42 ideas
9542 | The best known axiomatization of PL is Whitehead/Russell, with four axioms and two rules [Russell/Whitehead, by Hughes/Cresswell] |
Full Idea: The best known axiomatization of PL is Whitehead/Russell. There are four axioms: (p∨p)→p, q→(p∨q), (p→q)→(q∨p), and (q→r)→((p∨q)→(p∨r)), plus Substitution and Modus Ponens rules. | |
From: report of B Russell/AN Whitehead (Principia Mathematica [1913]) by GE Hughes/M Cresswell - An Introduction to Modal Logic Ch.1 |
21720 | Russell saw Reducibility as legitimate for reducing classes to logic [Linsky,B on Russell/Whitehead] |
Full Idea: The axiom of Reducibility ...is crucial in the reduction of classes to logic, ...and seems to be a quite legitimate logical notion for Russell. | |
From: comment on B Russell/AN Whitehead (Principia Mathematica [1913]) by Bernard Linsky - Russell's Metaphysical Logic 6.4 | |
A reaction: This is an unusual defence of the axiom, which is usually presumed to have been kicked into the long grass by Quine. If one could reduce classes to logic, that would destroy the opposition to logicism in a single neat coup. |
10044 | Russell denies extensional sets, because the null can't be a collection, and the singleton is just its element [Russell/Whitehead, by Shapiro] |
Full Idea: Russell adduces two reasons against the extensional view of classes, namely the existence of the null class (which cannot very well be a collection), and the unit classes (which would have to be identical with their single elements). | |
From: report of B Russell/AN Whitehead (Principia Mathematica [1913]) by Stewart Shapiro - Structure and Ontology p.459 | |
A reaction: Gödel believes in the reality of classes. I have great sympathy with Russell, when people start to claim that sets are not just conveniences to help us think about things, but actual abstract entities. Is the singleton of my pencil is on this table? |
18208 | We regard classes as mere symbolic or linguistic conveniences [Russell/Whitehead] |
Full Idea: Classes, so far as we introduce them, are merely symbolic or linguistic conveniences, not genuine objects. | |
From: B Russell/AN Whitehead (Principia Mathematica [1913], p.72), quoted by Penelope Maddy - Naturalism in Mathematics III.2 |
8204 | Lewis's 'strict implication' preserved Russell's confusion of 'if...then' with implication [Quine on Russell/Whitehead] |
Full Idea: Russell call 'if...then' implication, when the material conditional is a much better account; C.I.Lewis (in founding modern modal logic) preserved Russell's confusion by creating 'strict implication', and called that implication. | |
From: comment on B Russell/AN Whitehead (Principia Mathematica [1913]) by Willard Quine - Reply to Professor Marcus p.177 | |
A reaction: [A compession of Quine's paragraph]. All of this assumes that logicians can give an accurate account of what if...then means, when ordinary usage is broad and vague. Strict implication seems to drain all the normal meaning out of 'if...then'. |
9359 | Russell's implication means that random sentences imply one another [Lewis,CI on Russell/Whitehead] |
Full Idea: In Mr Russell's idea of implication, if twenty random sentences from a newspaper were put in a hat, and two of them drawn at random, one will certainly imply the other, and it is an even bet the implication will be mutual. | |
From: comment on B Russell/AN Whitehead (Principia Mathematica [1913]) by C.I. Lewis - A Pragmatic Conception of the A Priori p.366 | |
A reaction: This sort of lament leads modern logicians to suggest 'relevance' as an important criterion. It certainly seems odd that so-called 'classical logic' should contain a principle so at variance with everyday reasoning. |
21707 | Russell unusually saw logic as 'interpreted' (though very general, and neutral) [Russell/Whitehead, by Linsky,B] |
Full Idea: Russell did not view logic as an uninterpreted calculus awaiting interpretations [the modern view]. Rather, logic is a single 'interpreted' body of a priori truths, of propositions rather than sentence forms - but maximally general and topic neutral. | |
From: report of B Russell/AN Whitehead (Principia Mathematica [1913]) by Bernard Linsky - Russell's Metaphysical Logic 1 | |
