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The author or concept searched is found in the following 3 entries.
| Disputed term/author/ism | Author |
Entry |
Reference |
|---|---|---|---|
| Number Theory | Kripke | III 383f Substitutional quantification/sQ/number theory/KripkeVsWallace: the object language should be written substitutionally: the substitution class then consists of the number names: 0,0",0""... . The meta language needs a referential variable about the expressions of the object language - could we replace it with Goedel numbers? No, we cannot do this because the question was whether an ontology of numbers was used in the meta language, in addition to the ontology of expressions. We cannot even ask this question if we identify expressions with numbers. The two have asked the wrong question twice: 1) by having treated the object language variables as referential about numbers rather than as a substitutional with number names as substitutes, and 2) by interpreting the referential variables of the meta language as Goedel numbers instead of as symbol chains of the object language. Usually, identification of expressions with their Goedel numbers is harmless but here we must distinguish numbers and expressions. >Substitutional quantification, >Meta language, >Object language, >Gödel numbers, >Referential quantification. |
Kripke I S.A. Kripke Naming and Necessity, Dordrecht/Boston 1972 German Edition: Name und Notwendigkeit Frankfurt 1981 Kripke II Saul A. Kripke "Speaker’s Reference and Semantic Reference", in: Midwest Studies in Philosophy 2 (1977) 255-276 In Eigennamen, Ursula Wolf Frankfurt/M. 1993 Kripke III Saul A. Kripke Is there a problem with substitutional quantification? In Truth and Meaning, G. Evans/J McDowell Oxford 1976 Kripke IV S. A. Kripke Outline of a Theory of Truth (1975) In Recent Essays on Truth and the Liar Paradox, R. L. Martin (Hg) Oxford/NY 1984 |
| Number Theory | Quine | IX 81 Elementary Number Theory/Quine: this is the theory that can only be expressed with the terms "zero, successor, sum, power, product, identity" and with the help of connections from propositional logic and quantification using natural numbers. One can omit the first four of these points or the first two and the fifth. But the more detailed list is convenient, because the classical axiom system fits directly to it. Quine: our quantifiable variables allow other objects than numbers. However, we will now tacitly introduce a limitation to "x ε N". Elementary Number Theory/Quine: less than/equal to: superfluous here. "Ez(x + z = y)" - x ε N > Λ + x = x. - x,y ε N >{x} + y = {x+y}. IX 239 Relative Strength/Proof Theory/Theory/Provability/Quine: Goedel, incompleteness theorem (1931)(1). Since number theory can be developed in set theory, this means that the class of all theorems IX 239 (in reality, all the Goedel numbers of theorems) of an existing set theory can be defined in that same set theory, and different things can be proved about it in it. >Set Theory/Quine. Incompleteness Theorem: as a consequence, however, Goedel showed that set theory (if it is free of contradiction) cannot prove one thing through the class of its own theorems, namely that it is consistent, i.e., for example, that "0 = 1" does not lie within it. If the consistency of one set theory can be proved in another, then the latter is the stronger (unless both are contradictory). Zermelo's system is stronger than type theory. >Type theory, >Strength of theories, >Set theory, >Provability. 1.Kurt Gödel: Über formal unentscheidbare Sätze der Principia Mathematica und verwandter Systeme I. In: Monatshefte für Mathematik und Physik. 38, 1931, S. 173–198, doi:10.1007/BF01700692 II 178 Elementary number theory is the modest part of mathematics that deals with the addition and multiplication of integers. It does not matter if some true statements will remain unprovable. This is the core of Goedel's theorem. He has shown how one can form a sentence with any given proof procedure purely in the poor notation of elementary number theory, which can be proved then and only then if it is wrong. But wait! The sentence cannot be proved and still be wrong. So it is true, but not provable. Quine: we used to believe that mathematical truth consists in provability. Now we see that this view is untenable to mathematics as a whole. II 179 Goedel's incompleteness theorem (the techniques applied there) has proved useful in other fields: Recursive number theory, or recursion theory for short. Or hierarchy theory. >Goedel/Quine. III 311 Elementary Number Theory/Quine: does not even have a complete proof procedure. Proof: reductio ad absurdum: suppose we had it with which to prove every true sentence in the spelling of the elementary number theory, III 312 then there would also be a complete refutation procedure: to refute a sentence one would prove its negation. But then we could combine the proof and refutation procedure of page III 247 to a decision procedure. V 165 Substitutional Quantification/Referential Quantification/Numbers/Quine: Dilemma: the substitutional quantification does not help elementary number theory to any ontological thrift, for either the numbers run out or there are infinitely many number signs. If the explanatory speech of an infinite number sign itself is to be understood again in the sense of insertion, we face a problem at least as serious as that of numbers - if it is to be understood in the sense of referential quantification, then one could also be satisfied from the outset uncritically with object quantification via numbers. >Quantification/Quine. V 166 Truth conditions: if one now assumes substitutional quantification, one can actually explain the truth conditions for them by numbers by speaking only of number signs and their insertion. Problem: if numerals are to serve their purpose, they must be as abstract as numbers. Expressions, of which there should be an infinite number, could be identified by their Goedel numbers. No other approach leads to a noticeable reduction in abstraction. Substitutional quantification: forces to renounce the law that every number