Philosophy Dictionary of ArgumentsHome
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| Incompleteness: whether something is incomplete can only be determined in relation to something that allows for a closer determination. For this purpose, e.g. a continuation rule, a type description, or a categorization must be specified. Objects, which are also parts of something, can then be fully described as an object if they do not need the context of which they are a part. See also indeterminacy, determination, context/context dependency, description, description levels, completeness._____________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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Logic Texts on Incompleteness - Dictionary of Arguments
Read III 61 Incompleteness Theorem/Gödel/Read: the compact inference produces too little: there are intuitively valid inferences that mark it as invalid. For example, the most famous example is the Omega theory: assuming a formula is true for any natural number. Then "for every n, A(n) is true". This is not a classical logical conclusion from them, because it does not follow from any finite subset of any set. The Omega rule would allow us to deduce from the premises A(0),A(1)... etc. "for every n A(n)." But this is a rule that could never be applied, it would require that a proof be an infinite object. Def Omega model: the natural numbers, as well as the zero, with the operations of the successor, addition, multiplication and exponentiation. The Omega rule is not accepted as a rule of orthodox classical proof theory. How can I do this? According to classical representation, a rule is valid if it is not possible to make the premises true and the conclusion false by any interpretation over any range of definition. How can the premises A(0),A(1) etc. was, but be false for each n,A(n)? III 61/62 The explanation lies in the limitation of the expressiveness. >Compactness/Logic texts, >2nd order logic. III 64 The Omega rule requires an extra premise: "and these are all numbers". This extra premise is arithmetically true, but the non-standard models show that, as far as logic is concerned, it has to be formulated explicitly (in 1st level terms, i.e. logical terms). III 65 Two ways to see that this answer is not appropriate as a defense of classical logic and its compactness. >Compactness/Logic texts. 1. the extra provision "and these are all numbers " cannot be expressed in terms of 1st level terms. 2. a proposal by Wittgenstein: a long conjunction for "each F is G": "this is G and that is G and that other is G... RussellVs: these two statements are not equivalent, because the long conjunction needs a final clause "and these are all F's". ReadVsRussell: Error: if a conjunction is exhaustive, then the two statements are equivalent. If not, the extra clause has no effect, because it is wrong. It does not do extra work. >Second order logic._____________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. |
Logic Texts Me I Albert Menne Folgerichtig Denken Darmstadt 1988 HH II Hoyningen-Huene Formale Logik, Stuttgart 1998 Re III Stephen Read Philosophie der Logik Hamburg 1997 Sal IV Wesley C. Salmon Logic, Englewood Cliffs, New Jersey 1973 - German: Logik Stuttgart 1983 Sai V R.M.Sainsbury Paradoxes, Cambridge/New York/Melbourne 1995 - German: Paradoxien Stuttgart 2001 Re III St. Read Thinking About Logic: An Introduction to the Philosophy of Logic. 1995 Oxford University Press German Edition: Philosophie der Logik Hamburg 1997 |
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> Counter arguments against Logic Texts
> Counter arguments in relation to Incompleteness
