragement. Real life is, to most men, a long second-best, a perpetual compromise between the ideal and the possible; but the world of pure reason knows no compromise, no practical limitations, no barrier to creative activity embodying in splendid edifices  the passionate aspiration after the perfect form, remote even from the pitiful facts of nature, the generations have gradually created an ordered cosmos, where pure thought can dwell as in its natural home, and where one, at least, of our nobler impulses can escape from the dreary exile of the actual world.”

Despite Russell’s own belief that mathematics is “removed from passions” he was certainly not. His passionate love, his deep belief in sublimely pure science which knows “no compromise” between “practical limitations” and the “ideal” had a definite impact on his philosophical works.



1.  Our Knowledge of the External World (Ref 2) was published in 1914. Russell was 42 years old. By this time he raised the concept of logical analysis higher than his mathematics. He credited his achievements, regarding the theory of classes and functional analysis, to his philosophical logical analysis. Mathematics became a derivative of logical analysis. “I have been made aware of the importance of this problem by my friend a»d col1abor Dr. Whitehead to whom are due almost all the differences between views advocated here and those suggested in the problems of philosophy.” (Ref. 2., p. 8) 


2.  He begins by giving credit to George Cantor’s theory and solution of the mathematical infinities. “But the positive and detailed solution” of mathematical- symbolic representation of data of the sensible world was the next step, which Russell was to undertake. He believed that without ‘mathematical logic’ “it is practically impossible to manipulate ideas of the requisite abstractness and complexity.” ( Ref. 2.,  p. 81  )


3.  The Duality faith in the infallibility of mathematical construction on the one hand and his subcutaneous fear about some flaws in his new discoveries, he cautiously added are “tentative and incomplete.” (Ref. 2, p.9) These discoveries are described in advance as predicts of “pure logic.” However, he didn’t give any idea what he meant by these words, at this point. Instead he proceeded to point out that “logic is never used in the same sense by two different philosophers.” (Ref 2, p42) Logic since Aristotle was ‘syllogistic’ consisting of ‘subjects’ and ‘predicates’  “meant no more than a collection of technical terms and rules of syllogistic inference.” (Ref 2, p. 42)


4.  I am surprised to witness Russell's ferocity of words, projecting more than his objective, cool mathematical reasoning, by these remarks. It shows to me that he could not detach his feelings from his logic. Perhaps no one can. But, their strength is surprising and unwarranted in my opinion. At this point, it is also seems proper to note, that Russell did not get to his point directly. As we saw, he did not define what he meant by his 'mathematical construction' at the beginning, neither did he state what he means by 'logic' nor did he advance d how he is to progress systematically from the beginning of his thesis to its conclusion.


5.  I made these remarks to illustrate his method, as instances of focus on some particular, seemingly breaking the flow of continuity of the subject. This seems to be Russell’s method, which to me, is a reflection of duality of supreme intellect, interrupted by equally strong emotions.


6.  Another glimpse may be obtained into what Russell’s analytical method of presentation was is described by him in his introductory notes to Human Knowledge; Russell uses an analogy that is approaching a mountain in a haze; “we have indistinct boundaries, but gradually more detail becomes visible and the edges become sharper. (Ref. 4, pp. V, VI, XII) It is noticeable that he considers the WHOLE scenario, the mountain, its peaks, and not the details of its terrain. The details of the mountain emerge as we go closer and closer to the whole. Russell’s analysis begins therefore with the examination of the whole which may be broken down into smaller and smaller details. until they are seen clearly. It is perhaps an early indication of Russell's thinking, oriented towards starting with sets and collections of things, the large overall hazy picture, rather than building up theories from simple concepts toward more complex entities.


7.  The problem remaining for me is, that Russell have made discoveries of important particulars, which I understood. However, he did not put all the particulars; he has dealt with, in a summary overview, at the end, or a preview at the beginning. So using his own term I cannot account for seeing Russell's 'mountain' in the whole, except some extremely sharp details of its structure.




