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Ideas of Bertrand Russell, by Text

[British, 1872 - 1970, Born at Trelleck. Professor at Cambridge (Trinity). Taught Wittgenstein. Imprisoned for pacificism. Campaigner against nuclear weapons. Died in N. Wales.]

1897 Foundations of Geometry
p.109 Geometry is united by the intuitive axioms of projective geometry
Intro vii p.109 Geometrical axioms imply the propositions, but the former may not be true
1900 The Philosophy of Leibniz
p.8 p.119 All philosophy should begin with an analysis of propositions
1901 Mathematics and the Metaphysicians
p.76 p.76 In mathematic we are ignorant of both subject-matter and truth
p.78 p.78 Self-evidence is often a mere will-o'-the-wisp
p.81 p.81 Zeno achieved the statement of the problems of infinitesimals, infinity and continuity
p.86 p.86 A collection is infinite if you can remove some terms without diminishing its number
p.89 p.89 To solve Zeno's paradox, reject the axiom that the whole has more terms than the parts
1902 Letters to Frege
1904 p.92 I take Mont Blanc to be an actual part of any assertion about it
1903 The Principles of Mathematics
p.3 Russell discovered the paradox suggested by Burali-Forti's work
p.3 Russell invented the naïve set theory usually attributed to Cantor
p.131 Being is what belongs to every possible object of thought
p.132 Russell tried to replace Peano's Postulates with the simple idea of 'class'
p.271 Russell's approach had to treat real 5/8 as different from rational 5/8
Pref p.-19 What is true or false is not mental, and is best called 'propositions'
§001 p.3 Pure mathematics is the class of propositions of the form 'p implies q'
§006 p.6 Constants are absolutely definite and unambiguous
§012 p.11 There seem to be eight or nine logical constants
§016 p.14 Implication cannot be defined
§026 p.23 Terms are identical if they belong to all the same classes
§037 p.33 It would be circular to use 'if' and 'then' to define material implication
§046 p.42 The study of grammar is underestimated in philosophy
§047 p.43 I call an object of thought a 'term'. This is a wide concept implying unity and existence.
§051 p.47 Proposition contain entities indicated by words, rather than the words themselves
§054 p.50 A proposition is a unity, and analysis destroys it
§070 p.69 A set has some sort of unity, but not enough to be a 'whole'
§079 p.81 The null class is a fiction
§093 p.94 Variables don't stand alone, but exist as parts of propositional functions
§108 p.111 Definition by analysis into constituents is useless, because it neglects the whole
§109 p.112 Numbers were once defined on the basis of 1, but neglected infinities and +
§109 p.113 Numbers are properties of classes
§109 p.113 We can define one-to-one without mentioning unity
§118 p.122 We do not currently know whether, of two infinite numbers, one must be greater than the other
§129 p.133 Counting explains none of the real problems about the foundations of arithmetic
§135 p.138 The part-whole relation is ultimate and indefinable
§138 p.141 Analysis gives us nothing but the truth - but never the whole truth
§150 p.158 Some quantities can't be measured, and some non-quantities are measurable
§157 p.166 The principle of Abstraction says a symmetrical, transitive relation analyses into an identity
§157 p.166 Abstraction principles identify a common property, which is some third term with the right relation
§157 p.167 A certain type of property occurs if and only if there is an equivalence relation
§168 n* p.181 Axiom of Archimedes: a finite multiple of a lesser magnitude can always exceed a greater
§183 p.192 Finite numbers, unlike infinite numbers, obey mathematical induction
§204 p.216 Order rests on 'between' and 'separation'
§208 p.218 Order depends on transitive asymmetrical relations
§209 p.219 'Reflexiveness' holds between a term and itself, and cannot be inferred from symmetry and transitiveness
§209 p.219 Symmetrical and transitive relations are formally like equality
§230 p.241 Some claim priority for the ordinals over cardinals, but there is no logical priority between them
§232 p.243 Ordinals presuppose two relations, where cardinals only presuppose one
§241 p.248 Induction is proved in Dedekind, an axiom in Peano; the latter seems simpler and clearer
§242 p.249 Ordinals can't be defined just by progression; they have intrinsic qualities
§243 p.251 Properties of numbers don't rely on progressions, so cardinals may be more basic
§243 p.251 Dedekind's ordinals are just members of any progression whatever
§249 p.259 There are cardinal and ordinal theories of infinity (while continuity is entirely ordinal)
§250 p.260 Infinite numbers are distinguished by disobeying induction, and the part equalling the whole
§258 p.270 Real numbers are a class of rational numbers (and so not really numbers at all)
§284 p.305 'Any' is better than 'all' where infinite classes are concerned
§288 p.311 You can't get a new transfinite cardinal from an old one just by adding finite numbers to it
§290 p.312 For every transfinite cardinal there is an infinite collection of transfinite ordinals
§290 p.312 Transfinite ordinals don't obey commutativity, so their arithmetic is quite different from basic arithmetic
§290 p.312 Ordinals are defined through mathematical induction
§290 p.312 Ordinals are types of series of terms in a row, rather than than the 'nth' instance
§291 p.313 ω names the whole series, or the generating relation of the series of ordinal numbers
§293 p.317 Ordinals result from likeness among relations, as cardinals from similarity among classes
