Ideas from 'The Theory of Logical Types' by Bertrand Russell [1910], by Theme Structure

[found in 'Essays in Analysis' by Russell,Bertrand (ed/tr Lackey,Douglas) [George Braziller 1973,0-8076-0699-5]].

green numbers give full details    |     back to texts     |     unexpand these ideas

---

5. Theory of Logic / E. Structures of Logic / 5. Functions in Logic

21566

'Propositional functions' are ambiguous until the variable is given a value

			Full Idea: By a 'propositional function' I mean something which contains a variable x, and expresses a proposition as soon as a value is assigned to x. That is to say, it differs from a proposition solely by the fact that it is ambiguous.
			From: Bertrand Russell (The Theory of Logical Types [1910], p.216)
			A reaction: This is Frege's notion of a 'concept', as an assertion of a predicate which still lacks a subject.

5. Theory of Logic / L. Paradox / 6. Paradoxes in Language / a. The Liar paradox

21567

'All judgements made by Epimenedes are true' needs the judgements to be of the same type

			Full Idea: Such a proposition as 'all the judgements made by Epimenedes are true' will only be prima facie capable of truth if all his judgements are of the same order.
			From: Bertrand Russell (The Theory of Logical Types [1910], p.227)
			A reaction: This is an attempt to use his theory of types to solve the Liar. Tarski's invocation of a meta-language is clearly in the same territory.

6. Mathematics / C. Sources of Mathematics / 6. Logicism / b. Type theory

23457

Type theory cannot identify features across levels (because such predicates break the rules)

			Full Idea: Russell's theory of types meant that features common to different levels of the hierarchy became uncapturable (since any attempt to capture them would involve a predicate which disobeyed the hierarchy restrictions).
			From: comment on Bertrand Russell (The Theory of Logical Types [1910]) by Michael Morris - Guidebook to Wittgenstein's Tractatus 2H
			A reaction: I'm not clear whether this is the main reason why type theory was abandoned. Ramsey was an important critic.

21556

Classes are defined by propositional functions, and functions are typed, with an axiom of reducibility

			Full Idea: In Russell's mature 1910 theory of types classes are defined in terms of propositional functions, and functions themselves are regimented by a ramified theory of types mitigated by the axiom of reducibility.
			From: report of Bertrand Russell (The Theory of Logical Types [1910]) by Douglas Lackey - Intros to Russell's 'Essays in Analysis' p.133

6. Mathematics / C. Sources of Mathematics / 10. Constructivism / d. Predicativism

21568

A one-variable function is only 'predicative' if it is one order above its arguments

			Full Idea: We will define a function of one variable as 'predicative' when it is of the next order above that of its arguments, i.e. of the lowest order compatible with its having an argument.
			From: Bertrand Russell (The Theory of Logical Types [1910], p.237)
			A reaction: 'Predicative' just means it produces a set. This is Russell's strict restriction on which functions are predicative.