Find counter arguments by entering NameVs… or …VsName.
The author or concept searched is found in the following 12 entries.
| Disputed term/author/ism | Author |
Entry |
Reference |
|---|---|---|---|
| Ambiguity | Tarski | Berka I 451 Semantically ambiguous/Russell/Tarski: E.g. "name", "designating": a) in the case of objects b) in relation to classes, relations, etc.(1) >Description levels, >Definiteness, >Reference, >Object, >Classes, >Relations, >Class name, >Levels (Order), >Unambiguity. 1. A.Tarski, Der Wahrheitsbegriff in den formalisierten Sprachen, Commentarii Societatis philosophicae Polonorum. Vol 1, Lemberg 1935 |
Tarski I A. Tarski Logic, Semantics, Metamathematics: Papers from 1923-38 Indianapolis 1983 Berka I Karel Berka Lothar Kreiser Logik Texte Berlin 1983 |
| Classes | Mill | I 48 Class names/Mill: sometimes one tries to explain general terms as "class names". MillVs: better reverse: Definiton Class/Mill: an undefined set of particulars, which is denoted by a generic name (general term). Difference to "collectives" (> distribution): collectives only belong to wholenesses E.g. "76th regiment". This is not a general term but a singular term for a whole. E.g "a regiment": is at the same time a singular term and a general term. |
Mill I John St. Mill A System of Logic, Ratiocinative and Inductive, London 1843 German Edition: Von Namen, aus: A System of Logic, London 1843 In Eigennamen, Ursula Wolf Frankfurt/M. 1993 Mill II J. St. Mill Utilitarianism: 1st (First) Edition Oxford 1998 Mill Ja I James Mill Commerce Defended: An Answer to the Arguments by which Mr. Spence, Mr. Cobbett, and Others, Have Attempted to Prove that Commerce is Not a Source of National Wealth 1808 |
| Classes | Russell | I XIV Classes/Concepts/Gödel: can be construed as real objects, namely as "multiplicities of things" and concepts as properties or relations of things that exist independently of our definitions and constructions - which is just as legitimate as the assumption of physical bodies - they are as necessary for mathematics as they are for physics. >Platonism, >Universals, >Mathematical entities, cf. >Hartry Field's Antiplatonism. I XVIII Set/Gödel: realistic: classes exist, circle fault no fault, not even if it is seen constructivistically. But Gödel is a non-constructivist. Russell: classes are only facon de parler, only class names, term, no real classes. I XVIII Class names/Russell: eliminate through translation rules. I XVIII Classes/Principia Mathematica(1)/PM/Russell/Gödel: Principia do without classes, but only if one assumes the existence of a concept whenever one wants to construct a class - E.g. "red" or "colder" must be regarded as real objects. I 37 Class/Principia Mathematica/Russell: The class formed by the function jx^ is to be represented by z^ (φ z) - E.g. if φ x is an equation, z^ (φ z) will be the class of its roots - Example if φ x means: "x has two legs and no feathers", z^ (φ z) will be the class of the humans. I 120 Class/Principia Mathematica/Russell: incomplete symbol. >Incomplete symbols. Function: Complete Symbol - therefore no transitivity when classes are inserted for variables - E.g. x = y . x = z . > . y = z (transitivity) is a propositional function which always applies. But not if we insert a class for x and functions for y and z. - E.g. "z^ (φ z) = y ! z^" is not a value of "x = y" - because classes are incomplete symbols. 1. Whitehead, A.N. and Russel, B. (1910). Principia Mathematica. Cambridge: Cambridge University Press. Flor III 117 Classes/sets/things/objects/Russell/Flor: sets must not be seen as things - otherwise, we would always have also 2n things at n things (combinations - i.e. we would have more things than we already have - Solution: Eliminate class symbols from expressions - instead designations for propositional functions. >Quine: Class Abstraction. |
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 Flor I Jan Riis Flor "Gilbert Ryle: Bewusstseinsphilosophie" In Philosophie im 20. Jahrhundert, A. Hügli/P. Lübcke Reinbek 1993 Flor II Jan Riis Flor "Karl Raimund Popper: Kritischer Rationalismus" In Philosophie im 20. Jahrhundert, A.Hügli/P.Lübcke Reinbek 1993 Flor III J.R. Flor "Bertrand Russell: Politisches Engagement und logische Analyse" In Philosophie im 20. Jahrhundert, A. Hügli/P.Lübcke (Hg) Reinbek 1993 Flor IV Jan Riis Flor "Thomas S. Kuhn. Entwicklung durch Revolution" In Philosophie im 20. Jahrhundert, A. Hügli/P. Lübcke Reinbek 1993 |
