Disputed term/author/ism | Author![]() |
Entry![]() |
Reference![]() |
---|---|---|---|
Attributes | Quine | VII (d) 75ff Attribute/Quine: an attribute may eventually be introduced in a second step: e.g. "squareness" according to geometrical definition, but then the name also requires substitutability, i.e. an abstract entity > Universals. X 7ff Attribute/Quine: an attribute corresponds to properties, predicates are not the same as attributes. >Predicates/Quine. IX 178ff Attribute/(s): an attribute corresponds to the quantity of those x for which a particular condition applies: {x: x ε a} all objects that are mortal. Predicate: "x is mortal", is not a quantity, but a propositional function. The denomination forms refer "φx", "φ(x,y)" to the attribution. >Propositional Function/Quine. XII 38 Attributary Attitude/Quine: E.g. hunting, needing, catching, fearing, missing. Important to note here is that e.g. "lion hunt" does not require lions as individuals but as a species - > Introduction of properties. IX 177 Attributes/Ontology/Russell: for Russell, the universe consisted of individuals, attributes and relations of them, attributes and relations of such attributes and relations, etc. IX 178f Extensionality/Quine: extensionality is what distinguishes attributes and classes. >Extensionality/Quine So Russell has more to do with attributes than with classes. Two attributes can be of different order and are therefore certainly different, and yet the things that each have one or the other attribute are the same. For example the attribute "φ(φ^x <> φy) where "φ" has the order 1, an attribute only from y. For example the attribute ∀χ(χ^x <> χy), where "χ" has order 2, again one attribute only from y, but one attribute has order 2, the other has order 3. (> Classes/ >Quantities/ >Properties). XIII 22 Class/set/property/Quine: whatever you say about a thing seems to attribute a property to it. Property/Attribute/Tradition/Quine: in earlier times one used to say that an attribute is only called a property if it is specific to that thing. (a peculiarity of this object is...). New: today these two expressions (attribute, property) are interchangeable. "Attribute"/Quine: I do not use this term. Instead I use "property". Identity/equality/difference/properties/Quine: if it makes sense to speak of properties, then it also makes sense to speak of their equality or difference. Problem: but it does not make sense! Problem: if everything that has this one property, also has the other. Shall we say that it is simply the same quality? Very well. But people do not talk like that. For example to have a heart/kidney: is not the same, even if it also applies to the same living beings. Coextensivity/Quine: two properties are not sufficient for their identity. Identity/properties/possible solution: is there a necessary coextensiveness? >Coextensive/Quine Vs: Necessity is too unclear as a term. Properties/Quine: We only get along so well with the term property because identity is not so important for their identification or differentiation. XIII 23 Solution/Quine: we are talking about classes instead of properties, then we have also solved the problem e.g. heart/kidneys. Classes/Quine: are defined by their elements. That is the way of saying it, but unwisely, because the misunderstanding might arise that the elements cause the classes in a different way than objects cause their. Def Singleton/Singleton/Single Class: class with only one element. Def Class/Quine: (in useful use of the word): is simply a property in the everyday sense, without distinguishing coextensive cases. XIII 24 Class/Russell/Quine: it struck like a bomb when Russell discovered the platitude that each containment condition (condition of containment, element relationship) establishes a class. (see paradoxes, see impredictiveness). Russell's Paradox/Quine: applies to classes as well as to properties. It also shatters the platitude that anything said about a thing attributes a property. Properties/Classes/Quine: all restrictions we impose on classes to avoid paradoxes must also be imposed on properties. Property/Quine: we have to tolerate the term in everyday language. Mathematics: here we can talk about classes instead, because coextensiveness is not the problem. (see Definition, > Numbers). Properties/Science/Quine: in the sciences we do not talk about properties. |
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 |
Classes | Quine | I 289 Class abstraction is attributed to singular descriptions: (iy)(x)(x from y iff ..x..). Instead: x^(..x..). This does not work for intensional abstraction. Difference classes/properties: classes are identical with the same elements. Properties are not yet identical if they are assigned to the same things. >Properties/Quine. II 29 Classes: one could reinterpret all classes in their complement: "no element of .." and you would never notice anything. At the bottom layer every relative clause, every general term determines a class. II 100 Russell (Principia Mathematica(1)) classes are things: they must not be confused with the concept of classes. However: paradoxes also apply to class terms and propositional functions are not only for classes. Incomplete symbols (explanation by use) are used to explain away classes. 1. Whitehead, A.N. and Russel, B. (1910). Principia Mathematica. Cambridge: Cambridge University Press. VII (a) 18 Classes/Quine: simplify our access to physics but are still a myth. VII (f) 114 Classes/Quine: classes are no accumulations or collections! E.g. the class of stones in a pile cannot be identified with the pile: otherwise another class could also be identified with the same pile: e.g. the class of stone molecules in the pile. The validity theory applies to classes, but not to the individual sentences - predicates are not names of classes, classes are the extension of predicates - classes are assumed to be pre-existent. IX 21 Classes/Relations/Quine: classes are real objects if values of bound variables. IX 23 Class/Individuals/Quine: everything is class! If we understand individuals to be identical to their class of one (i.e. not elementless). IX 223 Classes/Quine: quantification through classes allows for terms that would otherwise be beyond our reach. XIII 24 Class/Quantity/Quine: we humans are stingy and so predisposed that we never use two words for the same thing, or we demand a distinction that should underlie it. XIII 25 Example ape/monkey: we distinguish them by size, while French and Germans have only one word for them. Problem: how is the dictionary supposed to explain the difference between "beer, which is rightly called so" and "ale, which is rightly called so"? Example Sets/Classes/Quine: here this behaves similarly. Class/Mathematics: some mathematicians treat classes as something of the same kind as properties (Quine pro, see above): sets as something more robust, though still abstract. >Properties/Quine. Classes: can contain sets as elements, but not other classes. (see impredicativity). Paradox/Paradoxes/Quine: lead to some element relationships not being able to define sets. Nevertheless, they can still define classes! von Neumann: established such a system in 1925. It simplifies evidence and strengthens the system, albeit at the risk of paradoxes. >Paradoxes/Quine. Problem: it requires imaginative distinctions and doublings, e.g. for every set there must be a coextensive class. Solution/Quine. (Quine 1940): simply identify the sets with the coextensive classes. XIII 26 Def Classes/Def Sets/QuineVsNeuman: new: sets are then classes of a certain type: a class is a set if it is an element of a class. A class is a Def outermost class/Quine: if it is not an element of a class. Russell's Paradox/Quine: some authors thought that by distinguishing between classes and sets, it showed that Russell's antinomy was mere confusion. Solution/some authors: classes themselves are not such substantial objects that they would come into question as candidates for elements according to a condition of containment. But sets can be. On the other hand: Sets: had never been understood as defined by conditions of abstinence. And from the beginning they had been governed by principles that Zermelo later made explicit. QuineVs: these are very perishable assumptions! In reality, sets were classes from the beginning, no matter what they were called. Vagueness of one word was also vagueness of the other word. Sets/Cantor/Quine: sure, the first sets at Cantor were point sets, but that does not change anything. QuineVsTradition/Quine: it is a myth to claim that sets were conceived independently of classes, and were later confused with them by Russell. That again is the mistake of seeing a difference in a difference between words. Solution/Quine: we only need sets and outermost classes to enjoy the advantages of von Neumann. >Sets/Quine. |
