Find counter arguments by entering NameVs… or …VsName.
The author or concept searched is found in the following 13 entries.
| Disputed term/author/ism | Author |
Entry |
Reference |
|---|---|---|---|
| Continuum | Quine | XIII 45 Def Discrete/Discreteness/Quine: an order of numbers or other objects is discrete if each object has an immediate predecessor or successor or both. For example, the integers are discrete. One the other hand: Def dense: fractions are dense and not discrete. Real numbers: are more than dense: they are continuous. Discrete/continuous: here we compare them as opposites. Discreteness: we need it to learn to count by distinguishing the objects. Irrational Numbers/Cantor: Theorem: most will always escape us. Number Theory: deals with integers. Real numbers: are used by the sciences. The contrast of both is illuminated by a pair of theorems: Goedel's Theorem: no proof procedure can encompass all truths of elementary number theory. Tarski's theorem: truth in the parallel theory of real numbers can be checked routinely, e.g. by a computer. N.B.: both systems are identical in their notation! The difference lies in the different interpretation of the variables (or their domains, one time the positive integers with the 0, the other time the positive real numbers with the 0). XIII 47 This leads to a difference in the truth of the formulas! a) real numbers: here the true formulas are a set that can be handled b) elementary number theory: not here. Continuity/Discreteness/Language/Quine: the interaction of the two terms is not limited to mathematics, it also exists in language: phonemes impose discreteness on the phonetic continuum. Discreteness/Quine: also allows yellowed or damaged manuscripts to be returned to a fresh state. The discreteness of the alphabet helps that the small deviations (e.g. yellowing) add up to larger ones. Continuum/Continuity: images, on the other hand, are a continuous medium: i.e. there are no standards for repairing a damaged copy or correcting a poor copy. Technology: here discreteness is often combined with continuity. Example clock: it should give the impression of moving continuously. XIII 48 Film/Quine: Continuity here is due to the weakness of our perception. Similar to the clock or our thinking about atoms over the millennia. Planck time/Quine: here we have the next approach of nature to continuity. |
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 |
| Disjunction | Logic Texts | Read III 79 Disjunction / tautology / Read: In a sense, "A or B" follows from A alone - but then is not equivalent to "if ~ A, then B". Logical Constants. Undecidability: Re III 262 Not constructive: e.g. the proof that there are two irrational numbers a and b, so that a is highly b rational (the disjunction of alternatives is constructively unacceptable here). We have no construction by which we can determine whether root 2 to the power of root 2 is rational or not). The excluded third party is therefore intuitionistic and not a substantial assertion. >Undecidability, >Intuitionism. Goldbach's conjecture: every even number greater than two should be the sum of two prime numbers. Not decidable. But we must not claim that it is either true or not. Theorem of the Excluded Middle/Constructivism/Read: Constructivists often present so-called "weak counterexamples" against the Excluded Third. If a is a real number, "a= 0" is not decidable. Consequently, the constructivist cannot claim that all real numbers are either identical with zero or not. (But this is more a question of representation). >Excluded middle, >Goldbach's conjecture. |
Logic Texts Me I Albert Menne Folgerichtig Denken Darmstadt 1988 HH II Hoyningen-Huene Formale Logik, Stuttgart 1998 Re III Stephen Read Philosophie der Logik Hamburg 1997 Sal IV Wesley C. Salmon Logic, Englewood Cliffs, New Jersey 1973 - German: Logik Stuttgart 1983 Sai V R.M.Sainsbury Paradoxes, Cambridge/New York/Melbourne 1995 - German: Paradoxien Stuttgart 2001 Re III St. Read Thinking About Logic: An Introduction to the Philosophy of Logic. 1995 Oxford University Press German Edition: Philosophie der Logik Hamburg 1997 |
| Formalism | d’Abro | A. d'Abro Die Kontroversen über das Wesen der Mathematik 1939 in Kursbuch 8 Mathematik 1967 33 Formalism: the formalist sees arithmetic and logic as complementary. A certain agreement between the two doctrines results from the impossibility of defining the number and, in particular, the whole number (VsFrege). The formalists, however, assert an indirect possibility on the basis of axioms. >Formalism/Frege, cf. >Formalism/Heyting. 50 Intuitionism/formalism/d'Abro: The intuitionist is a rigorist, insofar as he considers definitions and proofs accepted by the formalist to be inadequate. It should be admitted that they are not given by logic, but by intuition. E.g. Zermelo's (formalist) proof that the continuum is an ordered set. I.e., the points can be placed one after the other, with a successor for each point. >Intuitionism. PoincaréVsZermelo: he invented a typical argument: the pragmatist rejected Zermelo's proof because it would take too much time to carry it out, and the number of operations to be performed would be even greater than Aleph0, not to be expressed with a finite number of words. The pragmatist will conclude that the theorem is pointless. Camps: Formalists: Cantor, Hilbert, Zermelo, Russell - Intuitionists: Poincaré, Weyl >G. Cantor, >D. Hilbert, >E. Zermelo, >B. Russell, >H. Poincaré. 