| Distribution | Lyons | I 72 Def Distribution/Linguistics/Lyons: each linguistic unit has a characteristic distribution, namely the set of contexts in which it can stand. >Context. Distribution equivalent: two expressions can be in the same context. Correspondingly distribution-complementary or distribution-overlapping. If two units are at least partially distribution equivalent, they cannot contrast with each other. I 145 Distribution/Grammar/Lyons: We can take distribution as a starting point for a grammatical description: expressions have meaning when they are used in an appropriate context. >Expressions/Lyons, >Grammar, >Meaning. I 147 Distribution analysis/grammar/Lyons: a list would not be the most direct description of a text: in a sufficiently large language sample there will be a considerable overlap in the distribution of different words. >Lists. Distribution classes: Classes of words that can be used for each other in a sentence. Example "I drink beer": liquor, milk, water... and so forth. >Word classes, cf. >P. Gärdenfors. General/formal: e.g. we assume a material corpus of 17 "sentences": ab,ar,,pr,qab,dpb,aca,pca,pcp,qar,daca,dacp,dacqa,dacdp,acqp,acdp. Letters: stand here for words. I 148 Problem: we still have no distinction between "grammatically correct" and "meaningful" (useful). >Meaningful/meaningless, >Acceptability/Lyons. In our example, a and p have certain contexts in common (namely -r,pc-, dac-), b and r (a-,qa) and d and q (dac-a,-aca, ac-p) c: is unique in its distribution (a-a,p-c,p-p,qa-a,da-a,da-p), because no other word can be found in the same context as c. X: we now merge a and p in the class CX and insert this class name everywhere where either a or p occur. Sentences that differ only in that o is where the other sentence has a are thus grouped into a class. ((s) "disjunctive"): Xb,Xr,(ar,pr), qXb,dXb,XcX, (aca, pca, pcp), qXr, qXcX,dXcX (daca,dacp), dXcqX,dXcdX,qXcdX,XcdX,XcqX,XcdX,XcdX. Y: we set Y for b and r, Z: for d and q. Then we get 1. XY, (Xb,Xr) 2. ZXY (qXb, qXr, dXb) 3. XcX, 4. ZXcX (qXcX, dXcX) 5. ZXcZX (dXcqX, dXcdX, qXcdX) 6. XcZX (XcqX, XcdX). N.B.: with this we can capture the 17 sentences of our corpus through six structural formulas. (c is a one-membered class). They specify which consequences of word classes are acceptable. The consequences are linear. (see below). Grammatically correct: are sentences in our example, that result from these structure rules. This is only achieved by the fact that the sentences that occur are regarded as links in a superset of 48 sentences. (The number 48 is obtained by applying the syntagmatic length formulas to each of the six sentence types and adding the results). I 149 Generative/generative grammar/Lyons: the "grammar" in our example is generative in that it assigns a certain structural description to each sentence that appears in the "sample", for example pr is a sentence with the structure XY, pcda is a sentence with the structure XcZY, etc. >Generative Grammar. Grammar/Lyons: as it is understood here, it is nothing else than the description of the sentences of a language as combinations of words and word groups due to their affiliation to distribution classes. It is a kind of "algebra" in which the variables are the word classes and the constants or the values assumed by the variables in certain sentences are the individual words. >Grammar, cf. >Syntax. |
Ly II John Lyons Semantics Cambridge, MA 1977 Lyons I John Lyons Introduction to Theoretical Lingustics, Cambridge/MA 1968 German Edition: Einführung in die moderne Linguistik München 1995 |
| Formal Language | Tarski | Berka I 458 Formal language/Tarski: in a formal language the meaning of each term is uniquely determined by its shape. I 459 Variables: variables have no independent meaning. - Statements remain statements after translation into everyday language. Variable/Tarski: variables represent for us always names of classes of individuals. >Class name. Berka I 461 Formal language/terminology/abbreviations/spelling/Tarski: here: the studied language (object language). Symbols: N, A, I, P: negation, alternation, inclusion, quantifier - metalanguage: Symbols ng (negation), sm (sum = alternation), in (inclusion) - this is the language in which the examination is performed. ng, sm, etc. correspond to the colloquial expressions ((s) the formal symbols N, A, etc. do not). I 464 E.g. object language: Example expression: Nixi, xll: - meta language: translation of this expression: (structural-descriptive name, symbolic expression): name: "((ng ^ in) ^ v1) ^ v2" - but: see below: difference name/translation.(1) >Structural-descriptive name, >Quotation name, >Metalanguage. 1. A.Tarski, Der Wahrheitsbegriff in den formalisierten Sprachen, Commentarii Societatis philosophicae Polonorum. Vol 1, Lemberg 1935 --- Horwich I 112 Formal language/Tarski: in it all assertible sentences are theorems. - There may be a language with exactly specified structure, which is not formalized. - Then the assertibility may depend on extra-linguistic factors.(2) >Assertibility. 2. A. Tarski, The semantic Conceptions of Truth, Philosophy and Phenomenological Research 4, pp. 341-75 |