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 |
Conceptualism | Quine | XI 136 Intuitionism/Quine/Lauener: he compares it with ancient conceptualism: universals are created by the mind. VII (f) 125 ConceptualismVsPlatonism/Quine: treats classes as constructions, not as discoveries - Problem: Poincaré's "impredicative" definition: Def impredicative/Def Poincaré: "impredicative" means the specification of a class through a realm of objects, within which that class is located. VII (f) 126 Classes/Conceptualism/Quine: for him, classes only exist if they originate from an ordered origin. Classes/Conceptualism/Quine: conceptualism does not require classes to exist beyond conditions of belonging to elements that can be expressed. Cantor's proof: would entail something else: It appeals to a class h of those elements of class k which are not elements of the subclasses of k to which they refer. VII (f) 127 But this is how the class h is specified impredicatively! h is itself one of the partial classes of k. >Classes/Quine. Thus a theorem of classical mathematics goes overboard in conceptualism. The same fate strikes Cantor's proof of the existence of supernumerary infinity. QuineVsConceptualism: this is a welcome relief, but there are problems with much more fundamental and desirable theorems of mathematics: e.g. the proof that every limited sequence of numbers has an upper limit. VII (a) 14 Universals Dispute/Middle Ages/Quine: the old groups reappear in modern mathematics: Realism: Logicism Conceptualism: Intuitionism Nominalism: Formalism. Conceptualism/Middle Ages/Quine: holds on to universals, but as mind-dependent. ConceptualismVsReduceability Axiom: because the reduceability axiom reintroduces the whole platonistic class logic. >Universals/Quine. |
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 |
Description Levels | Field | II 345 Indefiniteness 2nd order: it is unclear whether an undecidable sentence has a particular truth value. >Platonism. II 354 Logic/second order logic/Field/(s): excludes non-standard models better than theory 1st order. - 2nd order has no impredicative comprehension scheme. >Second order logic, >Comprehension, >Unintended models, >Models, >Model theory. --- III 33 Theory of the 1st order/Field: E.g. the theory of the space-time points. - (s) E.g.a theory which only uses functions but does not quantify over them. >Quantification. Theory 2nd order/Field: E.g. theory of real numbers, because it quantifies over functions. - Quantities of higher order: are used for the definition of continuity and differentiability. III 37 Theory of 1st order/2nd order/Hilbert/Field: Variables 1st order: for points, lines, surfaces. 2nd order: Quantities of ... Solution/Field: quantification 2nd order in Hilbert's geometry as quantification over regions. Only axiom 2nd order: Dedekind's continuity axiom. III 95 f Logic 2nd order/Field: E.g. Quantifiers like "there are only finitely many". - Also not: E.g. "There are less Fs than Gs". >Quantifiers. III 98 Extension of the logic: preserves us from a huge range of additionally assumed entities. - E.g., what obeys the theory of gravity. QuineVs: we should rather accept abstract entities than to expand the logic. (Quine in this case pro Platonism). III 96 Platonism 1st order/Field: accepts abstract entities, but no logic 2nd order. Problem: but it needs this (because of the power quantifiers). |
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 |
Impredicativeness | Impredicativeness: Impredicatives are concepts which are defined only by means of the propositional sets to which they themselves belong. Problems arise in connection with possible circular conclusions. To avoid paradoxes, the demand is sometimes made to avoid impredicative concepts. See also Paradoxes, Russellian Paradoxy, Poincaré. |
||
Impredicativeness | Dummett | II 77/8 Def impredicative/Dummett: an extension is impredicative if it allows the predicate "true" also for sentences of the extended language (by adding the predicate "true"). >Expressiveness, >Metalanguage, >Description levels, >Richness. |
Dummett I M. Dummett The Origins of the Analytical Philosophy, London 1988 German Edition: Ursprünge der analytischen Philosophie Frankfurt 1992 Dummett II Michael Dummett "What ist a Theory of Meaning?" (ii) In Truth and Meaning, G. Evans/J. McDowell Oxford 1976 Dummett III M. Dummett Wahrheit Stuttgart 1982 Dummett III (a) Michael Dummett "Truth" in: Proceedings of the Aristotelian Society 59 (1959) pp.141-162 In Wahrheit, Michael Dummett Stuttgart 1982 Dummett III (b) Michael Dummett "Frege’s Distiction between Sense and Reference", in: M. Dummett, Truth and Other Enigmas, London 1978, pp. 116-144 In Wahrheit, Stuttgart 1982 Dummett III (c) Michael Dummett "What is a Theory of Meaning?" in: S. Guttenplan (ed.) Mind and Language, Oxford 1975, pp. 97-138 In Wahrheit, Michael Dummett Stuttgart 1982 Dummett III (d) Michael Dummett "Bringing About the Past" in: Philosophical Review 73 (1964) pp.338-359 In Wahrheit, Michael Dummett Stuttgart 1982 Dummett III (e) Michael Dummett "Can Analytical Philosophy be Systematic, and Ought it to be?" in: Hegel-Studien, Beiheft 17 (1977) S. 305-326 In Wahrheit, Michael Dummett Stuttgart 1982 |
Impredicativeness | Field | I 214 Def impredicative/Field: completely impredicative properties: are not at all derived from previously available properties. - In particular, there is no property to be a property. Quasi-impredicative: also allows "property to be a property". >Self-reference, >Predication. I 216 Classic example for impredicative definition: E.g. What is it for an ordinal number to be finite? Fin (ON) P [P is inductive & P (0)> P (ON)] whereby P is inductive is defined as: b [P(b) > P(b + 1)] ((s) All successors have the same property (to be a number)). The invalid objection against the impredicative definition (> VsImpredicativity) is that one cannot know that a given number, e.g. 2 is finite because, in order to show this, we must be able to show that 2 has every inductive property of 0. To show that 2 is finite, we must show first that exactly this 2 is finite (circular). Solution/Field: the solution is simple: if finiteness is an inductive property, then 2 is finite. - No circle. >Induction, >Deduction, >Circular reasoning, >Predicativeness. |
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 |
Impredicativeness | Flusser | Fl I V. Flusser Kommunikologie Mannheim 1996 |
|
Impredicativeness | Lorenzen | Berka I 266 LorenzenVsHerbrand/ConstructivismVs"Impredicativeness": fable realm of the "Impredicative". Berka I 269 LorenzenVsImpredicativeness: this condition is the one that excludes the impredicative definitions in the analysis, thus requiring the branching of the types.(1) >Type theory, >Ramified type theory, >Constructivism, >J. Herbrand. 1. P. Lorenzen, Ein dialogisches Konstruktivitätskriterium, in: Infinitistic Methods, (1961), 193-200 |
Lorn I P. Lorenzen Constructive Philosophy Cambridge 1987 Berka I Karel Berka Lothar Kreiser Logik Texte Berlin 1983 |
Impredicativeness | Poincaré | Thiel I 324 Impredicativeness/Paradoxes/Poincaré: Poincaré believed with this that the decisive criterion had been found: illegitimate, "non-predicative" conditions are those that contain such a circle. >impredicative/Russell. At first, it seemed sufficient to require expressions for the relation between element and set that in "x ε y" the second relation term y should belong to exactly one step higher than x (>type theory), thus the requirement that each permissible expression should be formed not only "predicatively" itself (i.e. not impredicatively) but also all arguments occurring in it must meet this condition, to form a >"ramified type theory". |
T I Chr. Thiel Philosophie und Mathematik Darmstadt 1995 |