53 According to Weyl, the concept of the irrational number must either be abandoned, or thoroughly modified. >Irrational numbers. Brouwer: when dealing with infinite quantities, the law of the excluded middle does not apply. >Excluded MIddle. The intuitionists assert with Poincaré that antinomies without any infinity are lopish. Poincaré: The antinomies of certain logicians are simply circular. >Paradoxes, >Circularity. 54 Formalism/d'Abro: E.g. d'Abro sees no obstacle to define x in the following way: (a) x has this and this relation to all members of type G. 55 (b) x is a term of G. For an intuitionist, according to Poincaré, such a definition is circular. For example, controversy about definitions that cannot be expressed in a finite number of words. It is refused by the intuitionists. >Definitions, >Definability. 1 + 1/2 + 1/4 + 1/8... This series, according to the intuitionists, is capable of being expressed in a finite number of words, since a rule can be formulated. It should be noted that the difference is theoretical and not practically important, a proof that e.g. could be formulated in a trillion words would be acceptable. |
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| Generalization | Thiel | I 180 Generality/Generalization/Infinity/Mathematics/Statements/Thiel: Who would have ever doubted that the Pythagorean theorem can be applied to an infinite number of cases? I 181 Problem: that in the formulation of the theorem irrational numbers are allowed as measures for the cathets of the right-angled triangles, for which we do not know any counting as for the rational numbers. If there were one, we could count the totality of the real numbers by combining them with a count of the rational numbers. Cantor's merit was to show the impossibility of this by his diagonal method. I 181 Table with columns and columns cut by diagonals. I 182 Def Dual Sequence/(s): Sequence of (binary) decisions as to whether a point is on the left or right half of a halved course. This leads to any rational number. I 184 But it leads to a contradiction. Then 1 bii = bii . the assumption that the dual sequence constructed as "negative" of its diagonals already occurs in the (arbitrary) list considered leads to an absurdity. After that, however, even the totality of all real numbers in the interval 0.1 cannot be recorded in a list (as Cantor also shows, not in an infinite list), it cannot be counted. I 185 So also not outside the interval. I 186 Continuum/Russell: (e.g.) sees an arithmetic term in the continuum, others a geometric one. I 189 New: Modern Mathematics I 189 (Topology) has defined the concept of the "border" of a set of points corresponding to the Aristotelian "border" in such a way that a point can be its own border. |
T I Chr. Thiel Philosophie und Mathematik Darmstadt 1995 |
| Geometry | Bigelow | I 360 Golden Section/Bigelow/Pargetter: this relation is all too real. Nevertheless, it is not a ratio in our sense. >Ontology, >Existence, >Relations. For example, if we create lines by stringing DF together we will never get a matching point with multiples of DC. Logical form/General/Incommensurability/Bigelow/Pargetter: n times DF will never be = m times DC. This also applies to the solution of Wiener (see above). Proportion: here is 2: (1 + √5), therefore it cannot be represented as ratio a:b for integers a and b. Incommensurability/proof: can be proven by raa: assuming DF and DC would be commensurable, i.e. there is a distance d that divides both DF and DC. Let's take a look at the rectangle (in the graphic above) FDC, d divides DF and DF equals EC. This divides both DC and EC. Therefore, it must also divide DE. Then the same size must divide both the larger and the smaller rectangle, which is not possible. d would then also have to divide the sides of the third rectangle in the drawing etc. ad infinitum. Therefore, no finite length can divide both sides of a golden rectangle. I 360 VsBigelow: incommensurability seems to be against our theory. >Irrational numbers. BigelowVsVs: Solution: we redefine "ratio" a little bit: we need a third relation: Definition Incommensurability/logical form/Bigelow/Pargetter: if two relations R and S are income-survable, then whenever x Rn y, follows that not: x Sm y, for whichever values of n and m are used. Repetition of n applications of R will never result in a match with m applications of S. N.B.: nevertheless, we can determine that the results of repeated applications of R and S are in a certain relation to each other. They are arranged in a linear order "<" ("smaller"). I.e. it can be, for an n and an