Find counter arguments by entering NameVs… or …VsName.
| Disputed term/author/ism | Author Vs Author |
Entry |
Reference |
|---|---|---|---|
| Hilbert | Quine Vs Hilbert | IX 187 Notation/Set Theory/Terminology/Hilbert/Ackermann: (1938, 1949): still lean to the old theory of Russell's statements functions (AF): for classes and relations: "F", "G", etc. with repressible indices, in stead of "x ε a" and "xRy" (Russell: "φx" and "ψ(x,y)" Hilbert "F(x)" and "G(x,y)". Quine: the similarity is misleading: The values of "F", "G", etc. are not AF, but classes and extensional relations and so for the only criterion that those are identical with the same extension. QuineVsHilbert: disadvantage that the attention is drawn away from essential differences between ML and logic. IX 188 It encourages us (incorrectly) to just consider the theory of classes and relations as a continuation of the QL, in which the thus far schematic predicate letters are re-registered in quantifiers and other places which were previously reserved for "x", "y" I.e. "F", "EG", "H(F,G)". The existence assumptions become too inconspicuous, although they are far-reaching! Just implicitly by quantification. Therefore every Comprehension assertion, e.g. EF∀x(FX >> ... x ...) by such insertions simply follows from "G EF ∀x (Fx Gx) which in turn follows from "∀x(Gx Gx)". This had escaped Hilbert and Ackermann, they also took on Comprehension axioms, they realized that they could have taken a primitive concept of abstraction instead (like Russell). Predicate Calculus/Functions Calculus/Church/Quine: (nth order): type theory breaking of after n types, fusion of set theory and logic (QuineVs). E.g. PK 2nd Stage: Theory of individuals and classes of individuals. It was simply seen as a QL where predicate letters are approved quantifiers. The actual QL then became a first stage PK. This trend also contained an erroneous distinction between TT and ML, as if one did not contain as good as assumptions about the other. On the other, hand he nourished the idea that the Ql itself already contained a theory of classes or attributes and relations in its "F" and "G". QuineVs: the vital distinction between schematic letters and quantifiable variables is neglected. X 96 Logic 2nd Stage/Hilbert's Successor/Quine: "higher-level PK": the values of these variables are in fact sets. This type of introduction makes them deceptively similar to logic. But it is wrong that only a few quantifiers are applied to existing predicate letters. E.g. the hypothesis "(Ey)(x)((x ε y) Fx)": here the existence of a set is asserted: {x:Fx}. This must be restricted to avoid antinomies. QuineVsHilbert: in the so-called higher order PK this assumption moves out of sight. The assumption is: "(EG) (x) (Gx Fx)" and follows from the purely logical triviality (x)(Fx Fx)" As long as we keep the scope of the values of "x" and "G" apart there is no risk of an antinomy. Nevertheless, a large piece of set theory has crept in unnoticed. XI 136 Mathematics/QuineVsHilbert/Lauener: more than mere syntax. Quine reluctantly professes Platonism. |
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 |
| Intuitionism | Poincaré Vs Intuitionism | Wessel I 236 PoincaréVsIntuitionism/VsConstruktivism/Wessel: (Poincaré calls the intuitionists pragmatists): "The pragmatist should take the position of the extension, the Cantorian that of comprehension (compréhension). The objects, however, are there before the inscriptions, and the set itself would exist if there was no one who would undertake to organize it." I 237 Intuitionism/Logic/Wessel: the intuitionists reject not only the concept of the actual infinite, but they also believe that they have to limit logic: Brouwer: the law of excluded third only applies within a certain finite main system, since it is possible to come to an empirical confirmation here. BrouwerVsLogic: as foundation of mathematics. Instead: vice versa! I 238 (s) It is about the practice of the mathematician, therefore the limits of the constructive possibilities are not random or can be overcome easily by logical considerations.) Constructivism/Brouwer/Heyting: examines the construction as such, without inquiring after the nature of the objects, e.g. whether they exist! Law of Excluded Third/Intuitionism/Heyting/Wessel: (a) k is the biggest prime number such that k-1 is also one; if there is no such number, k = 1 (s) "the only prime that is adjacent to another". (b) l is the biggest prime such that l-2 is also one; if there is no such number, l = 1. Wessel: k can really be determined (k = 3), while we do not have any methods to determine l. This leads to the rejection of the law of excluded third: for if the sequence of prime twins was either finite or infinite, then (b) would define an integer. Intuitionism/Logic/Logical Operators/Wessel: because certain laws of logic do not apply here, the different logics are various complexes of operators. But the intuitionists have the same claim, to comprehend the meaning of "and", "not", "or" in the everyday language. Def Conjunction/Intuitionism/Wessel: p u q can be claimed exactly then when both p and q can be claimed. |
Wessel I H. Wessel Logik Berlin 1999 |
| Neumann, J. von | Quine Vs Neumann, J. von | IX 227 Neumann/Set Theory/Quine: took (1925) extreme classes to extend Zermelo's system. QuineVsNeumann: halted before the full force of the axiom scheme "^ uFu ε θ" or (3) (Chapter 42) unfolded. His system provides "^uFu ε θ" when the bound variables in the formula "Fu" are restricted to all sets, otherwise it does not apply in general. --- IX 228 If the sets should be exactly Zermelo's classes, they could be specified by relativizing Zermelo's comprehension axioms to "Uθ". In particular, every abstraction term from the Zermelo's separation scheme "X n a ε θ" x ε Uθ> x n a ε Uθ, which we would use for "a", would be relativized to "Uθ" additionally. (That is, the "universal class of sets" or "there are only quantities"). Equivalent to this: Since such a relativization is guaranteed that a = z for a certain z we could just take the single axiom (1) x ε Uθ> x n z ε Uθ. |
Quine XIII Willard Van Orman Quine Quiddities Cambridge/London 1987 |
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