Ideas from 'Mathematical logic and theory of types' by Bertrand Russell [1908], by Theme Structure

[found in 'Logic and Knowledge' by Russell,Bertrand (ed/tr Marsh,Robert Charles) [Routledge 1956,0-415-09074-1]].

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4. Formal Logic / F. Set Theory ST / 8. Critique of Set Theory
Classes can be reduced to propositional functions
                        Full Idea: Russell held that classes can be reduced to propositional functions.
                        From: report of Bertrand Russell (Mathematical logic and theory of types [1908]) by Robert Hanna - Rationality and Logic 2.4
                        A reaction: The exact nature of a propositional function is disputed amongst Russell scholars (though it is roughly an open sentence of the form 'x is red').
5. Theory of Logic / L. Paradox / 5. Paradoxes in Set Theory / d. Russell's paradox
The class of classes which lack self-membership leads to a contradiction
                        Full Idea: The class of teaspoons isn't a teaspoon, so isn't a member of itself; but the class of non-teaspoons is a member of itself. The class of all classes which are not members of themselves is a member of itself if it isn't a member of itself! Paradox.
                        From: report of Bertrand Russell (Mathematical logic and theory of types [1908]) by A.C. Grayling - Russell Ch.2
                        A reaction: A very compressed version of Russell's famous paradox, often known as the 'barber' paradox. Russell developed his Theory of Types in an attempt to counter the paradox. Frege's response was to despair of his own theory.
6. Mathematics / C. Sources of Mathematics / 6. Logicism / b. Type theory
Type theory seems an extreme reaction, since self-exemplification is often innocuous
                        Full Idea: Russell's reaction to his paradox (by creating his theory of types) seems extreme, because many cases of self-exemplification are innocuous. The property of being a property is itself a property.
                        From: comment on Bertrand Russell (Mathematical logic and theory of types [1908]) by Chris Swoyer - Properties 7.5
                        A reaction: Perhaps it is not enough that 'many cases' are innocuous. We are starting from philosophy of mathematics, where precision is essentially. General views about properties come later.
Russell's improvements blocked mathematics as well as paradoxes, and needed further axioms
                        Full Idea: Unfortunately, Russell's new logic, as well as preventing the deduction of paradoxes, also prevented the deduction of mathematics, so he supplemented it with additional axioms, of Infinity, of Choice, and of Reducibility.
                        From: report of Bertrand Russell (Mathematical logic and theory of types [1908]) by Alan Musgrave - Logicism Revisited 2
                        A reaction: The first axiom seems to be an empirical hypothesis, and the second has turned out to be independent of logic and set theory.
6. Mathematics / C. Sources of Mathematics / 6. Logicism / c. Neo-logicism
Ramified types can be defended as a system of intensional logic, with a 'no class' view of sets
                        Full Idea: A defence of the ramified theory of types comes in seeing it as a system of intensional logic which includes the 'no class' account of sets, and indeed the whole development of mathematics, as just a part.
                        From: report of Bertrand Russell (Mathematical logic and theory of types [1908]) by Bernard Linsky - Russell's Metaphysical Logic 6.1
                        A reaction: So Linsky's basic project is to save logicism, by resting on intensional logic (rather than extensional logic and set theory). I'm not aware that Linsky has acquired followers for this. Maybe Crispin Wright has commented?
6. Mathematics / C. Sources of Mathematics / 10. Constructivism / d. Predicativism
A set does not exist unless at least one of its specifications is predicative
                        Full Idea: The idea is that the same set may well have different canonical specifications, i.e. there may be different ways of stating its membership conditions, and so long as one of these is predicative all is well. If none are, the supposed set does not exist.
                        From: report of Bertrand Russell (Mathematical logic and theory of types [1908]) by David Bostock - Philosophy of Mathematics 8.1
Russell is a conceptualist here, saying some abstracta only exist because definitions create them
                        Full Idea: It is a conceptualist approach that Russell is relying on. ...The view is that some abstract objects ...exist only because they are definable. It is the definition that would (if permitted) somehow bring them into existence.
                        From: report of Bertrand Russell (Mathematical logic and theory of types [1908]) by David Bostock - Philosophy of Mathematics 8.1
                        A reaction: I'm suddenly thinking that predicativism is rather interesting. Being of an anti-platonist persuasion about abstract 'objects', I take some story about how we generate them to be needed. Psychological abstraction seems right, but a bit vague.
Vicious Circle says if it is expressed using the whole collection, it can't be in the collection
                        Full Idea: The Vicious Circle Principle says, roughly, that whatever involves, or presupposes, or is only definable in terms of, all of a collection cannot itself be one of the collection.
                        From: report of Bertrand Russell (Mathematical logic and theory of types [1908], p.63,75) by David Bostock - Philosophy of Mathematics 8.1
                        A reaction: This is Bostock's paraphrase of Russell, because Russell never quite puts it clearly. The response is the requirement to be 'predicative'. Bostock emphasises that it mainly concerns definitions. The Principle 'always leads to hierarchies'.