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Ideas from 'Introduction to Mathematical Philosophy' by Bertrand Russell [1919], by Theme Structure

[found in 'Introduction to Mathematical Philosophy' by Russell,Bertrand [George Allen and Unwin 1975,0-04-510020-9]].

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1. Philosophy / F. Analytic Philosophy / 5. Linguistic Analysis
14456 'Socrates is human' expresses predication, and 'Socrates is a man' expresses identity
                        Full Idea: The is of 'Socrates is human' expresses the relation of subject and predicate; the is of 'Socrates is a man' expresses identity. It is a disgrace to the human race that it employs the same word 'is' for these entirely different ideas.
                        From: Bertrand Russell (Introduction to Mathematical Philosophy [1919], XVI)
                        A reaction: Does the second one express identity? It sounds more like membership to me. 'Socrates is the guy with the hemlock' is more like identity.
2. Reason / D. Definition / 3. Types of Definition
14426 A definition by 'extension' enumerates items, and one by 'intension' gives a defining property
                        Full Idea: The definition of a class or collection which enumerates is called a definition by 'extension', and one which mentions a defining property is called a definition by 'intension'.
                        From: Bertrand Russell (Introduction to Mathematical Philosophy [1919], II)
                        A reaction: In ordinary usage we take intensional definitions for granted, so it is interesting to realise that you might define 'tiger' by just enumerating all the tigers. But all past tigers? All future tigers? All possible tigers which never exist?
2. Reason / F. Fallacies / 8. Category Mistake / a. Category mistakes
8468 The sentence 'procrastination drinks quadruplicity' is meaningless, rather than false
                        Full Idea: Russell proposed (in his theory of types) that sentences like 'The number two is fond of cream cheese' or 'Procrastination drinks quadruplicity' should be regarded as not false but meaningless.
                        From: report of Bertrand Russell (Introduction to Mathematical Philosophy [1919]) by Alex Orenstein - W.V. Quine Ch.3
                        A reaction: This seems to be the origin of the notion of a 'category mistake', which Ryle made famous. The problem is always poetry, where abstractions can be reified, or personified, and meaning can be squeezed out of almost anything.
3. Truth / F. Semantic Truth / 1. Tarski's Truth / b. Satisfaction and truth
14454 An argument 'satisfies' a function φx if φa is true
                        Full Idea: We say that an argument a 'satisfies' a function φx if φa is true.
                        From: Bertrand Russell (Introduction to Mathematical Philosophy [1919], XV)
                        A reaction: We end up with Tarski defining truth in terms of satisfaction, so we shouldn't get too excited about what he achieved (any more than he got excited).
4. Formal Logic / A. Syllogistic Logic / 2. Syllogistic Logic
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?
                        Full Idea: Some moods of the syllogism are fallacious, e.g. 'Darapti': 'All M is S, all M is P, therefore some S is P', which fails if there is no M.
                        From: Bertrand Russell (Introduction to Mathematical Philosophy [1919], XV)
                        A reaction: This critique rests on the fact that the existential quantifier entails some existence, but the universal quantifier does not.
4. Formal Logic / F. Set Theory ST / 3. Types of Set / d. Infinite Sets
14427 We can enumerate finite classes, but an intensional definition is needed for infinite classes
                        Full Idea: We know a great deal about a class without enumerating its members …so definition by extension is not necessary to knowledge about a class ..but enumeration of infinite classes is impossible for finite beings, so definition must be by intension.
                        From: Bertrand Russell (Introduction to Mathematical Philosophy [1919], II)
                        A reaction: Presumably mathematical induction (which keeps apply the rule to extend the class) will count as an intension here.
4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / b. Axiom of Extensionality I
14428 Members define a unique class, whereas defining characteristics are numerous
                        Full Idea: There is only one class having a given set of members, whereas there are always many different characteristics by which a given class may be defined.
                        From: Bertrand Russell (Introduction to Mathematical Philosophy [1919], II)
4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / f. Axiom of Infinity V
14440 We may assume that there are infinite collections, as there is no logical reason against them
                        Full Idea: There is no logical reason against infinite collections, and we are therefore justified, in logic, in investigating the hypothesis that there are such collections.
                        From: Bertrand Russell (Introduction to Mathematical Philosophy [1919], VIII)
14447 Infinity says 'for any inductive cardinal, there is a class having that many terms'
                        Full Idea: The Axiom of Infinity may be enunciated as 'If n be any inductive cardinal number, there is at least one class of individuals having n terms'.