Impredicativeness | Quine | XIII 93 Impredicativeness/Quine: Previously it was said that you had specified a class without knowing anything about it if you could name the containment condition. Russell's Antinomy: showed that there had to be exceptions. Problem: was to specify a class by a containment condition by directly or indirectly referring to a set of classes that contained the class in question. >Classes/Quine. Russell's Antinomy: here the problematic containment condition was the non-self elementary. Example x is not an element of x. Paradox: arises from letting the x of the containment condition, among other things, be just the class defined by this containment condition. Def impredicative/Poincaré/Russell: is just this condition of containment for a class that exists in the class itself. This must be forbidden to avoid paradoxes. Circular Error Principle/QuineVsRussell: but that was too harsh a term: Specification/Class/Sets/Existence/Quine: specifying a class does not mean creating it! XIII 94 Specification/Circle/Introduce/QuineVsRussell: by specifying something it is not wrong to refer to a domain to which it has always belonged to. For example, statistical statements about a typical inhabitant by statements about the total population that contains this inhabitant. Introduction/Definition/linguistic/Quine: all we need is to equate an unfamiliar expression with an expression that is formed entirely with familiar expressions. Russell's Antinomy/Quine: is still perfectly fine as long as the class R is defined by its containment condition: "class of all objects x, so that x is not an element of x". Paradox/Solution/Russell/Quine: a solution is to distort familiar expressions so that they are no longer familiar in order to avoid a paradox. This was Russell's solution. Finally, "x is an element of x" ("contains itself") to be banished from the language. >Paradoxes/Quine. Solution/Zermelo/Quine: better: leave the language as it is, but New: for classes it should apply that not every containment condition defines a class. For example the class "R" remains well defined, but "Pegasus" has no object. I.e. there is no (well-defined) class like R. Circle/George Homans/Quine: true circularity: For example, a final club is one into which you can only be elected if you have not been elected to other final clubs. Quine: if this is the definition of an unfamiliar expression, then especially the definition of the last occurrence of "final club". Circle/Circularity/Quine: N.B.: yet it is understandable! Impredicativeness/impredicative/Russell/Quine: the real merit was to make it clear that not every containment condition determines a class. Formal: we need a hierarchical notation. Similar to the hierarchy of truth predicates we needed in the liar paradox. XIII 95 Variables: contain indexes: x0,y0: about individuals, x1,y2 etc. about classes, but classes of this level must not be defined by variables of this level. For example, for the definition of higher-level classes x2, y2 only variables of the type x0 and x1 may be used. Type Theory/Russell/Quine/N.B.: classes of different levels can be of the same type! Classes/Sets/Existence/Quine: this fits the metaphor that classes do not exist before they are determined. I.e. they are not among the values of the variables needed to specify them. ((s) And therefore the thing is not circular). Problem/QuineVsRussell: this is all much stricter than the need to avoid paradoxes and it is so strict that it prevents other useful constructions. For example, to specify the union of several classes of the same level, e.g. level 1 Problem: if we write "Fx1" to express that x1 is one of the many classes in question, then the Containment condition: for a set in this union: something is element of it iff it is an element of a class x1, so Fx1. Problem: this uses a variable of level 1, i.e. the union of classes of a level cannot be counted on to belong to that level. Continuity hypothesis: for its proof this means difficulties. Impredicativeness/Continuum/Russell/Quine: consequently he dropped the impredicativeness in the work on the first volume of Principia Mathematica(1). But it remains interesting in the context of constructivism. It is interesting to distinguish what we can and cannot achieve with this limitation. XIII 96 Predicative set theory/QuineVsRussell/Quine: is not only free of paradoxes, but also of unspecifiable classes and higher indeterminacy, which is the blessing and curse of impredicative theory. (See "infinite numbers", "classes versus sets"). Predicative set theory/Quine: is constructive set theory today. Predicative Set Theory/Quine: is strictly speaking exactly as described above, but today it does not matter which conditions of containment one chooses to specify a class. 1. Whitehead, A.N. and Russel, B. (1910). Principia Mathematica. Cambridge: Cambridge University Press. |
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 |
Mathematics | Brouwer | Thiel I 242/243 Brouwer/Thiel: For Brouwer, all laws of formal logic are only extrapolations from ratios of finite sets. Some fail in infinite wholes. >Sets, >Set theory, >Infinity, >Finiteness. Following Jacques Herbrand there are the following criteria for the procedure of meta-mathematics (Hilbert himself has no catalog) of driteria: 1. Operate only with a finite number of objects and functions. In particular, each rule of forming expressions and each conclusion rule may contain only a finite number of premises. >Premises, >J. Herbrand. 2. The value of each function used for each argument must be unambiguously calculabe. >Unambiguity, >Functions, >Calculability. 3. Never must the set of all objects belonging to an infinite set be considered. Accordingly, the definition of a mathematical object must not be Definition > impredicative, in the sense that in the defining condition a set containing this object ("later") as an element occurs. >Impredicativity, >Predicativity. 4. The existence of an object is to be asserted only by demonstrating the same or a constructive procedure. >Existence Assertion. 5. Any assertion of a statement about "all x" of a domain must be accompanied by an instruction, how a statement can be proved for an arbitrarily presented xo from the domain A(xo). >Universal statement. I 242 Definition finite: Prohibition of the (carefree) dealing with infinite wholes. >cf. >Finitism. Hilbert accepted the new starting situation provoked by Brouwer. There have been prominent examples of errors in the history which have occured through false transfers from finite to infinite wholes. >Finiteness. |
T I Chr. Thiel Philosophie und Mathematik Darmstadt 1995 |
Paradoxes | Poincaré | Thiel I 322 Russell's Antinomy/solution: an attempt to avoid the Russellian paradox would be to say instead of "all" always "all, which". So now the suspicion is centered on "all". >Russell's paradox, >"All", >Universal quantification. Poincaré saw this suspicion confirmed and claimed: Conditions such as "~ (x ε x)" are unsuitable to determine a set, for they require a circulus vitiosus. >Sets, >Set theory, >Classes, cf. >Outermost class, >Circularity, cf. >Self-reference. He did not get to this diagnosis with the help of the Russellian antinomy, but with the antinomy constructed by Jules Richard: I 323 Richard's Antinomy: The totality E of all the decimal fractions which can be defined by a finite number of words (from the letters of a finite alphabet). Also, the totality E of the decimal fractions is countable. But then we can define a new decimal fraction d by the rule: Is the n-th number of the n-th decimal fraction of E 0,1,2,3,4,5,6,7,8,9, so the the corresponding number of d is 1,2,3,4,5,6,7,8,1,1. Since, by definition, d differs from the n-th decimal fraction from E at the n-th place, and this applies to an arbitrary n, d is different from every decimal fraction from E, and therefore does not belong to E. On the other hand, d must lie in E because we have defined it with finitely many words, and E was the totality of all such decimal fractions. Solution/Poincaré: he generalized the solution provided by Richard himself that E can be correctly explained only as the totality of not all but the decimal fractions which can be defined with a finite number of words without already introducing the concept of the totality E itself. >Definition, >Definability, >Introduction. Burali-Forti/Poincaré: Poincaré transferred this explanation also to other antinomies e.g. the antinomy of Burali-Forti: of the "set Ω of all ordinals". They can only be applied correctly to the set of all ordinals which can be defined without the introduction of the set Ω. (Otherwise, Ω + 1 always results). Thiel I 324 Poincaré: believed that he had found the decisive criterion: illegitimate, "non-predicative" conditions are those that contain such a circle. >impredicative/Russell. At first, it seemed sufficient to demand of expressions the relation between element and set that in "x ∈ y" the second relation term y should belong to exactly one step higher than x (simple > type theory), thus the requirement that every permissible expression should be formed not only "predicatively" (i.e. not impredicatively), but also all arguments occurring in it must satisfy this condition, to lead to a "ramifieded type theory", (ramified hierarchy). VsType Theory: Its complications included not only the fact that such a theory