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Single Idea 6409
[filed under theme 6. Mathematics / C. Sources of Mathematics / 6. Logicism / b. Type theory
]
Full Idea
The idea that there should be something like a distinction of levels among properties is captured in Ramsey's 'simple theory of types'.
Gist of Idea
The 'simple theory of types' distinguishes levels among properties
Source
report of Frank P. Ramsey (works [1928]) by A.C. Grayling - Russell
Book Ref
Grayling,A.C.: 'Russell' [OUP 1996], p.31
A Reaction
I merely report this, though it is not immediately obvious how anyone would decide which 'level' a type belonged on.
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]
|
17901
|
Type theory prohibits (oddly) a set containing an individual and a set of individuals
[George/Velleman]
|
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]
|
16308
|
Set theory was liberated early from types, and recent truth-theories are exploring type-free
[Halbach]
|