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Single Idea 21558

[filed under theme 6. Mathematics / C. Sources of Mathematics / 10. Constructivism / d. Predicativism ]

Full Idea

Norms (containing one variable) which do not define classes I propose to call 'non-predicative'; those which do define classes I shall call 'predicative'.

Gist of Idea

'Predicative' norms are those which define a class

Source

Bertrand Russell (Difficulties of Transfinite Numbers and Types [1905], p.141)

Book Ref

Russell,Bertrand: 'Essays in Analysis', ed/tr. Lackey,Douglas [George Braziller 1973], p.141

Related Ideas

Idea 21559 We need rules for deciding which norms are predicative (unless none of them are) [Russell]

Idea 21568 A one-variable function is only 'predicative' if it is one order above its arguments [Russell]

The 581 ideas from Bertrand Russell

6410 The only real proper names are 'this' and 'that'; the rest are really definite descriptions. [Russell, by Grayling]
6417 In 1921 Russell abandoned sense-data, and the gap between sensation and object [Russell, by Grayling]
6475 In perception, the self is just a logical fiction demanded by grammar [Russell]
6474 Seeing is not in itself knowledge, but is separate from what is seen, such as a patch of colour [Russell]
6476 We cannot assume that the subject actually exists, so we cannot distinguish sensations from sense-data [Russell]
2792 It is possible the world came into existence five minutes ago, complete with false memories [Russell]
22326 Knowledge needs more than a sensitive response; the response must also be appropriate [Russell]
21706 At first matter is basic and known by sense-data; later Russell says matter is constructed [Russell, by Linsky,B]
6418 Russell rejected phenomenalism because it couldn't account for causal relations [Russell, by Grayling]
6402 In 1927, Russell analysed force and matter in terms of events [Russell, by Grayling]
14732 A perceived physical object is events grouped around a centre [Russell]
14733 An object produces the same percepts with or without a substance, so that is irrelevant to science [Russell]
23154 We divide mankind into friend and foe, and cooperate with one and compete with the other [Russell]
23153 Gradually loyalty to a creed increased, which could even outweigh nationality [Russell]
23152 Increasingly war expands communities, and unifies them through fear [Russell]
23151 A monarch is known to everyone in the group, and can thus unite large groups [Russell]
23155 In early societies the leaders needed cohesion, but the rest just had to obey [Russell]
23156 The economic and political advantages of great size seem to have no upper limit [Russell]
23157 Government has a negative purpose, to prevent trouble, and a positive aim of realising our desires [Russell]
23162 Managers are just as remote from workers under nationalisation as under capitalism [Russell]
23159 Scarce goods may be denied entirely, to avoid their unequal distribution [Russell]
23161 Slaves can be just as equal as free people [Russell]
23160 Being a slave of society is hardly better than being a slave of a despot [Russell]
23158 Slavery began the divorce between the work and the purposes of the worker [Russell]
23165 Socialists say economic justice needs some state control of industries, and of foreign trade [Russell]
23163 In an anarchy universities, research, books, and even seaside holidays, would be impossible [Russell]
23164 Modern justice is seen as equality, apart from modest extra rewards for exceptional desert [Russell]
23167 Power should be with smaller bodies, as long as it doesn't restrict central powers [Russell]
23168 A state is essential, to control greedy or predatory impulses [Russell]
23166 In democracy we are more aware of being governed than of our tiny share in government [Russell]
20179 Happiness involves enduring boredom, and the young should be taught this [Russell]
20178 Life is now more interesting, but boredom is more frightening [Russell]
20177 Boredom always involves not being fully occupied [Russell]
20180 A happy and joyous life must largely be a quiet life [Russell]
20176 Boredom is an increasingly strong motivating power [Russell]
10052 Geometry is united by the intuitive axioms of projective geometry [Russell, by Musgrave]
10053 Geometrical axioms imply the propositions, but the former may not be true [Russell]
2609 If God's decrees are good, and this is not a mere tautology, then goodness is separate from God's decrees [Russell]
4786 Russell's 'at-at' theory says motion is to be at the intervening points at the intervening instants [Russell, by Psillos]
16491 If we define 'this is not blue' as disbelief in 'this is blue', we eliminate 'not' as an ingredient of facts [Russell]
16489 Is it possible to state every possible truth about the whole course of nature without using 'not'? [Russell]
16490 Some facts about experience feel like logical necessities [Russell]
16488 It is hard to explain how a sentence like 'it is not raining' can be found true by observation [Russell]
16478 A mother cat is paralysed if equidistant between two needy kittens [Russell]
16476 For simple words, a single experience can show that they are true [Russell]
16485 Perception can't prove universal generalisations, so abandon them, or abandon empiricism? [Russell]
16482 All our knowledge (if verbal) is general, because all sentences contain general words [Russell]
16479 'Or' expresses hesitation, in a dog at a crossroads, or birds risking grabbing crumbs [Russell]
16480 A disjunction expresses indecision [Russell]
16483 Disjunction may also arise in practice if there is imperfect memory. [Russell]
16481 'Or' expresses a mental state, not something about the world [Russell]
16487 Maybe the 'or' used to describe mental states is not the 'or' of logic [Russell]
16486 The physical world doesn't need logic, but the mental world does [Russell]
16477 Asserting not-p is saying p is false [Russell]
16484 There are four experiences that lead us to talk of 'some' things [Russell]
16475 A 'heterological' predicate can't be predicated of itself; so is 'heterological' heterological? Yes=no! [Russell]
2947 Questions wouldn't lead anywhere without the law of excluded middle [Russell]
4758 Naïve realism leads to physics, but physics then shows that naïve realism is false [Russell]
21563 The 'no classes' theory says the propositions just refer to the members [Russell]
21564 Vicious Circle: what involves ALL must not be one of those ALL [Russell]
21565 Richard's puzzle uses the notion of 'definition' - but that cannot be defined [Russell]
13414 For Russell, numbers are sets of equivalent sets [Russell, by Benacerraf]
8469 Russell's proposal was that only meaningful predicates have sets as their extensions [Russell, by Orenstein]
8468 The sentence 'procrastination drinks quadruplicity' is meaningless, rather than false [Russell, by Orenstein]
14420 Infinity and continuity used to be philosophy, but are now mathematics [Russell]