must also consider orders in addition to types, but also the more than annoying fact that now, for example, the upper limit of a non-empty set of real numbers (whose existence is presupposed in all continuity considerations in classical analysis) is of higher order than the real numbers whose upper limit it is. The consequence is that one can no longer quantify simply via "all real numbers", but only via all real numbers of a certain order. This is unacceptable for the field mathematics, and a huge obstacle to the "arithmetic program" of classical basic research. Especially for the logicism which follows. >Logicism. I 325 Poincaré's analysis carries even further than he himself presumed. E.g. (1) (1) is wrong With the variant "the only sentence numbered on this page is wrong". Or in the form of "I lie (now)". one accepts the necessary empirical regressions on book pages and "now", this leads to formal contradictions. The "liar" is weaker, originally in the letter of the apostle Paul to Titus, verse 12 of the first chapter. Luther: Z "One of them always said, their own prophet: the Cretans are always liars, evil beasts, and idle bellies." A <> "All Cretans lie (always)" Synonymous with the statement: "for this statement applies: if it is made by a Cretan, its opposite is true." --- I 326 K(A) > ~A (>separation rule: A, A > B >> B I 92) According to the separation rule, the statement ~ A becomes a true statement. This implies, however, that A is false, while we have derived this demand from the assumption that A is true. Since this is only assumed hypothetically, the reasoning (also I 315 Zermelo-Russell's antinomy) shows, with reference to the reductio ad absurdum: (A> A)> A, that A is indeed false. This does not lead to any formal contradiction, if there is a Cretan who makes at least one single true statement, A is then simply wrong. Nevertheless, Poincaré would dispute the admissibility: the definition of the abbreviation sign A is a universal statement, in which the variability range of the quantifier consists of all propositions, and therefore also contains the statement A itself, A is therefore impredicatively defined and therefore inadmissible. The applicability of the Poincaré criterion comes unexpectedly because the liar antinomy, due to the occurrence of metalogical terms such as "true" and "false" belongs to another, actually non-mathematical, type of conclusions that Peano classified as "linguistic". |
T I Chr. Thiel Philosophie und Mathematik Darmstadt 1995 |
Paradoxes | Thiel | I 321 False Conclusions/Thiel: only of interest if they are intentionally induced as "fallacies", or if they smuggle supposedly legitimate conclusions into an argument in the form of "sophisms", or as in Kant's case so-called "paralogisms" which have their reason "in the nature of human reason" and are therefore "inevitably though not indissoluble". Example: arithmetic fallacy: 5 = 7 (I 321 +). Example: Syllogism with a quaternia terminorum (hidden occurrence of four instead of three allowed terms in a final schema) Flying elephants are fantasy imaginations. Imaginations are part of our reality. So, flying elephants are part of our reality. Paradoxes are something contrary to ordinary opinion (doxa). Other form: fact wrapped in a puzzle solution. For example, that a strap placed tightly around the equator would suddenly protrude by 1/2π, i.e. by about 16cm, after being extended by only one meter. I 322 In everyday use, paradoxes are often only corny things, like the hypochondriac who only imagines himself to have delusions (question of definition) or "Murphy's law" that everything lasts longer, even if one has already considered it. Since the English scientific literature "paradoxically" compromises both paradoxes (not real antinomies) and antinomies, a distinction has not yet prevailed. I 327 Example "crocodile conclusion" (already known in ancient times): a crocodile has robbed a child, the mother begs to give it back. The crocodile places the task of guessing what it will do next. The mother (logically preformed) says: you won't give it back to me. Hence stalemate. Because the mother now argues that the crocodile must give the child back, because if the statement is true, she gets it back on the basis of the agreement, but if it is false, then it is just wrong that she does not get the child back, so because it is true that she gets it. The crocodile, on the other hand, argues that there is no need to give the child back, because if the mother's statement is false, she will not get it back because of the agreement, but if it is true, it means that she will not get the child back. Only a careful analysis reveals that the agreement made does not yet provide a rule for action. If "z" stands for giving back, "a" for the mother's answer (which is still indefinite and can therefore only be represented schematically by a), the agreement does not yet provide a rule system that can be followed, but rather the rule schema. "a" ε true >> z "a" ε false >> ~z If the range of variability of a is not restricted, then one can also make choices of a that are incompatible with Tarski's condition of adequacy for truth definitions. >Adequacy/Tarski, >Convention T. I 328 This states that for a predicate of truth "W" and any statement p, from which it can be meaningfully stated, always "p" ε W <> p has to apply. In the crocodile conclusion, the mother selects ~z for a, thereby turning the rule scheme into the rule system. (R1) "~z" ε true >> z (R2) "~z" ε false >> ~z The crocodile now concludes to R2 and Tarski (with ~z for p) to ~z. The mother, on the one hand, deduces after R1 and on the other hand metalogically from the falsity of "~z" and from there (after Tarski) further to z. Since the argumentation makes use of a predicate of truth and a predicate of falsehood as well as the connection between both, the crocodile conclusion is usually counted among the "semantic" antinomies. One can see in it a precursor of Russell's antinomy. >Russellean Paradox. I 328 One should not hastily deduce from this that the antinomies and paradoxes have no meaning for mathematics. Both Poincaré's criterion (predictiveness) and type theory force a restriction of the so-called comprehension axiom, which determines the conditions permissible as defining conditions for sets of forms of statement. >Impredicativeness, >Comprehension, >Type theory. |
T I Chr. Thiel Philosophie und Mathematik Darmstadt 1995 |
Platonism | Quine | XII 44 Platonic Idea/Quine: is not the same as a mental idea. >Ideas/Quine. XI 136 Mathematics/QuineVsHilbert/Lauener: is more than just syntax. Quine reluctantly professes Platonism. XI 155 CarnapVsPlatonism/CarnapVsNominalism: is a metaphysical pseudo discussion. Solution: it is about choosing a language. >Language/Quine. VII(f) 125 Conceptualism VsPlatonism/Quine: treats classes as constructions, not as discoveries. Problem: Poincaré's impredicative definition: Def Impredicative Definition/Poincaré/Quine: the specification of a class by a realm of objects within which this class is located. >Classes/Quine. VII (f) 126 Classes/Platonism/Quine: when classes are considered pre-existing, there is no objection to picking one of them by a move that presupposes their existence. Classes/Conceptualism/Quine: for him, however, classes only exist if they originate from an ordered origin. Of course, this should not be interpreted in terms of time. VII (f) 127 Platonism/Conceptualism/Quine: both allow universals and classes as irreducible. Conceptualism: allows fewer classes. But rests on a rather metaphorical reason: "Origin". >Conceptualism/Quine. V 126 Platonism/Quine: is opened by form words, not by color words! Reason: a union of color spots has the same color, but a union of spots of a certain shape does not necessarily have the same shape. |
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 |
Predicativeness | Russell | I 96 Predicative functions: f (x) - non-predicative: f(x^): Function as opposed to its values! - Predicative(s): function that is one step higher than its argument. >Levels/Order, >Function/Russell. I 109 Class/predicative/totality/Principia Mathematica(1)/Russell: each class can be defined by a predictive function - hence the totality of classes, of which one can sensibly say that a given term belongs to them or not, a legitimate entirety, although the entirety of functions, of which one can say that a given term fulfils them or not, is not a legitimate entirety. >Totality. I 116 Definition predicative function/Principia Mathematica/Russell: notation: j ! (x,y) (predicative function of x and y.) - Def non-predicative/impredicative function/Principia Mathematica/Russell: function as opposed to its values: notation: j ! (x^, y^). 1. Whitehead, A.N. and Russel, B. (1910). Principia Mathematica. Cambridge: Cambridge University Press. E. Picardi Alfred North Whitehead/Bertrand Russell: Principia Mathematica aus "Hauptwerke der Philosophie 20. Jahrhundert" , Eva Picardi u.a. Stuttgart 1992 p. 18 Non-predicative properties: E.g. having all the properties of a great commander - predicative: E.g. to be born in Corsica. |