1. 'Our Knowledge of the External World' moves from Aristotle to the 17th century, to Bacon and Galileo. To Russell their contributions were the first beginnings of the new logical approach to both science and mathematics. However, it was “still an extension of the old logic,” and “In the final form of a perfected science it would seem, everything ought o be deductive.” But the works of science, which are based upon induction, are not purely logical to him, therefore he predicts: “Induction, if it remains at all, remains merely as one of the principles according to which deductions are affected.” (Ref. 2 p. 44)

2. From induction he progressed to investigate how to justify 'inferences.' What is the principle of inferring that the 'sun will rise tomorrow?' Instead of giving his own direct reply he describes Mill's Ideas. Accordingly, it is the 'the law of causation' which enables us to infer from frequent, statistically valid, known events, to future occurrences of the same. Same causes create the same effects. Russell’s question is; “what is the reason to believe in causation?”

(1) First, causation may be a priori

(2) Second, it may be postulated and

(3) Thirdly, “That it is an empirical generalization from past instances, in which it has been found to hold" (Ref. 2 , p.44)


After dismissing the first two, although not completely, he gives some credibility to Mill’s empirical generalization. Since the method involves generalizations. Since the method involves ’enumeration’ Russell asks, how is generalization justifiable. It took 45 pages for Russell to declare his views, still in an unclear manner to me; “We shall have to say, at most, that the data rendered the results probable. Causation holds, we shall say in every instance we have been able to test; therefore it probably holds in untested instances...we thus have at least maybe a logical principle, since it is without exception.” (Ref. 2, p.45)

3. Since the results are empirical, and only probable, Russell is not satisfied. He is looking for logical principles, which would enable him to find 'invariable deductive truths' methods other than empirical that are faultless. He did not offer one, instead he has given this definition; “If a proposition is true in every instance that we happen to know and if the instances are very numerous, than we shall say, it becomes probable on the data, that it will be true in any future instance.” (Ref. 2 p. 45) At this point, the topics of causation, inference, and the search for invariable truth are dropped. He returned briefly to the critique of syllogism.


The old logic reduces everything to subject-predicate form. Russell's main concern was, that traditional logic is unable to admit "the reality of relations” Seeming1y, Russell lets his reader to conclude why relations are important. He simply presents his classifications of  he simply presents his classification of relations  (Ref 2. p.58)

            SYMMETRICAL RELATION is which holds between A and B and also between Band A. like ‘relatives'

            ASYMMETRICAL RELATION is, which holds between A and B, but never holds between B and A, like ‘father and son.'

            NON-SYMMETRICAL RELATIONS are "All relations that are not symmetrical called non-symmetrical." like 'brother', since the assertion if A is a male brother of B sister; than A is a brother of B is true, but B is a brother of A is not true.

The terms put into single comma quotation marks signify the relations in the above sentences. Asymmetrical relations are; husband, father, wife, before, after, greater, above, etc.

            TRANSITIVE RELATIONS exist whenever it holds between A and Band also between Band C. like 'before'

            INTRANSITIVE RELATIONS exist whenever A has a relation to B and B to C, but A never has it to C, like 'father'

            NON-TRANSITIVE RELATIONS exist, whenever the relations are not transitive.

It took 58 pages for Russell, to underscore the significance of relations; “Asymmetrical relations, such as before, after, greater, and less, etc. the attempt to reduce them to properties becomes obviously impossible.”  (Ref. 2, p.58)

2. Russell achieved two important goals. First, he showed the limited usefulness of syllogistic propositions, which do not include relations such as, “All men are mortals, Socrates is a man, therefore Socrates is mortal,” requires empirical proof. We don’t know all men and we are not certain there are no immortals. Second, he was convinced that by accepting the reality of relations it proves that the world of sense is real.


l. The logical proof that relations exist was not sufficient for Russell. Reality needs other than logical proof, by facts. I speak of a ‘fact’...I mean that certain things have ‘quality’...or...‘relations.’ A complete description of the world is impossibility. It not only would require a catalogue of all things, but also one for their qualities and relations. However, the exploration of relationships of 'facts' and 'qualities' in order to find a logical generalization is possible.