§298 p.321 For Cantor ordinals are types of order, not numbers
§300 p.323 We aren't sure if one cardinal number is always bigger than another
§310 p.332 Denying mathematical induction gave us the transfinite
§321 p.350 The Achilles Paradox concerns the one-one correlation of infinite classes
§329 p.348 Infinite regresses have propositions made of propositions etc, with the key term reappearing
§337 p.355 Plato's 'Parmenides' is perhaps the best collection of antinomies ever made
§352 p.372 Pure geometry is deductive, and neutral over what exists
§353 p.373 In geometry, Kant and idealists aimed at the certainty of the premisses
§353 p.373 In geometry, empiricists aimed at premisses consistent with experience
§353 p.374 Geometry throws no light on the nature of actual space
§362 p.382 Two points have a line joining them (descriptive), a distance (metrical), and a whole line (projective)
§387 p.403 Mathematicians don't distinguish between instants of time and points on a line
§405 p.419 Quantity is not part of mathematics, where it is replaced by order
§412 p.429 In mathematics definitions are superfluous, as they name classes, and it all reduces to primitives
§423 p.445 Space is the extension of 'point', and aggregates of points seem necessary for geometry
§427 p.449 Many things have being (as topics of propositions), but may not have actual existence
§434 p.458 Mathematics doesn't care whether its entities exist
§437 p.465 Four classes of terms: instants, points, terms at instants only, and terms at instants and points
§439 p.466 Analysis falsifies, if when the parts are broken down they are not equivalent to their sum
§439 p.466 The only unities are simples, or wholes composed of parts
§439 p.467 Unities are only in propositions or concepts, and nothing that exists has unity
§440 p.467 The only classes are things, predicates and relations
§442 p.469 Occupying a place and change are prior to motion, so motion is just occupying places at continuous times
§442 p.470 The 'universe' can mean what exists now, what always has or will exist
§443 p.471 Change is obscured by substance, a thing's nature, subject-predicate form, and by essences
§448 p.474 Force is supposed to cause acceleration, but acceleration is a mathematical fiction.
§449 p.476 Moments and points seem to imply other moments and points, but don't cause them
§449 p.476 What exists has causal relations, but non-existent things may also have them
§459 p.485 The laws of motion and gravitation are just parts of the definition of a kind of matter
§460 p.486 We can drop 'cause', and just make inferences between facts
§478 p.478 "The death of Caesar is true" is not the same proposition as "Caesar died"
App B:523 p.8 As well as a truth value, propositions have a range of significance for their variables
App B:524 p.8 In 'x is a u', x and u must be of different types, so 'x is an x' is generally meaningless
p.249 p.175 Dedekind failed to distinguish the numbers from other progressions
p.42 p.100 Russell said the proposition must explain its own unity - or else objective truth is impossible
1904 To Frege (12/12/04)
p.97 Russell refuted Frege's principle that there is a set for each property
p.166 We don't assert private thoughts; the objects are part of what we assert
1905 On Denoting
p.5 Russell rejected sense/reference, because it made direct acquaintance with things impossible
p.6 The theory of descriptions eliminates the name of the entity whose existence was presupposed
p.12 Russell's theory explains non-existents, negative existentials, identity problems, and substitutivity
p.17 "Nobody" is not a singular term, but a quantifier
p.20 Existence is entirely expressed by the existential quantifier
p.20 'Elizabeth = Queen of England' is really a predication, not an identity-statement
p.25 Russell can't attribute existence to properties
p.27 Russell says names are not denotations, but definite descriptions in disguise
p.27 By eliminating descriptions from primitive notation, Russell seems to reject 'sense'
p.29 The meaning of a logically proper name is its referent, but most names are not logically proper
p.36 Critics say definite descriptions can refer, and may not embody both uniqueness and existence claims
p.37 Russell implies that 'the baby is crying' is only true if the baby is unique
p.42 Russell explained descriptions with quantifiers, where Frege treated them as names
p.43 Referring is not denoting, and Russell ignores the referential use of definite descriptions
p.60 Russell avoids non-existent objects by denying that definite descriptions are proper names
p.67 'Sense' is superfluous (rather than incoherent)
p.78 Russell assumes that expressions refer, but actually speakers refer by using expressions
p.101 Non-count descriptions don't threaten Russell's theory, which is only about singulars
p.129 Russell's theory must be wrong if it says all statements about non-existents are false
p.131 The theory of definite descriptions aims at finding correct truth conditions
p.155 Russell says a name contributes a complex of properties, rather than an object
p.155 Names don't have a sense, but are disguised definite descriptions
p.175 Denoting is crucial in Russell's account of mathematics, for identifying classes
p.183 The Theory of Description dropped classes and numbers, leaving propositions, individuals and universals
p.226 A definite description 'denotes' an entity if it fits the description uniquely
p.268 Are names descriptions, if the description is unknown, false, not special, or contains names?