m If x Rn y and x Sm z, then y < z. Golden Section/Bigelow/Pargetter: is clearly defined by the list of numbers n and m for which the above scheme applies. I 362 General: each proportion between two relations R and S can be unambiguously characterized by a list of natural numbers n and m for which the scheme applies. >Proportions. Proportion/Bigelow/Pargetter: this theory of proportions is based on Eudoxo's contribution to Euclid's Elements (Book 5 Def 5). Real Numbers/Bigelow/Pargetter: this theory of proportions as a theory of real numbers was developed by Dedekind and others at the end of the 19th century. >Real numbers. I 364 Geometry/Bigelow/Pargetter: geometry has to do with spatially instantiated universals. Therefore, it is vulnerable through empirical discoveries about space. It could be that we discover that space does not instantiate the geometric shapes that we had previously assumed to be instantiated like this. >Discoveries, >Space. Aristotle/Bigelow/Pargetter: according to him the forms would then be discarded. Plato/Bigelow/Pargetter: he allows first the acceptance of a non-Euclidean space. ((s) But if it is not directly perceptible to us and if it is instantiated in the universe, for example, it is not a problem for Aristotle either.) I 365 Universals/Platonism/Bigelow/Pargetter: actually he doesn't believe in uninstantiated universals either, but he will find them or invent them. Above all, he will say that pure mathematics is autonomous. >Ideas/Plato. |
Big I J. Bigelow, R. Pargetter Science and Necessity Cambridge 1990 |
| Imaginary Numbers | Quine | XIII 30 Imaginary Numbers/Quine: are actually of the same type as real numbers, they were only introduced later. They were only used to be able to draw roots from negative numbers. Equation: always has n solutions if the highest exponent is n. Real numbers: are the positive numbers and the 0. XIII 30 Negative real numbers/Quine: in order to get them in the first place, we first need a new kind of proportions (ratios) together with irrational numbers. Solution: we use excellent real numbers (positive and negative) to distinguish them from (positive) real numbers. Notation: excellent (signed, designated) real numbers: are notated as ordered pairs (gP) '0,x' and 'x, 0'. Ordered pairs/gP/Order/Quine: an artificial way to construct an ordered pair is for example {{x,y},x}... Here x is element of both elements. ((s) Thus, the order is determined). Then we can easily get y out as well. Imaginary unit: notation i: = √-1. Def imaginary number: is any product yi, where y is a signed real number. Def complex number: is any sum x + yi, where x and y are signed real numbers (called positive or negative signed). Because of the "indigestibility" of i, the sum is not commutative. I.e. the sum cannot be broken up differently. Example 5 = 3 + 2 = 4 + 1. This is the reason why complex numbers are often used to represent points of a plane. XIII 31 Complex Number/Tradition: previously (in the 19th century) they were assumed to be ordered pairs of two designated real numbers. Proportions/Ratio/Rational Numbers/Quine: have two senses. Positive Integers: have three senses. Complex numbers: the same thing happens here. Example a) √2, as originally constructed, b) the positive real number + √2, c) the complex number √2, thus √2 + 0i, thus <√2,0>. Real number: can always be represented as a complex number with the imaginary part = 0. N.B.: now the rational numbers have four senses and the positive integers have five senses! But that does not matter in practice. Also not as philosophical constructions. In "set theory and its logic" I have almost completely eliminated these doublings. Complex numbers with the imaginary part 0 become marked real numbers and these become unmarked normal real numbers etc. Numbers/Quine: (set theory and its logic): at the end all these numbers (complex, imaginary, real, rational) become natural numbers. Only the latter are doubled, only once, from the natural number n to the rational number 1/n. >Numbers/Quine, >Number Theory/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 |
| Intuitionism | d’Abro | A. d'Abro Die Kontroversen über das Wesen der Mathematik 1939 in Kursbuch 8 Mathematik 1967 50 Intuitionism/formalism/d'Abro: The intuitionist is a rigorist, insofar as he considers definitions and proofs accepted by the formalist to be inadequate. It should be admitted that they are not given by logic, but by intuition. >Formalism. E.g. Zermelo's (formalist) proof that the continuum is an ordered set. I.e., that the points can be placed one after the other, with a successor for each point. PoincaréVsZermelo: he invented a typical argument: the pragmatist rejected Zermelo's proof because it would take too much time to carry it out, and the number of operations to be performed would be even greater than Aleph zero, it cannot be expressed with a finite number of words. The pragmatist will conclude that the theorem is pointless. Camps: Formalists: Cantor, Hilbert, Zermelo, Russell - Intuitionists: Poincaré, Weyl >G. Cantor, >D. Hilbert, >E. Zermelo, >B. Russell, >H. Poincaré. 53 According to Weyl, the concept of the irrational number must either be abandoned, or thoroughly modified. >Irrational numbers. Brouwer: when dealing with infinite quantities, the law of the excluded middle does not apply. >Law of the excluded middle. The intuitionists assert with Poincaré that antinomies without any infinity are lopish. Poincaré: The antinomies of certain logicians are simply circular. >Circularity, >Paradoxes, >Infinity. |
