When I(Russell) say that my logic is atomistic, I mean that I share the common-sense belief that there are many separate things(lk178)

Propositions are not names for facts. I(Russell) had never realized it until pointed out to me by Ludwig Wittgenstein(lk187)

I(Russell) could no longer believe that knowing makes any difference to what is known. I prefer to describe my philosophy as 'logical atomism', rather than as 'realism'(lk323)

I(Russell) do not believe that there are complexes or unities in the same sense in which there are simples. I did believe this when I wrote The Principles of Mathematics, but, on account the doctrine of Types, I have since abandoned this view. To speak loosely, I regard simples and complexes as always of different types(lk336)

I must beg the reader to be indulgent if what I say is not exactly what I mean, and to try to see what I mean in spite of unavoidable linguistic obstacles to clear expression. In the language of mathematical Logic it is much easier to say what I want to say, but much harder to induce people to understand what I mean when I say it(lk336)

Immanuel Kant's theory of knowledge cannot be disentangled from his belief that mathematical propositions are both synthetic and a priori. My(Russell's) own work on the principles of mathematics was to me, at first, mainly interesting as a refutation of the view that mathematical propositions assert more than can be justified by deductive logic(lk368)

Having become convinced that the Georg Hegelian arguments against this and that were invalid, I(Russell) reacted to the opposite extreme and began to believe in the reality of whatever could not be disproved- e.g. points and instants and particles and Platonic universals(mpd10)

I(Russell) had regarded perception as a two-term relation of subject and object, as this had made it comparatively easy to understand how perception could give knowledge of something other than the subject. But under the influence of William James, I came to think this view mistaken, or at any rate an undue simplification. Sensations, at least, even those that are visual or auditory, came to seem to me not in their own nature relational occurrences(mpd10)

Whitehead persuaded Russell that one could do physics without supposing points and instants to be part of the stuff of the world. He considered - and in this Russell came to agree with him - that the stuff of the physical world could consist of events, each occupying a finite amount of space-time(mpd10)

If a relation is asymmetrical, transitive, and connected, then it arranges the terms of its field in a series. Russell believes he was the first to provide the word 'series' with an exact meaning in the form of the above statement(mpd69)

I(Russell) no longer think that the laws of logic are laws of things, on the contrary, I now regard them as purely linguistic. I no longer think of points, instants, and particles as part of the raw material of the world. What I said about induction in Our Knowledge of the External World now seems to me very crude. I spoke about universals and our knowledge of them with a confident assurance which I no longer feel, though I have no new opinions on the subject which I feel prepared to advocate with equal confidence(mpd77)

I(Russell) have done no definitely logical work since the second edition of the Principia in 1925, except the discussion of the principles of extensionality and atomicity and excluded middle in the 'Inquiry into Meaning and Truth'(mpd94)

I(Russell) thought when I wrote 'The Analysis of Mind' that it was possible to give a behavioristic account of desire, but as to this I now feel very doubtful(mpd111)

In 'The Analysis of Mind' I(Russell) argued the thesis that the 'stuff' of mental occurrences consists entirely of sensations and images. I do not know if this thesis was sound, but I am still quite convinced that many uses of language are inexplicable except by introducing images(mpd111)

A sentence has the property of duality of truth and falsehood. The problems involved in explaining the significance of sentences are both more difficult and more important than those involved in defining the meaning of object-words. In 'The Analysis of Mind' I(Russell) did not deal at all fully with these problems, but in 'An Inquiry into Meaning and Truth' I endeavored to offer adequate explanations in this region(mpd112)

The essential point on which I(Russell) differ from pragmatism is this: pragmatism holds that a belief is to be judged true if it has certain kinds of effects, whereas I hold that an empirical belief is to be judged true if it has certain kinds of causes(mpd131)

The solipsist will have to say 'I do not know whether I had a past or shall have a future, for these things are just as doubtful as the existence of other people or of the physical world.' No solipsist has ever gone as far as this, and therefore every solipsist has been inconsistent in accepting inferences about himself which have no better warrant than inferences about other people and things(mpd144)

As regards simples, I(Russell) can see no reason either to assert or deny that they may be reached by analysis. Ludwig Wittgenstein, in the Tractatus and I on occasion spoke of 'atomic facts' as the final reside in analysis, but it was never an essential part of the analytic philosophy to suppose that such facts were attainable(mpd164)

The central problem by which I(Russell) have sought to illustrate method is the problem of the relation between the crude data of sense and the space, time, and matter of mathematical physics. I have been made aware of the importance of this problem by my friend and collaborator Dr. Whitehead, to whom are due almost all the differences between the views advocated here and those suggested by The Problems of Philosophy. I owe to him the definition of points, the suggestion for the treatment of instants and "things," and the whole conception of the world of physics as a construction rather than an inference. What is said on these topics here is, in fact, a rough preliminary account of the more precise results which he is giving in the fourth volume of our Principia Mathematica. It will be seen that if his way of dealing with these topics is capable of being successfully carried through, a wholly new light is thrown on the time-honored controversies of realists and idealists, and a method is obtained of solving all that is soluble in their problem(okew11)