p.411 Definite descriptions fail to refer in three situations, so they aren't essentially referring
p.535 Logically proper names introduce objects; definite descriptions introduce quantifications
p.42 p.42 The idea of a variable is fundamental
p.43 p.43 Denoting phrases are meaningless, but guarantee meaning for propositions
p.46 p.46 In 'Scott is the author of Waverley', denotation is identical, but meaning is different
p.54 p.54 The ontological argument begins with an unproven claim that 'there exists an x..'
1907 Regressive Method for Premises in Mathematics
p.272 p.272 It seems absurd to prove 2+2=4, where the conclusion is more certain than premises
p.272 p.272 Arithmetic was probably inferred from relationships between physical objects
p.273 p.273 Which premises are ultimate varies with context
p.273 p.273 The sources of a proof are the reasons why we believe its conclusion
p.274 p.274 Non-contradiction was learned from instances, and then found to be indubitable
p.274 p.274 Induction is inferring premises from consequences
p.275 p.275 The law of gravity has many consequences beyond its grounding observations
p.276 p.276 Peano axioms not only support arithmetic, but are also fairly obvious
p.276 p.276 Arithmetic can have even simpler logical premises than the Peano Axioms
p.279 p.279 If one proposition is deduced from another, they are more certain together than alone
p.279 p.279 The most obvious beliefs are not infallible, as other obvious beliefs may conflict
p.279 p.279 Believing a whole science is more than believing each of its propositions
p.282 p.282 Finding the axioms may be the only route to some new results
p.283 p.283 Discoveries in mathematics can challenge philosophy, and offer it a new foundation
1908 Mathematical logic and theory of types
p.30 The class of classes which lack self-membership leads to a contradiction
p.38 Classes can be reduced to propositional functions
p.44 Type theory seems an extreme reaction, since self-exemplification is often innocuous
p.102 Russell's improvements blocked mathematics as well as paradoxes, and needed further axioms
p.230 A set does not exist unless at least one of its specifications is predicative
p.233 Russell is a conceptualist here, saying some abstracta only exist because definitions create them
p.63,75 p.225 Vicious Circle says if it is expressed using the whole collection, it can't be in the collection
1910 On the Nature of Truth and Falsehood
p.106 For Russell, both propositions and facts are arrangements of objects, so obviously they correspond
p.188 In 1906, Russell decided that propositions did not, after all, exist
1911 Knowledge by Acquaintance and Description-1
p.211 My 'acquaintance' with sense-data is nothing like my knowing New York
1911 On Relations of Universals and Particulars
p.160 Trope theorists cannot explain how tropes resemble each other
1912 On the Notion of Cause
p.173 p.173 The law of causality is a source of confusion, and should be dropped from philosophy
p.175 p.175 'Necessary' is a predicate of a propositional function, saying it is true for all values of its argument
p.177 p.177 If causes are contiguous with events, only the last bit is relevant, or the event's timing is baffling
p.178 p.178 Philosophers usually learn science from each other, not from science
p.179 p.179 In causal laws, 'events' must recur, so they have to be universals, not particulars
p.185 p.185 Striking a match causes its igniting, even if it sometimes doesn't work
p.186 p.186 The constancy of scientific laws rests on differential equations, not on cause and effect
1912 Problems of Philosophy
p. If Russell rejects innate ideas and direct a priori knowledge, he is left with a tabula rasa
p.33 Russell started philosophy of language, by declaring some plausible sentences to be meaningless
p.43 After 1912, Russell said sense-data are last in analysis, not first in experience
p.176 Russell (1912) said phenomena only resemble reality in abstract structure
p.213 Russell's representationalism says primary qualities only show the structure of reality
Ch. 1 p.1 It is natural to begin from experience, and presumably that is the basis of knowledge
Ch. 1 p.4 'Sense-data' are what are immediately known in sensation, such as colours or roughnesses
Ch. 2 p.8 Descartes showed that subjective things are the most certain
Ch. 2 p.9 Philosophers must get used to absurdities
Ch. 2 p.10 It is not illogical to think that only myself and my mental events exist
Ch. 2 p.10 If the cat reappears in a new position, presumably it has passed through the intermediate positions