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| Meaning | Flusser | I 134 Meaning/Flusser: the apparent meaning of texts is based on "grammatical errors". The fact that the books that mediate begin to form walls is a sign that the information contained in the texts becomes unimaginable. It is a mistake to try to get an idea when reading texts. >Text, >Writing, >Understanding, >Imagination, >Signs, >Grammar, >Information. I 135 Meaning/Fiction/Flusser: no argument can hide the fact that unimaginable texts mean nothing. >Meaning, >Conceivability, >Sense, >Argumentation. VsFlusser: there are many objections to that. For example, that "root 2" is unimaginable, but it is an enrichment of meaning. >Numbers, >Irrational numbers. FlusserVsVs: Misunderstanding: The image that one learns to make oneself of concepts is not the meaning that the concept means, but rather it gives it its meaning. >Concepts, >Images. I 165 Meaning/Image/Flusser: Photography from the telescope is an image that means the terms "star" and "astronomer", by making them conceivable. >Photography. It would therefore be wrong to believe that in an astronomy book the texts describe the images, on the contrary: the texts arise from the photos. I. e.: in such astronomical texts, technical images are used correctly. >Terminology/Flusser, >Techno-images. The problem of the telescope has not yet been solved, however: the telescope was made for observation purposes, i. e. it is based on the belief that stars exist independently of it, and at the same time it doubts this belief when used. >Observation, >Reality, >World/Thinking, >Theory, >Measuring. |
Fl I V. Flusser Kommunikologie Mannheim 1996 |
| Object | Quine | I 102 Goodman: "Rabbitness": is a discontinuous space-time segment, which consists of rabbits. --- I 372f Objects of propositional attitude eliminated: "Thomas believes (Cicero has): no longer the form" Fab" a = Thomas, b = () - but: "Fa" where "F" is a complex expression - no longer "believes" term, but operator. I 402 Existence: does not arise from dichotomy "single thing" - "universal" - it does not matter whether they do exist. "Equator", "North Pole" - linking with stimuli is weak argument for primacy of physical objects, but makes terms accessible for all positions. >Existence/Quine. I 412 Object: name which is denoted by singular terms, accepts it as values - (but the singular term is eliminated!) - E.g. "glimmer", but not "glimmeriness". I 438 Ideal objects are not permitted - geometric objects are permitted (no identity without localization). I 435 Relativity: additional dimension: space-time: point moments are absolutely different, independent of relative movement of the viewpoint. II 30 Object/Quine: space-time piece can also be distributed or scattered. (Nominalism, Goodman). II 23 Physical object is deceptive - better space-time pieces - "space" and "places as such" untenable, otherwise there would be absolute standstill and absolute movement - 4-digit coordinates suffice - ontology of pure set theory - no more physical object. II 156 ff Object (physical)/Quine: arbitrarily scattered and arbitrarily singled out - pocket contents, single coin at various points in time, combination with the Eiffel Tower, space-time points, anything - are not so strongly body-oriented - identification like from one possible world to another: without content as long as no instructions are given - value of a variable. VI 32 Object/Ontology/Quine: bodies constitute themselves as ideal nodes in the centers of overlapping observation sentences - problem: observation sentences are not permanent - therefore the objectification (reification) is always already a theory. VI 34 Question: what should be considered real objectification and not just a theoretically useful one (like classes). VI 35 Abstract objects: it is pointless to speak of permanent stimulus phases - solution: pronouns and bound variables - Vs singular term: are often not referring - there must be unspecifiable irrational numbers - Solution: bound variable instead of singular term. VI 38f Objectification/Reification/Quine: for the first time in predicative connection of observation sentences - instead of their mere conjunction - "This is a blue pebble": calls for embedding pebble into the blue. VI 41 Abstract objects/Modal/Putnam/Parsons: modal operators can save abstract objects - QuineVsModal logic: instead quantification (postulation of objects) - so we can take the slack out of the truth function. >Modal Logic/Quine. VII (d) 69 