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 |
Relations | Russell | Geach I 320ff Relation/Principia Mathematica(1)/Russell/Geach: sentences of the form "Fab" must be treated as individual copies of the form "Ya", i.e. a sentence that says how A is related to B is a particular type of the predication of a. Quine: E.g. Edith envies everyone who is happier than Edith - Herbert is not happier than anyone who envies Herbert; so we prove that Herbert is not happier than Edith. Solution: addition of assumptions: either A "Edith envies Herbert" or B ..."does not". Problem: in A "envies Herbert" it is a term, in B "happier than Edith". We cannot form a predicate with a name as we need it. Therefore the relations must be predicative. >Predicativeness, >Impredicativeness. Relational propositions make predications about the related things A and B. - Then it makes sense to say that there is something in A which answers the predication, but if we apply the same sentence to B, there is nothing that answers the relation. It is unnatural to regard the state of "being envied" as a property of Herbert. Russell I 48 Relation/Russell: D"R: class of all terms that have the relation R to this or that thing. - R"y": "the R of y": "the father of y". 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 Gea I P.T. Geach Logic Matters Oxford 1972 |
Set Theory | Lorenzen | Berka I 269 Inductive Definition/Set Theory/LorenzenVsSet Theory: For example, an inductive definition of a set M by a(y) › y ε M, x ε M u b(x,y) › y ε M whereby a (x) and b(x,y) are already defined formulas in which M does not occur, is "explained" set theoretically that M should be the average of all sets N satisfying these implications with N instead of M. Lorenzen: whoever wants to defend a claim n ε M (sic) will hardly attempt all these sets N. As P, he will rather defend O against either directly a(n), or he will first give an m which he will defend b (m, n) and m ∈ M. >Dialogical logic. Step number/Lorenzen: in order to determine this procedure as the the dialogical sense of the inductive definition of M, we must also require of P to indicate the number of steps required for complete proof for each assertion of the form x ∈ M. >Step number. E.g. suppose, for example, he traces n ε M back to the assertion m ∈ M and has stated the step number v for n ε M... I 270 ...so he must specify a step number μ ‹ v for m ε M. Without such information, P could assert "smaller" ‹ for the integers in the following inductive definition 0 ‹ y for positive numbers y x ‹ y _› x +/ 1 ‹ y +/ 1 e.g. 1 ‹ 0, and begin a "proof" with the aid of 0 ‹ 1, 1 ‹ 2, 2 ‹ 3 .... Of course, the proof could not be finished, but O could not prove this. Dialogical logic/Lorenzen: in these dialogues, it is never permitted to intervene suddenly in the "free speech" of the opponent. If, on the other hand, P has to specify a step number v, he will have lost his assertion at the latest after v steps. Step number: the steps are, of course, natural numbers. If one wants to give infinite inductive definitions, i.e. such with an infinite number of premisses, a dialogical meaning, one must allow transfinite ordinal numbers as the step numbers. Inductive Definition/LorenzenVsHerbrand: For example, a function sequence f1, f2 ... is already defined and the induction scheme a(y) › y ε M (x)fx(y) ε M › y ε M is adressed. This definition is by no means "impredicative". >Imprecativeness. But it is also not really constructive either. We have infinitely many premises here f1 (y) ∈ M, f2 (y) ε M ... which are necessary to prove y ∈ M. Infinite: in dialogue one cannot defend every premise, one will therefore allow O to select an fm(y) e M. This must then be claimed and defended by P. In addition, P must specify a generally transfinite ordinal number as the step number. Step number: the step number of a premise must always be specified as less than the step number of the conclusion. Winning strategy: of P: must provide the step numbers for all opponent's elections. II. Number-class/second/Lorenzen: set-theoretically one can prove easily the existence of suitable ordinal numbers of the II. number class. One can define transfinite recursion through this: y ε M0 ‹› a(y) y ε Mλ ‹› (x)fx(y) ε Ux x ‹ λ Mx. . Then M = Ul l › μ Ml for a suitable μ and if M is to be a set of natural numbers, μ can be taken from the II. number class. Constructively, if the inductive definition is to be constructive, the ordinal numbers used must also be "constructive". Here it is obvious to limit oneself to the recursive ordinal numbers of Church and Kleene.(1) >Constructivism, >Intuitionism, >Recursion, >Recursivity. 1. P. Lorenzen, Ein dialogisches Konstruktivitätskriterium, in: Infinitistic Methods, (1961), 193-200 |
Lorn I P. Lorenzen Constructive Philosophy Cambridge 1987 Berka I Karel Berka Lothar Kreiser Logik Texte Berlin 1983 |
Truth Functions | Tugendhat | I 286 Def truth-functional/Tugendhat: depends on other sentences, not on the situation - (contrast: deictic). - Brandom: inferential structure, >Deixis. I 315 Truth-functional/Tugendhat: "true" occurs several times. Contrast: predicative: "true" does not occur more than once. >Predicativity, >Impredicativity, >Circular reasoning. |
Tu I E. Tugendhat Vorlesungen zur Einführung in die Sprachanalytische Philosophie Frankfurt 1976 Tu II E. Tugendhat Philosophische Aufsätze Frankfurt 1992 |
![]() |
Disputed term/author/ism | Author Vs Author![]() |
Entry![]() |
Reference![]() |
---|---|---|---|
Conceptualism | Quine Vs Conceptualism | VII (f) 126 Classes/Conceptualism/Quine: does not require classes to exist beyond expressible conditions of membership of elements. ((s) VsPlatonism: Quasi requires that there should also be classes without such conditions, as classes should be independent of speakers.) Cantor's proof: would lead to something else: He namely appeals to a class h of those members of the class k that are not elements of the subclasses of k to which they refer. VII (f) 127 But thus the class h is specified impredicatively! h is in fact itself part of the subclass of k. Thus a theorem of classical mathematics goes overboard in conceptualism. The same fate also applies to Cantor's proof of the existence of hyper-countable infinities. QuineVsConceptualism: which is indeed a welcome relief, but there are problems with much more fundamental and desirable theorems of mathematics: Ex proof that every limited sequence of numbers has an upper limit. ConceptualismVsReducibility Axiom: because it reintroduces the entire Platonist class logic. |
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 |
Counterfactual Conditional | Field Vs Counterfactual Conditional | I 220 Problem of Quantities/PoQ/Modality/Field: but this does not exclude a possible modal solution to the PoQ: perhaps other operators can help? Anyway, I do not know how that could be excluded, even if I do not know what these operators should look like. I 221 Counterfactual Conditional/Co.Co./PoQ/Field: one suggestion is to use Counterfactual Conditional to solve the PoQ: FieldVsCounterfactual Conditional: 1) they are known to be extremely vague. Therefore, you should not rely on them when formulating a physical theory. Neither should we use Counterfactual Conditional for the development of geometrical concepts. 2) DummettVsCounterfactual conditionals: They cannot be "barely true": if a Counterfactual Conditional is supposed to be true, then there must be some facts (known or unknown facts) that can be determined without Counterfactual Conditional, and by virtue of which the Counterfactual Conditional are true. (Dummett, 1976, p.89). Then the relationism cannot use the Counterfactual Conditional for the PoQ, because in that case the principle requires: if distance relations are counterfactually defined, then situations that differ in their distance relations (like situations A and B) must also differ in non-counterfactual respects!. Substantivalism: can guarantee that. Relationism: cannot, and if it could, it would need no Counterfactual Conditional. 3) VsCounterfactual conditionals: does not work for very similar reasons for which the version with impredicative properties (P3) did not work: no theory about counterfactually defined relations works if these relations cannot also be counterfactually defined, (This is the formal reason for the metaphysical argument of Dummett, for why Counterfactual Conditional cannot be "barely true"). E.g. In order to prove the incompatibility of "double distance" and "triple distance" (given that z and w do not occupy the same point, i.e. given that zw is not congruent with zz - (logical form: local equality) - then you would need the incompatibility of the following: a) if there were a point u in the middle between x and y, then uy would be congruent with zw. b) if there were a point s between x and y, and a point t between s and y, so that xs, st and ty were all congruent, then ty would be congruent with zw. If these Counterfactual Conditional were somehow derivable from non-counterfactual statements, E.g. statements about spacetime points (ST points), then you could probably, and by way of derivation. I 222 Together with the demonstrable relations between the non-counterfactual statements win an argument for the incompatibility of (a) and (b). But if we have no non-counterfactual support, we would have to consider them as bare facts. That would not be so bad if you only needed a small amount of them, but we would need a very large number of them. |