Ch. 2 p.11 Belief in real objects makes our account of experience simpler and more systematic
Ch. 2 p.11 It is hard not to believe that speaking humans are expressing thoughts, just as we do ourselves
Ch. 2 p.11 Dreams can be explained fairly scientifically if we assume a physical world
Ch. 2 p.11 We have an 'instinctive' belief in the external world, prior to all reflection
Ch. 2 p.12 Philosophy verifies that our hierarchy of instinctive beliefs is harmonious and consistent
Ch. 3 p.13 It is rational to believe in reality, despite the lack of demonstrative reasons for it
Ch. 3 p.14 Space is neutral between touch and sight, so it cannot really be either of them
Ch. 3 p.16 Because we depend on correspondence, we know relations better than we know the items that relate
Ch. 3 p.18 There is no reason to think that objects have colours
Ch. 4 p.19 'Idealism' says that everything which exists is in some sense mental
Ch. 4 p.23 Knowledge of truths applies to judgements; knowledge by acquaintance applies to sensations and things
Ch. 4 p.23 I can know the existence of something with which nobody is acquainted
Ch. 5 p.25 'Acquaintance' is direct awareness, without inferences or judgements
Ch. 5 p.26 All knowledge (of things and of truths) rests on the foundations of acquaintance
Ch. 5 p.27 If we didn't know our own minds by introspection, we couldn't know that other people have minds
Ch. 5 p.27 In perceiving the sun, I am aware of sun sense-data, and of the perceiver of the data
Ch. 5 p.28 In seeing the sun, we are acquainted with our self, but not as a permanent person
Ch. 5 p.28 A universal of which we are aware is called a 'concept'
Ch. 5 p.28 We are acquainted with outer and inner sensation, memory, Self, and universals
Ch. 5 p.28 Every complete sentence must contain at least one word (a verb) which stands for a universal
Ch. 5 p.28 The phrase 'a so-and-so' is an 'ambiguous' description'; 'the so-and-so' (singular) is a 'definite' description
Ch. 5 p.29 Proper names are really descriptions, and can be replaced by a description in a person's mind
Ch. 5 p.30 It is pure chance which descriptions in a person's mind make a name apply to an individual
Ch. 5 p.32 Knowledge by descriptions enables us to transcend private experience
Ch. 6 p.32 Every understood proposition is composed of constituents with which we are acquainted
Ch. 6 p.35 Science aims to find uniformities to which (within the limits of experience) there are no exceptions
Ch. 6 p.35 Chickens are not very good at induction, and are surprised when their feeder wrings their neck
Ch. 6 p.36 It doesn't follow that because the future has always resembled the past, that it always will
Ch. 6 p.36 We can't know that our laws are exceptionless, or even that there are any laws
Ch. 6 p.38 We can't prove induction from experience without begging the question
Ch. 7 p.40 Three Laws of Thought: identity, contradiction, and excluded middle
Ch. 7 p.40 Demonstration always relies on the rule that anything implied by a truth is true
Ch. 7 p.41 The rationalists were right, because we know logical principles without experience
Ch. 7 p.42 Judgements of usefulness depend on judgements of value
Ch. 7 p.43 Maths is not known by induction, because further instances are not needed to support it
Ch. 7 p.43 In any possible world we feel that two and two would be four
Ch. 7 p.45 The mortality of Socrates is more certain from induction than it is from deduction
Ch. 8 p.46 Kant showed that we have a priori knowledge which is not purely analytic
Ch. 9 p.6 Russell claims that universals are needed to explain a priori knowledge (as their relations)
Ch. 9 p.50 The law of contradiction is not a 'law of thought', but a belief about things
Ch. 9 p.53 Every sentence contains at least one word denoting a universal, so we need universals to know truth
Ch. 9 p.54 Propositions express relations (prepositions and verbs) as well as properties (nouns and adjectives)
Ch. 9 p.54 Confused views of reality result from thinking that only nouns and adjectives represent universals
Ch. 9 p.55 All universals are like the relation "is north of", in having no physical location at all
Ch. 9 p.55 'Resemblance Nominalism' won't work, because the theory treats resemblance itself as a universal
Ch. 9 p.56 That Edinburgh is north of London is a non-mental fact, so relations are independent universals
Ch. 9 p.57 Normal existence is in time, so we must say that universals 'subsist'
Ch. 9 p.57 If we identify whiteness with a thought, we can never think of it twice; whiteness is the object of a thought