Object/Quine: may be unconnected: E.g. USA Alaska. XII 36 Properties/Identity/Quine: Problem: (unlike objects) they are ultimately based on synonymy within a language - more language-specific identity. >Properties/Quine. V 39 Ultimately we do without rigorous individuation of properties and propositions. (different term scheme) - Frege dito: (Basic Laws): do not extend identity to terms. XII 68 Object/Theory/Quine: what is an object, ultimately, cannot be stated - only in terms of a theory - (ultimately overall theory, i.e. language use) - but wrong: to say that talk about things would only make sense within a wider range - that would correspond to the false thesis that no predicate applied to all things - there are universal predicates. >Mention, >use, >word, >object. |
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 |
| Operationalism | Barrow | I 37 Operationalism: Science is a system of rules for the exploration of the world in the laboratory - linguistically: how are the words used? - InstrumentalismVsEmpiricism: useful concepts are not only those which can be traced back to the sense data. >Empiricism, >Instrumentalism, >Sense data. Theories and laws of nature are only instruments to make the environment understandable. True/false do not exist as properties of theories. Idealism: since all knowledge is filtered through our minds, we are never sure if there is a connection to reality. >Idealism. I 42 OperationalismVsEmpiricism: theories may also be invented - therefore, the observer receives a more important role. >Observation, >Ideal observer, >Theories. I 41f VsOperationalism/Barrow: asks what is measurable. Therefore, he must exclude complex and irrational numbers. >Numbers, >Measurements. Fragmentation of science: every time we use a different method of measurement, we need to consider a number as a different variable. Circular reasoning: operationalism presumes that we know what an permissible operation is. Problem: certain concepts may only be used when sensitive devices allow accurate measurements. >Fine-grained/coarse-grained. |
B I John D. Barrow Warum die Welt mathematisch ist Frankfurt/M. 1996 B II John D. Barrow The World Within the World, Oxford/New York 1988 German Edition: Die Natur der Natur: Wissen an den Grenzen von Raum und Zeit Heidelberg 1993 B III John D. Barrow Impossibility. The Limits of Science and the Science of Limits, Oxford/New York 1998 German Edition: Die Entdeckung des Unmöglichen. Forschung an den Grenzen des Wissens Heidelberg 2001 |
| Proper Names | Quine | I 230 Ambiguity: The name Paul is not ambiguous, it is not a general term but a singular term with dissemination. - Ambiguity action/habit: delivery (action, object). >Ambiguity/Quine I 316 Name: is a general term that applies only to one object - Ryle: x itself is not a property! - Middle Ages: Socrates, human, mortal: were on the same level - closes truth value gaps, claims no synonymy. >General Terms/Quine, >Truth Value Gap/Quine VII (a) 12ff Name/Quine: is always eliminated - language does not need it. VII (d) 75ff Name/Quine: Frege: a name must be substitutable - this is even possible with abstract entities. VII (i) 167 Proper Names/Quine: can be analyzed as descriptions. Then we can eliminate all singular terms as far as theory is concerned. >Descriptions/Quine VIII 24ff Name/Quine: names are constant substitutions of variables. X 48 A name always refers to only one object - predicate: refers to many. We replace them in the standard grammar by predicates: first: a= instead of a, then predicate A. - The sentence Fa then becomes Ex(Ax.Fx). >Predicates/Quine X 48 Name/Quine: it is not possible to quantify about them, so they are a different category than variables - names can be replaced by variables, but not always vice versa. X 124 Name/logic/Substitutional Quantification/Quine: problem: there are never enough names for all objects of the world: e.g. if a set is not determined by any open sentence, it has no name either. - Otherwise E.g. Name a determination: x ε a - E.g. irrational numbers cannot be traced back to integers. ((s)>substitution class). Lauener XI 39 Name/General Term/Quine/Lauener: names are eliminated by being reconstructed as a general term. As = a - then: Pegasus/truth value: then "Pegasus flies". (Ex)(X = c u Fx) is wrong, because Pegasus does not exist. (There is no pegasus, the conjunction is wrong). (>unicorn example). - The logical status of a proper name does not depend on the type of introduction, but only on the relation to other expressions. XII 78 Name/Quine: is distinguished by the fact that they may be inserted for variables. |
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 Q XI H. Lauener Willard Van Orman Quine München 1982 |
| Sets | Mates | I 49 Sets/Mates: for any propositional function there is a set, but not vice versa - basic: there are more sets than propositional functions. >Power set, >irrational numbers, >Sets, >Set theory, >Propositional functions. |
Mate I B. Mates Elementare Logik Göttingen 1969 Mate II B. Mates Skeptical Essays Chicago 1981 |
