more from this thinker     |     more from this text

Single Idea 18003

[filed under theme 6. Mathematics / C. Sources of Mathematics / 6. Logicism / b. Type theory ]

Full Idea

Russell argues that in a statement of the form 'x is a u' (and correspondingly, 'x is a not-u'), 'x must be of different types', and hence that ''x is an x' must in general be meaningless'.

Gist of Idea

In 'x is a u', x and u must be of different types, so 'x is an x' is generally meaningless

Source

report of Bertrand Russell (The Principles of Mathematics [1903], App B:524) by Ofra Magidor - Category Mistakes 1.2

Book Ref

Magidor,Ofra: 'Category Mistakes' [OUP 2013], p.8

A Reaction

" 'Word' is a word " comes to mind, but this would be the sort of ascent to a metalanguage (to distinguish the types) which Tarski exploited. It is the simple point that a classification can't be the same as a member of the classification.

Related Idea

Idea 18002 As well as a truth value, propositions have a range of significance for their variables [Russell]

The 23 ideas with the same theme [theory that maths is a hierarchy of set types]:

10607 Frege's logic has a hierarchy of object, property, property-of-property etc. [Frege, by Smith,P]
10093 The ramified theory of types used propositional functions, and covered bound variables [Russell/Whitehead, by George/Velleman]
8691 The Russell/Whitehead type theory was limited, and was not really logic [Friend on Russell/Whitehead]
21555 For 'x is a u' to be meaningful, u must be one range of individuals (or 'type') higher than x [Russell]
18003 In 'x is a u', x and u must be of different types, so 'x is an x' is generally meaningless [Russell, by Magidor]
23457 Type theory cannot identify features across levels (because such predicates break the rules) [Morris,M on Russell]
21556 Classes are defined by propositional functions, and functions are typed, with an axiom of reducibility [Russell, by Lackey]
10418 Type theory seems an extreme reaction, since self-exemplification is often innocuous [Swoyer on Russell]
10047 Russell's improvements blocked mathematics as well as paradoxes, and needed further axioms [Russell, by Musgrave]
23478 Type theory means that features shared by different levels cannot be expressed [Morris,M on Russell]
6409 The 'simple theory of types' distinguishes levels among properties [Ramsey, by Grayling]
21557 Russell confused use and mention, and reduced classes to properties, not to language [Quine, by Lackey]
18127 Simple type theory has 'levels', but ramified type theory has 'orders' [Bostock]
10265 Chihara's system is a variant of type theory, from which he can translate sentences [Chihara, by Shapiro]
8759 We can replace type theory with open sentences and a constructibility quantifier [Chihara, by Shapiro]
21703 Types are 'ramified' when there are further differences between the type of quantifier and its range [Linsky,B]
21714 The ramified theory subdivides each type, according to the range of the variables [Linsky,B]
21721 Higher types are needed to distinguished intensional phenomena which are coextensive [Linsky,B]
10092 In the unramified theory of types, the types are objects, then sets of objects, sets of sets etc. [George/Velleman]
10094 The theory of types seems to rule out harmless sets as well as paradoxical ones. [George/Velleman]
10095 Type theory has only finitely many items at each level, which is a problem for mathematics [George/Velleman]
17901 Type theory prohibits (oddly) a set containing an individual and a set of individuals [George/Velleman]
16308 Set theory was liberated early from types, and recent truth-theories are exploring type-free [Halbach]