8745 Classes are logical fictions, and are not part of the ultimate furniture of the world [Russell]
14423 '0', 'number' and 'successor' cannot be defined by Peano's axioms [Russell]
14421 Discovering that 1 is a number was difficult [Russell]
14424 Numbers are needed for counting, so they need a meaning, and not just formal properties [Russell]
14422 Any founded, non-repeating series all reachable in steps will satisfy Peano's axioms [Russell]
14425 A number is something which characterises collections of the same size [Russell]
14427 We can enumerate finite classes, but an intensional definition is needed for infinite classes [Russell]
14428 Members define a unique class, whereas defining characteristics are numerous [Russell]
14426 A definition by 'extension' enumerates items, and one by 'intension' gives a defining property [Russell]
14430 If a relation is symmetrical and transitive, it has to be reflexive [Russell]
14429 Classes are logical fictions, made from defining characteristics [Russell]
14431 The definition of order needs a transitive relation, to leap over infinite intermediate terms [Russell]
14441 The formal laws of arithmetic are the Commutative, the Associative and the Distributive [Russell]
10450 Russell admitted that even names could also be used as descriptions [Russell, by Bach]
13510 Could a number just be something which occurs in a progression? [Russell, by Hart,WD]
14432 'Asymmetry' is incompatible with its converse; a is husband of b, so b can't be husband of a [Russell]
14433 Mathematically expressed propositions are true of the world, but how to interpret them? [Russell]
14435 The essence of individuality is beyond description, and hence irrelevant to science [Russell]
14434 What matters is the logical interrelation of mathematical terms, not their intrinsic nature [Russell]
14436 A series can be 'Cut' in two, where the lower class has no maximum, the upper no minimum [Russell]
14439 A complex number is simply an ordered couple of real numbers [Russell]
14438 New numbers solve problems: negatives for subtraction, fractions for division, complex for equations [Russell]
14440 We may assume that there are infinite collections, as there is no logical reason against them [Russell]
14442 If straight lines were like ratios they might intersect at a 'gap', and have no point in common [Russell]
14443 The British parliament has one representative selected from each constituency [Russell]
14445 Choice shows that if any two cardinals are not equal, one must be the greater [Russell]
14444 Choice is equivalent to the proposition that every class is well-ordered [Russell]
14446 We can pick all the right or left boots, but socks need Choice to insure the representative class [Russell]
14447 Infinity says 'for any inductive cardinal, there is a class having that many terms' [Russell]
14449 There is always something psychological about inference [Russell]
12197 Inferring q from p only needs p to be true, and 'not-p or q' to be true [Russell]
14450 All forms of implication are expressible as truth-functions [Russell]
14451 Propositions are mainly verbal expressions of true or false, and perhaps also symbolic thoughts [Russell]
14454 An argument 'satisfies' a function φx if φa is true [Russell]
14453 The Darapti syllogism is fallacious: All M is S, all M is P, so some S is P' - but if there is no M? [Russell]
14452 All the propositions of logic are completely general [Russell]
12444 Logic is concerned with the real world just as truly as zoology [Russell]
14458 Asking 'Did Homer exist?' is employing an abbreviated description [Russell]
14457 Names are really descriptions, except for a few words like 'this' and 'that' [Russell]
7311 The only genuine proper names are 'this' and 'that' [Russell]
14455 'I met a unicorn' is meaningful, and so is 'unicorn', but 'a unicorn' is not [Russell]
14456 'Socrates is human' expresses predication, and 'Socrates is a man' expresses identity [Russell]
14459 Reducibility: a family of functions is equivalent to a single type of function [Russell]
14461 Propositions about classes can be reduced to propositions about their defining functions [Russell]
14460 If something is true in all possible worlds then it is logically necessary [Russell]
14462 In modern times, logic has become mathematical, and mathematics has become logical [Russell]
10057 Logic can only assert hypothetical existence [Russell]
14464 Logic can be known a priori, without study of the actual world [Russell]
14465 Maybe numbers are adjectives, since 'ten men' grammatically resembles 'white men' [Russell]
14463 Existence can only be asserted of something described, not of something named [Russell]
21570 Numbers are just verbal conveniences, which can be analysed away [Russell]
8854 My 'acquaintance' with sense-data is nothing like my knowing New York [Williams,M on Russell]
21710 We know a universal in 'yellow differs from blue' or 'yellow resembles blue less than green does' [Russell]
21571 Philosophical systems are interesting, but we now need a more objective scientific philosophy [Russell]
21572 Philosophical disputes are mostly hopeless, because philosophers don't understand each other [Russell]
21573 When problems are analysed properly, they are either logical, or not philosophical at all [Russell]
21576 With asymmetrical relations (before/after) the reduction to properties is impossible [Russell]
21575 When we attribute a common quality to a group, we can forget the quality and just talk of the group [Russell]
21577 Empirical truths are particular, so general truths need an a priori input of generality [Russell]
21574 Hegel's confusions over 'is' show how vast systems can be built on simple errors [Russell]
21579 Objects are treated as real when they connect with other experiences in a normal way [Russell]
21578 Global scepticism is irrefutable, but can't replace our other beliefs, and just makes us hesitate [Russell]
6416 Other minds seem to exist, because their testimony supports realism about the world [Russell, by Grayling]
21581 We never experience times, but only succession of events [Russell]
21582 Physicists accept particles, points and instants, while pretending they don't do metaphysics [Russell]
21580 Science condemns sense-data and accepts matter, but a logical construction must link them [Russell]
21583 When sense-data change, there must be indistinguishable sense-data in the process [Russell]
21584 A sense of timelessness is essential to wisdom [Russell]
21585 The tortoise won't win, because infinite instants don't compose an infinitely long time [Russell]
21586 The logical connectives are not objects, but are formal, and need a context [Russell]
21588 Logic gives the method of research in philosophy [Russell]
21587 Philosophers sometimes neglect truth and distort facts to attain a nice system [Russell]
21684 Atomic facts may be inferrable from others, but never from non-atomic facts [Russell]
22316 A positive and negative fact have the same constituents; their difference is primitive [Russell]