Ch. 9 p.57 If we consider whiteness to be merely a mental 'idea', we rob it of its universality
Ch.10 p.59 All a priori knowledge deals with the relations of universals
Ch.10 p.59 I learn the universal 'resemblance' by seeing two shades of green, and their contrast with red
Ch.10 p.62 We can know some general propositions by universals, when no instance can be given
Ch.11 p.65 Some propositions are just self-evident, but some proven propositions are also self-evident
Ch.11 p.65 Particular instances are more clearly self-evident than any general principles
Ch.11 p.66 Images are not memory, because they are present, and memories are of the past
Ch.11 p.67 As shown by memory, self-evidence comes in degrees
Ch.11 p.68 If self-evidence has degrees, we should accept the more self-evident as correct
Ch.12 p. Truth as congruence may work for complex beliefs, but not for simple beliefs about existence
Ch.12 p.70 The coherence theory says falsehood is failure to cohere, and truth is fitting into a complete system of Truth
Ch.12 p.70 A good theory of truth must make falsehood possible
Ch.12 p.70 Truth is a property of a belief, but dependent on its external relations, not its internal qualities
Ch.12 p.70 Truth and falsehood are properties of beliefs and statements
Ch.12 p.70 In a world of mere matter there might be 'facts', but no truths
Ch.12 p.71 More than one coherent body of beliefs seems possible
Ch.12 p.71 If we suspend the law of contradiction, nothing will appear to be incoherent
Ch.12 p.71 Coherence is not the meaning of truth, but an important test for truth
Ch.12 p.72 In order to explain falsehood, a belief must involve several terms, not two
Ch.12 p.73 Belief relates a mind to several things other than itself
Ch.12 p.74 Truth is when a mental state corresponds to a complex unity of external constituents
Ch.12 p.75 Beliefs are true if they have corresponding facts, and false if they don't
Ch.13 p.76 True belief is not knowledge when it is deduced from false belief
Ch.13 p.76 A true belief is not knowledge if it is reached by bad reasoning
Ch.13 p.78 Knowledge cannot be precisely defined, as it merges into 'probable opinion'
Ch.14 p.82 Metaphysics cannot give knowledge of the universe as a whole
Ch.14 p.87 Philosophy is similar to science, and has no special source of wisdom
1913 The Theory of Knowledge
p.129 p.127 Logical truths are known by their extreme generality
1914 On the Nature of Acquaintance
p.35 The only real proper names are 'this' and 'that'; the rest are really definite descriptions.
1914 Our Knowledge of the External World
p.45 Other minds seem to exist, because their testimony supports realism about the world
1914 The Relation of Sense-Data to Physics
p.3 Russell held that we are aware of states of our own brain
p.131 Sense-data are qualities devoid of subjectivity, which are the basis of science
§II p.142 We do not know whether sense-data exist as objects when they are not data
§II p.142 Individuating sense-data is difficult, because they divide when closely attended to
§III p.143 Ungiven sense-data can no more exist than unmarried husbands
§III p.143 'Sensibilia' are identical to sense-data, without actually being data for any mind
§III p.144 Sense-data are not mental, but are part of the subject-matter of physics
§IV p.146 Sense-data are objects, and do not contain the subject as part, the way beliefs do
§IV p.146 Sense-data are usually objects within the body, but are not part of the subject
§IX p.158 Matter is the limit of appearances as distance from the object diminishes
§V p.149 We need not deny substance, but there seems no reason to assert it
§VI p.149 Where possible, logical constructions are to be substituted for inferred entities
§VII p.152 No sensibile is ever a datum to two people at once
§VII p.153 There is 'private space', and there is also the 'space of perspectives'
§VIII p.157 Sense-data may be subjective, if closing our eyes can change them
§XI p.162 The assumption by physicists of permanent substance is not metaphysically legitimate
§XI p.164 Continuity is a sufficient criterion for the identity of a rock, but not for part of a smooth fluid
§XI p.165 Physical things are series of appearances whose matter obeys physical laws
1915 The Ultimate Constituents of Matter
p.123 p.123 Visible things are physical and external, but only exist when viewed
p.124 p.124 A man is a succession of momentary men, bound by continuity and causation
p.125 p.125 Matter requires a division into time-corpuscles as well as space-corpuscles
p.125 p.125 Classes, grouped by a convenient property, are logical constructions
