# Philosophy Dictionary of Arguments

Home 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.

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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 Item Summary Meta data

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.
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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)}.

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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 [Author1]Vs[Author2] or [Author]Vs[term] 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

> Counter arguments against Quine
> Counter arguments in relation to Recursion

Ed. Martin Schulz, access date 2021-06-20