Philosophy Dictionary of ArgumentsHome
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| Systems, philosophy of science: systems are compilations of rules for the formation of statements on a previously defined subject domain. Apart from the - usually recursive - rules for the combination of expressions or signs, the specification of the vocabulary or sign set of the system is also required. See also axioms, axiom systems, theories, strength of theories, expressiveness, rules, order, recursion, models, structure, system 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 Systems - Dictionary of Arguments
VII (e) 91 Abbreviations/Quine: defining abbreviations are always outside of a formal system - that's why we need to get an expression in simple notation before we examine it in relation to hierarchy. - - - IX 190 System/Quine: a new system is not introduced by new definitions, but by new distinctions. ((s) Example (s): if I always have to note "n + 1" to mark the difference between real and rational numbers, I did not eliminate the real numbers, but kept the old difference. I only changed the notation, not the ontology.) IX 232 Theory/Enlargement/Extension/System/Quine: an enlargement is not an extension! Extension: addition of axioms, can create contradictions. Magnification/Quine: means to relativize an added scheme to already existing axioms of a system, e.g. to "Uϑ", (s) so if something exists in "Uϑ", it must be a set. Such a magnification never creates a contradiction. IX 237 Theory/stronger/weaker/Quine: if a deductive system is an extension of another in the sense that its theorems include all of the other and others, then in a certain way one is stronger than the other. But this basis of comparison is weak: 1. It fails if each of the two systems has theorems that are not found in the other. (Comparability). 2. It depends on randomness of interpretation and not simply on structural properties. Example: suppose we would have exactly "=" and "R" as primitive two-digit predicates with an ordinary identity axiom and transitivity. Now we extend the system by adding the reflexivity "x(xRx)". The extended system is only stronger if we equate its "R" with the original "R". But if we reinterpret its "xRy" as "x = y v x R y" using the original "R", then all its theorems are provable in the non-extended system. (>Löwenheim, >Provability), Example (less trivial): Russell's method ((1) to (4), Chapter 35) to ensure extensionality for classes without having to accept them for attributes. Given is a set theory without extensionality. We could extend it by adding this axiom, and yet we could show that all theorems of the extended system could be reinterpreted with Russell's method as theorems already provable in the non-extended system. Stronger/weaker/Quine: a better standard for the comparison of strength is the "comparison by reinterpretation": if we can reinterpret the primitive logical signs (i.e. in set theory only "e") in such a way that all theorems of this system become translations of the theorems of the other system, then the latter system is at least as strong as the first one. IX 238 If this is not possible in the other direction, one system is stronger than the other. Def "ordinal strength"/Quine: another meaningful sense of strength of a system is the following surprising numerical measure: the smallest transfinite ordinal number, whose existence can no longer be proven in the system. Any normal set theory can, of course, prove the existence of infinitely many transfinite numbers, but that does not mean that you get them all. Transfinite/Quine: what is so characteristic about it is that we then iterate the iteration further and iterate the iteration of iterations until our apparatus somehow blocks. The smallest transfinite number after blocking the apparatus then indicates how strong the apparatus was. An axiom that can be added to a system with the visible goal of increased ordinal strength is the axiom that there is an unattainable number beyond w (omega). (End of Chapter 30). An endless series of further axioms of this kind is possible. Strength of systems/Ordinal Numbers/Quine: another possibility to use ordinal numbers for strength: we can extend the theory of cumulative types to transfinite types by accrediting to the x-th type for each ordinal number x, all classes whose elements all have a type below x. So the universe of the theory of cumulative types in chapter 38, which lacks the transfinite types, is even the ω-th type. Def "Natural Model"/Montague/Vaught/Quine: this is what they call this type, if the axioms of set theory are fulfilled, if one takes their universe as such a type. So Zermelo's set theory without infinity axiom has the ω-th type as a natural model. (We have seen this in chapter 38). So the ordinal strength of this system is at most ω, obviously not smaller. With infinity axiom: ω + ω. Strength of the system of von Neumann-Bernays: one more than the first unattainable number after w. - - - XII 33 Object/existence/system/Quine: systematic considerations can lead us to reject certain objects XII 34 or to declare certain terms as non-referring. Occurrence: also individual occurrences of terms. This is Frege's point of view: an event can refer to something on one occasion, not on another (referential position). Example "Thomas believes that Tullius wrote the Ars Magna". In reality he confuses Tullius with Lullus._____________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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Ed. Martin Schulz, access date 2024-04-16