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The author or concept searched is found in the following 10 entries.
| Disputed term/author/ism | Author |
Entry |
Reference |
|---|---|---|---|
| Continuum Hypothesis | Russell | I XXIII Def continuum hypothesis/Gödel: (generalized): no cardinal number exists between the potency of some arbitrary set and power set of the set of its subsets. >Power set, >Set, >Set theory, >Subsets, >Continuum. |
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 |
| Individuals | Quine | IX 22 Individuals/Quine: Problem: if y and z are elementless, we obtain y = z due to the extensionality law as it is formulated here - i.e. there is only one single elementless thing - if it is the empty class, then individuals are not elementless - if individuals are elementless, and if they exist at all, then there is only one individual and not an empty class - solution: we do not have to regard individuals as elementless. They are identical with their class of one - and also with the class of one of this class, etc. >Classes/Quine. IX 199 Individuals/QuineVsFraenkel: individuals are not elementless - solution: individuals are identical with their classes of one - therefore "Tnx" is no longer needed to protect y from the flood of individuals. And now that the zero classes of all types are identified with each other, "Tnx" is no longer required to fend off the tide of zero classes - power set axiom/cumulative theoretical terms: we can now accept it without protection: Ey∀x(x ε y ↔ x ⊆ z). With this we are even more liberal than Zermelo. |
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 |
| Propositions | Lewis | Frank I 17 Proposition/Lewis: the number of possible worlds in which this proposition is true. >Possible world/Lewis. Def property/Lewis: the number of (actual or non-actual) beings that have this property. >Properties/Lewis. Proposition/Lewis/Frank: now a one-to-one correspondence can be established between each proposition and the property to inhabit a world in which the proposition applies. It makes it possible to dispense with propositions as the objects of the attitudes. But there are now attitudes that cannot be analyzed as an attitude toward a proposition: where we locate ourselves in space and time. E.g. memory loss: someone bumps into their own biography and can still not fit themselves in. - ((s) Because proposition = number of possible worlds, then - e.g. I’m true here in every possible worlds. - Therefore no knowledge). Frank I 329 Proposition: number of possible worlds in which they are true (extensional). Advantage: non-perspectivic access. - ((s) Not everyone has their own possible worlds.) Frank I 355 Propositions: have nothing intersubjective per se. - Problematic therefore is the subjectivity of reference of the first person. >First Person, >Subjectivity, >Centered world. Hector-Neri Castaneda (1987b): Self-Consciousness, Demonstrative Reference, and the Self-Ascription View of Believing, in: James E. Tomberlin (ed) (1987a): Critical Review of Myles Brand's "Intending and Acting", in: Nous 21 (1987), 45-55 James E. Tomberlin (ed.) (1986): Hector-Neri.Castaneda, (Profiles: An International Series on Contemporary Philosophers and Logicians, Vol. 6), Dordrecht 1986 --- Lewis IV 137 Proposition/Lewis: divides the population into inhabitants of such worlds in which it applies and those in which it does not apply - one assigns oneself to one of the worlds through belief and localizes oneself in a region of logical space - if quantification over several possible worlds is possible (cross-world), there is a large population across worlds and times. IV 142 E.g. Heimson thinks I’m Hume/Perry/Lewis: self-attribution of a property, not an empty proposition Heimson is Hume - all propositions that are true for Hume, are also true for Heimson, because both live in the same world. - Lewis: So Heimson believes the same things as Hume by believing a true proposition - the predicate -believes to be Hume - applies to both. E.g. of HeimsonVsPropositions as objects of belief - otherwise "I am Hume" would either be true both times or false both times - ((s) difference > proposition / > statement). IV 145 Proposition: in a divided world any proposition is either true or false - hence individual objects of desire are more likely properties (that can be self-attributed) than propositions. IV 146 Proposition: No Proposition: E.g. - there is something that I