Tarski I A. Tarski Logic, Semantics, Metamathematics: Papers from 1923-38 Indianapolis 1983 Berka I Karel Berka Lothar Kreiser Logik Texte Berlin 1983 Horwich I P. Horwich (Ed.) Theories of Truth Aldershot 1994 |
| Incomplete Symbols | Russell | I 64f Def incomplete Symbol/Principia Mathematica(1)/Russell: E.g. proposition or sentence. (Contrast: Judgment that is already complete because it brings together perception and sentence about the perception). >Proposition, >Sentence, >Judgment, >Perception. I 65 E.g. the proposition Socrates is a human being requires some kind of supplement. - But if I judge the same wording, the corresponding judgement is complete - although no explicit amendment was made to the proposition. I 64 Incomplete Symbol/Principia Mathematica/Russell: sentence or proposition - complete: Judgment (brings together sentence and perception). >Description, >Name. I 95 Description/name/proper name/Principia Mathematica/Russell: E.g. round square is a description, not a proper name. - ((s) so names are not abbreviated descriptions.) - Description: incomplete. - Name/Russell: complete, complete symbol. I 95 Incomplete symbols: formulas in mathematics: only useful in use - descriptions, remain undetermined. - Symbol for classes. I 95f Complete symbols: proper names: e.g. Socrates. 1. Whitehead, A.N. and Russel, B. (1910). Principia Mathematica. Cambridge: Cambridge University Press. Flor III 117 Def incomplete symbols/Russell/Flor: class names, or expressions that contain class names and thus can be replaced by other symbols. Example "all humans" "some people": - (logical fictions). |
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 Flor I Jan Riis Flor "Gilbert Ryle: Bewusstseinsphilosophie" In Philosophie im 20. Jahrhundert, A. Hügli/P. Lübcke Reinbek 1993 Flor II Jan Riis Flor "Karl Raimund Popper: Kritischer Rationalismus" In Philosophie im 20. Jahrhundert, A.Hügli/P.Lübcke Reinbek 1993 Flor III J.R. Flor "Bertrand Russell: Politisches Engagement und logische Analyse" In Philosophie im 20. Jahrhundert, A. Hügli/P.Lübcke (Hg) Reinbek 1993 Flor IV Jan Riis Flor "Thomas S. Kuhn. Entwicklung durch Revolution" In Philosophie im 20. Jahrhundert, A. Hügli/P. Lübcke Reinbek 1993 |
| Language Acquisition | Quine | V 68 Language Learning/Quine: the success depends on whether the similarity standards match. - What do the episodes have in common? V 71 Approval/learning/Quine: instead of reward: is more general. There are not enough situations for rewards, because not everything is pronounced. - Language learning: is not only done by linguistic statements, but also through non-verbal responses. - Even animals. - Approval: leads to voicing one's own sentences -> Gavagai: if you just wait until the parents say rabbit, you will not find out that everything that is called a rabbit is also referred to as an animal. Radical Interpretation: question and answer game is essential here. - Approval must be obtained. ((s) reward has to be confirmed otherwise). V 75 1) red from the child, yes from the mother - 2) vice versa - generalization: by previously learned expressions - criterion for approval: readiness, to express an observation sentence on one’s own initiative. >Observation Sentences V 77 Good/language learning/QuineVsMoore/QuineVsTradition: two factors: perception similarity and desire. - Distinction between aesthetically good and morally well: the former feels good, the latter announces former. Moral/Quine: as flavor: community thing. V 113 Truth/language learning/Quine: somehow such a connection of meaning and truth is indicative for learning, regardless of the logical particles - we learn the use of declarative sentences by learning the truth conditions - but truth value is learned late. V 121 Compliance/language learning/Quine: in casual conversation - not hidden meanings. - ((s) internal objects). V 147 Set theory/language learning/Quine: set theory/language learning/Quine: by imagining the substitutional quantification as a simulation of the referential quantification, we imagine the general term as a simulation of abstract singular terms, of names of attributes or names of classes. - Class name: is an abstract singular term, not a general term. >Classes/Quine. VI 89f Whole sentence/holophrastic/language learning/Quine: we need whole sentences to define that e.g. a mirrored object is meant - or reflection. - ((s) or mirroring). |
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 |
