Philosophy Dictionary of ArgumentsHome | |||
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Recursion, theory of science, philosophy: recursion is a certain form in which rules are formulated, and which makes it possible to produce infinitely many possible cases from the application of a finite system of rules. See also inserting, embedding, infinity, systems, models, theories._____________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 Recursion - Dictionary of Arguments
IX 58 Recursive definition/recursion/sum/product/potency/arithmetic/Quine: recursion scheme: x + 0 = x - x + S°y = S°(x + y); - x times 0 = 0; - x times (S°y) = x + x times y - (s) difference to the successor for x u y equal)›; - x0 = S°0 (=1) ; - x S°y = x times x y. - "Plus"/plus sign/Quine: so we can eliminate "+" completely from "x + 3": "S°(S°(S°x))" - but not from "x + y" (Because we do not know how often we need the successor of x) - multiplication: we can eliminate the "times" from "x 3 times": "x + (x + (x + 0))" but not from "x times y" - recursions are real definitions if we regard the characters as scheme letters for numbers, not as bound variables. --- IX 126 Transfinite recursion/sum/product/potency/Quine: x * 0 = 0. x * (S 'z) = x + x * z - transformed into a real or direct definition: x * y = (λv(x + v))Iy'0 - general divice: a'0 = k, a'(S'z) = b'(a'z) - a'y = b Iy'k - from the last element: a = U{w: w ε Seq u ‹k,0› ε w u w I S ^w ≤ b}. - Advanced, liberal recursion: not only from the last previous element. - instead totality of the previous elements a = U{w: w ε Seq u ∀y(y ε ^w''ϑ ›› ‹w'y, w re {z:z ‹ y}› ε g)}._____________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 |