2. Facts are usually complex, and have several constituents. We may 'assign' a quality to a single ‘thing’; it now has only two constituents; the thing and the quality the thing and the quality. When the classification is extended to the

the thing and the quality. When the classification is extended to the relations between two or more things, there is always one more relationship than the number of things involved. These are the ‘terms’ of the relationship. Russell concludes, “that there are single facts consisting of  a simple relation and more than two things,” which leads to the generalization and the term, ‘form’ that is the ‘logical form of facts.’ (Ref. 2, p.62) Since the propositions consist of multiple facts, they should be examined through these simple entities, which are the ‘logical form of facts.’

3. The fact itself is objective and independent of human thought, but ‘the assertion of fact’ involves thought. Thoughts and believes of the mind may be positive or negative assertions. Therefore, “A form of words, which maybe true or false, I shall call a proposition.” That is a sentence composed of facts, their relationships and qualities which maybe asserted or denied. A proposition may be true or false.


  1. The analysis of facts lead to the classification of ‘logical form of facts’ and the concept of ‘propositions.’ These details can now be further generalized as “Atomic Propositions’ which are a ‘proposition’ which express what we call a ‘fact,’ i.e. which when asserted, asserts a certain ‘thing,’ has a certain ‘quality,’ or that certain things have certain ‘relations’ will be called ‘atomic propositions.’ (Ref. 2, P.62. My underlining)

2. This process of breaking down ‘things’ into elementary ‘terms’ as underlined above, is an implicit testimony what Russell’s term ‘logical analysis’ meant.

3. ‘Atomic facts’ are empirical, and since, what is the basis for judgments regarding atomic propositions? Atomic assertions express facts, true or false, we can only asset their validity by empirically verifying the existence of things, their qualities and relations. “Atomic facts are what determine whether atomic propositions are asserted or denied.” (Ref. 2, p.62) Atomic facts are elementary empirical facts with certain qualities, terms in atomic propositions with relations. For example, ‘this is red’ or ‘this is before that,’ are simple term atomic propositions in which the quality ‘red’ and the relation ‘before’ are ‘atomic facts.’ “It follows that, if atomic facts are to be known at all, some at least must be known without inference. (Ref. 2, p. 62, my boldface)

4. What do atomic statements reveal? This is his own doubting question, which seems to be part of Russell’s analytical method. Do they reveal all the facts, relations, and qualities? He concluded that we don’t know all the atomic facts, composing atomic propositions’ but “If we did know them all than the truth of propositions could be derived logically.” But we don’t know ‘all’ atomic facts, and in any case “facts are empirical. Therefore he concluded: “But in the first acquisition of knowledge concerning atomic facts, logic is useless.”




1. His initial classification was unyielding, in a sense, ‘atomic facts' did not fulfill his criteria to create something independent of empirical evidence. In an implicit manner Russell reveals, at this point, what he means by 'pure logic;' “Pure logic is independent of atomic facts” and conversely, atomic facts are independent of logic. Pure logic and atomic facts are the two poles, the wholly a priori and the wholly empirical.” (Ref. 1, p.63)


2. Russell’s mathematical predisposition now is in full view. He believes in a totally non-empirical ‘pure logic,’ ‘wholly a priori.’ He provides no evidence as yet I brought up this issue here because Russell interrupted his classification process and entered into the discussion of logic within absolute purity. I fail to understand it why? But how am I to discriminate between analytical and pure logic? He is not giving, seemingly, any reason for it.