wish now and I will also want it even when I have it, only I will be happier then - no proposition, because it applies to the time before and after - one time of me will not be happy to live in a world where it will happen at some time. - Solution: the wish for the property to be located later in time - localization in logical space instead of proposition: E.g. The Crusader wants a region in logical space without avoidable misfortune - these are properties. V 160 Proposition: no linguistic entity - no language has enough sentences to express all the propositions - truth functional operations with propositions are Boolean operations about sets of possible worlds. - > inclusion, overlapping. --- ad Stechow 42 Language/Infinite/Lewis/(s): number of propositions is greater than the number of sentences, because power set of the possible worlds). |
Lewis I David K. Lewis Die Identität von Körper und Geist Frankfurt 1989 Lewis I (a) David K. Lewis An Argument for the Identity Theory, in: Journal of Philosophy 63 (1966) In Die Identität von Körper und Geist, Frankfurt/M. 1989 Lewis I (b) David K. Lewis Psychophysical and Theoretical Identifications, in: Australasian Journal of Philosophy 50 (1972) In Die Identität von Körper und Geist, Frankfurt/M. 1989 Lewis I (c) David K. Lewis Mad Pain and Martian Pain, Readings in Philosophy of Psychology, Vol. 1, Ned Block (ed.) Harvard University Press, 1980 In Die Identität von Körper und Geist, Frankfurt/M. 1989 Lewis II David K. Lewis "Languages and Language", in: K. Gunderson (Ed.), Minnesota Studies in the Philosophy of Science, Vol. VII, Language, Mind, and Knowledge, Minneapolis 1975, pp. 3-35 In Handlung, Kommunikation, Bedeutung, Georg Meggle Frankfurt/M. 1979 Lewis IV David K. Lewis Philosophical Papers Bd I New York Oxford 1983 Lewis V David K. Lewis Philosophical Papers Bd II New York Oxford 1986 Lewis VI David K. Lewis Convention. A Philosophical Study, Cambridge/MA 1969 German Edition: Konventionen Berlin 1975 LewisCl Clarence Irving Lewis Collected Papers of Clarence Irving Lewis Stanford 1970 LewisCl I Clarence Irving Lewis Mind and the World Order: Outline of a Theory of Knowledge (Dover Books on Western Philosophy) 1991 Fra I M. Frank (Hrsg.) Analytische Theorien des Selbstbewusstseins Frankfurt 1994 |
| Ramsey Sentence | Schurz | I 213 Ramsey-sentene/RS/Theoretical Terms/Schurz: Here Theoretical Terms are not eliminated completely, but existentially quantified over them. Given a theory , which we now take to be a single theorem T(τ1,...τn,) (the conjunction of all axioms of T. Theoretical terms: τ1,...τn. Moreover, there are various non-theoretical terms π which are not written on separately. Then the Ramsey theorem of T is: (5.8 1) R(T): EX1,...Xn: T(X1,...Xn) Everyday language translation: there are theoretical entities X1,..Xn which satisfy the assertions of the theory. Pointe: an empirical (not theoretical) proposition follows from T exactly if it follows from R(T). ((s) It follows from the theory if it follows from the Ramsey theorem of the theory, i.e., from the assumption that the theoretical entities exist.) Thus, it holds: (5.8 -2) E(R(T)) = E(T) Notation: E(T): empirical proposition that follows from theory T. Schurz: i.e. a theory and its Ramsey theorem have the same empirical content. >Carnap-sentence/Schurz, >Empirical content. Ramsey-sentence: Here no more theoretical terms occur! Instead of it: "theoretical" variables. Therefore many, including Ramsey, saw the Ramsey theorem as an empirical theorem (not as a theoretical one. Ramsey theorem: should thus be the sought empirically equivalent non-theoretical axiomatization of the theory. HempelVs/MaxwellVs/Schurz: this is problematic because the RS asserts the existence of certain entities that we call "theoretical". Ramsey theorem/interpretation/realism/instrumentalism/Schurz: the interpretation of the RS as theoretical or non-theoretical depends on whether one interprets 2nd level quantifiers realistically or instrumentally. (a) instrumentalist interpretation: here one assumes that the range of individuals D consists of empirically accessible individuals, and runs the variables Xi over arbitrary subsets of D. (There are no theoretical individuals here). >Instrumentalism/Schurz. Whether these extensions correspond to certain theoretical real properties or not is inconsequential. (Sneed 1971(1), Ketland 2004(2), 291) I 214 Ramsey-sentence/instrumentalism: is then model-theoretically