## Philosophy Dictionary of ArgumentsHome | |||

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Impredicativeness: Impredicatives are concepts which are defined only by means of the propositional sets to which they themselves belong. Problems arise in connection with possible circular conclusions. To avoid paradoxes, the demand is sometimes made to avoid impredicative concepts. See also Paradoxes, Russellian Paradoxy, Poincaré._____________ 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. | |||

Author | Concept | Summary/Quotes | Sources |
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W.V.O. Quine on Impredicativeness - Dictionary of Arguments XIII 93 Impredicativeness/Quine: Previously it was said that you had specified a class without knowing anything about it if you could name the containment condition. Russell's Antinomy: showed that there had to be exceptions. Problem: was to specify a class by a containment condition by directly or indirectly referring to a set of classes that contained the class in question. >Classes/Quine. Russell's Antinomy: here the problematic containment condition was the non-self elementary. Example x is not an element of x. Paradox: arises from letting the x of the containment condition, among other things, be just the class defined by this containment condition. Def impredicative/Poincaré/Russell: is just this condition of containment for a class that exists in the class itself. This must be forbidden to avoid paradoxes. Circular Error Principle/QuineVsRussell: but that was too harsh a term: Specification/Class/Sets/Existence/Quine: specifying a class does not mean creating it! XIII 94 Specification/Circle/Introduce/QuineVsRussell: by specifying something it is not wrong to refer to a domain to which it has always belonged to. For example, statistical statements about a typical inhabitant by statements about the total population that contains this inhabitant. Introduction/Definition/linguistic/Quine: all we need is to equate an unfamiliar expression with an expression that is formed entirely with familiar expressions. Russell's Antinomy/Quine: is still perfectly fine as long as the class R is defined by its containment condition: "class of all objects x, so that x is not an element of x". Paradox/Solution/Russell/Quine: a solution is to distort familiar expressions so that they are no longer familiar in order to avoid a paradox. This was Russell's solution. Finally, "x is an element of x" ("contains itself") to be banished from the language. >Paradoxes/Quine. Solution/Zermelo/Quine: better: leave the language as it is, but New: for classes it should apply that not every containment condition defines a class. For example the class "R" remains well defined, but "Pegasus" has no object. I.e. there is no (well-defined) class like R. Circle/George Homans/Quine: true circularity: For example, a final club is one into which you can only be elected if you have not been elected to other final clubs. Quine: if this is the definition of an unfamiliar expression, then especially the definition of the last occurrence of "final club". Circle/Circularity/Quine: N.B.: yet it is understandable! Impredicativeness/impredicative/Russell/Quine: the real merit was to make it clear that not every containment condition determines a class. Formal: we need a hierarchical notation. Similar to the hierarchy of truth predicates we needed in the liar paradox. XIII 95 Variables: contain indexes: x ^{0},y^{0}: about individuals, x^{1},y^{2} etc. about classes, but classes of this level must not be defined by variables of this level. For example, for the definition of higher-level classes x^{2}, y^{2} only variables of the type x^{0} and x^{1} may be used. Type Theory/Russell/Quine/N.B.: classes of different levels can be of the same type! Classes/Sets/Existence/Quine: this fits the metaphor that classes do not exist before they are determined. I.e. they are not among the values of the variables needed to specify them. ((s) And therefore the thing is not circular). Problem/QuineVsRussell: this is all much stricter than the need to avoid paradoxes and it is so strict that it prevents other useful constructions. For example, to specify the union of several classes of the same level, e.g. level 1 Problem: if we write "Fx ^{1}" to express that x^{1} is one of the many classes in question, then the Containment condition: for a set in this union: something is element of it iff it is an element of a class x1, so Fx ^{1}. Problem: this uses a variable of level 1, i.e. the union of classes of a level cannot be counted on to belong to that level. Continuity hypothesis: for its proof this means difficulties. Impredicativeness/Continuum/Russell/Quine: consequently he dropped the impredicativeness in the work on the first volume of Principia Mathematica ^{(1)}. But it remains interesting in the context of constructivism. It is interesting to distinguish what we can and cannot achieve with this limitation. XIII 96 Predicative set theory/QuineVsRussell/Quine: is not only free of paradoxes, but also of unspecifiable classes and higher indeterminacy, which is the blessing and curse of impredicative theory. (See "infinite numbers", "classes versus sets"). Predicative set theory/Quine: is constructive set theory today. Predicative Set Theory/Quine: is strictly speaking exactly as described above, but today it does not matter which conditions of containment one chooses to specify a class. 1. Whitehead, A.N. and Russel, B. (1910). Principia Mathematica. Cambridge: Cambridge University Press. _____________ 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 InFrom a Logical Point of View, , Cambridge, MA 1953 Quine VII (b) W. V. A. Quine Two dogmas of empiricism InFrom a Logical Point of View, , Cambridge, MA 1953 Quine VII (c) W. V. A. Quine The problem of meaning in linguistics InFrom a Logical Point of View, , Cambridge, MA 1953 Quine VII (d) W. V. A. Quine Identity, ostension and hypostasis InFrom a Logical Point of View, , Cambridge, MA 1953 Quine VII (e) W. V. A. Quine New foundations for mathematical logic InFrom a Logical Point of View, , Cambridge, MA 1953 Quine VII (f) W. V. A. Quine Logic and the reification of universals InFrom a Logical Point of View, , Cambridge, MA 1953 Quine VII (g) W. V. A. Quine Notes on the theory of reference InFrom a Logical Point of View, , Cambridge, MA 1953 Quine VII (h) W. V. A. Quine Reference and modality InFrom a Logical Point of View, , Cambridge, MA 1953 Quine VII (i) W. V. A. Quine Meaning and existential inference InFrom 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 InZur 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 |

Ed. Martin Schulz, access date 2024-03-01