| Modal Logic | Quine | II 152 Modal Logic/Quine: The entire modal logic is context-dependent - what is the role of someone or something? It is on the same level as essential properties. (Essentialism). VII (h) 151 Modal Logic/ontology/Quine: instead of Venus as a material object we now have three objects: Venus term, morning star term, evening star term - avoiding opaque contexts: class names as objects rather than classes, numerical names as objects instead of numbers - number concept/number of planets concept: a term is not larger/smaller than another one - reason: necessity is not satisfied by physical objects (> Necessity/Hume). - Necessity/possibility: is only introduced by way of reference, not by the objects - necessity concerns relations, not objects (not existence) - Frege: "sense (meaning) of names" Quine: Problem: individuation requires analyticity and synonymy - E.g. (s) "The term Morning Star necessarily includes the appearance on the morning sky. VII (h) 151f Modal Logic/Quine: makes essentialism necessary, i.e. you cannot do without necessary traits of the objects themselves, because you cannot do without quantification - QuineVsModal Logic: actually there is nothing necessary to the objects "themselves", but only in the way of reference. VII (h) 151 Modal Logic/Ontology/Quine: the condition that two names for x must be synonymous is not a condition for objects, but for singular terms - no necessity de re - Venus does not decide about morning star/evening star. - ((s) The conditions are equivalent not the objects. > necessity. VII (h) 154 Modal Logic/Church/Quine: quantified variables should be limited to intensional values - Proposition: complex names of intensional objects - then instead of necessity operator for whole sentences: Necessity predicate is based on complex names ("propositions") - no modal logic in the narrower sense. >Propositions/Quine. VII (h) 154 Modal Logic/Smullyan/Quine: there is a strict separation of proper names and (overt or covert) descriptions - names which denote the same objects are always synonymous (if x = y, then nec. x = y.) - In this case, sentences like (number of the planets = 9) which do not have a substitutable identity must be analyzed by descriptions rather than through proper names (Quine pro). - QuineVs: one must still consider opaque contexts, even if descriptions and other singular terms are eliminated all together. >Proper Names/Quine. VII (h) 154 Modal Logic/Necessity/Planet Example/Quine: the only hope is to accept the situation as described in (33): there are exactly x planets) and still insist that the object x in question is necessarily more than 7! (> Essentialism). - An object itself, regardless by what it is named or not named, must be considered in a way that it has some traits necessarily and others by chance! And notwithstanding the fact that the random traits stem from a way of reference, as well as the necessary ones from other modes of reference - ~nec. [p. (x = x)] where "p" stands for any random truth. VII (h) 156 Modal Logic/Quine: one must accept an Aristotelian essentialism, if one wants to permit quantified modal logic. VII (h) 156 Modal Logic/planet/Quine: the property of being bigger than 9 = the property of being bigger than 9 - but wrong: the property of exceeding the number of planets = the property of being bigger than 9 (s) New: although now the number is the same, the property is not the same - (E.g.) (x = The property of being greater than x = the property to be greater than 9) - any non-truth-functional language leads to opaque contexts. X 107 Modality/modal/Quine: Problem: extension-identical (coextensive) predicates are not interchangeable salva veritate. >Modalities/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 |
| Object | Russell | I 106 Object/combinations/Principia Mathematica(1)/PM/Russell: combinations are not an object, also not a whole as "12 apostles" as a property that belongs to every apostle. Solution: Difference intensional/extensional functions. >Intension, >Extension. 1. Whitehead, A.N. and Russel, B. (1910). Principia Mathematica. Cambridge: Cambridge University Press. Flor III 125 Things/objects/Russell: two theories: A. defined as logical constructions - e.g. series of classes of sensory data. B. known only from descriptions. Flor III 126 Object/Person/Russell: People are also physical objects! Problem: therefore they need to be understood by Russell as hidden descriptions or class names - a subject name must be an incomplete symbol - by no means a name. >Person, >Name, >Description, >Incomplete symbol. |
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 Flor I Jan Riis Flor "Gilbert Ryle: Bewusstseinsphilosophie" In Philosophie im 20. Jahrhundert, A. Hügli/P. Lübcke Reinbek 1993 Flor II Jan Riis Flor "Karl Raimund Popper: Kritischer Rationalismus" In Philosophie im 20. Jahrhundert, A.Hügli/P.Lübcke Reinbek 1993 Flor III J.R. Flor "Bertrand Russell: Politisches Engagement und logische Analyse" In Philosophie im 20. Jahrhundert, A. Hügli/P.Lübcke (Hg) Reinbek 1993 Flor IV Jan Riis Flor "Thomas S. Kuhn. Entwicklung durch Revolution" In Philosophie im 20. Jahrhundert, A. Hügli/P. Lübcke Reinbek 1993 |