p.126 p.126 If my body literally lost its mind, the object seen when I see a flash would still exist
p.131 p.131 We could probably, in principle, infer minds from brains, and brains from minds
p.132 p.132 Matter is a logical construction
p.134 p.134 Six dimensions are needed for a particular, three within its own space, and three to locate that space
p.138 p.138 Sense-data are purely physical
1918 The Philosophy of Logical Atomism
p.18 'Existence' means that a propositional function is sometimes true
p.35 Not only atomic truths, but also general and negative truths, have truth-makers
p.50 Treat description using quantifiers, and treat proper names as descriptions
p.54 Russell asserts atomic, existential, negative and general facts
p.68 Russell uses 'propositional function' to refer to both predicates and to attributes
p.83 Russell's atomic facts are actually compounds, and his true logical atoms are sense data
§I p.33 Logical atomism aims at logical atoms as the last residue of analysis
§I p.36 Facts make propositions true or false, and are expressed by whole sentences
§I p.41 Propositions don't name facts, because each fact corresponds to a proposition and its negation
§II p.52 In a logically perfect language, there will be just one word for every simple object
§II p.53 The names in a logically perfect language would be private, and could not be shared
§III p.70 An inventory of the world does not need to include propositions
§III p.71 The business of metaphysics is to describe the world
§IV.2 p.79 I no longer believe in propositions, especially concerning falsehoods
§IV.3 p.81 The theory of error seems to need the existence of the non-existent
§IV.4 p.84 Perception goes straight to the fact, and not through the proposition
§V p.88 Modal terms are properties of propositional functions, not of propositions
§V p.93 Once you have enumerated all the atomic facts, there is a further fact that those are all the facts
§VI p.100 Romulus does not occur in the proposition 'Romulus did not exist'
§VI p.102 You can understand 'author of Waverley', but to understand 'Scott' you must know who it applies to
§VI p.108 You can discuss 'God exists', so 'God' is a description, not a name
§VII p.126 Normally a class with only one member is a problem, because the class and the member are identical
§VIII p.129 Logical atoms aims to get down to ultimate simples, with their own unique reality
§VIII p.129 Numbers are classes of classes, and hence fictions of fictions
§VIII p.140 Reducing entities and premisses makes error less likely
24th pg p.394 There are a set of criteria for pinning down a logically proper name
66th pg p.399 A name has got to name something or it is not a name
1919 Introduction to Mathematical Philosophy
p.58 Russell's proposal was that only meaningful predicates have sets as their extensions
p.58 The sentence 'procrastination drinks quadruplicity' is meaningless, rather than false
p.168 For Russell, numbers are sets of equivalent sets
Pref p.-6 Infinity and continuity used to be philosophy, but are now mathematics
Ch.18 p.123 Classes are logical fictions, and are not part of the ultimate furniture of the world
I p.3 Discovering that 1 is a number was difficult
I p.8 Any founded, non-repeating series all reachable in steps will satisfy Peano's axioms
I p.9 '0', 'number' and 'successor' cannot be defined by Peano's axioms
I p.10 Numbers are needed for counting, so they need a meaning, and not just formal properties
II p.12 A number is something which characterises collections of the same size
II p.12 A definition by 'extension' enumerates items, and one by 'intension' gives a defining property
II p.13 We can enumerate finite classes, but an intensional definition is needed for infinite classes
II p.13 Members define a unique class, whereas defining characteristics are numerous
II p.16 If a relation is symmetrical and transitive, it has to be reflexive
II n1 p.14 Classes are logical fictions, made from defining characteristics
IV p.38 The definition of order needs a transitive relation, to leap over infinite intermediate terms
IX p.94 The formal laws of arithmetic are the Commutative, the Associative and the Distributive
p.175 p.536 Russell admitted that even names could also be used as descriptions
p.8 p.148 Could a number just be something which occurs in a progression?
V p.42 'Asymmetry' is incompatible with its converse; a is husband of b, so b can't be husband of a
VI p.55 Mathematically expressed propositions are true of the world, but how to interpret them?