                        From: Bertrand Russell (Introduction to Mathematical Philosophy [1919], XIII)
                        A reaction: So for every possible there exists a set of terms for it. Notice that they are 'terms', not 'objects'. We must decide whether we are allowed terms which don't refer to real objects.
4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / j. Axiom of Choice IX
14443 The British parliament has one representative selected from each constituency
                        Full Idea: We have a class of representatives, who make up our Parliament, one being selected out of each constituency.
                        From: Bertrand Russell (Introduction to Mathematical Philosophy [1919], XII)
                        A reaction: You can rely on Russell for the clearest illustrations of these abstract ideas. He calls the Axiom of Choice the 'Multiplicative' Axiom.
14444 Choice is equivalent to the proposition that every class is well-ordered
                        Full Idea: Zermelo has shown that [the Axiom of Choice] is equivalent to the proposition that every class is well-ordered, i.e. can be arranged in a series in which every sub-class has a first term (except, of course, the null class).
                        From: Bertrand Russell (Introduction to Mathematical Philosophy [1919], XII)
                        A reaction: Russell calls Choice the 'Multiplicative' Axiom.
14445 Choice shows that if any two cardinals are not equal, one must be the greater
                        Full Idea: The [Axiom of Choice] is also equivalent to the assumption that of any two cardinals which are not equal, one must be the greater.
                        From: Bertrand Russell (Introduction to Mathematical Philosophy [1919], XII)
                        A reaction: It is illuminating for the uninitiated to learn that this result can't be taken for granted (with infinite cardinals).
14446 We can pick all the right or left boots, but socks need Choice to insure the representative class
                        Full Idea: Among boots we distinguish left and right, so we can choose all the right or left boots; with socks no such principle suggests itself, and we cannot be sure, without the [Axiom of Choice], that there is a class consisting of one sock from each pair.
                        From: Bertrand Russell (Introduction to Mathematical Philosophy [1919], XII)
                        A reaction: A deservedly famous illustration of a rather tricky part of set theory.
4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / p. Axiom of Reducibility
14459 Reducibility: a family of functions is equivalent to a single type of function
                        Full Idea: The Axiom of Reducibility says 'There is a type of a-functions such that, given any a-function, it is formally equivalent to some function of the type in question'. ..It involves all that is really essential in the theory of classes. But is it true?
                        From: Bertrand Russell (Introduction to Mathematical Philosophy [1919], XVII)
                        A reaction: I take this to say that in the theory of types, it is possible to reduce each level of type down to one type.
4. Formal Logic / F. Set Theory ST / 5. Conceptions of Set / c. Logical sets
14461 Propositions about classes can be reduced to propositions about their defining functions
                        Full Idea: It is right (in its main lines) to say that there is a reduction of propositions nominally about classes to propositions about their defining functions.
                        From: Bertrand Russell (Introduction to Mathematical Philosophy [1919], XVII)
                        A reaction: The defining functions will involve the theory of types, in order to avoid the paradoxes of naïve set theory. This is Russell's strategy for rejecting the existence of sets.
4. Formal Logic / F. Set Theory ST / 7. Natural Sets
8469 Russell's proposal was that only meaningful predicates have sets as their extensions
                        Full Idea: Russell's solution (in the theory of types) consists of restricting the principle that every predicate has a set as its extension so that only meaningful predicates have sets as their extensions.
                        From: report of Bertrand Russell (Introduction to Mathematical Philosophy [1919]) by Alex Orenstein - W.V. Quine Ch.3
                        A reaction: There might be a chicken-and-egg problem here. How do you decide the members of a set (apart from ostensively) without deciding the predicate(s) that combine them?
4. Formal Logic / F. Set Theory ST / 8. Critique of Set Theory
8745 Classes are logical fictions, and are not part of the ultimate furniture of the world
                        Full Idea: The symbols for classes are mere conveniences, not representing objects called 'classes'. Classes are in fact logical fictions; they cannot be regarded as part of the ultimate furniture of the world.
                        From: Bertrand Russell (Introduction to Mathematical Philosophy [1919], Ch.18), quoted by Stewart Shapiro - Thinking About Mathematics 5.2
                        A reaction: I agree. For 'logical fictions' read 'abstractions'. To equate abstractions with fictions is to underline the fact that they are a human creation. They are either that or platonic objects - there is no middle way.
5. Theory of Logic / A. Overview of Logic / 4. Pure Logic
14452 All the propositions of logic are completely general
                        Full Idea: It is part of the definition of logic that all its propositions are completely general.