an empirical theorem! Because the models that determine the truth value of R(T) are purely empirical models (D, e1,...em). " ei": extensions of the empirical terms, pi: empirical terms of T. Structuralism: calls these empirical models "partial" models (Balzer et al. 1987(3),57). Empirical model/Schurz: is easily extendible to a full model (D, e1,...em, t1,..tn), ti: are the extensions of the theoretical terms. Pointe: this does not yet mean that R(T) is logically equivalent to E(T). Because R(T) is a 2nd level proposition and E(T) contains 1st level propositions. >Structuralism/Schurz. Def Ramsey-eliminable: if there is a 1st level empirical proposition equivalent to a RS L, then the theortical term is called Ramsey-eliminable. (Sneed 1971(1), 53). b) Realist interpretation: (Lewis, 1970(4), Papineau 1996(5)): assumes that the existence quantified variables denote real theoretical entities. The models are then no longer simple realist models: >Realism/Schurz. 1. New theoretical individuals are added to the individual domain. New: Dt. 2. not every subset of Dt corresponds to a real property. En. Ex In the simplest case, one must assume a set Et of extensions of "genuine" theoretical properties over which 2nd level variables run. Realism/Ramsey-sentence: new: now not every empirical model of instrumentalistically interpreted RS is extensible to a model of realistically interpreted Ramsey-sentence, because the quantifiers (Exi) of R(T) can have satisfactions in the power set of Det but no satisfactions in Et. In philosophical words: an empirical model, which fulfills the RS instrumentalistically, cannot be read off whether the respective theoretical entities, whose existence is postulated by R(T), are merely useful fictions or real existing entities. Instrumentalism: Proposition: Theoretical entities are useful fictions. Realism/Ramsey Theorem: here R(T) contains more than just the empirical content of a theory, it also contains the total synthetic content: if we assume that the meaning of Theoretical Terms is not determined by anything other than this theory itself, then the assertion that T makes about the world seems to be precisely that of R(T): there are unobservable entities X1,...Xn that satisfy the total assertion of the theory T(X1,...Xn). >Carnap-sentence/Schurz. 1. Sneed, J. D. (1971). The Logical Structure of Mathematical Physics. Dordrecht: Reidel. 2. Ketland, J. (2004). "Empirical Adequacy and Ramsification", British Journal for the Philosoph y of Science 55, 287-300. 3. Balzer, W. et al (1987). An Architectonic for Science. Dordrecht: Reidel. 4. Lewis, D. (1970). "How to definie Theoretical Terms", wiederabgedruckt in ders. Philosophical Papers Vol I. Oxford: Oxford University Press. 5. Papineau, D. (1996). "Theory-dependent Terms", >Philosophy of Science 63, 1- 20. |
Schu I G. Schurz Einführung in die Wissenschaftstheorie Darmstadt 2006 |
| Russell’s Paradox | Logic Texts | Sainsbury V 163 Russell s paradox/properties/Sainsbury: basic problem: cases in which properties can be applied to themselves - most do not. E.g. the property of being a person is a property and not a person! So it does not apply to the property of being a person. But some properties are true of themselves. - E.g. the property of being a man is not a man. - But the property of being a non-man, is itself a non-man. >Self-reference, >Heterology, >Paradoxes. V 165 There is a relationship with Cantor's proof that the power set of each class has more elements than the class itself, but you can block Russell's paradox, and still allow the proof of Cantor. |
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 Sai I R.M. Sainsbury Paradoxes, Cambridge/New York/Melbourne 1995 German Edition: Paradoxien Stuttgart 1993 |
| Russell’s Paradox | Thiel | I 313 ff Russell's Paradox/Thiel: discovered independently and earlier by Zermelo, but not published. Example: While the set of birds itself is not a bird, i.e. not a thing of the kind that it itself deals with as elements among itself, the set of all sets is obviously a set (e.g. the set of all terms is also a term). This is the exceptional case, while sets that do not contain themselves as elements are the normal case. So a set A can be "normal" or not. Therefore, we can ask in particular whether the set of all sets itself is normal or not. >Sets. If we note the explanation of the element relationship by x ε {y|B(y)} <> B(x), it follows from the assumption that