VI p.59 What matters is the logical interrelation of mathematical terms, not their intrinsic nature
VI p.61 The essence of individuality is beyond description, and hence irrelevant to science
VII p.69 A series can be 'Cut' in two, where the lower class has no maximum, the upper no minimum
VII p.71 Dedekind's axiom that his Cut must be filled has the advantages of theft over honest toil
VII p.74 New numbers solve problems: negatives for subtraction, fractions for division, complex for equations
VII p.75 A complex number is simply an ordered couple of real numbers
VIII p.77 We may assume that there are infinite collections, as there is no logical reason against them
X p.101 If straight lines were like ratios they might intersect at a 'gap', and have no point in common
XII p.119 The British parliament has one representative selected from each constituency
XII p.123 Choice is equivalent to the proposition that every class is well-ordered
XII p.124 Choice shows that if any two cardinals are not equal, one must be the greater
XII p.126 We can pick all the right or left boots, but socks need Choice to insure the representative class
XIII p.131 Infinity says 'for any inductive cardinal, there is a class having that many terms'
XIV p.149 There is always something psychological about inference
XIV p.153 Inferring q from p only needs p to be true, and 'not-p or q' to be true
XIV p.154 All forms of implication are expressible as truth-functions
XV p.155 Propositions are mainly verbal expressions of true or false, and perhaps also symbolic thoughts
XV p.159 All the propositions of logic are completely general
XV p.164 The Darapti syllogism is fallacious: All M is S, all M is P, so some S is P' - but if there is no M?
XV p.164 An argument 'satisfies' a function φx if φa is true
XVI p.169 Logic is concerned with the real world just as truly as zoology
XVI p.170 'I met a unicorn' is meaningful, and so is 'unicorn', but 'a unicorn' is not
XVI p.172 'Socrates is human' expresses predication, and 'Socrates is a man' expresses identity
XVI p.178 The only genuine proper names are 'this' and 'that'
XVI p.178 Names are really descriptions, except for a few words like 'this' and 'that'
XVI p.179 Asking 'Did Homer exist?' is employing an abbreviated description
XVII p.191 Reducibility: a family of functions is equivalent to a single type of function
XVII p.193 Propositions about classes can be reduced to propositions about their defining functions
XVII p.193 If something is true in all possible worlds then it is logically necessary
XVIII p.194 In modern times, logic has become mathematical, and mathematics has become logical
XVIII p.203 Existence can only be asserted of something described, not of something named
XVIII p.204 Logic can only assert hypothetical existence
XVIII p.204 Logic can be known a priori, without study of the actual world
XVIII p.205 Maybe numbers are adjectives, since 'ten men' grammatically resembles 'white men'
1919 On Propositions: What they are,and Meaning
§II p.294 If we object to all data which is 'introspective' we will cease to believe in toothaches
§II p.299 There are distinct sets of psychological and physical causal laws
§III p.304 The three questions about belief are its contents, its success, and its character
§III p.307 Our important beliefs all, if put into words, take the form of propositions
§III p.308 A proposition expressed in words is a 'word-proposition', and one of images an 'image-proposition'
§IV p.319 Propositions of existence, generalities, disjunctions and hypotheticals make correspondence tricky
§IV p.320 In its primary and formal sense, 'true' applies to propositions, not beliefs
p.285 p.285 The truth or falsehood of a belief depends upon a fact to which the belief 'refers'
p.285 p.285 A proposition is what we believe when we believe truly or falsely
1921 The Analysis of Mind
p.46 In 1921 Russell abandoned sense-data, and the gap between sensation and object
Lec. VIII p.141 In perception, the self is just a logical fiction demanded by grammar
Lec. VIII p.141 Seeing is not in itself knowledge, but is separate from what is seen, such as a patch of colour
Lec. VIII p.142 We cannot assume that the subject actually exists, so we cannot distinguish sensations from sense-data
p.159 p.159 It is possible the world came into existence five minutes ago, complete with false memories
1923 Vagueness
p.9 Since natural language is not precise it cannot be in the province of logic
p.62 p.62 Vagueness is only a characteristic of representations, such as language
1924 Logical Atomism
p.17 Russell gave up logical atomism because of negative, general and belief propositions
p.143 p.143 It is logic, not metaphysics, that is fundamental to philosophy
p.145 p.145 Maths can be deduced from logical axioms and the logic of relations
p.145 p.145 Some axioms may only become accepted when they lead to obvious conclusions
p.151 p.151 Subject-predicate logic (and substance-attribute metaphysics) arise from Aryan languages
p.152 p.152 As propositions can be put in subject-predicate form, we wrongly infer that facts have substance-quality form
p.153 p.153 Meaning takes many different forms, depending on different logical types.