                        From: Bertrand Russell (Introduction to Mathematical Philosophy [1919], XV)
5. Theory of Logic / A. Overview of Logic / 8. Logic of Mathematics
14462 In modern times, logic has become mathematical, and mathematics has become logical
                        Full Idea: Logic has become more mathematical, and mathematics has become more logical. The consequence is that it has now become wholly impossible to draw a line between the two; in fact, the two are one.
                        From: Bertrand Russell (Introduction to Mathematical Philosophy [1919], XVIII)
                        A reaction: This appears to be true even if you reject logicism about mathematics. Logicism is sometimes rejected because it always ends up with a sneaky ontological commitment, but maybe mathematics shares exactly the same commitment.
5. Theory of Logic / C. Ontology of Logic / 1. Ontology of Logic
14464 Logic can be known a priori, without study of the actual world
                        Full Idea: Logical propositions are such as can be known a priori, without study of the actual world.
                        From: Bertrand Russell (Introduction to Mathematical Philosophy [1919], XVIII)
                        A reaction: This remark constrasts strikingly with Idea 12444, which connects logic to the actual world. Is it therefore a priori synthetic?
10057 Logic can only assert hypothetical existence
                        Full Idea: No proposition of logic can assert 'existence' except under a hypothesis.
                        From: Bertrand Russell (Introduction to Mathematical Philosophy [1919], XVIII)
                        A reaction: I am prepared to accept this view fairly dogmatically, though Musgrave shows some of the difficulties of the if-thenist view (depending on which 'order' of logic is being used).
12444 Logic is concerned with the real world just as truly as zoology
                        Full Idea: Logic is concerned with the real world just as truly as zoology, though with its more abstract and general features.
                        From: Bertrand Russell (Introduction to Mathematical Philosophy [1919], XVI)
                        A reaction: I love this idea and am very sympathetic to it. The rival view seems to be that logic is purely conventional, perhaps defined by truth tables etc. It is hard to see how a connective like 'tonk' could be self-evidently silly if it wasn't 'unnatural'.
5. Theory of Logic / F. Referring in Logic / 1. Naming / b. Names as descriptive
10450 Russell admitted that even names could also be used as descriptions
                        Full Idea: Russell clearly anticipated Donnellan when he said proper names can also be used as descriptions, adding that 'there is nothing in the phraseology to show whether they are being used in this way or as names'.
                        From: report of Bertrand Russell (Introduction to Mathematical Philosophy [1919], p.175) by Kent Bach - What Does It Take to Refer? 22.2 L1
                        A reaction: This seems also to anticipate Strawson's flexible and pragmatic approach to these things, which I am beginning to think is correct.
14458 Asking 'Did Homer exist?' is employing an abbreviated description
                        Full Idea: When we ask whether Homer existed, we are using the word 'Homer' as an abbreviated description.
                        From: Bertrand Russell (Introduction to Mathematical Philosophy [1919], XVI)
                        A reaction: It is hard to disagree with Russell over this rather unusual example. It doesn't seem so plausible when Ottiline refers to 'Bertie'.
14457 Names are really descriptions, except for a few words like 'this' and 'that'
                        Full Idea: We can even say that, in all such knowledge as can be expressed in words, with the exception of 'this' and 'that' and a few other words of which the meaning varies on different occasions - no names occur, but what seem like names are really descriptions.
                        From: Bertrand Russell (Introduction to Mathematical Philosophy [1919], XVI)
                        A reaction: I like the caveat about what is expressed in words. Russell is very good at keeping non-verbal thought in the picture. This is his famous final reduction of names to simple demonstratives.
5. Theory of Logic / F. Referring in Logic / 1. Naming / f. Names eliminated
7311 The only genuine proper names are 'this' and 'that'
                        Full Idea: In all knowledge that can be expressed in words - with the exception of "this" and "that", and a few other such words - no genuine proper names occur, but what seem like genuine proper names are really descriptions
                        From: Bertrand Russell (Introduction to Mathematical Philosophy [1919], XVI)
                        A reaction: This is the terminus of Russell's train of thought about descriptions. Suppose you point to something non-existent, like a ghost in a misty churchyard? You'd be back to the original problem of naming a non-existent!
5. Theory of Logic / F. Referring in Logic / 2. Descriptions / a. Descriptions
14455 'I met a unicorn' is meaningful, and so is 'unicorn', but 'a unicorn' is not
                        Full Idea: In 'I met a unicorn' the four words together make a significant proposition, and the word 'unicorn' is significant, …but the two words 'a unicorn' do not form a group having a meaning of its own. It is an indefinite description describing nothing.