R is not normal that R is an element of the set R of normal sets, i.e. normal. On the other hand, according to the condition defining R, the normality of R means that ~(R ε R) applies, i.e. that R is not an element of the set of all normal sets, i.e. not normal. From (R ε R) follows ~(R ε R) and from ~(R ε R) follows ~(~(R ε R)). I 316 The mental operation was, among other things, the transition from the statement that a thing had a property to the statement that it belonged to the set of all things with this property. This led to the "basic crisis" in mathematics. How to react? I 317 1. One could deny that such antinomies affect mathematics at all. One could simply claim that the concepts and conclusions involved do not occur in the analysis, for example. 2. One could describe the terms and conclusions used in the analysis as incorrect. I 318 3. Would it be conceivable that the antinomies would not be regarded as serious "sentences" at all? But only as imagined without evidence. Then they would be on a par with the paradoxes. If antinomies were not taken seriously in earlier times, it was also because mathematics was far from a deductive approach. I 319 In the beginning, it seemed as if the antinomies had a common error, the avoidance of which did not seem difficult any more: If one forms the sets of all sets, then this is apparently the largest set at all. But as to any set we can also form to it (e.g. "A") the power set PA which is greater than A - a contradiction. Sets of all sets: Problem: cannot be the largest, because its power set must be larger. >Power set. If one forms the set of all ordinal numbers, then this set itself determines an order type with the ordinal number Ω. From this it can be shown that it is 1 larger than the largest ordinal number in the series of all ordinal numbers. Since in this series also Ω itself must occur that would be a contradiction again. It seems that the formation of all sets goes too far and, like the formation of a top genus (summum genus) in traditional terminology, should be excluded by limiting the size of permissible sets. I 320 At first, such a ban did not seem to solve all problems, e.g. Zermelo-Russell's antinomy, at all, unless one wanted to ban all summaries without exception, which would have already impaired the structure of the analysis. Wasn't there simply a false conclusion, as in the paradoxes of the infinite of the type of infinite sets that are in a genuine part whole and yet equal in power? I 322 Russell's Antinomy/Solution: an attempt to avoid Russell's paradox: instead of always saying "all" saying "all who". Thus now the suspicion falls on the "all". Poincaré saw this suspicion confirmed and claimed: Conditions like "~(x x) are inappropriate to determine a set, because they require a circulus vitiosus. He had found this diagnosis not from Russell's antinomy, but from the antinomy constructed by Jules Richard. |
T I Chr. Thiel Philosophie und Mathematik Darmstadt 1995 |
| Set Theory | Set Theory: set theory is the system of rules and axioms, which regulates the formation of sets. The elements are exclusively numbers. Sets contain individual objects, that is, numbers as elements. Furthermore, sets contain sub-sets, that is, again sets of elements. The set of all sub-sets of a set is called the power set. Each set contains the empty set as a subset, but not as an element. The size of sets is called the cardinality. Sets containing the same elements are identical. See also comprehension, comprehension axiom, selection axiom, infinity axiom, couple set axiom, extensionality principle. |
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| Set Theory | Basieux | Basieux I 86 Axioms of the Set Theory/Halmos(1)/Basieux: 1) extensionality axiom: two sets are only equal iff they have the same elements >Extensionality. 2) selection axiom: for every set A and every condition (or property) E(x) there is a set B, whose elements are exactly every x of A, for which E(x) applies >Selection axiom 3) pairing axiom: for every two sets there is always one set that contains those two as elements 4) combination axiom: for every set system there is a set that contains all elements that belong to at least one set of the given system 5) power set axiom: for every quantity there is a set system that contains all the subsets of the given set among its elements >Power set. 6) infinity axiom: there is a set that contains the empty set and with each of its elements also its successor >Infinity axiom 7) choice axiom: the Cartesian product of a (non-empty) system of non-empty sets is non-empty 8) replacement axiom: S(a,b) be a statement of the kind that for each element a of a set A the set {b I S (a,b)} can be formed. Then there is a function F with domain A such that F(a) = {b I S(a,b)} for every a in A. >Axioms. 1. Halmos, Paul (1974). Naive set theory, Santa Clara University. |