| Objects (Material Things) | Russell | Geach I 314 Definition Object/Definition Person/Russell: (logical atomism): an object is a set of classes of particulars, and therefore a logical fiction. "Real things (= sense data) only last a very short time". GeachVsRussell: he tried to apply two theories of classes at once: 1. the "no-classes theory" that classes are only fictions 2. the "composition theory": that classes are composed of their elements. Flor III 125 Things/objects/Russell: two theories: A. defined as logical constructions - e.g. series of classes of sensory data. B. known only from descriptions. Flor III 126 Object/Person/Russell: People are also physical objects! Problem: therefore they need to be understood by Russell as hidden descriptions or class names - a subject name must be an incomplete symbol - by no means a name. >Person, >Name, >Description, >Incomplete symbol. |
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 Flor I Jan Riis Flor "Gilbert Ryle: Bewusstseinsphilosophie" In Philosophie im 20. Jahrhundert, A. Hügli/P. Lübcke Reinbek 1993 Flor II Jan Riis Flor "Karl Raimund Popper: Kritischer Rationalismus" In Philosophie im 20. Jahrhundert, A.Hügli/P.Lübcke Reinbek 1993 Flor III J.R. Flor "Bertrand Russell: Politisches Engagement und logische Analyse" In Philosophie im 20. Jahrhundert, A. Hügli/P.Lübcke (Hg) Reinbek 1993 Flor IV Jan Riis Flor "Thomas S. Kuhn. Entwicklung durch Revolution" In Philosophie im 20. Jahrhundert, A. Hügli/P. Lübcke Reinbek 1993 |
| Relative Clauses | Quine | II 203 Relative clauses: (x:Fx) - but not: (Ey)(y=(x:Fx)) - that would testify existence of a class - here, mathematics goes beyond logic. >Existence statements. II 199 Class Name: is a singular term, indicating a class - placeholder class name: real. bindable variable whose values are classes - a relative clause is no class name. >Classes, >Proper names, >Proxy, >Singular terms. --- V 129 Relative clause/Quine: function: separating the object from what the sentence says of it. - A relative clause becomes a general term, if the relative pronoun is put in front (which stands for the name of the object): E.g. which I bought from the man who had found it. - The general term says the same as the original sentence. >General terms. GeachVs: instead understanding relative pronouns as "and he" or "if he" or "since he". Geach’s donkey: Whoever has a donkey, beats it: Solution/Geach: analysis of the relative pronoun who with "if he": every human being, if he has a donkey, be beats it. >Donkey sentences. V 133 Relative clause: can make a predication of the form a is P from every sentence on an object - E.g. Fido is such that I bought him from a man who had found him. Relative clause: has adjectival function here - substantivic: with thing, E.g. Fido is a thing such that ...-’ original form: useful when a relative clause functions as a general term. - E.g. in the universal categorical sentence (Construction) [an a is a b], [each a is a b]. >Predication, >Adjectives. Universal categorical sentence: no predication but a coupling of two general terms. |
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 |
| Set Theory | Prior | I 165 Nouns/Prior: nouns are no names, no class names. >Names, >Singular Terms, >General Terms, >Terms, >Words, >Word Meaning, >Reference, >Predication, >Naming, >Designation, >Denotation, >Comprehension. Epsilon/Principia Mathematica(1)/Russell: "x ε a": Translation: "A is an element of the class of humans" seems to be a relation between a concrete and an abstract object. >Element relation. Vs: better "x is a": "Russell is a man" - Prior: "is a" is no real verb that makes a sentence of a name, rather a sentence of name and noun - "ε" is not a real predicate. >Predicates, >Predication, >Is, >Type theory. 1. Whitehead, A.N. and Russel, B. (1910). Principia Mathematica. Cambridge: Cambridge University Press. |
Pri I A. Prior Objects of thought Oxford 1971 Pri II Arthur N. Prior Papers on Time and Tense 2nd Edition Oxford 2003 |
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| Disputed term/author/ism | Author Vs Author |
Entry |
Reference |