                        From: Bertrand Russell (Introduction to Mathematical Philosophy [1919], XVI)
6. Mathematics / A. Nature of Mathematics / 2. Geometry
14442 If straight lines were like ratios they might intersect at a 'gap', and have no point in common
                        Full Idea: We wish to say that when two straight lines cross each other they have a point in common, but if the series of points on a line were similar to the series of ratios, the two lines might cross in a 'gap' and have no point in common.
                        From: Bertrand Russell (Introduction to Mathematical Philosophy [1919], X)
                        A reaction: You can make a Dedekind Cut in the line of ratios (the rationals), so there must be gaps. I love this idea. We take for granted intersection at a point, but physical lines may not coincide. That abstract lines might fail also is lovely!
6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / b. Types of number
14438 New numbers solve problems: negatives for subtraction, fractions for division, complex for equations
                        Full Idea: Every generalisation of number has presented itself as needed for some simple problem. Negative numbers are needed to make subtraction always possible; fractions to make division always possible; complex numbers to make solutions of equations possible.
                        From: Bertrand Russell (Introduction to Mathematical Philosophy [1919], VII)
                        A reaction: Doesn't this rather suggest that we made them up? If new problems turn up, we'll invent another lot. We already have added 'surreal' numbers.
6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / c. Priority of numbers
13510 Could a number just be something which occurs in a progression?
                        Full Idea: Russell toyed with the idea that there is nothing to being a natural number beyond occurring in a progression
                        From: report of Bertrand Russell (Introduction to Mathematical Philosophy [1919], p.8) by William D. Hart - The Evolution of Logic 5
                        A reaction: How could you define a progression, without a prior access to numbers? - Arrange all the objects in the universe in ascending order of mass. Use scales to make the selection. Hence a finite progression, with no numbers!
6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / i. Reals from cuts
14436 A series can be 'Cut' in two, where the lower class has no maximum, the upper no minimum
                        Full Idea: There is no maximum to the ratios whose square is less than 2, and no minimum to those whose square is greater than 2. This division of a series into two classes is called a 'Dedekind Cut'.
                        From: Bertrand Russell (Introduction to Mathematical Philosophy [1919], VII)
6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / j. Complex numbers
14439 A complex number is simply an ordered couple of real numbers
                        Full Idea: A complex number may be regarded and defined as simply an ordered couple of real numbers
                        From: Bertrand Russell (Introduction to Mathematical Philosophy [1919], VII)
6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / m. One
14421 Discovering that 1 is a number was difficult
                        Full Idea: The discovery that 1 is a number must have been difficult.
                        From: Bertrand Russell (Introduction to Mathematical Philosophy [1919], I)
                        A reaction: Interesting that he calls it a 'discovery'. I am tempted to call it a 'decision'.
6. Mathematics / A. Nature of Mathematics / 4. Using Numbers / c. Counting procedure
14424 Numbers are needed for counting, so they need a meaning, and not just formal properties
                        Full Idea: We want our numbers to be such as can be used for counting common objects, and this requires that our numbers should have a definite meaning, not merely that they should have certain formal properties.
                        From: Bertrand Russell (Introduction to Mathematical Philosophy [1919], I)
                        A reaction: Why would just having certain formal properties be insufficient for counting? You just need an ordered series of unique items. It isn't just that we 'want' this. If you define something that we can't count with, you haven't defined numbers.
6. Mathematics / A. Nature of Mathematics / 4. Using Numbers / f. Arithmetic
14441 The formal laws of arithmetic are the Commutative, the Associative and the Distributive
                        Full Idea: The usual formal laws of arithmetic are the Commutative Law [a+b=b+a and axb=bxa], the Associative Law [(a+b)+c=a+(b+c) and (axb)xc=ax(bxc)], and the Distributive Law [a(b+c)=ab+ac)].
                        From: Bertrand Russell (Introduction to Mathematical Philosophy [1919], IX)
6. Mathematics / A. Nature of Mathematics / 5. The Infinite / a. The Infinite
14420 Infinity and continuity used to be philosophy, but are now mathematics
                        Full Idea: The nature of infinity and continuity belonged in former days to philosophy, but belongs now to mathematics.
                        From: Bertrand Russell (Introduction to Mathematical Philosophy [1919], Pref)
                        A reaction: It is hard to disagree, since mathematicians since Cantor have revealed so much about infinite numbers (through set theory), but I think it remains an open question whether philosophers have anything distinctive to contribute.