1. After a brief excursion into ‘pure logic’ Russell returned to his process of classification. The introduction of 'molecular propositions' follows: “Molecular propositions are such as contain conjunctions – if, or, and, unless, etc. - and such words are the marks of a molecular proposition.” For example, “If it rains, I shall bring my umbrella.” “This assertion is just as capable of truth or falsehood as the assertion of atomic proposition.” (Ref.2, p.64) What is the difference and the significance between molecular and atomic verification? There are two atomic assertions given in the example above, ‘conjoined’ by a conditional term ‘if,’ “thus the connection of two propositions, which does not depend upon whether they are asserted or denied, but only the second being inferable from the first.” (Ref.2, p.64)... The truth value of the second atomic proposition – “I shall bring my umbrella” - entirely depends upon the empirical truth of the first proposition and the conjunction ‘if,’ in this case.



l. These were the type of propositions with which the textbooks of Russell’s young years were loaded with. Russell classified ‘general propositions’ into two types; positive and negative assertions. Unlike atomic and molecular propositions general propositions such as 'all men are mortal’ “cannot be known by inference from atomic facts alone.” (Ref.2, p.65)


2. Russell’s argument is that a collection of things inferred ‘al1 men,' in this case, cannot be asserted, unless the whole universe is known to have only mortal men. In general; “unless we know, all things belong to this collection of things I have examined, that each separate thing was not an immortal men.” (Ref.2, p.65) Therefore the assertion and the inference is not warranted.



l. Since no one can explore all the collections of the universe, therefore general truth cannot be inferred from particular truths alone. Can we know 'general truth' at all? Russell’s summary is; “all empirical evidence is of particular truth. Hence, if there is any knowledge of general truths at all, there must be some knowledge of general truth which is independent of empirical evidence, i.e. does not depend upon data of the sense.” (Ref.2, p.65-66)

2. The above conclusion is a refutation of the older empiricists, who “believed that our knowledge is derived from the senses.” Russell must admit that there is general knowledge not derived from sense and that some of this knowledge is not derived from sense and that some of this knowledge is not obtained by inference, but is “primitive.” He did not give any hint what he called primitive not inferred logic is, at this point. Instead he had found it in logic, which is to be explained next.



1. Here I preferred to quote Russell in the original:

“Such general knowledge is to be found in logic. Whether there is such knowledge not derived from logic, I do not know; but in logic, at any rate, we have such knowledge. It will be remembered that we excluded from pare logic such propositions as ‘Socrates is a man’, all men are mortal, therefore Socrates is mortal,’ because Socrates and man are empirical terms, only to be understood through particular experience. The corresponding proposition in pure logic is: If anything has a certain property, and whatever this property has a certain other property, than the thing in question has another property. This proposition is absolutely general: it applies to all things and all properties. And it is quite self-evident. Thus in such propositions of pure logic we have the self-evident general propositions of which we in search.”  


The proposition was re-written; “If Socrates is a man, and all men are mortal, then Socrates is mortal,” and in this form, it conforms to the pure criteria of writing general propositions, and it is true in virtue of its form alone. This is hypothetical representation in a purely logical form. The subjects and predicates are instances only , in this manner, they may be replaced by other terms, while the truth expressed by the proposition will not be violated.




1. According to Weitz "Although Russell, unlike Moore and others, has never stated what he means by analysis, it is quite clear from his uses of the term that he conceives it as form of definition - either real or contextual, non linguistic,  or linguistic.”  (Ref. 7, p.99) Referring to other works, Weitz lists such phrases as “true analysis,” “complete analysis,” “faulty analysis” used by Russell, “phrases which make sense only on a view on analysis as real definition, but is also primarily the enumeration of the empirical constituents of the given complex.” (Ref. 7, p. 99)


2. W. Barrett describes the friendship between Russell and Wittgenstein. Russell gave credit to Wittgenstein for hi~ original idea of logical atomism, which both men gave different interpretation. Interestingly, later in life they both rejected logical atomism. But, Wittgenstein's rejection seems to be more profound then Russell's. ”Wittgenstein seemed to have at long last established the final and clear distinction between factual propositions and the propositions of mathematics and logic. The latter were tautologies, and hence said nothing about the world. This allowed positivists to escape from the ancient view of semi-sacred character of mathematics as a body of eternal truth about eternal objects only imperfectly manifested in sensory experience.” (Ref. 5, vol. 2, p. 4)  

3. Kenny quotes Wittgenstein, whose verdict is brief and devastating. “In the Prototractatus he wrote that all analytical propositions are tautologies.” (PTL 4.44602) and in the Tractatus “The propositions of logic are tautologies. Therefore the propositions of logic say nothing (They are the analytic propositions.)” (TPL 6.1-6.II)