10968 Russell gave up logical atomism because of negative, general and belief propositions [Russell, by Read]
6107 It is logic, not metaphysics, that is fundamental to philosophy [Russell]
6108 Maths can be deduced from logical axioms and the logic of relations [Russell]
6109 Some axioms may only become accepted when they lead to obvious conclusions [Russell]
6110 Subject-predicate logic (and substance-attribute metaphysics) arise from Aryan languages [Russell]
6111 As propositions can be put in subject-predicate form, we wrongly infer that facts have substance-quality form [Russell]
6112 Meaning takes many different forms, depending on different logical types [Russell]
6113 To mean facts we assert them; to mean simples we name them [Russell]
6114 'Simples' are not experienced, but are inferred at the limits of analysis [Russell]
6115 Vagueness, and simples being beyond experience, are obstacles to a logical language [Russell]
6116 A logical language would show up the fallacy of inferring reality from ordinary language [Russell]
6117 Philosophy should be built on science, to reduce error [Russell]
21722 Better to construct from what is known, than to infer what is unknown [Russell]
6118 Philosophy is logical analysis, followed by synthesis [Russell]
10059 In mathematic we are ignorant of both subject-matter and truth [Russell]
7554 Self-evidence is often a mere will-o'-the-wisp [Russell]
7556 A collection is infinite if you can remove some terms without diminishing its number [Russell]
7557 To solve Zeno's paradox, reject the axiom that the whole has more terms than the parts [Russell]
21532 Full empiricism is not tenable, but empirical investigation is always essential [Russell]
21533 Contingency arises from tensed verbs changing the propositions to which they refer [Russell]
21534 The only thing we can say about relations is that they relate [Russell]
21535 Objects only exist if they 'occupy' space and time [Russell]
21536 When I perceive a melody, I do not perceive the notes as existing [Russell]
21537 I assume we perceive the actual objects, and not their 'presentations' [Russell]
21538 If two people perceive the same object, the object of perception can't be in the mind [Russell]
21539 Excluded middle can be stated psychologically, as denial of p implies assertion of not-p [Russell]
21540 Relational propositions seem to be 'about' their terms, rather than about the relation [Russell]
21541 The complexity of the content correlates with the complexity of the object [Russell]
21542 Do incorrect judgements have non-existent, or mental, or external objects? [Russell]
21543 If p is false, then believing not-p is knowing a truth, so negative propositions must exist [Russell]
21544 It seems that when a proposition is false, something must fail to subsist [Russell]
21531 Common sense agrees with Meinong (rather than Russell) that 'Pegasus is a flying horse' is true [Lackey on Russell]
21545 I prefer to deny round squares, and deal with the difficulties by the theory of denoting [Russell]
21546 We can't sharply distinguish variables, domains and values, if symbols frighten us [Russell]
21547 On Meinong's principles 'the existent round square' has to exist [Russell]
21550 Science reduces indexicals to a minimum, but they can never be eliminated from empirical matters [Russell]
21551 Empirical words need ostensive definition, which makes them egocentric [Russell]
21552 Common speech is vague; its vocabulary and syntax must be modified, for precision [Russell]
6420 Only by analysing is progress possible in philosophy [Russell]
6419 In 1899-1900 I adopted the philosophy of logical atomism [Russell]
6426 Intuitionism says propositions are only true or false if there is a method of showing it [Russell]
7528 Leibniz bases everything on subject/predicate and substance/property propositions [Russell]
6423 We tried to define all of pure maths using logical premisses and concepts [Russell]
6427 Unverifiable propositions about the remote past are still either true or false [Russell]
6424 Formalists say maths is merely conventional marks on paper, like the arbitrary rules of chess [Russell]
6425 Formalism can't apply numbers to reality, so it is an evasion [Russell]
6432 Analysis gives new knowledge, without destroying what we already have [Russell]
6430 In epistemology we should emphasis the continuity between animal and human minds [Russell]
6431 Empiricists seem unclear what they mean by 'experience' [Russell]
6433 Behaviourists struggle to explain memory and imagination, because they won't admit images [Russell]
6435 You can believe the meaning of a sentence without thinking of the words [Russell]
6434 Facts are everything, except simples; they are either relations or qualities [Russell]
6438 Complex things can be known, but not simple things [Russell]
6439 Names are meaningless unless there is an object which they designate [Russell]
6437 The theory of types makes 'Socrates and killing are two' illegitimate [Russell]
6436 I gradually replaced classes with properties, and they ended as a symbolic convenience [Russell]
6440 Universals can't just be words, because words themselves are universals [Russell]
6444 True belief about the time is not knowledge if I luckily observe a stopped clock at the right moment [Russell]
6443 Surprise is a criterion of error [Russell]
6441 Pragmatism judges by effects, but I judge truth by causes [Russell]
6442 Truth belongs to beliefs, not to propositions and sentences [Russell]
4396 The law of causality is a source of confusion, and should be dropped from philosophy [Russell]
8375 'Necessary' is a predicate of a propositional function, saying it is true for all values of its argument [Russell]
8376 If causes are contiguous with events, only the last bit is relevant, or the event's timing is baffling [Russell]
8378 Philosophers usually learn science from each other, not from science [Russell]
8379 In causal laws, 'events' must recur, so they have to be universals, not particulars [Russell]
8380 Striking a match causes its igniting, even if it sometimes doesn't work [Russell]
8381 The constancy of scientific laws rests on differential equations, not on cause and effect [Russell]
16385 A definite description 'denotes' an entity if it fits the description uniquely [Russell, by Recanati]
4567 Russell argued with great plausibility that we rarely, if ever, refer with our words [Russell, by Cooper,DE]
5810 Referring is not denoting, and Russell ignores the referential use of definite descriptions [Donnellan on Russell]
16349 Russell rejected sense/reference, because it made direct acquaintance with things impossible [Russell, by Recanati]
7313 'Sense' is superfluous (rather than incoherent) [Russell, by Miller,A]
16987 By eliminating descriptions from primitive notation, Russell seems to reject 'sense' [Russell, by Kripke]
4570 Russell assumes that expressions refer, but actually speakers refer by using expressions [Cooper,DE on Russell]
