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Single Idea 10607
[filed under theme 6. Mathematics / C. Sources of Mathematics / 6. Logicism / b. Type theory
]
Full Idea
Frege's general logical system involves a type hierarchy, distinguishing objects from properties from properties-of-properties etc., with every item belonging to a determinate level.
Gist of Idea
Frege's logic has a hierarchy of object, property, property-of-property etc.
Source
report of Gottlob Frege (Begriffsschrift [1879]) by Peter Smith - Intro to Gödel's Theorems 14.1
Book Ref
Smith,Peter: 'An Introduction to Gödel's Theorems' [CUP 2007], p.118
A Reaction
The Theory of Types went on to apply this hierarchy to classes, where Frege's disastrous Basic Law V flattens the hierarchy of classes, putting them on the same level (Smith p.119)
The
23 ideas
with the same theme
[theory that maths is a hierarchy of set types]:
10607
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Frege's logic has a hierarchy of object, property, property-of-property etc.
[Frege, by Smith,P]
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10093
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The ramified theory of types used propositional functions, and covered bound variables
[Russell/Whitehead, by George/Velleman]
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8691
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The Russell/Whitehead type theory was limited, and was not really logic
[Friend on Russell/Whitehead]
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21555
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For 'x is a u' to be meaningful, u must be one range of individuals (or 'type') higher than x
[Russell]
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18003
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In 'x is a u', x and u must be of different types, so 'x is an x' is generally meaningless
[Russell, by Magidor]
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23457
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Type theory cannot identify features across levels (because such predicates break the rules)
[Morris,M on Russell]
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21556
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Classes are defined by propositional functions, and functions are typed, with an axiom of reducibility
[Russell, by Lackey]
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10418
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Type theory seems an extreme reaction, since self-exemplification is often innocuous
[Swoyer on Russell]
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10047
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Russell's improvements blocked mathematics as well as paradoxes, and needed further axioms
[Russell, by Musgrave]
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23478
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Type theory means that features shared by different levels cannot be expressed
[Morris,M on Russell]
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6409
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The 'simple theory of types' distinguishes levels among properties
[Ramsey, by Grayling]
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21557
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Russell confused use and mention, and reduced classes to properties, not to language
[Quine, by Lackey]
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18127
|
Simple type theory has 'levels', but ramified type theory has 'orders'
[Bostock]
|
10265
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Chihara's system is a variant of type theory, from which he can translate sentences
[Chihara, by Shapiro]
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8759
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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]
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21714
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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]
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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]
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16308
|
Set theory was liberated early from types, and recent truth-theories are exploring type-free
[Halbach]
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