| Substitutional Quantification | Quine | V 140 Substitutional quantification/Quine: is open for other grammatical categories than just singular term but has other truth function. - Referential quantification: here, the objects do not even need to be specifiable by name. >Referential quantification, >Truth functions, >Singular terms. --- V 141 Language learning: first substitution quantification: from relative pronouns. - Later: referential quantification: because of categorical sentences. Substitution quantification: would be absurd: that every inserted name that verifies Fx also verifies Gx - absurd: that each apple or rabbit would have to have a name or a singular description. - Most objects do not have names. --- V 140 Substitutional Quantification/Referential Quantification/Truth Function/Quine: referential universal quantification: can be falsified by one single object, even though this is not specifiable by a name. - The same substitutional universal quantification: in contrast, remains true. - Existential quantification: referential: may be true due to a non-assignable value. - The same in substitutional sense: does not apply for lack of an assignable example. --- V 146f Substitutional Quantification/Quine: Problem: Blind spot: substitutional universal quantification: E.g. none of the substitution cases should be rejected, but some require abstention. - Existential quantification: E.g. none of the cases is to be approved, but some abstention is in order.- then neither agree nor abstain. (Equivalent to the alternation). --- Ad V 170 Substitutional Quantification/(s): related to the quantification over apparent classes in Quine’s meta language? --- V 175 Numbers/Classes/Quantification/Ontology/Substitutional quantification/Quine: first substitutional quantification through numbers and classes. - Problem: Numbers and classes can then not be eliminated. - Can also be used as an object quantification (referential quantification) if one allows every number to have a successor. - ((s) with substitution quantification each would have to have a name.) Class quantifier becomes object quantifier if one allows the exchange of the quantifiers (AQU/AQU/ - EQu/EQu) - so the law of the partial classes of one was introduced. --- X 124 Substitutional quantification/Quine: requires name for the values of the variables. Referential quantification/(s) speaks of objects at most. - Definition truth/Substitutional Quantification/Barcan/Quine: applying-Quantification - is true iff at least one of its cases, which is obtained by omitting the quantifier and inserting a name for the variable, is true. - Problem: almost never enough names for the objects in a not overly limited world. - E.g. No Goedel numbers for irrational numbers. - Then substitutional quantification can be wrong, because there is no name for the object, but the referential quantification can be true at the same time - i.e. both are not extensionally equal. X 124 Names/logic/substitutional quantification/Quine: Problem: never enough names for all objects in the world: e.g. if a set is not determined by an open sentence, it also has no name. - Otherwise E.g. Name a, Determination: x ε a - E.g. irrational numbers cannot be attributed to integers. - (s) > substitution class. --- XII 79f Substitutional Quantification/Quine: Here the variables are placeholders for words of any syntactic category (except names) - Important argument: then there is no way to distinguish names from the rest of the vocabulary and real referential variables. ((s) Does that mean that one cannot distinguish fragments like object and greater than, and that structures like "there is a greater than" would be possible?). XII 80 Substitutional Quantification/Quine: Problem: Assuming an infinite range of named objects. - Then it is possible to show for each substitution result of a name the truth of a formula and simultaneously to refute the universal quantification of the formula. - (everyone/all). - Then we have shown that the range has at least one unnamed object. - ((s) (> not enough names). - Therefore QuineVsSubstitutional Quantification. E.g. assuming the range contained the real name - Then not all could be named, but the unnamed cannot be separated. - The theory can always be strengthened to name a certain number, but not all - referential quantification: attributes nameless objects to itself. - Trick: (see above) every substitution result with a name is true, but makes universal quantification false. ((s) Thus an infinite number of objects secured). - A theory of real names must be based on referential quantification. |
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 |
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| Disputed term/author/ism | Author Vs Author |
Entry |
Reference |
|---|---|---|---|
