Philosophy Dictionary of ArgumentsHome
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| 2nd order Logic: Predicate logic of the 2nd order goes beyond predicate logic of the 1st level allowing quantification over properties and relations, and not just objects. Thus comparisons of the powerfulness of sets become possible. Problems which are expressed in everyday terms with terms such as "greater", "between", etc., and e.g. the specification of all the properties of an object require predicate logic of the 2nd order. Since the 2nd level logic is not complete (because there are, for example, an infinite number of properties of properties), one often tries to get on with the logic of the 1st order._____________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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Hartry Field on Second Order Logic, HOL - Dictionary of Arguments
I 37 Second Order Logic/Second Order Logic/Higher Order Logic/HOL/Field: Here, the the quantifiers have no recursive method of evidence. >Quantifiction, >Quantifiers, >Logic, >Recursion. Quantification/Field: therefore it is vague and indeterminate, but even then applies: (A > logically true (A)) & (~ A > logically true (~ A)) is always true. The vagueness refers to the A. --- II 238 Referential indeterminacy/logical operators/2nd order Logic/Field: special case: Question: can complex logical operators - e.g., unrestricted 2nd order quantifiers ((s) via properties) have any particular truth conditions? No: e.g. everything that you express with them can be reformulated (reduced) with a more restricted quantification (via sets). It does not help to say e.g. "with "for all properties" I mean for all properties". >"Everything he said"). >Truth conditions, >Sets, >Extensions, >Extensionality. All/Field: the use of "all" without quotes is itself the subject of a reinterpretation. >All/Field. ((s) There could be a contradictory, still undiscovered property which should not be included under "all properties.") Field: E.g. Acceleration near speed of light - here the definitive operator would again help. VsDeflationism: Deflationism could simply say ".. all .. " is true iff all ... Vs: in addition one needs the definitive-operator (dft-operator), which demands conditions - but it does not specify them. Field: dito with Higher Order Quantification (HOL). --- III 39 First order Logic/2nd order/stronger/weaker/attenuation/Field: to weaken the second order logic to the 1st order, we can attenuate the second-order axioms to the axiom-schemata of first-order , namely the schema of separation. Problem: not many non-standard models come in. Namely, models in which quantities that are in reality infinite, satisfy the formula which usually defines straight finiteness. >unintended models. III 92 2nd Order Logic/Field: we have it at two places: 1. At the axiomatization of the geometry of the spacetime and at the scalar order of spacetime points we have III 93 The "complete logic of the part-whole relation", or the "complete logic of the Goodman sums". 2. The binary quantifier "less than". But we do not need this if we have Goodman's sums: Goodman's sum: it's logic is sufficient to give comparisons of powerfulness. For heuristic reasons, however, we want to keep an extra logic for powerfulness ("less than")._____________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. |
Field I H. Field Realism, Mathematics and Modality Oxford New York 1989 Field II H. Field Truth and the Absence of Fact Oxford New York 2001 Field III H. Field Science without numbers Princeton New Jersey 1980 Field IV Hartry Field "Realism and Relativism", The Journal of Philosophy, 76 (1982), pp. 553-67 In Theories of Truth, Paul Horwich, Aldershot 1994 |
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> Counter arguments against Field
> Counter arguments in relation to Second Order Logic, HOL
Ed. Martin Schulz, access date 2024-04-28