p.156 p.156 To mean facts we assert them; to mean simples we name them
p.158 p.158 'Simples' are not experienced, but are inferred at the limits of analysis
p.159 p.159 Vagueness, and simples being beyond experience, are obstacles to a logical language
p.159 p.159 A logical language would show up the fallacy of inferring reality from ordinary language
p.160 p.160 Philosophy should be built on science, to reduce error
p.162 p.162 Philosophy is logical analysis, followed by synthesis
1927 The Analysis of Matter
p.24 In 1927, Russell analysed force and matter in terms of events
p.61 Russell rejected phenomenalism because it couldn't account for causal relations
23 p.244 An object produces the same percepts with or without a substance, so that is irrelevant to science
23 p.244 A perceived physical object is events grouped around a centre
1940 An Inquiry into Meaning and Truth
5 p.75 A 'heterological' predicate can't be predicated of itself; so is 'heterological' heterological? Yes=no!
5 p.76 For simple words, a single experience can show that they are true
5 p.77 Asserting not-p is saying p is false
5 p.79 'Or' expresses hesitation, in a dog at a crossroads, or birds risking grabbing crumbs
5 p.79 A mother cat is paralysed if equidistant between two needy kittens
5 p.80 A disjunction expresses indecision
5 p.81 'Or' expresses a mental state, not something about the world
5 p.82 All our knowledge (if verbal) is general, because all sentences contain general words
5 p.83 Disjunction may also arise in practice if there is imperfect memory.
5 p.83 There are four experiences that lead us to talk of 'some' things
5 p.86 Perception can't prove universal generalisations, so abandon them, or abandon empiricism?
5 p.88 The physical world doesn't need logic, but the mental world does
5 p.88 Maybe the 'or' used to describe mental states is not the 'or' of logic
c.p.88 p.185 Questions wouldn't lead anywhere without the law of excluded middle
p.13 p.13 Naïve realism leads to physics, but physics then shows that naïve realism is false
1948 Human Knowledge: its scope and limits
p.113 Russell's 'at-at' theory says motion is to be at the intervening points at the intervening instants
9 p.111 Is it possible to state every possible truth about the whole course of nature without using 'not'?
9 p.111 It is hard to explain how a sentence like 'it is not raining' can be found true be observation
9 p.113 Some facts about experience feel like logical necessities
9 p.114 If we define 'this is not blue' as disbelief in 'this is blue', we eliminate 'not' as an ingredient of facts
1954 Human Society in Ethics and Politics
p.48 p.226 If God's decrees are good, and this is not a mere tautology, then goodness is separate from God's decrees
1959 My Philosophical Development
Ch.1 p.9 In 1899-1900 I adopted the philosophy of logical atomism
Ch.1 p.11 Only by analysing is progress possible in philosophy
Ch.2 p.82 Intuitionism says propositions are only true or false if there is a method of showing it
Ch.5 p.48 Leibniz bases everything on subject/predicate and substance/property propositions
Ch.7 p.57 We tried to define all of pure maths using logical premisses and concepts
Ch.10 p.82 Formalists say maths is merely conventional marks on paper, like the arbitrary rules of chess
Ch.10 p.82 Formalism can't apply numbers to reality, so it is an evasion
Ch.10 p.83 Unverifiable propositions about the remote past are still either true or false
Ch.11 p.95 In epistemology we should emphasis the continuity between animal and human minds
Ch.11 p.97 Empiricists seem unclear what they mean by 'experience'
Ch.11 p.98 Analysis gives new knowledge, without destroying what we already have
Ch.13 p.111 Behaviourists struggle to explain memory and imagination, because they won't admit images
Ch.13 p.112 Facts are everything, except simples; they are either relations or qualities
Ch.13 p.114 You can believe the meaning of a sentence without thinking of the words
Ch.14 p.117 I gradually replaced classes with properties, and they ended as a symbolic convenience
Ch.14 p.119 The theory of types makes 'Socrates and killing are two' illegitimate
Ch.14 p.123 Complex things can be known, but not simple things
Ch.14 p.125 Names are meaningless unless there is an object which they designate
Ch.14 p.128 Universals can't just be words, because words themselves are universals
Ch.15 p.131 Pragmatism judges by effects, but I judge truth by causes
Ch.15 p.136 Surprise is a criterion of error
Ch.15 p.136 Truth belongs to beliefs, not to propositions and sentences
Ch.15 p.140 True belief about the time is not knowledge if I luckily observe a stopped clock at the right moment