7767 The theory of definite descriptions aims at finding correct truth conditions [Russell, by Lycan]
11006 Russell started a whole movement in philosophy by providing an analysis of descriptions [Read on Russell]
7758 'Elizabeth = Queen of England' is really a predication, not an identity-statement [Russell, by Lycan]
18944 Russell's theories aim to preserve excluded middle (saying all sentences are T or F) [Sawyer on Russell]
15159 The meaning of a logically proper name is its referent, but most names are not logically proper [Russell, by Soames]
18941 Names don't have a sense, but are disguised definite descriptions [Russell, by Sawyer]
4945 Russell says names are not denotations, but definite descriptions in disguise [Russell, by Kripke]
18942 Russell says a name contributes a complex of properties, rather than an object [Russell, by Sawyer]
7745 Are names descriptions, if the description is unknown, false, not special, or contains names? [McCullogh on Russell]
6411 Critics say definite descriptions can refer, and may not embody both uniqueness and existence claims [Grayling on Russell]
10433 Definite descriptions fail to refer in three situations, so they aren't essentially referring [Russell, by Sainsbury]
2612 Russell rewrote singular term names as predicates [Russell, by Ayer]
7757 "Nobody" is not a singular term, but a quantifier [Russell, by Lycan]
18943 Russell implies that all sentences containing empty names are false [Sawyer on Russell]
10449 Logically proper names introduce objects; definite descriptions introduce quantifications [Russell, by Bach]
6061 Existence is entirely expressed by the existential quantifier [Russell, by McGinn]
11009 Russell's theory must be wrong if it says all statements about non-existents are false [Read on Russell]
1608 The theory of descriptions eliminates the name of the entity whose existence was presupposed [Russell, by Quine]
7754 Russell's theory explains non-existents, negative existentials, identity problems, and substitutivity [Russell, by Lycan]
21529 Russell showed how to define 'the', and thereby reduce the ontology of logic [Russell, by Lackey]
6333 The theory of definite descriptions reduces the definite article 'the' to the concepts of predicate logic [Russell, by Horwich]
6412 Russell implies that 'the baby is crying' is only true if the baby is unique [Grayling on Russell]
7743 Russell explained descriptions with quantifiers, where Frege treated them as names [Russell, by McCullogh]
7310 Russell avoids non-existent objects by denying that definite descriptions are proper names [Russell, by Miller,A]
12006 Denying definite description sentences are subject-predicate in form blocks two big problems [Russell, by Forbes,G]
4569 Russell says apparent referring expressions are really assertions about properties [Russell, by Cooper,DE]
21549 The theory of descriptions lacks conventions for the scope of quantifiers [Lackey on Russell]
12796 Non-count descriptions don't threaten Russell's theory, which is only about singulars [Laycock on Russell]
7532 Denoting is crucial in Russell's account of mathematics, for identifying classes [Russell, by Monk]
11988 Russell's analysis means molecular sentences are ambiguous over the scope of the description [Kaplan on Russell]
18775 Russell showed that descriptions may not have ontological commitment [Russell, by Linsky,B]
7533 The Theory of Description dropped classes and numbers, leaving propositions, individuals and universals [Russell, by Monk]
18777 If the King of France is not bald, and not not-bald, this violates excluded middle [Linsky,B on Russell]
6063 Russell can't attribute existence to properties [McGinn on Russell]
5772 The idea of a variable is fundamental [Russell]
5774 Denoting phrases are meaningless, but guarantee meaning for propositions [Russell]
5775 In 'Scott is the author of Waverley', denotation is identical, but meaning is different [Russell]
5773 The ontological argument begins with an unproven claim that 'there exists an x..' [Russell]
21726 In graspable propositions the constituents are real entities of acquaintance [Russell]
5779 There are distinct sets of psychological and physical causal laws [Russell]
5778 If we object to all data which is 'introspective' we will cease to believe in toothaches [Russell]
5781 Our important beliefs all, if put into words, take the form of propositions [Russell]
5782 A proposition expressed in words is a 'word-proposition', and one of images an 'image-proposition' [Russell]
5780 The three questions about belief are its contents, its success, and its character [Russell]
5783 Propositions of existence, generalities, disjunctions and hypotheticals make correspondence tricky [Russell]
5784 In its primary and formal sense, 'true' applies to propositions, not beliefs [Russell]
5776 A proposition is what we believe when we believe truly or falsely [Russell]
5777 The truth or falsehood of a belief depends upon a fact to which the belief 'refers' [Russell]
22891 We could be aware of time if senses briefly vibrated, extending their experience of movement [Russell, by Bardon]
21744 Legally curbing people's desires is inferior to improving their desires [Russell]
21741 'You ought to do p' primarily has emotional content, expressing approval [Russell]
21746 Unlike hate, all desires can be satisfied by love [Russell]
21747 Goodness is a combination of love and knowledge [Russell]
21743 In wartime, happiness is hating the enemy, because it gives the war a purpose [Russell]
21742 Originally virtue was obedience, to gods, government, or custom [Russell]
21745 Act so as to produce harmonious rather than discordant desires [Russell]
21740 I doubt whether ethics is part of philosophy [Russell]
22306 To explain false belief we should take belief as relating to a proposition's parts, not to the whole thing [Russell]
22308 Only the actual exists, so possibilities always reduce to actuality after full analysis [Russell]
22307 Propositions don't name facts, because two opposed propositions can match one fact [Russell]
21702 In 1918 still believes in nonlinguistic analogues of sentences, but he now calls them 'facts' [Russell, by Quine]
9022 Russell uses 'propositional function' to refer to both predicates and to attributes [Quine on Russell]
18348 Not only atomic truths, but also general and negative truths, have truth-makers [Russell, by Rami]
7744 Treat description using quantifiers, and treat proper names as descriptions [Russell, by McCullogh]
5465 Modern trope theory tries, like logical atomism, to reduce things to elementary states [Russell, by Ellis]
18376 Russell asserts atomic, existential, negative and general facts [Russell, by Armstrong]
6060 'Existence' means that a propositional function is sometimes true [Russell]
21708 Russell's new logical atomist was of particulars, universals and facts (not platonic propositions) [Russell, by Linsky,B]
19051 Russell's atomic facts are actually compounds, and his true logical atoms are sense data [Russell, by Quine]