A reaction: This is the view which Wittgenstein challenged, saying logic is just conventional. Linsky claims that Russell's logicism is much more plausible, once you understand his view of logic. |
10036 | In 'Principia' a new abstract theory of relations appeared, and was applied [Russell/Whitehead, by Gödel] |
Full Idea: In 'Principia' a young science was enriched with a new abstract theory of relations, ..and not only Cantor's set theory but also ordinary arithmetic and the theory of measurement are treated from this abstract relational standpoint. | |
From: report of B Russell/AN Whitehead (Principia Mathematica [1913]) by Kurt Gödel - Russell's Mathematical Logic p.448 | |
A reaction: I presume this is accounting for relations in terms of ordered sets. |
18248 | A real number is the class of rationals less than the number [Russell/Whitehead, by Shapiro] |
Full Idea: For Russell the real number 2 is the class of rationals less than 2 (i.e. 2/1). ...Notice that on this definition, real numbers are classes of rational numbers. | |
From: report of B Russell/AN Whitehead (Principia Mathematica [1913]) by Stewart Shapiro - Thinking About Mathematics 5.2 |
18152 | Russell takes numbers to be classes, but then reduces the classes to numerical quantifiers [Russell/Whitehead, by Bostock] |
Full Idea: Although Russell takes numbers to be certain classes, his 'no-class' theory then eliminates all mention of classes in favour of the 'propositional functions' that define them; and in the case of the numbers these just are the numerical quantifiers. | |
From: report of B Russell/AN Whitehead (Principia Mathematica [1913]) by David Bostock - Philosophy of Mathematics 9.B.4 |
10025 | Russell and Whitehead took arithmetic to be higher-order logic [Russell/Whitehead, by Hodes] |
Full Idea: Russell and Whitehead took arithmetic to be higher-order logic, ..and came close to identifying numbers with numerical quantifiers. | |
From: report of B Russell/AN Whitehead (Principia Mathematica [1913]) by Harold Hodes - Logicism and Ontological Commits. of Arithmetic p.148 | |
A reaction: The point here is 'higher-order'. |
8683 | Russell and Whitehead were not realists, but embraced nearly all of maths in logic [Russell/Whitehead, by Friend] |
Full Idea: Unlike Frege, Russell and Whitehead were not realists about mathematical objects, and whereas Frege thought that only arithmetic and analysis are branches of logic, they think the vast majority of mathematics (including geometry) is essentially logical. | |
From: report of B Russell/AN Whitehead (Principia Mathematica [1913]) by Michčle Friend - Introducing the Philosophy of Mathematics 3.1 | |
A reaction: If, in essence, Descartes reduced geometry to algebra (by inventing co-ordinates), then geometry ought to be included. It is characteristic of Russell's hubris to want to embrace everything. |
10037 | 'Principia' lacks a precise statement of the syntax [Gödel on Russell/Whitehead] |
Full Idea: What is missing, above all, in 'Principia', is a precise statement of the syntax of the formalism. | |
From: comment on B Russell/AN Whitehead (Principia Mathematica [1913]) by Kurt Gödel - Russell's Mathematical Logic p.448 |
10093 | The ramified theory of types used propositional functions, and covered bound variables [Russell/Whitehead, by George/Velleman] |
Full Idea: Russell and Whitehead's ramified theory of types worked not with sets, but with propositional functions (similar to Frege's concepts), with a more restrictive assignment of variables, insisting that bound, as well as free, variables be of lower type. | |
From: report of B Russell/AN Whitehead (Principia Mathematica [1913]) by A.George / D.J.Velleman - Philosophies of Mathematics Ch.3 | |
A reaction: I don't fully understand this (and no one seems much interested any more), but I think variables are a key notion, and there is something interesting going on here. I am intrigued by ordinary language which behaves like variables. |
8691 | The Russell/Whitehead type theory was limited, and was not really logic [Friend on Russell/Whitehead] |
Full Idea: The Russell/Whitehead type theory reduces mathematics to a consistent founding discipline, but is criticised for not really being logic. They could not prove the existence of infinite sets, and introduced a non-logical 'axiom of reducibility'. | |