4. Ayer describes Russell's motivation, to seek philosophical truths, basically believed in the truth of mathematics. He did attempt to find "ulterior justification,” independent of empirical inferences, both for mathematical and philosophical propositions "not just contingently but necessarily true.” (Ref. 8, p. 22) His ideals were to express fundamental propositions “in purely logical terms and secondly, the development of a system of logic.” (Ref. 8, .p. 23) Ayer thought Russell’s work did not give a picture of reality "I should not mind saying that it was a play of our fancy, just the sense that its propositions were not descriptive of the world, but did no more than express rules of inference in accordance with a system of counting that we have chosen to adopt.” (Ref. 8, p. 24)


1. When Russell's History of Western Philosophy (Ref. 3) was published in 1946, it was more than three decades after he wrote that emotional testimony about his love and respect of mathematics, quoted in the Introduction. In this great summary work he wrote, as a commentary to John Locke's An Essay Concerning Human Understanding. "No one has yet succeeded in inventing a philosophy at once credible and self-consistent (Ref.3, p. 613)

2. The last great summary work of Russell was written 34 years after of Our Knowledge of The External World (Ref. 2) under the title: Human Knowledge (Ref. 4) He was 76 years old, and his views changed considerably. He retracted on the atomic theory of propositions, but in my opinion his greatest effort was to reconsider his views about the purity and absoluteness of his philosophy. Philosophy, for him now, became just one of the sciences, and absoluteness, which is total purity, is not achievable “...I have endeavored to philosophy can become a science, something perfectly definite, capable of embodiment of maxims, and adequate in all branches of philosophy, to yield whatever objective scientific knowledge to attain.” (Ref. 4, p.7) It is significant for me to notice the changes in the terms now he used; a change from the ethereal to the empirical.

3. He was disappointed with the fact that twentieth century philosophy becomes too abstract and mathematical. Perhaps he did not feel nor did he express his regret to be instrumental of that outcome. Perhaps, he did not recognize the enormous effect he had on the philosophy of his contemporaries. Toward the end of his life he probably felt, as Wittgenstein did, that words can only be understood within the meaning of linguistic human activities. “Words plus their behavioural surroundings make up the language game.” (Ref. 9, p.14)



References are in brackets following the quotations in the text, in which the first digits indicate the work referenced, followed by a comma, and other numerals after the p. indicate page numbers.

1.  B.G. Russell, Mysticism and Logic And Other Essays,  First (1910) 11th" impression 1959, George Allen and Goodwin Ltd., Ruskin House,  40 Museum St. London, G.B.

2.  B.G. Russell, Our Knowledge Of The External World, As A Field for Scientific Method in Philosophy, George Allen & Unwin Ltd. Ruskin House, 40 Museum St" London G"D. First (1914) 5th pub. 1961

3.  B.G. Russell: A 3. B. G Russell, History of Western Philosophy, Counterpoint Unwin Paperbacks, London, 1946

4. B. G Russell, Human Knowledge; Its Scope and Limits, Simon and Schuster, New York, 1948.

5.  W. Barrett, H.D. Aiken (ed) Philosophy In The Twentieth, Century, 2 Volumes, Random House, New York, 1962

6.  B. Blanshard, Reason and Belief, New Haven, Yale Univ. Press, 1975, Ch. XV. Goodness and the Absolute. On Bradley. (pp. 525-6)

7.  Weitz, Analysis, Philosophical in Edwards (ed) Encyclopedia of Philosophy, Vol. 1 (pp. 97-105)

8.  A. J. Ayer, Philosophy in the Twentieth Century, Vintage books, New York, 1st (1982) Random House, 2nd 1984

9.  A. Kenny,   L. Wittgenstein (Ed) Harvard U. Press, Cambridge Mass. (1973) 4th printing 1981

10.  Anthony Flew, A Dictionary of Philosophy (Ed)First Pub. 1979, Pan Books Ltd. 1984, Laurence Urlang Asst. Aylesburry, London