Philosophy Lexicon of Arguments

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Type theory: The type theory is a restriction of formal systems to a kind of reference which prevents symbols of a level (of a type) from referring to symbols of the same level (the same type). This is intended to avoid paradoxes arising from a self-reference of the signs or expressions used. Original proposals for type theories are given by B. Russell (B. Russell, “Mathematical logic as based on the theory of types”, in American Journal of Mathematics, 30, 1908, pp. 222-262). See also self-reference, circularity, paradoxes, Russell's Paradox, branched type theory.

Annotation: The above characterizations of concepts are neither definitions nor exhausting presentations of problems related to them. Instead, they are intended to give a short introduction to the contributions below. – Lexicon of Arguments.

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VII 91ff
QuineVsType theory: 1) universal class: because the type theory only allows uniform types as elements of a class, the universal class V leads to an infinite series of quasi-universal classes, each for one type - 2) negation: ~x stops including all non-elements of x and only includes those non-elements that belong to the next lower level - 3) Zero class: even this accordingly leads to an infinite number of zero classes - 4) Boolean class algebra: is no longer applicable to classes in general, but is reproduced at each level - 5) Relational calculus: accordingly to be established new at every level - 6) arithmetic: the numbers cease to be uniform. At each level (type) there is a new 0, new 1, new 2, etc.
IX 186
Definition branched type theory/Russell/Quine: distinction of orders for statement functions whose arguments are of one single order - in order for two attributes with the same extension to be able to differ in terms of their orders, attributes with the same extension must be distinguished and be called attributes and not classes - new: this becomes superfluous when we drop the branching - Solution: context definition/Russell: we define class abstraction through context, thus "e" remains the only basic concept apart from quantifiers, variables and statement-logical links - context definition for class abstraction: "yn e {xn: Fxn}" stands for "Ez n + 1["xn(xn e z n+1 Fxn) u yn e z n + 1]".
IX 191~
Cumulative types/Set Theory/Quine: Type 0: Only L is of type 0 - type 1: L and {L} and nothing else - Type n: should generally include only this and the 2n sets that belong to type n-1 - in this way, every quantification only interprets a finite number of cases. Each closed expression can be mechanically tested on being true - that no longer works when the axiom of infinity is added.
IX 198
Cumulative types/Quine: advantages: if we equate the zero classes of all class types, (~T0x u ~T0y u "w(w e x w e y) u x e z) > y ez is a single axiom, no longer an axiom scheme - in int "~T0x u ~T0y" avoids that the individuals L are identified with one another - we need individuals, but we identify them with their classes of one (see above) - but one exception: if x is an individual, "x e x" shall be considered as true, (Above, "x e y" became false if both were not objects of sequential types).
IX 201
Cumulative Type Theory/Quine: individuals: identified with their classes of one - no longer elementless, have themselves as elements - therefore definite identity: a = b if a < b < a - zero classes of all types can now be identified (formerly: "No individuals" , "no classes", etc.)
IX 204
Natural numbers/QuineVsRussell: his type theory even has problems with Frege’s numbers: perhaps the successor relation does not bring something new always: Example 5 is then the class of all classes from five individuals, assuming that there are only five individuals in that universe. So 5 in type 2 equals {J1} ,then 6, or S"5, in type 5 equals {z1: Ey0(y0 e z1 u z1 n _{y0} = J1)}: this equals L², because "y e z u z n _{y} = J" is contradictory - but then 7, or S"6, equals S"L², which is reduced to L² - i.e. S"x = x when x equals 6 in type 2, provided that there are no more than five individuals - otherwise the theory of numbers would collapse.

Explanation of symbols: Roman numerals indicate the source, arabic numerals indicate the page number. The corresponding books are indicated on the right hand side. ((s)…): Comment by the sender of the contribution.

W.V.O. Quine
Wort und Gegenstand Stuttgart 1980

W.V.O. Quine
Theorien und Dinge Frankfurt 1985

W.V.O. Quine
Grundzüge der Logik Frankfurt 1978

W.V.O. Quine
Mengenlehre und ihre Logik Wiesbaden 1967

W.V.O. Quine
Die Wurzeln der Referenz Frankfurt 1989

W.V.O. Quine
Unterwegs zur Wahrheit Paderborn 1995

W.V.O. Quine
From a logical point of view Cambridge, Mass. 1953

W.V.O. Quine
Bezeichnung und Referenz
Zur Philosophie der idealen Sprache, J. Sinnreich (Hg), München 1982

W.V.O. Quine
Philosophie der Logik Bamberg 2005

W.V.O. Quine
Ontologische Relativität Frankfurt 2003

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Ed. Martin Schulz, access date 2017-11-21