From: comment on B Russell/AN Whitehead (Principia Mathematica [1913]) by Michčle Friend - Introducing the Philosophy of Mathematics 3.6 | |
A reaction: To have reduced most of mathematics to a founding discipline sounds like quite an achievement, and its failure to be based in pure logic doesn't sound too bad. However, it seems to reduce some maths to just other maths. |
10305 | In 'Principia Mathematica', logic is exceeded in the axioms of infinity and reducibility, and in the domains [Bernays on Russell/Whitehead] |
Full Idea: In the system of 'Principia Mathematica', it is not only the axioms of infinity and reducibility which go beyond pure logic, but also the initial conception of a universal domain of individuals and of a domain of predicates. | |
From: comment on B Russell/AN Whitehead (Principia Mathematica [1913], p.267) by Paul Bernays - On Platonism in Mathematics p.267 | |
A reaction: This sort of criticism seems to be the real collapse of the logicist programme, rather than Russell's paradox, or Gödel's Incompleteness Theorems. It just became impossible to stick strictly to logic in the reduction of arithmetic. |
8684 | Russell and Whitehead consider the paradoxes to indicate that we create mathematical reality [Russell/Whitehead, by Friend] |
Full Idea: Russell and Whitehead are particularly careful to avoid paradox, and consider the paradoxes to indicate that we create mathematical reality. | |
From: report of B Russell/AN Whitehead (Principia Mathematica [1913]) by Michčle Friend - Introducing the Philosophy of Mathematics 3.1 | |
A reaction: This strikes me as quite a good argument. It is certainly counterintuitive that reality, and abstractions from reality, would contain contradictions. The realist view would be that we have paradoxes because we have misdescribed the facts. |
8746 | To avoid vicious circularity Russell produced ramified type theory, but Ramsey simplified it [Russell/Whitehead, by Shapiro] |
Full Idea: Russell insisted on the vicious circle principle, and thus rejected impredicative definitions, which resulted in an unwieldy ramified type theory, with the ad hoc axiom of reducibility. Ramsey's simpler theory was impredicative and avoided the axiom. | |
From: report of B Russell/AN Whitehead (Principia Mathematica [1913]) by Stewart Shapiro - Thinking About Mathematics 5.2 | |
A reaction: Nowadays the theory of types seems to have been given up, possibly because it has no real attraction if it lacks the strict character which Russell aspired to. |
4986 | A weaker kind of reductionism than direct translation is the use of 'bridge laws' [Kirk,R] |
Full Idea: If multiple realisability means that psychological terms cannot be translated into physics, one weaker kind of reductionism resorts to 'bridge laws' which link the theory to be reduced to the reducing theory. | |
From: Robert Kirk (Mind and Body [2003], §3.8) | |
A reaction: It seems to me that reduction is all-or-nothing, so there can't be a 'weaker' kind. If they are totally separate but linked by naturally necessary laws (e.g. low temperature and ice), they are supervenient, but not reducible to one another. |
12033 | An object is identical with itself, and no different indiscernible object can share that [Russell/Whitehead, by Adams,RM] |
Full Idea: Trivially, the Identity of Indiscernibles says that two individuals, Castor and Pollux, cannot have all properties in common. For Castor must have the properties of being identical with Castor and not being identical with Pollux, which Pollux can't share. | |
From: report of B Russell/AN Whitehead (Principia Mathematica [1913], I p.57) by Robert Merrihew Adams - Primitive Thisness and Primitive Identity 2 | |
A reaction: I suspect that either the property of being identical with itself is quite vacuous, or it is parasytic on primitive identity, or it is the criterion which is actually used to define identity. Either way, I don't find this claim very illuminating. |
10040 | Russell showed, through the paradoxes, that our basic logical intuitions are self-contradictory [Russell/Whitehead, by Gödel] |
Full Idea: By analyzing the paradoxes to which Cantor's set theory had led, ..Russell brought to light the amazing fact that our logical intuitions (concerning such notions as truth, concept, being, class) are self-contradictory. | |