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 |
Poincaré, H. | Quine Vs Poincaré, H. | IX 176 Classes/existence/Quine: the basic idea rather states that they are there from the beginning, and are not created by description. impredicative/QuineVsPoincaré: if that is so, then there can be no obvious fallacy in impredicative description. It is reasonable to separate out a desired class, where one indicates a property of it, even if there is the danger, to quantify about it along with everything else in the universal class. ((s) classes/(s): determine only one property of each of their elements.) Quine: E.g. just like this one can call a certain person an ordinary consumer, based on average values, in which their own values have some influence. |
Quine XIII Willard Van Orman Quine Quiddities Cambridge/London 1987 |
Principia Mathematica | Gödel Vs Principia Mathematica | Russell I XIV Circular Error Principle/VsPrincipia Mathematica(1)/PM/Russell/Gödel: thus seems to apply only to constructivist assumptions: when a term is understood as a symbol, together with a rule to translate sentences containing the symbol into sentences not containing it. Classes/concepts/Gödel: can also be understood as real objects, namely as "multiplicities of things" and concepts as properties or relations of things that exist independently of our definitions and constructions! This is just as legitimate as the assumption of physical bodies. They are also necessary for mathematics, as they are for physics. Concept/Terminology/Gödel: I will use "concept" from now on exclusively in this objective sense. A formal difference between these two conceptions of concepts would be: that of two different definitions of the form α(x) = φ(x) it can be assumed that they define two different concepts α in the constructivist sense. (Nominalistic: since two such definitions give different translations for propositions containing α.) For concepts (terms) this is by no means the case, because the same thing can be described in different ways. For example, "Two is the term under which all pairs fall and nothing else. There is certainly more than one term in the constructivist sense that satisfies this condition, but there could be a common "form" or "nature" of all pairs. All/Carnap: the proposal to understand "all" as a necessity would not help if "provability" were introduced in a constructivist manner (..+...). Def Intensionality Axiom/Russell/Gödel: different terms belong to different definitions. This axiom holds for terms in the circular error principle: constructivist sense. Concepts/Russell/Gödel: (unequal terms!) should exist objectively. (So not constructed). (Realistic point of view). When only talking about concepts, the question gets a completely different meaning: then there seems to be no objection to talking about all of them, nor to describing some of them with reference to all of them. Properties/GödelVsRussell: one could surely speak of the totality of all properties (or all of a certain type) without this leading to an "absurdity"! ((s) > Example "All properties of a great commander". Gödel: this simply makes it impossible to construe their meaning (i.e. as an assertion about sense perception or any other non-conceptual entities), which is not an objection to someone taking the realistic point of view. Part/whole/Mereology/GödelVsRussell: neither is it contradictory that a part should be identical (not just the same) with the whole, as can be seen in the case of structures in the abstract sense. Example: the structure of the series of integers contains itself as a special part. I XVI/XVII Even within the realm of constructivist logic there are certain approximations to this self-reflectivity (self-reflexivity/today: self-similarity) of impredicative qualities, namely e.g. propositions, which as parts of their meaning do not contain themselves, but their own formal provability. There are also sentences that refer to a totality of sentences to which they themselves belong: Example: "Each sentence of a (given) language contains at least one relational word". This makes it necessary to look for other solutions to the paradoxes, according to which the fallacy does not consist in the assumption of certain self-reflectivities of the basic terms, but in other assumptions about them! The solution may have been found for the time being in simple type theory. Of course, all this refers only to concepts. Classes: one should think that they are also not created by their definitions, but only described! Then the circular error principle does not apply again. Zermelo splits classes into "levels", so that only sets of lower levels can be elements of sets of higher levels. Reducibility Axiom/Russell/Gödel: (later dropped) is now taken by the class axiom (Zermelo's "axiom of choice"): that for each level, for any propositional function φ(x) the set of those x of this level exists for which φ(x) is true. This seems to be implied by the concept of classes as multiplicities. I XVIII Extensionality/Classes: Russell: two reasons against the extensional view of classes: 1. the existence of the zero class, which cannot be well a collection, 2. the single classes, which should be identical with their only elements. GödelVsRussell: this could only prove that the zero classes and the single classes (as distinguished from their only element) are fictions to simplify the calculation, and do not prove that all classes are fictions! Russell: tries to get by as far as possible without assuming the objective existence of classes. According to this, classes are only a facon de parler. Gödel: but also "idealistic" propositions that contain universals could lead to the same paradoxes. Russell: creates rules of translation according to which sentences containing class names or the term "class" are translated into sentences not containing them. Class Name/Russell: eliminate by translation rules. Classes/Principia Mathematica/Russell/Gödel: the Principia Mathematica can do without classes, but only if you assume the existence of a concept whenever you want to construct a class. First, some of them, the basic predicates and relations like "red", "colder" must be apparently considered real objects. The higher terms then appear as something constructed (i.e. something that does not belong to the "inventory of the world"). I XIX Ramsey: said that one can form propositions of infinite length and considers the difference finite/infinite as not so decisive. Gödel: Like physics, logic and mathematics are based on real content and cannot be "explained away". Existence/Ontology/Gödel: it does not behave as if the universe of things is divided into orders and one is forbidden to speak of all orders, but on the contrary: it is possible to speak of all existing things. But classes and concepts are not among them. But when they are introduced as a facon de parler, it turns out that the extension of symbolism opens the possibility of introducing them in a more comprehensive way, and so on, to infinity. To maintain this scheme, however, one must presuppose arithmetics (or something equivalent), which only proves that not even this limited logic can be built on nothing. I XX Constructivist posture/constructivism/Russell/Gödel: was abandoned in the first edition, since the reducibility axiom for higher types makes it necessary that basic predicates of arbitrarily