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| Attributes | Quine Vs Attributes | III 262 General Terms/Quine: are represented by predicate letters such as "F", "G", etc. (§ 12.22, where they were simply called "Termini"). Universality/Quine: is not ambiguity! Ex ambiguous: is the singular term "Miller"! It can refer to different people in different contexts. Similarly: Singular Term: "the basement", "the President" ((s) >indefinite description). General Term: "basement", "President". Concrete Term: "Cerberus", "Unicorn" Abstract Term: "7", "3 + 4", "piety" terms for numbers, classes, attributes. Concrete General Term: "man", "red house", "house". Abstract General Term/Quine: "prime", "zoological genus", "virtue", because every virtue and every number and every species is an abstract object. ((s) then "piety" is an abstract singular term). Attribute/Quine: I do not care much for them as entities that are supposed to be different from classes. III 263 Attributes: can be considered different, even if they apply to the same things. E.g. "having a heart", "having kindneys". Classes/QuineVsAttributes: classes are easier to identify and to distinguish. If we must distinguish, then: Attribute/Quine: e.g. "human nature": Name of an attribute. Class Name/Name of a Class/Quine: "humanity". |
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 |
| Frege, G. | Quine Vs Frege, G. | Quine I 425 VsFrege: tendency to object orientation. Tendency to align sentences to names and then take the objects to name them. I 209 Identity/Aristotle/Quine. Aristotle, on the contrary, had things right: "Whatever is predicated by one should always be predicated by the other" QuineVsFrege: Frege also wrong in "Über Sinn und Bedeutung". QuineVsKorzybski: repeated doubling: Korzybski "1 = 1" must be wrong, because the left and right side of the equation spatially different! (Confusion of character and object) "a = b": To say a = b is not the same, because the first letter of the alphabet cannot be the second: confusion between the sign and the object. Equation/Quine: most mathematicians would like to consider equations as if they correlated numbers that are somehow the same, but different. Whitehead once defended this view: 2 + 3 and 3 + 2 are not identical, the different sequence leads to different thought processes (QuineVs). I 264 according to Russell "Propositional Attitudes": believes, says, strives to, that, argues, is surprised, feares, wishes, etc. ... I 265 Propositional attitudes create opaque contexts into which quantification is not allowed. (>) It is not permissible to replace a singular term by an equally descriptive term, without stretching the truth value here. Nor a general term by an equally comprehensive one. Also cross-references out of opaque contexts are prohibited. I 266 Frege: in a structure with a propositional attitude a sentence or term may not denote truth values, a class nor an individual, but it works as "name of a thought" or name of a property or as an "individual term". QuineVsFrege: I will not take any of these steps. I do not forbid the disruption of substitutability, but only see it as an indication of a non-designating function. II 201 Frege emphasized the "unsaturated" nature of the predicates and functions: they must be supplemented with arguments. (Objections to premature objectification of classes or properties). QuineVsFrege: Frege did not realize that general terms can schematized without reifying classes or properties. At that time, the distinction between schematic letters and quantifiable variables was still unclear. II 202 "So that" is ontologically harmless. Despite the sad story of the confusion of the general terms and class names, I propose to take the notation of the harmless relative clause from set theory and to write: "{x:Fx} and "ε" for the harmless copula "is a" (containment). (i.e.the inversion of "so that"). Then we simply deny that we are using it to refer to classes! We slim down properties, they become classes due to the well-known advantages of extensionality. The quantification over classes began with a confusion of the general with the singular. II 203 It was later realized that not every general term could be allocated its own class, because of the paradoxes. The relative clauses (written as term abstracts "{x: Fx}") or so-that sentences could continue to act in the property of general terms without restrictions, but some of them could not be allowed to exercise a dual function as a class name, while others could. What is crucial is which set theory is to be used. When specifying a quantified expression a variable may not be replaced by an abstraction such as: "x} Fx". Such a move would require a premise of the form (1), and that would be a higher form of logic, namely set theory: (1) (Ey)(y = {x:Fx}) This premise tells us that there is such a class. And at this point, mathematics goes beyond logic! III 98 Term/Terminology/Quine: "Terms", here as a general absolute terms, in part III single-digit predicates. III 99 Terms are never sentences. Term: is new in part II, because only here we are beginning to disassemble sentences. Applying: Terms apply. Centaur/Unicorn/Quine: "Centaur" applies to any centaur and to nothing else, i.e. it applies to nothing, since there are no