From: report of B Russell/AN Whitehead (Principia Mathematica [1913]) by Kurt Gödel - Russell's Mathematical Logic p.452 | |
A reaction: The main intuition that failed was, I take it, that every concept has an extension, that is, there are always objects which will or could fall under the concept. |
5001 | Maybe we should see intentionality and consciousness as a single problem, not two [Kirk,R] |
Full Idea: Many philosophers today have adopted the view that we can achieve an enormous simplification by reducing the two components of the mind-body problem - intentionality and consciousness - into one; ...consciousness is no more than representations. | |
From: Robert Kirk (Mind and Body [2003], §8.4) | |
A reaction: One would then see subjective experience and informational content as two consequences of a single mental activity. This strikes me as the correct route to go. We do, after all, learn BY experiencing. Hence concepts are tied in with qualia. |
4993 | If a bird captures a worm, we could say its behaviour is 'about' the worm [Kirk,R] |
Full Idea: When a bird pulls a worm from the ground, then swallows it piece by piece, there is a sense in which its behaviour can be said to be about the worm. | |
From: Robert Kirk (Mind and Body [2003], §5.4) | |
A reaction: This is preparing the ground for a possible behaviourist account of intentionality. Reply: you could say rain is about puddles, or you could say we have adopted Dennett's 'intentional stance' to birds, but it tells us nothing about their psychology. |
5000 | Behaviourism says intentionality is an external relation; language of thought says it's internal [Kirk,R] |
Full Idea: The conflict over whether intentionality is a matter of behavioural relations with the rest of the world, or of the internal states of the subject, is at its most dramatic in the contrast between behaviourism and the language of thought hypothesis. | |
From: Robert Kirk (Mind and Body [2003], §7.10) | |
A reaction: I just don't believe any behaviourist external account of intentionality, which ducks the question of how it all works. Personally I am more drawn to maps and models than to a language of thought. I plan my actions in an imagined space-time world. |
4982 | Dualism implies some brain events with no physical cause, and others with no physical effect [Kirk,R] |
Full Idea: If the mind causes brain events, then they are not caused by other brain events, and such causal gaps should be detectable by scientists; there should also be a gap of brain-events which cause no other brain events, because they are causing mind events. | |
From: Robert Kirk (Mind and Body [2003], §2.5) | |
A reaction: This is the double causation problem which Spinoza had spotted (Idea 4862). Expressed this way, it seems a screamingly large problem for dualism. We should be able to discover some VERY strange physical activity in the brain. |
4991 | Behaviourism seems a good theory for intentional states, but bad for phenomenal ones [Kirk,R] |
Full Idea: For many kinds of mental states, notably intentional ones such as beliefs and desires, behaviourism is appealing, ..but for sensations and experiences such as pain, it seems grossly implausible. | |
From: Robert Kirk (Mind and Body [2003], §5.1) | |
A reaction: The theory does indeed make a bit more sense for intentional states, but it still strikes me as nonsense that there is no more to my belief that 'Whales live in the Atlantic' than a disposition to say something. WHY do I say this something? |
4994 | Behaviourism offers a good alternative to simplistic unitary accounts of mental relationships [Kirk,R] |
Full Idea: There is a temptation to think that 'aboutness', and the 'contents' of thoughts, and the relation of 'reference', are single and unitary relationships, but behaviourism offers an alternative approach. | |
From: Robert Kirk (Mind and Body [2003], §5.5) | |
A reaction: Personally I wouldn't touch behaviourism with a barge-pole (as it ducks the question of WHY certain behaviour occurs), but a warning against simplistic accounts of intentional states is good. I am sure there cannot be a single neat theory of refererence. |
4992 | In 'holistic' behaviourism we say a mental state is a complex of many dispositions [Kirk,R] |