high type exist. From constructivism remains only 1. Classes as facon de parler 2. The definition of ~, v, etc. as valid for propositions containing quantifiers, 3. The stepwise construction of functions of orders higher than 1 (of course superfluous because of the R-Axiom) 4. the interpretation of definitions as mere typographical abbreviations (all incomplete symbols, not those that name an object described by the definition!). Reducibility Axiom/GödelVsRussell: this last point is an illusion, because of the reducibility axiom there are always real objects in the form of basic predicates or combinations of such according to each defined symbol. Constructivist posture/constructivism/Principia Mathematica/Gödel: is taken again in the second edition and the reducibility axiom is dropped. It is determined that all basic predicates belong to the lowest type. Variables/Russell/Gödel: their purpose is to enable the assertions of more complicated truth functions of atomistic propositions. (i.e. that the higher types are only a facon de parler.). The basis of the theory should therefore consist of truth functions of atomistic propositions. This is not a problem if the number of individuals and basic predicates is finite. Ramsey: Problem of the inability to form infinite propositions is a "mere secondary matter". I XXI Finite/infinite/Gödel: with this circumvention of the problem by disregarding the difference between finite and infinite a simpler and at the same time more far-reaching interpretation of set theory exists: Then Russell's Apercu that propositions about classes can be interpreted as propositions about their elements becomes literally true, provided n is the number of (finite) individuals in the world and provided we neglect the zero class. (..) + I XXI Theory of integers: the second edition claims that it can be achieved. Problem: that in the definition "those cardinals belonging to each class that contains 0 and contains x + 1 if it contains x" the phrase "each class" must refer to a given order. I XXII Thus whole numbers of different orders are obtained, and complete induction can be applied to whole numbers of order n only for properties of n! (...) The question of the theory of integers based on ramified type theory is still unsolved. I XXIII Theory of Order/Gödel: is more fruitful if it is considered from a mathematical point of view, not a philosophical one, i.e. independent of the question of whether impredicative definitions are permissible. (...) impredicative totalities are assumed by a function of order α and ω . Set/Class/Principia Mathematica(1)/Russell/Type Theory/Gödel: the existence of a well-ordered set of the order type ω is sufficient for the theory of real numbers. Def Continuum Hypothesis/Gödel: (generalized): no cardinal number exists between the power of any arbitrary set and the power of the set of its subsets. Type Theory/VsType Theory/GödelVsRussell: mixed types (individuals together with predications about individuals etc.) obviously do not contradict the circular error principle at all! I XXIV Russell based his theory on quite different reasons, similar to those Frege had already adopted for the theory of simpler types for functions. Propositional functions/statement function/Russell/Gödel: always have something ambiguous because of the variables. (Frege: something unsaturated). Propositional function/p.f./Russell/Gödel: is so to speak a fragment of a proposition. It is only possible to combine them if they "fit together" i.e. are of a suitable type. GödelVsRussell: Concepts (terms) as real objects: then the theory of simple types is not plausible, because what one would expect (like "transitivity" or the number two) to be a concept would then seem to be something that stands behind all its different "realizations" on the different levels and therefore does not exist according to type theory. I XXV Paradoxes in the intensional form/Gödel: here type theory brings a new idea: namely to blame the paradoxes not on the axiom that every propositional function defines a concept or a class, but on the assumption that every concept results in a meaningful proposition if it is claimed for any object as an argument. The objection that any concept can be extended to all arguments by defining another one that gives a false proposition whenever the original one was meaningless can easily be invalidated by pointing out that the concept "meaningfully applicable" does not always have to be meaningfully applicable itself. 1. Whitehead, A.N. and Russel, B. (1910). Principia Mathematica. Cambridge: Cambridge University Press. |
Göd II Kurt Gödel Collected Works: Volume II: Publications 1938-1974 Oxford 1990 |
substit. Quantific. | Quine Vs substit. Quantific. | V 158 VsSubstitutional Quantification/SQ/Quine: the SQ has been deemed unusable for the classic ML for a false reason: because of uncountability. The SQ does not accept nameless classes as values of variables. ((s) E.g. irrational numbers, real numbers, etc. do not have names, i.e. they cannot be Gödel numbered). I.e. SQ allows only a countable number of classes. Problem: Even the class of natural numbers has uncountably many sub-classes. And at some point we need numbers! KripkeVs: in reality there is no clear contradiction between SQ and hyper-countability! No function f lists all classes of natural numbers. Cantor shows this based on the class {n:~ (n e f(n))} which is not covered by the enumeration f. refQ: demands it in contrast to a function f enumerating all classes of natural numbers? It seems so at first glance: it seems you could indicate f by numbering all abstract terms for classes lexicographically. Vs: but the function that numbers the expressions is not quite the desired f. It is another function g. Its values are abstract terms, while the f, which would contradict the Cantor theorem, would have classes as values... V 159 Insertion character: does ultimately not mean that the classes are abstract terms! ((s) I.e. does not make the assumption of classes necessary). The cases of insertion are not names of abstract terms, but the abstract terms themselves! I.e. the alleged or simulated class names. Function f: that would contradict Cantor's theorem is rather the function with the property that f(n) is the class which is denoted by the n-th abstract term g(n). Problem: we cannot specify this function in the notation of the system. Otherwise we end up with Grelling's antinomy or that of Richard. That's just the feared conflict with Cantor's theorem. This can be refute more easily: by the finding that there is a class that is not denoted by any abstract term: namely the class (1) {x.x is an abstract term and is not a member of the class it denotes}. That leaves numbers and uncountability aside and relates directly to expressions and classes of expressions. (1) is obviously an abstract expression itself. The antinomy is trivial, because it clearly relies on the name relation. ((s) x is "a member of the class of abstract expressions and not a member of this class"). V 191 Substitutional Quantification/SQ/Nominalism/Quine: the nominalist might reply: alright, let us admit that the SQ does not clean the air ontologically, but still we win something with it: E.g. SQ about numbers is explained based on expressions and their insertion instead of abstract objects and reference. QuineVsSubstitutional Quantification: the expressions to be inserted are just as abstract entities as the numbers themselves. V 192 NominalismVsVs: the ontology of real numbers or set theory could be reduced to that of elementary number theory by establishing truth conditions for the sQ based on Gödel numbers. QuineVs: this is not nominalistic, but Pythagorean. This is not about the extrapolation of the concrete and abhorrence of the abstract, but about the acceptance of natural numbers and the refutal of the most transcendent nnumbers. As Kronecker says: "The natural numbers were created by God, the others are the work of man." QuineVs: but even that does not work, we have seen above that the SQ about classes is, as a matter of principle, incompatible with the object quantification over objects. V 193 VsVs: the quantification over objects could be seen like that as well. QuineVs: that was not possible because there are not enough names. Zar could be taught RZ coordination, but that does not explain language learning. Ontology: but now that we are doing ontology, could the coordinates help us? QuineVs: the motivation is, however, to re-interpret the SQ about objects to eliminate the obstacle of SQ about classes. And why do we want to have classes? The reason was quasi nominalistic, in the sense of