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The author or concept searched is found in the following 4 entries.
| Disputed term/author/ism | Author |
Entry |
Reference |
|---|---|---|---|
| All | Russell | I 80 "All properties of" is illegitimate! -> Reducibility axiom: only "property 2nd order of ..." - of the whole of predicates, not of Napoleon! - Reducibility-axioms are necessary for identity (GoedelVs). I 81f All properties of a great emperor/Principia Mathematica(1)/Russell: Solution: (j): f (! j ^ z) implies j! (Example of Napoleon) - because the reference to a set of predicates is not itself a predicate of Napoleon. I 143f Principia Mathematica, 2nd Edition: "Napoleon had all the properties": New: variable function! 1. Whitehead, A.N. and Russel, B. (1910). Principia Mathematica. Cambridge: Cambridge University Press. |
Russell I B. Russell/A.N. Whitehead Principia Mathematica Frankfurt 1986 Russell II B. Russell The ABC of Relativity, London 1958, 1969 German Edition: Das ABC der Relativitätstheorie Frankfurt 1989 Russell IV B. Russell The Problems of Philosophy, Oxford 1912 German Edition: Probleme der Philosophie Frankfurt 1967 Russell VI B. Russell "The Philosophy of Logical Atomism", in: B. Russell, Logic and KNowledge, ed. R. Ch. Marsh, London 1956, pp. 200-202 German Edition: Die Philosophie des logischen Atomismus In Eigennamen, U. Wolf (Hg) Frankfurt 1993 Russell VII B. Russell On the Nature of Truth and Falsehood, in: B. Russell, The Problems of Philosophy, Oxford 1912 - Dt. "Wahrheit und Falschheit" In Wahrheitstheorien, G. Skirbekk (Hg) Frankfurt 1996 |
| Compactness | Logic Texts | Read III 59 Compactness: the classic logical conclusion is compact. To understand this, we must acknowledge that the set of premises can be infinite. Classically, every logical truth (of which there are infinite numbers) is a conclusion from any statement. This can be multiplied, by double negation, the conjunction of itself with its double negation, and so on. III 60 The classical compactness does not mean that a conclusion cannot have an infinite number of premises, it can. But classically it is valid exactly when the conclusion follows from a finite subset of the premises. Compactness limits the expressiveness of a logic. Proof: is performed purely syntactically. In itself, the proof has no meaning. Its correctness is defined based on its form and structure. >Proof. III 61 The counterpart of proof is completeness: there should be a derivation. >Incompleteness/logic texts. III 61 The Omega rule (>Incompleteness/logic texts) 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 the premises are true and the conclusion is false by no 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. In non-compact logic, there may be a categorical set of formulas for arithmetic, but the proof methods require compactness. For expressiveness: >Richness, >Meta language, >Object language. Difference compact/non compact: classical logic is a 1st order logic. A categorical set of axioms for arithmetic must be a second order logic. ((s) quantifiers also for properties). >Quantifier, cf. >Schematic letters. For example, Napoleon had all the properties of a great general: "for every quality f, if for every person x, if x was a great general, then x had f, then Napoleon had f". In reality it is a little more subtle. For syntactically one cannot distinguish whether a formula is like the 1st or 2nd level above. >2nd order logic. |
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 |
| Functions | Russell | I XII / XIII Function/Russell/Gödel: Axiom: functions can only occur "through their values", i.e. they are extensional. >Extensionality, >Extension. I 58 Function/Russell: presupposes values, but values do not presuppose a function - ((s) In order for 16 to be a square number, there must be a natural number 16 first, etc.) I 69 Function/Principia Mathematica(1)/Russell: no object, since ambiguous - "values of j z^" are assigned to the j and not to the z. I 72 Def A-Functions/Principia Mathematica/Russell: functions that make sense for a given argument a - ((s) E.g. reversal of function: for example, y = x² can give the value y = 4 for x = 2). - A-function: now we can conversely search for functions that give the value 4 E.g. root of - 16, 2² and any number of others - E.g. "A satisfies all functions that belong to the selection in question": we replace a by a variable and get an a-function. However, and according to the circle fault principle, it may not be an element of this selection, since it refers to the totality of this selection - the selection consists of all those functions that satisfy f(jz^) - then the function is (j). ({f(jz^)) implies jx} where x is the argument - such that there are other a-functions for any possible selection of a-functions that are outside of the selection - ((s) > "Everythingl he said"). I 107 Derived function/notation/Principia Mathematica/Russell: (derived from a predicative function). "f{z^(q,z)}" - defined as follows: if a function f(y ! z^) is given, our derived function must be: "there is a predicative function, which is formally equivalent to j z^ and satisfies f" - always extensional. I 119 Function/Truth/Principia Mathematica/Russell: a function that is always true, can still be false for the argument (ix)( j x) - if this object does not exist. I 119 Function/Waverley/Identity/Equivalence/Principia Mathematica/Russell: the functions x = Scott and x = author of Waverley are formally equivalent - but not identical, because George IV did not want to know if Scott = Scott. I 144 Varying function/variable function/variability/Principia Mathematica/Russell: old: only transition from e.g. "Socrates is mortal" to "Socrates is wise" (from f ! x to f ! y) (sic) - new: (Second Edition): now the transition to "Plato is mortal" is also possible - (from j ! a to y ! a) - "notation: Greek letters: stand for individuals, Latin ones for predicates -> E.g. "Napoleon had all the properties of a great emperor" - Function as variable. 1. Whitehead, A.N. and Russel, B. (1910). Principia Mathematica. Cambridge: Cambridge University Press. |
Russell I B. Russell/A.N. Whitehead Principia Mathematica Frankfurt 1986 Russell II B. Russell The ABC of Relativity, London 1958, 1969 German Edition: Das ABC der Relativitätstheorie Frankfurt 1989 Russell IV B. Russell The Problems of Philosophy, Oxford 1912 German Edition: Probleme der Philosophie Frankfurt 1967 Russell VI B. Russell "The Philosophy of Logical Atomism", in: B. Russell, Logic and KNowledge, ed. R. Ch. Marsh, London 1956, pp. 200-202 German Edition: Die Philosophie des logischen Atomismus In Eigennamen, U. Wolf (Hg) Frankfurt 1993 Russell VII B. Russell On the Nature of Truth and Falsehood, in: B. Russell, The Problems of Philosophy, Oxford 1912 - Dt. "Wahrheit und Falschheit" In Wahrheitstheorien, G. Skirbekk (Hg) Frankfurt 1996 |
| Second Order Logic, HOL | Logic Texts | Read III 62f "All the characteristics of a great emperor"/compactness/1st Level/2nd level/Read: a categorical set of axioms for arithmetic must be a 2nd level logic. Logical form: "for every quality f, if for every person x, if x was a great general, then x had f, then Napoleon had f. But: purely syntactically you cannot decide whether this is 1st or 2nd level. What distinguishes the two is their semantics! >Syntax, >Semantics. The definition area can be arbitrary, provided it is not empty - Russell: Addition: "... and these are all..." ReadVs: that is either superfluous in an explicitly specified conjunction or wrong. Omega rule: needs the addition, however it cannot be expressed in the 1st level logic to exclude non-standard models, but it should be formulated in the 1st level (i.e. in logical terms). III 152f Logic 1st order: individuals, 2nd order: variables for predicates, distribution of predicates by quantifiers. 1st order: allows restricted vocabulary of the 2nd level: existence and universal quantifier. Other properties 2nd level are not definable in the logic of the 1st order: e.g. to be finite, or to be true of most things. >Operator, >Level (Order), >Description level. |
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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