Full Idea: There is a non-reductive version of behaviourism ( which we can call 'global' or 'holistic') which says there is no more to having mental states than having a complex of certain kinds of behavioural dispositions. | |
From: Robert Kirk (Mind and Body [2003], §5.2) | |
A reaction: This is designed to meet a standard objection to behaviourism - that there is no straight correlation between what I think and how I behave. The present theory is obviously untestable, because a full 'complex' of human dispositions is never repeated. |
4990 | The inverted spectrum idea is often regarded as an objection to behaviourism [Kirk,R] |
Full Idea: The inverted spectrum idea is often regarded as an objection to behaviourism. | |
From: Robert Kirk (Mind and Body [2003], §4.5) | |
A reaction: Thus, my behaviour at traffic lights should be identical, even if I have a lifelong inversion of red and green. A good objection. Note that physicalists can believe in inverted qualia as well a dualists, as long as the brain states are also inverted. |
4984 | All meaningful psychological statements can be translated into physics [Kirk,R] |
Full Idea: All psychological statements which are meaningful, that is to say, which are in principle verifiable, are translatable into propositions which do not involve psychological concepts, but only the concepts of physics. | |
From: Robert Kirk (Mind and Body [2003], §3.8) | |
A reaction: This shows how eliminativist behaviourism arises out of logical positivism (by only allowing what is verifiable). The simplest objection: we can't verify the mental states of others, because they are private, but they are still the best explanation. |
4998 | Instead of representation by sentences, it can be by a distribution of connectionist strengths [Kirk,R] |
Full Idea: In a connectionist system, information is represented not by sentences but by the total distribution of connection strengths. | |
From: Robert Kirk (Mind and Body [2003], §7.6) | |
A reaction: Neither sentences (of a language of thought) NOR connection strengths strike me as very plausible ways for a brain to represent things. It must be something to do with connections, but it must also be to do with neurons, or we get bizarre counterexamples. |
4985 | If mental states are multiply realisable, they could not be translated into physical terms [Kirk,R] |
Full Idea: If psychological states are multiply realisable it is hard to see how they could possibly be translated into physical terms. | |
From: Robert Kirk (Mind and Body [2003], §3.8) | |
A reaction: Reductive funtionalism would do it. A writing iimplement is physical and multiply realisable. Personally I prefer the strategy of saying mental states are NOT multiply realisable. If frog brains differ from ours, they probably don't feel pain like us. |
21725 | The multiple relations theory says assertions about propositions are about their ingredients [Russell/Whitehead, by Linsky,B] |
Full Idea: The multiple relations theory of judgement proposes that assertions about propositions are dependent upon genuine facts involving belief and other attitude relations, subjects of those attitudes, and the constituents of the belief. | |
From: report of B Russell/AN Whitehead (Principia Mathematica [1913]) by Bernard Linsky - Russell's Metaphysical Logic 7.2 | |
A reaction: This seems to require a commitment to universals (especially relations) with which we can be directly acquainted. I prefer propositions, but as mental entities, not platonic entities. |
23474 | A judgement is a complex entity, of mind and various objects [Russell/Whitehead] |
Full Idea: When a judgement occurs, there is a certain complex entity, composed of the mind and the various objects of the judgement. | |
From: B Russell/AN Whitehead (Principia Mathematica [1913], p.44) | |
A reaction: This is Russell's multiple-relation theory of judgement, which replaced his earlier belief in unified propositions (now 'false abstractions'). He seems to have accepted Locke's view, that the act of judgement produces the unity. |
23455 | The meaning of 'Socrates is human' is completed by a judgement [Russell/Whitehead] |
Full Idea: When I judge 'Socrates is human', the meaning is completed by the act of judging. | |