6091 Propositions don't name facts, because each fact corresponds to a proposition and its negation [Russell]
6089 Logical atomism aims at logical atoms as the last residue of analysis [Russell]
6090 Facts make propositions true or false, and are expressed by whole sentences [Russell]
6093 The names in a logically perfect language would be private, and could not be shared [Russell]
6092 In a logically perfect language, there will be just one word for every simple object [Russell]
6094 An inventory of the world does not need to include propositions [Russell]
6095 The business of metaphysics is to describe the world [Russell]
6096 I no longer believe in propositions, especially concerning falsehoods [Russell]
6097 The theory of error seems to need the existence of the non-existent [Russell]
6098 Perception goes straight to the fact, and not through the proposition [Russell]
6099 Modal terms are properties of propositional functions, not of propositions [Russell]
6100 Once you have enumerated all the atomic facts, there is a further fact that those are all the facts [Russell]
6119 You can discuss 'God exists', so 'God' is a description, not a name [Russell]
6101 Romulus does not occur in the proposition 'Romulus did not exist' [Russell]
6102 You can understand 'author of Waverley', but to understand 'Scott' you must know who it applies to [Russell]
6103 Normally a class with only one member is a problem, because the class and the member are identical [Russell]
6106 Reducing entities and premisses makes error less likely [Russell]
6105 Logical atoms aims to get down to ultimate simples, with their own unique reality [Russell]
6104 Numbers are classes of classes, and hence fictions of fictions [Russell]
10423 There are a set of criteria for pinning down a logically proper name [Russell, by Sainsbury]
10426 A name has got to name something or it is not a name [Russell]
21712 I know longer believe in shadowy things like 'that today is Wednesday' when it is actually Tuesday [Russell]
21709 You can't name all the facts, so they are not real, but are what propositions assert [Russell]
23169 Democratic institutions become impossible in a fanatical democracy [Russell]
23170 Liberal opinions are tentative rather than dogmatic, and are always responsive to new evidence [Russell]
23171 Empiricism is ethically superior, because dogmatism favours persecution and hatred [Russell]
23172 Empiricist Liberalism is the only view for someone who favours scientific evidence and happiness [Russell]
22329 Logic is highly general truths abstracted from reality [Russell, by Glock]
21569 It is good to generalise truths as much as possible [Russell]
7529 All philosophy should begin with an analysis of propositions [Russell]
21520 That our heaven is a dull place reflects the misery of excessive work in life [Russell]
21521 Anarchy does not maximise liberty [Russell]
21522 Democracy is inadequate without a great deal of devolution [Russell]
21523 We would not want UK affairs to be settled by a world parliament [Russell]
21517 Individuals need creativity, reverence for others, and self-respect [Russell]
21518 Men unite in pursuit of material things, and idealise greed as part of group loyalty [Russell]
21519 We need security and liberty, and then encouragement of creativity [Russell]
21495 Theoretical and practical politics are both concerned with the best lives for individuals [Russell]
21524 The right to own land gives a legal right to a permanent income [Russell]
21525 When the state is the only employer, there is no refuge from the prejudices of other people [Russell]
21526 Unfortunately ordinary voters can't detect insincerity [Russell]
21528 Groups should be autonomous, with a neutral authority as arbitrator [Russell]
21527 On every new question the majority is always wrong at first [Russell]
21681 Given all true atomic propositions, in theory every other truth can thereby be deduced [Russell]
19164 If propositions are facts, then false and true propositions are indistinguishable [Davidson on Russell]
11010 Being is what belongs to every possible object of thought [Russell]
15895 Russell discovered the paradox suggested by Burali-Forti's work [Russell, by Lavine]
18254 Russell's approach had to treat real 5/8 as different from rational 5/8 [Russell, by Dummett]
7530 Russell tried to replace Peano's Postulates with the simple idea of 'class' [Russell, by Monk]
15894 Russell invented the naïve set theory usually attributed to Cantor [Russell, by Lavine]
18722 Negations are not just reversals of truth-value, since that can happen without negation [Wittgenstein on Russell]
11849 It at least makes sense to say two objects have all their properties in common [Wittgenstein on Russell]
14102 What is true or false is not mental, and is best called 'propositions' [Russell]
14103 Pure mathematics is the class of propositions of the form 'p implies q' [Russell]
14104 Constants are absolutely definite and unambiguous [Russell]
14105 There seem to be eight or nine logical constants [Russell]
14106 Implication cannot be defined [Russell]
14107 Terms are identical if they belong to all the same classes [Russell]
14108 It would be circular to use 'if' and 'then' to define material implication [Russell]
14109 The study of grammar is underestimated in philosophy [Russell]
7781 I call an object of thought a 'term'. This is a wide concept implying unity and existence. [Russell]
14110 Proposition contain entities indicated by words, rather than the words themselves [Russell]
14111 A proposition is a unity, and analysis destroys it [Russell]
14112 A set has some sort of unity, but not enough to be a 'whole' [Russell]
14113 The null class is a fiction [Russell]
14114 Variables don't stand alone, but exist as parts of propositional functions [Russell]
14115 Definition by analysis into constituents is useless, because it neglects the whole [Russell]
14116 Numbers were once defined on the basis of 1, but neglected infinities and + [Russell]
14117 Numbers are properties of classes [Russell]
14118 We can define one-to-one without mentioning unity [Russell]
14119 We do not currently know whether, of two infinite numbers, one must be greater than the other [Russell]
14120 Counting explains none of the real problems about the foundations of arithmetic [Russell]
14121 The part-whole relation is ultimate and indefinable [Russell]
14122 Analysis gives us nothing but the truth - but never the whole truth [Russell]
14123 Some quantities can't be measured, and some non-quantities are measurable [Russell]
10583 Abstraction principles identify a common property, which is some third term with the right relation [Russell]
10582 The principle of Abstraction says a symmetrical, transitive relation analyses into an identity [Russell]
10584 A certain type of property occurs if and only if there is an equivalence relation [Russell]
14124 Axiom of Archimedes: a finite multiple of a lesser magnitude can always exceed a greater [Russell]
14125 Finite numbers, unlike infinite numbers, obey mathematical induction [Russell]
