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 1973,0807606995]].
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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 metalanguage is clearly in the same territory.

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