Psychology Dictionary of ArgumentsHome
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| Leopold Löwenheim: Leopold Löwenheim (1878-1957) was a German mathematician who worked on mathematical logic. He is best known for the Löwenheim-Skolem theorem, which states that every first-order theory with an infinite model also has a countable model. See also Models, Model theory, Satisfaction, Satisfiability, Infinity, Countability, Real numbers, Numbers, Word meaning, Reference, Ambiguity._____________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 Loewenheim - Dictionary of Arguments
X 79 Validity/Sentence/Quantity/Schema/Quine: if quantities and sentences fall apart in this way, there should be a difference between these two definitions of validity (via schema with sentences) or models (with quantities). But it follows from the Loewenheim theorem that the two definitions of validity (via sentences or quantities) do not fall apart as long as the object language is not too weakly (poorly) expressive. Condition: the object language must be able to express (include) the elementary number theory. Object Language: in such a language, a scheme that remains true for all sentence implementations is also fulfilled by all models and vice versa. The demand of elementary number theory is quite weak. Def Elementary Number Theory/eZT/Quine: is about positive integers using addition, multiplication, identity, truth functions and quantification. >Number Theory/Quine. Standard Grammar/Quine: the standard grammar would express the addition, multiplication and identity functions by appropriate predicates. That is how we get the two sentences: (I) If a scheme remains true for all implentations of sentences of the elementary number theory sets, then it is fulfilled by all models. X 80 (II) If a scheme is fulfilled by each model, then e is true for all settings of sets. Quine: Sentence (I) goes back to Loewenheim 1915: Sentence of Loewenheim/Quine: every scheme that is ever fulfilled by a model is fulfilled by a model 'U,‹U,β,α...', where U contains only the positive integers. Loewenheim/Hilbert/Bernays: intensification: the quantities α, β,γ,...etc. may each be determined by a sentence of the elementary number theory: So: (A) If a scheme is fulfilled by a model at all, it is true when using sentences of the elementary number theory instead of its simple schemes. Prerequisite for the implentations: the quantifiable variables must have the positive integers in their value range. However, they may also have other values. (I) follows from (A) that: (A) is equivalent to its contraposition: if a schema is wrong in all the implementations of s of sentences of the elementary number theory, it is not fulfilled by any model. If we speak here about its negation instead of the schema, then "false2" becomes "true" and "from no model" becomes "from every model". This gives us (I). The sentence (II) is based on the theorem of the deductive completeness of the quantifier logic. II 29 Classes: one could reinterpret all classes in its complement, "not an element of ..." - you would never notice anything! Bottom layer: each relative clause, each general term determines a class. >Classes/Quine. V 160 Loewenheim/Quine: there is no reinterpretation of characters - but rather a change of terms and domains - the meanings of the characters for truth functions and for quantifiers remain constant. The difference is not that big and can only play a role with the help of a new term: "ε" or "countable". For quantifiers and truth functions only the difference finite/infinte plays a role. Uncountable is not a matter of opinion. Solution: it is all about which term is fundamental: countable or uncountable._____________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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> Counter arguments against Quine
> Counter arguments in relation to Loewenheim