It was while Russell was attempting to make sense out of Gottfried Leibniz's diverse writings that he first realized the importance of relations to logic(pliv)

If Russell interpreted Gottfried Leibniz as admitting only subject-predicate propositions, then the fantastic doctrines followed as direct consequences. This revelation applied equally to the systems of Benedict Spinoza and Georg Hegel and Bradley; they had all denied the existence of external relations, but, in their cases, the resulting philosophy was a monism and not a pluralistic monadism(plv)

At the time when I wrote "The Philosophy of Leibniz," I knew little of mathematical logic, or of George Cantor's theory of infinite numbers. I should not now say as is said in the following pages, that the propositions of pure mathematics are "synthetic." The important distinction is between proposition deducible from logic and propositions not so deducible; the former may advantageously be defined as "analytic," the latter as "synthetic." Leibniz held that, for God, all propositions are analytic; modern logicians, for the most part regard pure mathematics as analytic, but consider all knowledge of matters of fact to be synthetic. Again, I should not now say: "It is evident that not every monad can have an organic body, if this consists of other subordinate monads" (p. 150). This assumes that the number of monads must be finite, whereas Leibniz supposed the number to be infinite. "In every particle of the universe," he says, "a world of infinite creatures is contained" (Opuscules, p. 522). Thus it is possible for every monad to have a body composed of subordinate monads, just as every fraction is greater than an infinite number of other fractions(plxii)

Why Gottfried Leibniz thought that monads cannot interact; how he became persuaded of the identity of indiscernibles; what he meant by the law of sufficient reason -- these and many other questions seemed to demand an answer, but to find none(plxvii)

At the time when I(Russell) wrote the "Principles" I shared with Gottlob Frege a belief in the Platonic reality of numbers, which, in my imagination peopled the timeless realm of Being(pmx)

Broadly speaking, I(Russell) still think the Principles is right where it disagrees with what had been previously held, but where it agrees with older theories it is apt to be wrong(pmxiv)

I(Russell) had seen Frege's Grounding, but, owing to the great difficulty of his symbolism, I had failed to grasp its importance or to understand its contents(pmxvi)

The subjects treated in the appendices of the Principles are so difficult that I(Russell) feel little confidence in my present opinions, and regard any conclusions which may be advocated as essentially hypotheses(pmxvi)

About six years ago, I(Russell) began an investigation into the philosophy of dynamics. I was met by the difficulty that, when a particle is subject to several forces, no one of the component accelerations actually occurs, but only resultant accelerations of which they are not parts; this fact rendered illusory such causation of particulars by particulars as is affirmed, at first sight, by the law of gravitation. It appeared also that the difficulty in regard to absolute motion is insoluble on a relational theory of space. From these two questions I was led to a re-examination of the principles of geometry, thence to the philosophy of continuity and infinity and thence, with a view to discovering the meaning of the word any, to symbolic logic(pmxvii)

The basis of Principia is presumably adequate to the derivation of all codified mathematical theory, except for a fringe requiring the axiom of infinity and the axiom of choice as additional assumptions(flpov89)

Russell, in 1900, was working on the first volume of his Principles of Mathematics. Here he undertook to analyze the logical foundations of arithmetic and other parts of mathematics within the medium of ordinary language, without explicit recourse to Peano's symbols(slp14)

In Principia relations are not reduced to classes(slp123)

The foundational portion of Whitehead and Russell's Principia Mathematica, taking up barely the first two hundred pages, contrasts markedly with the main body of that work. Propositional functions are in evidence only in the foundational part; thereafter the work proceeds in terms rather of classes and relations-in-extension(stl251)

Russell had also a philosophical preference for attributes, and felt that in contextually defining classes on the basis of a theory of attributes he was explaining the obscurer in terms of the clearer. But this feeling was due to his failure to distinguish between propositional functions as predicates, or expressions, and propositional functions as attributes(stl256)

In Principles of Mathematics, 1903, Russell's ontology was unrestrained. Every word referred to something. If the word was a proper name, in Russell's somewhat deviant sense of that phrase, its object was a thing, otherwise a concept. He limited the term 'existence' to things, but reckoned things liberally, even including instants and points of empty space(tt74)

The atoms of Russell's Logical Atomism are not atomic facts but sense data(tt83)

In Our Knowledge of the External World Russell wrote as though he had eliminated classes, and not just reduced them to attributes(tt84)

Principia has been advanced to such a point as to suggest that no fundamental difficulties stand in the way of completing the process. The foundations of arithmetic are developed in Principia, and therewith those branches of mathematics which, like analysis and theory of number, spring from arithmetic. Abstract algebra proceeds readily from the relation theory of Principia. Only geometry remains untouched, and this field can be brought into line simply by identifying n-dimensional figures with those n-adic arithmetical relations with which they are correlated through analytic geometry(wp87)