6. Mathematics / B. Foundations for Mathematics / 4. Axioms for Number / a. Axioms for numbers
14431 The definition of order needs a transitive relation, to leap over infinite intermediate terms
                        Full Idea: Order must be defined by means of a transitive relation, since only such a relation is able to leap over an infinite number of intermediate terms. ...Without it we would not be able to define the order of magnitude among fractions.
                        From: Bertrand Russell (Introduction to Mathematical Philosophy [1919], IV)
6. Mathematics / B. Foundations for Mathematics / 4. Axioms for Number / d. Peano arithmetic
14423 '0', 'number' and 'successor' cannot be defined by Peano's axioms
                        Full Idea: That '0', 'number' and 'successor' cannot be defined by means of Peano's five axioms, but must be independently understood.
                        From: Bertrand Russell (Introduction to Mathematical Philosophy [1919], I)
14422 Any founded, non-repeating series all reachable in steps will satisfy Peano's axioms
                        Full Idea: Given any series which is endless, contains no repetitions, has a beginning, and has no terms that cannot be reached from the beginning in a finite number of steps, we have a set of terms verifying Peano's axioms.
                        From: Bertrand Russell (Introduction to Mathematical Philosophy [1919], I)
6. Mathematics / B. Foundations for Mathematics / 5. Definitions of Number / d. Hume's Principle
14425 A number is something which characterises collections of the same size
                        Full Idea: The number 3 is something which all trios have in common, and which distinguishes them from other collections. A number is something that characterises certain collections, namely, those that have that number.
                        From: Bertrand Russell (Introduction to Mathematical Philosophy [1919], II)
                        A reaction: This is a verbal summary of the Fregean view of numbers, which marks the arrival of set theory as the way arithmetic will in future be characterised. The question is whether set theory captures all aspects of numbers. Does it give a tool for counting?
6. Mathematics / B. Foundations for Mathematics / 7. Mathematical Structuralism / a. Structuralism
14434 What matters is the logical interrelation of mathematical terms, not their intrinsic nature
                        Full Idea: What matters in mathematics is not the intrinsic nature of our terms, but the logical nature of their interrelations.
                        From: Bertrand Russell (Introduction to Mathematical Philosophy [1919], VI)
                        A reaction: If they have an instrinsic nature, that would matter far more, because that would dictate the interrelations. Structuralism seems to require that they don't actually have any intrinsic nature.
6. Mathematics / C. Sources of Mathematics / 5. Numbers as Adjectival
14465 Maybe numbers are adjectives, since 'ten men' grammatically resembles 'white men'
                        Full Idea: 'Ten men' is grammatically the same form as 'white men', so that 10 might be thought to be an adjective qualifying 'men'.
                        From: Bertrand Russell (Introduction to Mathematical Philosophy [1919], XVIII)
                        A reaction: The immediate problem, as Frege spotted, is that such expressions can be rephrased to remove the adjective (by saying 'the number of men is ten').
6. Mathematics / C. Sources of Mathematics / 6. Logicism / a. Early logicism
13414 For Russell, numbers are sets of equivalent sets
                        Full Idea: Russell's own stand was that numbers are really only sets of equivalent sets.
                        From: report of Bertrand Russell (Introduction to Mathematical Philosophy [1919]) by Paul Benacerraf - Logicism, Some Considerations (PhD) p.168
                        A reaction: Benacerraf is launching a nice attack on this view, based on our inability to grasp huge numbers on this basis, or to see their natural order.
6. Mathematics / C. Sources of Mathematics / 10. Constructivism / e. Psychologism
14449 There is always something psychological about inference
                        Full Idea: There is always unavoidably something psychological about inference.
                        From: Bertrand Russell (Introduction to Mathematical Philosophy [1919], XIV)
                        A reaction: Glad to find Russell saying that. Only pure Fregeans dream of a logic that rises totally above the minds that think it. See Robert Hanna on the subject.
7. Existence / A. Nature of Existence / 1. Nature of Existence
14463 Existence can only be asserted of something described, not of something named
                        Full Idea: Existence can only be asserted of something described, not of something named.
                        From: Bertrand Russell (Introduction to Mathematical Philosophy [1919], XVIII)
                        A reaction: This is the motivation behind Russell's theory of definite descriptions, and epitomises the approach to ontology through language. Sounds wrong to me!
7. Existence / D. Theories of Reality / 7. Fictionalism
14429 Classes are logical fictions, made from defining characteristics
                        Full Idea: Classes may be regarded as logical fictions, manufactured out of defining characteristics.
