Philosophy Dictionary of ArgumentsHome
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| Type theory: Restriction of formal systems to a type of reference that prevents symbols of one level (of one type) from referring to symbols of the same level (of the same type). This is intended to avoid paradoxes arising from the self-referentiality of the symbols or expressions used. Original proposals for type theories come from 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, Stages, 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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W.V.O. Quine on Type Theory - Dictionary of Arguments
VII (e) 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level - 3) Zero class: even this accordingly leads to an infinite number of zero classes - 4) Boolean class algebra: is no longer applicable to classes in general, but is reproduced at each level - 5) Relational calculus: accordingly to be established new at every level - 6) arithmetic: the numbers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 ramified 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 "ε" remains the only basic concept apart from quantifiers, variables and statement-logical links. - Context definition for class abstraction: "yn ε {xn: Fxn}" stands for "∃z n + 1["xn(xn ε z n+1 Fxn) u yn ε z n + 1]". IX 191ff 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 ε x ↔ w ε y) u x ε z) › y ε z 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 ε x" shall be considered as true, (Above, "x ε 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.) >Classes, >Sets, >Set theory, >Paradoxes. 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 {ϑ1} ,then 6, or S"5, in type 5 equals {z1: ∃y0(y0 e z1 u z1 n _{y0} = ϑ1)}: this equals Λ², because "y ε z u z n _{y} = ϑ" is contradictory - but then 7, or S"6, equals S'Λ², which is reduced to Λ² - 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. >Numbers._____________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. Translations: Dictionary of Arguments The note [Concept/Author], [Author1]Vs[Author2] or [Author]Vs[term] resp. "problem:"/"solution:", "old:"/"new:" and "thesis:" is an addition from the Dictionary of Arguments. If a German edition is specified, the page numbers refer to this edition. |
Quine I W.V.O. Quine Word and Object, Cambridge/MA 1960 German Edition: Wort und Gegenstand Stuttgart 1980 Quine II W.V.O. Quine Theories and Things, Cambridge/MA 1986 German Edition: Theorien und Dinge Frankfurt 1985 Quine III W.V.O. Quine Methods of Logic, 4th edition Cambridge/MA 1982 German Edition: Grundzüge der Logik Frankfurt 1978 Quine V W.V.O. Quine The Roots of Reference, La Salle/Illinois 1974 German Edition: Die Wurzeln der Referenz Frankfurt 1989 Quine VI W.V.O. Quine Pursuit of Truth, Cambridge/MA 1992 German Edition: Unterwegs zur Wahrheit Paderborn 1995 Quine VII W.V.O. Quine From a logical point of view Cambridge, Mass. 1953 Quine VII (a) W. V. A. Quine On what there is In From a Logical Point of View, , Cambridge, MA 1953 Quine VII (b) W. V. A. Quine Two dogmas of empiricism In From a Logical Point of View, , Cambridge, MA 1953 Quine VII (c) W. V. A. Quine The problem of meaning in linguistics In From a Logical Point of View, , Cambridge, MA 1953 Quine VII (d) W. V. A. Quine Identity, ostension and hypostasis In From a Logical Point of View, , Cambridge, MA 1953 Quine VII (e) W. V. A. Quine New foundations for mathematical logic In From a Logical Point of View, , Cambridge, MA 1953 Quine VII (f) W. V. A. Quine Logic and the reification of universals In From a Logical Point of View, , Cambridge, MA 1953 Quine VII (g) W. V. A. Quine Notes on the theory of reference In From a Logical Point of View, , Cambridge, MA 1953 Quine VII (h) W. V. A. Quine Reference and modality In From a Logical Point of View, , Cambridge, MA 1953 Quine VII (i) W. V. A. Quine Meaning and existential inference In From a Logical Point of View, , Cambridge, MA 1953 Quine VIII W.V.O. Quine Designation and Existence, in: The Journal of Philosophy 36 (1939) German Edition: Bezeichnung und Referenz In Zur Philosophie der idealen Sprache, J. Sinnreich (Hg), München 1982 Quine IX W.V.O. Quine Set Theory and its Logic, Cambridge/MA 1963 German Edition: Mengenlehre und ihre Logik Wiesbaden 1967 Quine X W.V.O. Quine The Philosophy of Logic, Cambridge/MA 1970, 1986 German Edition: Philosophie der Logik Bamberg 2005 Quine XII W.V.O. Quine Ontological Relativity and Other Essays, New York 1969 German Edition: Ontologische Relativität Frankfurt 2003 Quine XIII Willard Van Orman Quine Quiddities Cambridge/London 1987 |
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> Counter arguments against Quine
> Counter arguments in relation to Type Theory
Ed. Martin Schulz, access date 2024-04-28