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| Sets | Bigelow | I 47 Sets/Quine/Goodman/Bigelow/Pargetter: we may no longer need any other universals if we allow sets. Because you can do almost anything with sets that mathematics needs. Armstrong: he believes in universals, but not in sets! >D. Armstrong, >Universals. BigelowVsQuine/BigelowVsGoodman: for science we need more universals than sets, for example probability and necessity. >Probability, >Necessity. I 95 Universals/Sets/Predicates/Bigelow/Pargetter: if a predicate does not correspond to a universal, e.g. dogs, we assume that they correspond to at least one set. Predicate/Bigelow/Pargetter: but even then we cannot assume that each predicate corresponds to a set! Set/Bigelow/Pargetter: For example, there is no set X containing all and only the pairs Universal Set/Universal Class/Bigelow/Pargetter: can also not exist. Predicate: "is a set" does not correspond to a set that contains all and only the things it applies to! (Paradox, because of the impossible amount of all sets). Set theory/Bigelow/Pargetter: we are still glad if we can assign something to most predicates, and therefore set theory (which originates from mathematics and not from semantics) is a stroke of luck for semantics. Reference/Semantics/Bigelow/Pargetter: set theory helps to impose more explanatory force on the reference in order to formulate a truth theory (WT). It remains open which role reference should play. I 371 Existence/sets/set theory/axiom/Bigelow/Pargetter: none of the following axioms secures the existence of sets: pair set axiom, extensionality axiom, union set axiom, power set axiom, separation axiom: they all only tell us what happens if there are already sets. Axioms/Zermelo-Fraenkel/Bigelow/Pargetter: their axioms are recursive: i.e. they create new things from old things. Based on two axioms: I 372 Infinity axiom/Zermelo-Fraenkel/Bigelow/Pargetter: (normally formalized to contain the empty set axiom). Stands for the existence of a set containing all natural numbers according to von Neumann. Omega/Bigelow/Pargetter: according to our mathematical realism, the sets in the sequence ω are not identical to natural numbers. They instantiate them. That is why the infinity axiom is so important. Infinity axiom/Ontology/Bigelow/Pargetter: the infinity axiom has real ontological significance. It ensures the existence of sufficient sets to instantiate the rich structures of mathematics. And physics. Question: is the axiom true? For example, suppose a quality of "being these things". And suppose there is an extra thing that is not included. Then it is very plausible that there will be the qualities of being "those things" that apply to all previous things plus extra things. To do this, these properties must first be available. Moreover, if we are realists about such properties, such a property can count as an "extra thing"! I 373 This ensures that if there is an initial segment of, the next element of the sequence also exists. Infinity: but requires more than that. We still have to make sure that the whole of ω exists! I.e. there must be the property "to be one of these things", whereby this is a property instantiated by all and only by Neumann numbers. That is plausible in our construction, because we use sets as plural essences (see above) to understand. Problem: we only have to guarantee a starting segment for the Neumann figures. That should be the empty set. Empty set/Bigelow/Pargetter: how plausible is their existence in our metaphysics? >Empty set, >Metaphysics. |
Big I J. Bigelow, R. Pargetter Science and Necessity Cambridge 1990 |
| Sets | Mates | I 49 Sets/Mates: for any propositional function there is a set, but not vice versa - basic: there are more sets than propositional functions. >Power set, >irrational numbers, >Sets, >Set theory, >Propositional functions. |
Mate I B. Mates Elementare Logik Göttingen 1969 Mate II B. Mates Skeptical Essays Chicago 1981 |
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