| Bostock, D. | Simons Vs Bostock, D. | I 86 Part/mereology/Bostock/Simons: (Bostock 1979): his mereology should be a basis for his theory of extensive measurement, the rational and irrational numbers. Part-Relation/Bostock: thesis: there is more than one part-relation! SimonsVsBostock: (see below, Part II): Bostock's assumptions are still too strong to be as minimal as he assumes. System P/mereology/Bostock: there are no sets, "‹" is a basic concept. I 87 Least upper bound/l.u.b./sum/product/mereology/Bostock: mereology takes as duality for the product not the sums + and σ but the least upper bound (l.u.b.) +’ and σ’. Compact Set/Bostock: the second axiom (...) tells us that when F-s exist and they are limited above, they then have a sum (σ, not l.u.b. σ'). The resulting system is a little weaker than the classical mereology: it does not force us to assume the existence of a universe. SimonsVsBostock: with this, his system is still very strong. Bostock: his system only provides 6 nonisomorphic models ((s) interpretations) for the 7-element model (see above). A binary least upper bound exists when two objects have an upper bound at all. Bostock needs this relative strength in order to be able to express the analogy between parts and subsets. Simons: that is just not the case for the classical mereology. Bostock: thesis: it is the analogy between part and subset that explains why the concept of the part is at all important to us. SimonsVsBostock: which cannot be denied but will be undermined in part II for other cases. BostockVsMereology/stronger/weaker: one should avoid its strongest theses because there are classes of objects that are unlimited above, or they could exist. The strong classical mereology boils down to that there should be sums that are, in a certain sense, too large or too heterogeneous. Sum/Bostock: we need an additional condition: sums should be formed exclusively of their summands. This is intended to exclude unintended interpretations of P that are not mereologically. E.g. the Hasse diagrams from §1.4: higher points are obviously not formed from the lower points. "To consist of"/mereology/Simons: this is itself a mereological term. The lower points do not form the higher because they are not parts of them! Part/Bostock/Simons: Bostock's informal condition that we should really understand "part" as part is nothing other than that we do not want unintended models. |
Simons I P. Simons Parts. A Study in Ontology Oxford New York 1987 |
| 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 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 |
| Tarski, A. | Barcan Vs Tarski, A. | Quine X 124 Substitutional Quantification/SQ/Truth Conditions//tr.cond./Barcan-MarcusVsTarski/Quine: the truth conditions for quantification where names replace the variables ((s) truth conditions: i.e. that names exist) were accepted by Ruth Barcan. They are in an interesting contrast to Tarski's truth conditions. (5) (see above X 67) (5) For all x, y and i: x satisfies the existential quantification of y in which variant (i) is quantified iff. y is fulfilled by an n tuple x', for which applies: xj = x’j for all j unequal i. The new truth conditions also have the circularity described in the middle of Chapter 3: the existential quantification is true if at least one of its cases is true ((s) "true" appears twice). old: the big difference is that (5) speaks only of the values of variables and uses no names. ((s) analogous to constructivism: the method of proof must be known like the objects here, which is to be proved by assigning names). (5): Is much more complicated than the new form (SQ). SQ: so far, we do not have any deviation, only different descriptions of the same quantification, as long as all objects have a name. Problem: in a not too limited world, there are never enough names for all objects, never as many names as there are objects. Even if there are infinitely many names. E.g. If a set is not determined by an open sentence, it does not have a name: otherwise, if the name was "a", for example, the name of the corresponding open sentence would be "x ε a". X 125 E.g. we cannot have many different names for the irrational numbers, because we cannot assign them to the integers. E.g. we cannot form Gödel numbers for each irrational number. |
Quine I W.V.O. Quine Word and Object, Cambridge/MA 1960 German Edition: Wort und Gegenstand Stuttgart 1980 |
| Various Authors | Frege Vs Various Authors | Brandom II 83 FregeVsBoole: no material contents, therefore unable to follow scientific concept formation. Boole: "scope equality". Frege I 32 Addition/Hankel: wants to define: "if a and b are arbitrary elements of the basic series, then the sum of a + b is understood to be that one member of the basic series for which the formula a + (b + e) = a + b + e is true." (e is supposed to be the positive unit here). Addition/Sum/FregeVsHankel: 1) thus, the sum is explained by itself. If you do not yet know what a + b is, you will not be able to understand a + (b + e). 