From: B Russell/AN Whitehead (Principia Mathematica [1913], p.44), quoted by Michael Morris - Guidebook to Wittgenstein's Tractatus | |
A reaction: Morris says this is Russell's multiple-relations theory of judgement. The theory accompanies the rejection of the concept of the unified proposition. When I hear 'Socrates had a mole on his shoulder' I get the meaning without judging. |
23480 | The multiple relation theory of judgement couldn't explain the unity of sentences [Morris,M on Russell/Whitehead] |
Full Idea: When Russell moved to his multiple relation theory of judgement …he then faced difficulties making sense of the unity of sentences. | |
From: comment on B Russell/AN Whitehead (Principia Mathematica [1913], p.44) by Michael Morris - Guidebook to Wittgenstein's Tractatus 3A | |
A reaction: Roughly, he seems committed to saying that there is only unity if you think there is unity; there is no unity in a sentence prior to the act of judgement. |
18275 | Only the act of judging completes the meaning of a statement [Russell/Whitehead] |
Full Idea: When I judge 'Socrates is human', the meaning is completed by the act of judging, and we no longer have an incomplete symbol. | |
From: B Russell/AN Whitehead (Principia Mathematica [1913], p.44), quoted by J. Alberto Coffa - The Semantic Tradition from Kant to Carnap | |
A reaction: Personally I would have thought that you needed to know the meaning properly before you could make the judgement, but then he is Bertrand Russell and I'm not. |
4997 | It seems unlikely that most concepts are innate, if a theory must be understood to grasp them [Kirk,R] |
Full Idea: It is widely accepted that for many concepts, if not all, grasping the concept requires grasping some theory, ...which makes difficulties for the view that concepts are not learned: for 'radical concept nativism', as Fodor calls it. | |
From: Robert Kirk (Mind and Body [2003], §7.3) | |
A reaction: Not a problem for traditional rationalist theories, where the whole theory can be innate along with the concept, but a big objection to modern more cautious non-holistic views (such as Fodor's). Does a bird have a concept AND theory of a nest? |
4999 | For behaviourists language is just a special kind of behaviour [Kirk,R] |
Full Idea: Behaviourists regard the use of language as just a special kind of behaviour. | |
From: Robert Kirk (Mind and Body [2003], §7.9) | |
A reaction: This is not an intuitively obvious view of language. We behave, and then we talk about behaviour. Performative utterances (like promising) have an obvious behavioural aspect, as do violent threats, but not highly theoretical language (such as maths). |
4995 | Behaviourists doubt whether reference is a single type of relation [Kirk,R] |
Full Idea: To most behaviourists it seems misguided to expect there to be a single relation that connects referring expressions with their referents. | |
From: Robert Kirk (Mind and Body [2003], §5.5) | |
A reaction: You don't need to be a behaviourist to feel this doubt. Think about names of real people, names of fictional people, reference to misunderstood items, or imagined items, or reference in dreams, or to mathematical objects, or negations etc. |
23453 | Propositions as objects of judgement don't exist, because we judge several objects, not one [Russell/Whitehead] |
Full Idea: A 'proposition', in the sense in which a proposition is supposed to be the object of a judgement, is a false abstraction, because a judgement has several objects, not one. | |
From: B Russell/AN Whitehead (Principia Mathematica [1913], p.44), quoted by Michael Morris - Guidebook to Wittgenstein's Tractatus 2E | |
A reaction: This is the rejection of the 'Russellian' theory of propositions, in favour of his multiple-relations theory of judgement. But why don't the related objects add up to a proposition about a state of affairs? |
7903 | The six perfections are giving, morality, patience, vigour, meditation, and wisdom [Nagarjuna] |
Full Idea: The six perfections are of giving, morality, patience, vigour, meditation, and wisdom. | |
From: Nagarjuna (Mahaprajnaparamitashastra [c.120], 88) | |
A reaction: What is 'morality', if giving is not part of it? I like patience and vigour being two of the virtues, which immediately implies an Aristotelian mean (which is always what is 'appropriate'). |