centaurs. III 100 Applying/Quine: Problem: "evil" does not apply to the quality of malice, nor to the class of evil people, but only to each individual evil person. Term/Extension/Quine: Terms have extensions, but a term is not the denotation of its extension. QuineVsFrege: one sentence is not the denotation of its truth value. ((s) Frege: "means" - not "denotes"). Quine: advantage. then we do not need to assume any abstract classes. VII (f) 108 Variables/Quine: "F", etc.: not bindable! They are only pseudo-predicates, vacancies in the sentence diagram. "p", "q", etc.: represent whole statements, they are sometimes regarded as if they needed entities whose names these statements are. Proposition: these entities are sometimes called propositions. These are rather hypothetical abstract entities. VII (f) 109 Frege: alternatively: his statements always denote one or the other of exactly two entities: "the true one" or "the false one". The truth values. (Frege: statements: name of truth values) Quine pro Frege: better suited to distinguish the indistinguishable. (see above: maxim, truth values indistinguishable in the propositional calculus (see above VII (d) 71). Propositions/Quine: if they are necessary, they should rather be viewed as names for statements. Everyday Language/Quine: it is best if we return to everyday language: Names are one kind of expression and statements are another! QuineVsFrege: sentences (statements) must not be regarded as names and "p", "q" is not as variables that assume entities as values that are entities denoted by statements. Reason: "p", "q", etc. are not bound variables! Ex "[(p>q). ~p]> ~p" is not a sentence, but a scheme. "p", "q", etc.: no variables in the sense that they could be replaced by values! (VII (f) 111) VII (f) 115 Name/QuineVsFrege: there is no reason to treat statements as names of truth values, or even as names. IX 216 Induction/Fregean Numbers: these are, other than those of Zermelo and of von Neumann, immune against the trouble with the induction (at least in the TT), and we have to work with them anyway in NF. New Foundations/NF: But NF is essentially abolishing the TT! Problem: the abolition of TT invites some unstratified formulas. Thus, the trouble with induction can occur again. NFVsFrege: is, on the other hand, freed from the trouble with the finite nature which the Fregean arithmetic touched in the TT. There, a UA was needed to ensure the uniqueness of the subtraction. Subtraction/NF: here there is no problem of ambiguity, because NF has infinite classes - especially θ - without ad-hoc demands. Ad 173 Note 18: Sentences/QuineVsFrege/Lauener: do not denote! Therefore, they can form no names (by quotation marks). XI 55 QuineVsFrege/Existence Generalisation/Modal/Necessary/Lauener: Solution/FregeVsQuine: this is a fallacy, because in odd contexts a displacement between meaning and sense takes place. Here names do not refer to their object, but to their normal sense. The substitution principle remains valid, if we use a synonymous phrase for ")". QuineVsFrege: 1) We do not know when names are synonymous. (Synonymy). 2) in formulas like e.g. "(9>7) and N(9>7)" "9" is both within and outside the modal operaotor. So that by existential generalization (Ex)((9>7) and N(9>7)) comes out and that's incomprehensible. Because the variable x cannot stand for the same thing in the matrix both times. |
W.V.O. Quine I Quine Wort und Gegenstand Stuttgart 1980, Reclam II Quine Theorien und Dinge Frankfurt/M 1985, Suhrkamp III Quine Grundzüge der Logik Frankfurt/M 1978 IV Oliver R. Scholz "Quine" aus Hügli (Hrsg) Philosophie im 20. Jahrh., Reinbek 1993 V Quine Wurzeln der Referenz Frankfurt 1989 VI Quine Unterwegs zur Wahrheit Paderborn (Schöningh) 1995 VII Quine Form al logical point of view Cambrinde 1953 IX Quine Mengenlehre und ihre Logik Vieweg 1967 X Quine Philosophie der Logik Bamberg 2005 XI Henri Lauener Quine München 1982 XII Quine Ontologische Relativität Sprechen über Gegenstände, Naturalisierte Erkenntnistheorie |
| Lesniewski, St. | Prior Vs Lesniewski, St. | I 43 Abstracts/Prior: Ontological Commitment/Quine: quantification of non-nominal variables nominalises them and thus forces us to believe in the corresponding abstract objects. Here is a more technical argument which seems to point into Quine's direction at first: Properties/Abstraction Operator/Lambda Notation/Church/Prior: logicians who believe in the real existence of properties sometimes introduce names for them. Abstraction Operator: should form names from corresponding predicates. Or from open sentences. Lambda: λ followed by a variable, followed by the open sentence in question. E.g. if φx is read as "x is red", I 44 then the property of redness is: λxφx. E.g. if Aφxψx: "x is red or x is green" (A: Here