2) if you’d like to object that not the sum, but the addition should be explained, then you could still argue that a + b would be a blank sign if there was no member of the basic series or several of them of the required type. Frege I 48 Numbers/FregeVsNewton: he wants to understand numbers as the ratio of each size to another of the same kind. Frege: it can be admitted that this appropriately describes the numbers in a broader sense including fractions and irrational numbers. But this requires the concepts of size and the size ratio!. I 49 It would also not be possible to understand numbers as quantities, because then the concept of quantity and the quantity ratios would be presumed. I 58 Number/Schlömilch: "Notion of the location of an object in a series". FregeVsSchlömilch: then always the same notion of a place in a series would have to appear when the same number occurs, and that is obviously wrong. This could be avoided if he liked to understand an objective idea as imagination, but then what difference would there be between the image and the place itself?. I 60 Frege: then arithmetic would be psychology. If two were an image, then it would initially only be mine. Then we could perhaps have many millions of twos. I 64 Unit/Baumann: Delimitation. FregeVsBaumann: E.g. if you say the earth has a moon, you do not want to declare it a delimited one, but you rather say it as opposed to what belongs to Venus or Jupiter. I 65 With respect to delimitation and indivisibility, the moons of Jupiter can compete with ours and are just as consistent as our moon in this sense. Unit/Number/Köpp: Unit should not only be undivided, but indivisible!. FregeVsKöpp: this is probably supposed to be a feature that is independent from arbitrariness. But then nothing would remain, which could be counted and thought as a unit! VsVs: then perhaps not indivisibility itself, but the be considering to be indivisible could be established as a feature. FregeVs: 1) Nothing is gained if you think the things different from what they are!. I 66 2) If you do not want to conclude anything from indivisibility, what use is it then? 3) Decomposabiltiy is actually needed quite often: E.g. in the problem: a day has 24 hours, how many hours have three days?. I 69 Unit/Diversity/Number/FregeVsJevons: the emphasis on diversity also only leads to difficulties. E.g. If all units were different, you could not simply add: 1 + 1 + 1 + 1..., but you would always have to write: 1" + 1"" + 1 """ + 1 """", etc. or even a + b + c + d... (although units are meant all the time). Then we have no one anymore!. I 78 ff: ++ Number neither description nor representation, abstraction not a definition - It must not be necessary to define equality for each case. Infinite/Cantor: only the finite numbers should be considered real. Just like negative numbers, fractions, irrational and complex numbers, they are not sense perceptible. FregeVsCantor: we do not need any sensory perceptions as proofs for our theorems. It suffices if they are logically consistent. I 117 - 127 ++ VsHankel: sign (2-3) is not empty, but determinate content! Signs are never a solution! - Zero Class/FregeVsSchröder: (> empty set) false definition of the zero class: there can be no class that is contained in all classes as an element, therefore it cannot be created by definition. (The term is contradictory). IV 14 VsSchröder: you cannot speak of "classes" without already having given a concept. - Zero must not be contained as an element in another class (Patzig, Introduction), but only "subordinate as a class". (+ IV 100/101). II 93 Euclid/FregeVsEuclid: makes use of implied conditions several times, which he states neither under his principles nor under the requirements of the special sentence. E.g. The 19th sentence of the first book of the elements (in each triangle the greater angle is located opposite the larger side) presupposes the following sentences: 1) If a distance is not greater than another, then it is equal to or smaller than the first one. 2) If an angle is equal to another, then it is not greater than the first one. 3) If an angle is less than another, it is not greater than the first one. Waismann II 12 FregeVsPostulates: why is it not also required that a straight line is drawn through three arbitrary points? Because this demand contains a contradiction. Well, then they should proof that those other demands do not contain any contradictions!. Russell: postulates offer the advantages of theft over honest work. Existence equals solvability of equations: the fact that √2 exists means that x² 2 = 0 is solvable. |
F I G. Frege Die Grundlagen der Arithmetik Stuttgart 1987 F II G. Frege Funktion, Begriff, Bedeutung Göttingen 1994 F IV G. Frege Logische Untersuchungen Göttingen 1993 Bra I R. Brandom Making it exlicit. Reasoning, Representing, and Discursive Commitment, Cambridge/MA 1994 German Edition: Expressive Vernunft Frankfurt 2000 Bra II R. Brandom Articulating reasons. An Introduction to Inferentialism, Cambridge/MA 2001 German Edition: Begründen und Begreifen Frankfurt 2001 Waismann I F. Waismann Einführung in das mathematische Denken Darmstadt 1996 Waismann II F. Waismann Logik, Sprache, Philosophie Stuttgart 1976 |
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