14126 Order rests on 'between' and 'separation' [Russell]
14127 Order depends on transitive asymmetrical relations [Russell]
10585 Symmetrical and transitive relations are formally like equality [Russell]
10586 'Reflexiveness' holds between a term and itself, and cannot be inferred from symmetry and transitiveness [Russell]
14128 Some claim priority for the ordinals over cardinals, but there is no logical priority between them [Russell]
14129 Ordinals presuppose two relations, where cardinals only presuppose one [Russell]
9977 Ordinals can't be defined just by progression; they have intrinsic qualities [Russell]
14132 Properties of numbers don't rely on progressions, so cardinals may be more basic [Russell]
14133 There are cardinal and ordinal theories of infinity (while continuity is entirely ordinal) [Russell]
14134 Infinite numbers are distinguished by disobeying induction, and the part equalling the whole [Russell]
14135 Real numbers are a class of rational numbers (and so not really numbers at all) [Russell]
14137 'Any' is better than 'all' where infinite classes are concerned [Russell]
14138 You can't get a new transfinite cardinal from an old one just by adding finite numbers to it [Russell]
14140 For every transfinite cardinal there is an infinite collection of transfinite ordinals [Russell]
14139 Transfinite ordinals don't obey commutativity, so their arithmetic is quite different from basic arithmetic [Russell]
14142 Ordinals are types of series of terms in a row, rather than the 'nth' instance [Russell]
14141 Ordinals are defined through mathematical induction [Russell]
14143 ω names the whole series, or the generating relation of the series of ordinal numbers [Russell]
14144 Ordinals result from likeness among relations, as cardinals from similarity among classes [Russell]
14145 For Cantor ordinals are types of order, not numbers [Russell]
14146 We aren't sure if one cardinal number is always bigger than another [Russell]
14147 Denying mathematical induction gave us the transfinite [Russell]
14149 The Achilles Paradox concerns the one-one correlation of infinite classes [Russell]
14148 Infinite regresses have propositions made of propositions etc, with the key term reappearing [Russell]
14151 Pure geometry is deductive, and neutral over what exists [Russell]
14152 In geometry, Kant and idealists aimed at the certainty of the premisses [Russell]
14154 Geometry throws no light on the nature of actual space [Russell]
14153 In geometry, empiricists aimed at premisses consistent with experience [Russell]
14155 Two points have a line joining them (descriptive), a distance (metrical), and a whole line (projective) [Russell, by PG]
14156 Mathematicians don't distinguish between instants of time and points on a line [Russell]
14158 Quantity is not part of mathematics, where it is replaced by order [Russell]
14159 In mathematics definitions are superfluous, as they name classes, and it all reduces to primitives [Russell]
14160 Space is the extension of 'point', and aggregates of points seem necessary for geometry [Russell]
14161 Many things have being (as topics of propositions), but may not have actual existence [Russell]
22303 It makes no sense to say that a true proposition could have been false [Russell]
14162 Mathematics doesn't care whether its entities exist [Russell]
14163 Four classes of terms: instants, points, terms at instants only, and terms at instants and points [Russell]
14165 Analysis falsifies, if when the parts are broken down they are not equivalent to their sum [Russell]
14166 Unities are only in propositions or concepts, and nothing that exists has unity [Russell]
14164 The only unities are simples, or wholes composed of parts [Russell]
14167 The only classes are things, predicates and relations [Russell]
14169 The 'universe' can mean what exists now, what always has or will exist [Russell]
14168 Occupying a place and change are prior to motion, so motion is just occupying places at continuous times [Russell]
14170 Change is obscured by substance, a thing's nature, subject-predicate form, and by essences [Russell]
14171 Force is supposed to cause acceleration, but acceleration is a mathematical fiction [Russell]
14172 Moments and points seem to imply other moments and points, but don't cause them [Russell]
14173 What exists has causal relations, but non-existent things may also have them [Russell]
14174 The laws of motion and gravitation are just parts of the definition of a kind of matter [Russell]
14175 We can drop 'cause', and just make inferences between facts [Russell]
14176 "The death of Caesar is true" is not the same proposition as "Caesar died" [Russell]
21555 For 'x is a u' to be meaningful, u must be one range of individuals (or 'type') higher than x [Russell]
18002 As well as a truth value, propositions have a range of significance for their variables [Russell]
18003 In 'x is a u', x and u must be of different types, so 'x is an x' is generally meaningless [Russell, by Magidor]
21341 Philosophers of logic and maths insisted that a vocabulary of relations was essential [Russell, by Heil]
18246 Dedekind failed to distinguish the numbers from other progressions [Shapiro on Russell]
19157 Russell said the proposition must explain its own unity - or else objective truth is impossible [Russell, by Davidson]
21724 The main aim of the multiple relations theory of judgement was to dispense with propositions [Russell, by Linsky,B]
13468 Russell started philosophy of language, by declaring some plausible sentences to be meaningless [Russell, by Hart,WD]
6510 Russell (1912) said phenomena only resemble reality in abstract structure [Russell, by Robinson,H]
7290 If Russell rejects innate ideas and direct a priori knowledge, he is left with a tabula rasa [Russell, by Thompson]
6514 Russell's representationalism says primary qualities only show the structure of reality [Russell, by Robinson,H]
6415 After 1912, Russell said sense-data are last in analysis, not first in experience [Russell, by Grayling]
5358 'Sense-data' are what are immediately known in sensation, such as colours or roughnesses [Russell]
5357 It is natural to begin from experience, and presumably that is the basis of knowledge [Russell]
5365 Dreams can be explained fairly scientifically if we assume a physical world [Russell]
5363 If the cat reappears in a new position, presumably it has passed through the intermediate positions [Russell]
5367 Belief in real objects makes our account of experience simpler and more systematic [Russell]
5364 It is hard not to believe that speaking humans are expressing thoughts, just as we do ourselves [Russell]
5359 Descartes showed that subjective things are the most certain [Russell]
5362 It is not illogical to think that only myself and my mental events exist [Russell]
5366 We have an 'instinctive' belief in the external world, prior to all reflection [Russell]
5361 Philosophers must get used to absurdities [Russell]