adjunction) "Property of being red or green": λx∀φxψx. To say that such a property characterizes an object, we just put the name of the property in front of the name of the object. Lambda Calculus/Prior: usually has a rule that says that an object y has the property of φ-ness iff. y φt. I.e. we can equate: (λy∀φxψx)y = ∀φyψy. ((s) y/x: because "for y applies: something (x) is...") One might think that someone who does not believe in the real existence of properties does not need such a notation. But perhaps we do need it if we want to be free for all types of quantification. E.g. all-quantification of higher order: a) C∏φCφy∑φyCAψyXy∑xAψxXx, i.e. If (1) for all φ, if y φt, then φt is something then (2) if y is either ψt or Xt, then something results in either ψ or X. That's alright. Problem: if we want to formulate the more general principle of which a) is a special case: first: b) C∏φΘφΘ() Where we want to insert in the brackets that which symbolizes the alternation of a pair of verbs "ψ" and "X". AψX does not work, because A must not be followed by two verbs, but only by two sentences. We could introduce a new symbol A', which allows: (A’ φψ)x = Aψxψx this turns the whole thing into: c) C∏φΘφΘA’ψX From this we obtain by instantiation: of Θ d) C∏φCφy∑xφxCA’ψXy∑xA’ψXx. And this, Lesniewski's definition of "A", results in a). This is also Lesniewski's solution to the problem. I 45 PriorVsLesniewski: nevertheless, this is somewhat ad hoc. Lambda Notation: gives us a procedure that can be generalized: For c) gives us e) C∏φΘφΘ(λzAψzXz) which can be instatiated to: f) C∏φCφy∑xφx(λzAψzXz)y∑x(λzAψzXz)y. From this, λ-conversion takes us back to a). Point: λ-conversion does not take us back from e) to a), because in e) the λ-abstraction is not bound to an individual variable. So of some contexts, "abstractions" cannot be eliminated. I 161 Principia Mathematica(1)/PM/Russell/Prior: Theorem 24.52: the universe is not empty The universal class is not empty, the all-class is not empty. Russell himself found this problematic. LesniewskiVsRussell: (Introduction to Principia Mathematica): violation of logical purity: that the universal class is believed to be not empty. Ontology/Model Theory/LesniewskiVsRussell: for him, ontology is compatible with an empty universe. PriorVsLesniewski: his explanation for this is mysterious: Lesniewski: types at the lowest level stand for name (as in Russell). But for him not only for singular names, but equally for general names and empty names! Existence/LesniewskiVsRussell: is then something that can be significantly predicted with an ontological "name" as the subject. E.g. "a exists" is then always a well-formed expression (Russell: pointless!), albeit not always true. Epsilon/LesniewskiVsRussell: does not only connect types of different levels for him, but also the same level! (Same logical types) E.g. "a ε a" is well-formed in Lesniewski, but not in Russell. I 162 Set Theory/Classes/Lesniewski/Prior: what are we to make of it? I suggest that we conceive this ontology generally as Russell's set theory that simply has no variables for the lowest logical types. Names: so-called "names" of ontology are then not individual names like in Russell, but class names. This solves the first of our two problems: while it is pointless to split individual names, it is not so with class names. So we split them into those that are applied to exactly one individual, to several, or to none at all. Ontology/Lesniewski/Russell/Prior: the fact that there should be no empty class still requires an explanation. Names/Lesniewski/Prior: Lesniewski's names may therefore be logically complex! I.e. we can, for example, use to form their logical sum or their logical product! And we can construct a name that is logically empty. E.g. the composite name "a and not-a". Variables/Russell: for him, on the other hand, individual variables are logically structureless. Set Theory/Lesniewski/Prior: the development of Russell's set theory but without variables at the lowest level (individuals) causes problems, because these are not simply dispensable for Russell. On the contrary; for Russell, classes are constructed of individuals. Thus he has, as it were, a primary (for individuals, functors) and a secondary language (for higher-order functors, etc.) Basic sentences are something like "x ε a". I 163 Def Logical Product/Russell: e.g. of the αs and βs: the class of xs is such that x is an α, and x is a β. 1. Whitehead, A.N. and Russel, B. (1910). Principia Mathematica. Cambridge: Cambridge University Press. |
Pri I A. Prior Objects of thought Oxford 1971 Pri II Arthur N. Prior Papers on Time and Tense 2nd Edition Oxford 2003 |
| 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 |
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