relative empiricism. Problem: if the relative empiricism SQ talks about classes, it also speaks for refQ about objects. This is because both views are closest to the genetic origins. Coordinates: this trick will be a poor basis for SQ about objects, just like (see above) SQ about numbers. Substitutional/Referential Quantification/Charles Parsons/Quine: Parsons has proposed a compromise between the two: according to this, for the truth of an existential quantification it is no longer necessary to have a true insertion, there only needs to be an insertion that contains free object variables and is fulfilled by any values of the same. Universal quantification: Does accordingly no longer require only the truth of all insertions that do not contain free variables. V 194 It further requires that all insertions that contain free object variables are fulfilled by all values. This restores the law of the single sub-classes and the interchangeability of quantifiers. Problem: this still suffers from impredicative abstract terms. Pro: But it has the nominalistic aura that the refQ completely lacks, and will satisfy the needs of set theory. XI 48 SQ/Ontology/Quine/Lauener: the SQ does not make any ontological commitment in so far as the inserted names do not need to designate anything. I.e. we are not forced to assume values of the variables. XI 49 QuineVsSubstitutional Quantification: we precisely obscure the ontology by that fact that we cannot get out of the linguistic. XI 51 SQ/Abstract Entities/Quine/Lauener: precisely because the exchange of quantifiers is prohibited if one of the quantifiers referential, but the other one is substitutional, we end up with refQ and just with that we have to admit the assumption of abstract entities. XI 130 Existence/Ontology/Quine/Lauener: with the saying "to be means to be the value of a bound variable" no language dependency of existence is presumed. The criterion of canonical notation does not suppose an arbitrary restriction, because differing languages - e.g. Schönfinkel's combinator logic containing no variables - are translatable into them. Ontological Relativity/Lauener: then has to do with the indeterminacy of translation. VsSubstitutional Quantification/Quine/Lauener: with it we remain on a purely linguistic level, and thus repeal the ontological dimension. But for the variables not singular terms are used, but the object designated by the singular term. ((s) referential quantification). Singular Term/Quine/Lauener: even after eliminating the singular terms the objects remain as the values of variables. XI 140 QuineVsSubstitutional Quantification: is ontologically disingenuous. |
Quine XIII Willard Van Orman Quine Quiddities Cambridge/London 1987 |
Tarski, A. | Kripke Vs Tarski, A. | III 337 Expansion/Language/Kripke: Here we need Set Theory, at least the sets of the expressions of L. (As Tarski, who is dealing, however,with referential language). DavidsonVsTarski/Kripke: he needs less ontology and less richness of metalanguage. III 367 Substitutional quantification/sQ/KripkeVsTarski: substitution quantification together with the formula Q(p,a) solves Tarski’s problem to define a "true sentence". III 410 Language/Kripke: When a language is introduced, an explicit definition of W is a necessary and sufficient condition that the language has mathematically defined (extensional) semantics. Otherwise, the language can be explained in informal English. The semantics is then intuitive. Before Tarski, semantics have generally been treated that way. Convention T/DavidsonVsTarski/Kripke: for Davidson the axioms must be finite in number. Kripke: his work is much more controversial than that of Tarski. Field I 245 Def disquotational truth/dW/Field: can be defined with the help of substitution quantification (∏/(s): for all sentences, not objects .... is valid) for all sentences, not objects") definiert werden. S is true iff ∏p(if S = "p", p). where "p" sentences are substituents. But which sentences?. Konjunctions/Understanding/Paradoxies/Field: Konjunctions of sentences: makes only sense if the sentences have been understood beforehand, i.e. that the conjunctions themselves (and sentences constructed from them) are not allowed as conjuncts. (>Semantic Paradoxes, (s) >Everything he said is true). Solution: Tarski similar hierarchy of T-predicates. Predicates: then the definition of the dW by substitutional quantification (sQ)is typically ambiguous: each element of the hierarchy is provided by the corresponding sQ. KripkeVsTarski: (Kripke 1975): he is to restrictive for our aim: as such we do not obtain all ueK that we need. Solution/Kripke: others, quasi imprädikative Interpretation von dW. Analog für Field I 246 Substitutional Quantification/sQ/Kripke: Authorizes sentences to be a part of themselves and things, which are build from those sentences, to be conjuncts. However, the truth value of those quasi impredicative conjuncts are to be objectively indeterminate until the truth value is assigned to a certain level. sQ/Field: Allows then ueK without semantic ascent. If we want to talk about the non-linguistic world, why should we use sentences which we do not need?. → sQ: Could then be used as a basic term. → Basic term/Field: This means that a) the basic term is not defined by even more basic termini. → b) the basic term does not try to explain even more basic terms in theory (Field for each a) and b). → If we accept a), we need, however, to explain how the term obtains its meaning. Perhaps from logical laws which regulate its use. If we accept a), it is not a problem to accept b) as well. → Explanation/Field: e.g. the issue regarding mentalistic terms is not to give a meaning, but to show that the term is not primitive (basal). The ideology in logical terms does not need to be reduced that much. |
Kripke I S.A. Kripke Naming and Necessity, Dordrecht/Boston 1972 German Edition: Name und Notwendigkeit Frankfurt 1981 Kripke II Saul A. Kripke "Speaker’s Reference and Semantic Reference", in: Midwest Studies in Philosophy 2 (1977) 255-276 In Eigennamen, Ursula Wolf Frankfurt/M. 1993 Kripke III Saul A. Kripke Is there a problem with substitutional quantification? In Truth and Meaning, G. Evans/J McDowell Oxford 1976 Kripke IV S. A. Kripke Outline of a Theory of Truth (1975) In Recent Essays on Truth and the Liar Paradox, R. L. Martin (Hg) Oxford/NY 1984 Field I H. Field Realism, Mathematics and Modality Oxford New York 1989 Field IV Hartry Field "Realism and Relativism", The Journal of Philosophy, 76 (1982), pp. 553-67 In Theories of Truth, Paul Horwich Aldershot 1994 |
Type Theory | Verschiedene Vs Type Theory | Thiel I 324 Poincaré: believed he had found the decisive criterion: illegitimate, "non-predictive" conditions are those that contain such a circle. (>impressive, Russell). At first it seemed sufficient to demand from expressions for the relationship between element and set that in "x ε y" the second relational element y must belong to an exactly 1 higher level than x (simple >type theory) so the demand that every admissible expression should not only be itself "predicative" (i.e. not impredicative), but also all arguments occurring in it must satisfy this condition leads to a ">branched type theory". VsType Theory: Among its complications was not only the fact that such a theory has to consider not only types but also orders, and also the more than annoying fact that now e.g. the upper limit of a non-empty set of real numbers (their existence in all steadiness considerations in the classical VsType Theory) is not only a question of the order, but also of the fact that the upper limit of a non-empty set of real numbers (their existence in all steadiness considerations is presupposed in the classical analysis) of higher order than the real numbers whose upper limit it is. The consequence of this is that one can no longer simply quantify using "all real numbers", but only using all real numbers of a certain order. Unacceptable for mathematics, and a huge obstacle for the "arithmetic program" of classical basic research. All the more so for the logicism that follows. |
T I Chr. Thiel Philosophie und Mathematik Darmstadt 1995 |
Various Authors | Lorenzen Vs Various Authors | Berka I 266 Overcountable/infinite/LorenzenVsSet Theory: fable realm of the "overcountable". ((s) not constructible at all, >constructivism). LorenzenVsHerbrand/ConstructivismVs "impredicative": fable realm of the "impredicative".(1) 1. P. Lorenzen, Ein dialogisches Konstruktivitätskriterium, in: Infinitistic Methods, (1961), 193-200 |
Lorn I P. Lorenzen Constructive Philosophy Cambridge 1987 Berka I Karel Berka Lothar Kreiser Logik Texte Berlin 1983 |
![]() |