5368 Philosophy verifies that our hierarchy of instinctive beliefs is harmonious and consistent [Russell]
5369 It is rational to believe in reality, despite the lack of demonstrative reasons for it [Russell]
5371 Because we depend on correspondence, we know relations better than we know the items that relate [Russell]
5370 Space is neutral between touch and sight, so it cannot really be either of them [Russell]
5372 There is no reason to think that objects have colours [Russell]
5375 Knowledge of truths applies to judgements; knowledge by acquaintance applies to sensations and things [Russell]
5373 'Idealism' says that everything which exists is in some sense mental [Russell]
5376 I can know the existence of something with which nobody is acquainted [Russell]
5387 It is pure chance which descriptions in a person's mind make a name apply to an individual [Russell]
5378 All knowledge (of things and of truths) rests on the foundations of acquaintance [Russell]
5379 If we didn't know our own minds by introspection, we couldn't know that other people have minds [Russell]
5380 In perceiving the sun, I am aware of sun sense-data, and of the perceiver of the data [Russell]
5381 In seeing the sun, we are acquainted with our self, but not as a permanent person [Russell]
5384 A universal of which we are aware is called a 'concept' [Russell]
5382 We are acquainted with outer and inner sensation, memory, Self, and universals [Russell, by PG]
5389 Knowledge by descriptions enables us to transcend private experience [Russell]
5377 'Acquaintance' is direct awareness, without inferences or judgements [Russell]
5383 Every complete sentence must contain at least one word (a verb) which stands for a universal [Russell]
5386 Proper names are really descriptions, and can be replaced by a description in a person's mind [Russell]
5385 The phrase 'a so-and-so' is an 'ambiguous' description'; 'the so-and-so' (singular) is a 'definite' description [Russell]
5393 We can't know that our laws are exceptionless, or even that there are any laws [Russell]
5391 Science aims to find uniformities to which (within the limits of experience) there are no exceptions [Russell]
5390 Chickens are not very good at induction, and are surprised when their feeder wrings their neck [Russell]
5394 We can't prove induction from experience without begging the question [Russell]
5392 It doesn't follow that because the future has always resembled the past, that it always will [Russell]
5388 Every understood proposition is composed of constituents with which we are acquainted [Russell]
5398 Judgements of usefulness depend on judgements of value [Russell]
5395 Demonstration always relies on the rule that anything implied by a truth is true [Russell]
5396 Three Laws of Thought: identity, contradiction, and excluded middle [Russell]
5401 The mortality of Socrates is more certain from induction than it is from deduction [Russell]
5399 Maths is not known by induction, because further instances are not needed to support it [Russell]
5400 In any possible world we feel that two and two would be four [Russell]
5397 The rationalists were right, because we know logical principles without experience [Russell]
4428 Propositions express relations (prepositions and verbs) as well as properties (nouns and adjectives) [Russell]
5406 Confused views of reality result from thinking that only nouns and adjectives represent universals [Russell]
4479 All universals are like the relation "is north of", in having no physical location at all [Russell, by Loux]
4030 Russell claims that universals are needed to explain a priori knowledge (as their relations) [Russell, by Mellor/Oliver]
5409 Normal existence is in time, so we must say that universals 'subsist' [Russell]
5408 If we identify whiteness with a thought, we can never think of it twice; whiteness is the object of a thought [Russell]
4441 'Resemblance Nominalism' won't work, because the theory treats resemblance itself as a universal [Russell]
4427 Every sentence contains at least one word denoting a universal, so we need universals to know truth [Russell]
4429 If we consider whiteness to be merely a mental 'idea', we rob it of its universality [Russell]
5407 That Edinburgh is north of London is a non-mental fact, so relations are independent universals [Russell]
5405 The law of contradiction is not a 'law of thought', but a belief about things [Russell]
5410 I learn the universal 'resemblance' by seeing two shades of green, and their contrast with red [Russell]
4430 All a priori knowledge deals with the relations of universals [Russell]
5411 We can know some general propositions by universals, when no instance can be given [Russell]
5412 Some propositions are self-evident, but their implications may also be self-evident [Russell]
5413 Particular instances are more clearly self-evident than any general principles [Russell]
5415 As shown by memory, self-evidence comes in degrees [Russell]
5416 If self-evidence has degrees, we should accept the more self-evident as correct [Russell]
5414 Images are not memory, because they are present, and memories are of the past [Russell]
21711 Russell's 'multiple relations' theory says beliefs attach to ingredients, not to propositions [Russell, by Linsky,B]
5427 Truth is when a mental state corresponds to a complex unity of external constituents [Russell]
5425 In order to explain falsehood, a belief must involve several terms, not two [Russell]
5426 Belief relates a mind to several things other than itself [Russell]
5419 Truth and falsehood are properties of beliefs and statements [Russell]
5420 Truth is a property of a belief, but dependent on its external relations, not its internal qualities [Russell]
5422 More than one coherent body of beliefs seems possible [Russell]
5423 If we suspend the law of contradiction, nothing will appear to be incoherent [Russell]
5424 Coherence is not the meaning of truth, but an important test for truth [Russell]
5417 A good theory of truth must make falsehood possible [Russell]
5421 The coherence theory says falsehood is failure to cohere, and truth is fitting into a complete system of Truth [Russell]
7395 Truth as congruence may work for complex beliefs, but not for simple beliefs about existence [Joslin on Russell]
5428 Beliefs are true if they have corresponding facts, and false if they don't [Russell]
5418 In a world of mere matter there might be 'facts', but no truths [Russell]
5430 A true belief is not knowledge if it is reached by bad reasoning [Russell]
5429 True belief is not knowledge when it is deduced from false belief [Russell]
5431 Knowledge cannot be precisely defined, as it merges into 'probable opinion' [Russell]
5434 Philosophy is similar to science, and has no special source of wisdom [Russell]
5432 Metaphysics cannot give knowledge of the universe as a whole [Russell]
17628 Arithmetic was probably inferred from relationships between physical objects [Russell]
17627 It seems absurd to prove 2+2=4, where the conclusion is more certain than premises [Russell]