                        From: Bertrand Russell (Introduction to Mathematical Philosophy [1919], II n1)
                        A reaction: I agree with this. The idea that in addition to the members there is a further object, the set containing them, is absurd. Sets are a tool for thinking about the world.
8. Modes of Existence / A. Relations / 4. Formal Relations / a. Types of relation
14430 If a relation is symmetrical and transitive, it has to be reflexive
                        Full Idea: It is obvious that a relation which is symmetrical and transitive must be reflexive throughout its domain.
                        From: Bertrand Russell (Introduction to Mathematical Philosophy [1919], II)
                        A reaction: Compare Idea 13543! The relation will return to its originator via its neighbours, rather than being directly reflexive?
14432 'Asymmetry' is incompatible with its converse; a is husband of b, so b can't be husband of a
                        Full Idea: The relation of 'asymmetry' is incompatible with the converse. …The relation 'husband' is asymmetrical, so that if a is the husband of b, b cannot be the husband of a.
                        From: Bertrand Russell (Introduction to Mathematical Philosophy [1919], V)
                        A reaction: This is to be contrasted with 'non-symmetrical', where there just happens to be no symmetry.
9. Objects / D. Essence of Objects / 3. Individual Essences
14435 The essence of individuality is beyond description, and hence irrelevant to science
                        Full Idea: The essence of individuality always eludes words and baffles description, and is for that very reason irrelevant to science.
                        From: Bertrand Russell (Introduction to Mathematical Philosophy [1919], VI)
                        A reaction: [context needed for a full grasp of this idea] Russell seems to refer to essence as much as to individuality. The modern essentialist view is that essences are not beyond description after all. Fundamental physics is clearer now than in 1919.
10. Modality / B. Possibility / 8. Conditionals / c. Truth-function conditionals
12197 Inferring q from p only needs p to be true, and 'not-p or q' to be true
                        Full Idea: In order that it be valid to infer q from p, it is only necessary that p should be true and that the proposition 'not-p or q' should be true.
                        From: Bertrand Russell (Introduction to Mathematical Philosophy [1919], XIV)
                        A reaction: Rumfitt points out that this approach to logical consequences is a denial of any modal aspect, such as 'logical necessity'. Russell observes that for a good inference you must know the disjunction as a whole. Could disjunction be modal?...
14450 All forms of implication are expressible as truth-functions
                        Full Idea: There is no need to admit as a fundamental notion any form of implication not expressible as a truth-function.
                        From: Bertrand Russell (Introduction to Mathematical Philosophy [1919], XIV)
                        A reaction: Note that this is from a book about 'mathematical' philosophy. Nevertheless, it seems to have the form of a universal credo for Russell. He wasn't talking about conditionals here. Maybe conditionals are not implications (in isolation, that is).
10. Modality / E. Possible worlds / 1. Possible Worlds / a. Possible worlds
14460 If something is true in all possible worlds then it is logically necessary
                        Full Idea: Saying that the axiom of reducibility is logically necessary is what would be meant by saying that it is true in all possible worlds.
                        From: Bertrand Russell (Introduction to Mathematical Philosophy [1919], XVII)
                        A reaction: This striking remark is a nice bridge between Leibniz (about whom Russell wrote a book) and Kripke.
14. Science / B. Scientific Theories / 1. Scientific Theory
14433 Mathematically expressed propositions are true of the world, but how to interpret them?
                        Full Idea: We know that certain scientific propositions - often expressed in mathematical symbols - are more or less true of the world, but we are very much at sea as to the interpretation to be put upon the terms which occur in these propositions.
                        From: Bertrand Russell (Introduction to Mathematical Philosophy [1919], VI)
                        A reaction: Enter essentialism, say I! Russell's remark is pretty understandable in 1919, but I don't think the situation has changed much. The problem of interpretation may be of more interest to philosophers than to physicists.
19. Language / D. Propositions / 1. Propositions
14451 Propositions are mainly verbal expressions of true or false, and perhaps also symbolic thoughts
                        Full Idea: We mean by 'proposition' primarily a form of words which expresses what is either true or false. I say 'primarily' because I do not wish to exclude other than verbal symbols, or even mere thoughts if they have a symbolic character.
                        From: Bertrand Russell (Introduction to Mathematical Philosophy [1919], XV)
                        A reaction: I like the last bit, as I think of propositions as pre-verbal thoughts, and I am sympathetic to Fodor's 'language of thought' thesis, that there is a system of representations within the brain.