17629 Which premises are ultimate varies with context [Russell]
17630 The sources of a proof are the reasons why we believe its conclusion [Russell]
17632 Non-contradiction was learned from instances, and then found to be indubitable [Russell]
17631 Induction is inferring premises from consequences [Russell]
17633 The law of gravity has many consequences beyond its grounding observations [Russell]
17639 Believing a whole science is more than believing each of its propositions [Russell]
17637 The most obvious beliefs are not infallible, as other obvious beliefs may conflict [Russell]
17638 If one proposition is deduced from another, they are more certain together than alone [Russell]
17640 Finding the axioms may be the only route to some new results [Russell]
17641 Discoveries in mathematics can challenge philosophy, and offer it a new foundation [Russell]
23025 Philosophers should be more inductive, and test results by their conclusions, not their self-evidence [Russell]
6483 Russell held that we are aware of states of our own brain [Russell, by Robinson,H]
8244 Sense-data are qualities devoid of subjectivity, which are the basis of science [Russell, by Deleuze/Guattari]
6458 Individuating sense-data is difficult, because they divide when closely attended to [Russell]
6459 We do not know whether sense-data exist as objects when they are not data [Russell]
6460 'Sensibilia' are identical to sense-data, without actually being data for any mind [Russell]
6461 Ungiven sense-data can no more exist than unmarried husbands [Russell]
6462 Sense-data are not mental, but are part of the subject-matter of physics [Russell]
6463 Sense-data are objects, and do not contain the subject as part, the way beliefs do [Russell]
6464 Sense-data are usually objects within the body, but are not part of the subject [Russell]
6470 Matter is the limit of appearances as distance from the object diminishes [Russell]
6465 We need not deny substance, but there seems no reason to assert it [Russell]
6466 Where possible, logical constructions are to be substituted for inferred entities [Russell]
6468 There is 'private space', and there is also the 'space of perspectives' [Russell]
6467 No sensibile is ever a datum to two people at once [Russell]
6469 Sense-data may be subjective, if closing our eyes can change them [Russell]
6471 The assumption by physicists of permanent substance is not metaphysically legitimate [Russell]
6473 Physical things are series of appearances whose matter obeys physical laws [Russell]
6472 Continuity is a sufficient criterion for the identity of a rock, but not for part of a smooth fluid [Russell]
18130 Axiom of Reducibility: there is always a function of the lowest possible order in a given level [Russell, by Bostock]
21560 Any linguistic expression may lack meaning when taken out of context [Russell]
21561 'The number one is bald' or 'the number one is fond of cream cheese' are meaningless [Russell]
21562 There is no complexity without relations, so no propositions, and no truth [Russell]
22315 There can't be a negative of a complex, which is negated by its non-existence [Potter on Russell]
23476 Logical constants seem to be entities in propositions, but are actually pure form [Russell]
23477 We use logical notions, so they must be objects - but I don't know what they really are [Russell]
18273 Logical truths are known by their extreme generality [Russell]
23457 Type theory cannot identify features across levels (because such predicates break the rules) [Morris,M on Russell]
21556 Classes are defined by propositional functions, and functions are typed, with an axiom of reducibility [Russell, by Lackey]
21566 'Propositional functions' are ambiguous until the variable is given a value [Russell]
21567 'All judgements made by Epimenedes are true' needs the judgements to be of the same type [Russell]
21568 A one-variable function is only 'predicative' if it is one order above its arguments [Russell]
18126 A set does not exist unless at least one of its specifications is predicative [Russell, by Bostock]
18128 Russell is a conceptualist here, saying some abstracta only exist because definitions create them [Russell, by Bostock]
21718 Ramified types can be defended as a system of intensional logic, with a 'no class' view of sets [Russell, by Linsky,B]
10418 Type theory seems an extreme reaction, since self-exemplification is often innocuous [Swoyer on Russell]
10047 Russell's improvements blocked mathematics as well as paradoxes, and needed further axioms [Russell, by Musgrave]
23478 Type theory means that features shared by different levels cannot be expressed [Morris,M on Russell]
6407 The class of classes which lack self-membership leads to a contradiction [Russell, by Grayling]
11064 Classes can be reduced to propositional functions [Russell, by Hanna]
18124 Vicious Circle says if it is expressed using the whole collection, it can't be in the collection [Russell, by Bostock]
13365 Russell's Paradox is a stripped-down version of Cantor's Paradox [Priest,G on Russell]
10711 Russell's paradox means we cannot assume that every property is collectivizing [Potter on Russell]
7531 We don't assert private thoughts; the objects are part of what we assert [Russell]
9127 Russell refuted Frege's principle that there is a set for each property [Russell, by Sorensen]
21559 We need rules for deciding which norms are predicative (unless none of them are) [Russell]
21558 'Predicative' norms are those which define a class [Russell]
7534 In 1906, Russell decided that propositions did not, after all, exist [Russell, by Monk]
6343 For Russell, both propositions and facts are arrangements of objects, so obviously they correspond [Horwich on Russell]
7545 Visible things are physical and external, but only exist when viewed [Russell]
7546 A man is a succession of momentary men, bound by continuity and causation [Russell]
7547 Matter requires a division into time-corpuscles as well as space-corpuscles [Russell]
7548 Classes, grouped by a convenient property, are logical constructions [Russell]
7549 If my body literally lost its mind, the object seen when I see a flash would still exist [Russell]
7550 We could probably, in principle, infer minds from brains, and brains from minds [Russell]
7551 Matter is a logical construction [Russell]
7552 Six dimensions are needed for a particular, three within its own space, and three to locate that space [Russell]
7553 Sense-data are purely physical [Russell]
14327 Trope theorists cannot explain how tropes resemble each other [Russell, by Mumford]
16045 General facts supervene on particular facts, but cannot be inferred from them [Russell, by Bennett,K]
9051 Since natural language is not precise it cannot be in the province of logic [Russell, by Keefe/Smith]
21597 Logical connectives have the highest precision, yet are infected by the vagueness of true and false [Russell, by Williamson]
9054 Vagueness is only a characteristic of representations, such as language [Russell]