has a successor. A number would be the last, but the substitutional quantification theorist would not know which one. It would depend on actual inscriptions in the present and future. (Quine/Goodman 1947). This would be similar to Esenin Volpin's theory of producible numbers: one would have an unknown finite bound. V 191 QuineVsSubstitutional Quantification: the expressions to be used are abstract entities as are the numbers themselves. V 192 NominalismVsVs: one could reduce the ontology of real numbers or set theory to that of elementary number theory by establishing truth conditions for substitutional quantification on the basis of Goedel numbers. >Goedel Numbers/Quine. QuineVs: this is not nominalistic, but Pythagorean. It is not about the high estimation of the concrete and disgust for the abstract, but about the acceptance of natural numbers and the rejection of most transcendent numbers. As Kronecker says: "The natural numbers were created by God, the others are human work". QuineVs: but even that is not possible, we saw above that the subsitutional quantification over classes is basically not compatible with the object quantification over objects. V 193 VsVs: one could also understand the quantification of objects in this way. QuineVs: that wasn't possible because there aren't enough names. You could teach space-time coordination, but that doesn't explain language learning. X 79 Validity/Sentence/Quantity/Schema/Quine: if quantities and sentences fall apart in this way, there should be a difference between these two definitions of validity about schema (with sentences) and models (with sentences). But it follows from the Löwenheim theorem that the two definitions of validity (using sentences or sets) do not fall apart as long as the object language is not too weak in expression. Condition: the object language must be able to express (contain) the elementary number theory. Object Language: In such a language, a scheme that remains true in all insertions of propositions is also fulfilled by all models and vice versa. >Object Language/Quine The requirement of elementary number theory is rather weak. Def Elementary Number Theory/Quine: speaks about positive integers by means of addition, multiplication, identity, truth functions and quantification. Standard Grammar/Quine: the standard grammar would express the functors of addition, multiplication, like identity, by suitable predicates. X 83 Elementary Number Theory/Quine: is similar to the theory of finite n-tuples and effectively equivalent to a certain part of set theory, but only to the theory of finite sets. XI 94 Translation Indeterminacy/Quine/Harman/Lauener: ("Words and Objections"): e.g. translation of number theory into the language of set theory by Zermelo or von Neumann: both versions translate true or false sentences of number theory into true or false sentences of set theory. Only the truth values of sentences like e.g. "The number two has exactly one element", which had no sense before translation, differ from each other in both systems. (XI 179: it is true in von Neumann's and false in Zermelo's system, in number theory it is meaningless). XI 94 Since they both serve all purposes of number theory in the same way, it is not possible to mark one of them as a correct translation. |
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 |
| Number Theory | Tarski | Berka I 532 Elementary number theory/Tarski: the science in which all variables represent names of natural numbers and as constants: (in addition to the characters of the propositional calculus and the functional calculus) the characters of zero, unity, equality, the sum and of the product may occur.(1) >Numbers, >Unity, >Equality, >Equal sign, >Variables, >Name of a number, >Natural numbers, >Real numbers, >One, >Zero. 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 |
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| Disputed term/author/ism | Author Vs Author |
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| Anti-Objectivism | Field Vs Anti-Objectivism | II 318 Undecidability/VsAnti-Objectivism/AO/Field: other examples are less favorable for the anti-objectivism: E.g. Gödel. Even very simple sentences may be undecidable. E.g. (*) for all natural numbers x, B(x) where B(x) is a decidable predicate, i.e. a predicate, so that for each numeral n we can either prove B(n) or ~B(n). (Through an uncontroversial proof). Problem: you may say now that every undecidable sentence must be objectively correct (see above, must follow from the axioms). Then proof of ~B(n) would be proof of the negation of (*), as opposed to its undecidability. So, because of the assumption about B(x) B(n) must be provable for each number n, thus presumably objectively correct. This seems to show, however, that the generalization (*) is also objectively correct. (This is not undisputed, because it requires as a final step that it is objectively the case that there are no other natural numbers than those for which there are names. ((s)> "not enough names"). FieldVs extreme Anti-Objectivism: if that can be believed, however, he must adopt a more moderate position. Elementary Number Theory/ENT/Undecidability/Field: in fact, almost everyone believes that the choice between an undecidable proposition and its negation is objective, also for the generalized ENT. That would be hard to give up, because many assertions about provability and consistency are actually undecidable number-theoretic assertions, so that the anti-objectivist would have to say that they lack objectivity. Only few of them want that. Nevertheless, it is not obvious that if the ENT is granted objectivity, it would also have to be conceded to the higher regions. I 347 Anti-Objectivism/Gödel/Field/Conclusion/(s): Gödel gives no reason to assume that some undecidable propositions have certain truth values. (pro extreme anti-objectivism, by Field). VsAnti-Objectivism/Gödel/Field: It may be objected that the Gödel sentences of the candidates for our most mathematical theory should not only have a certain truth value, but that they are true! The argument goes by. Induction: all logical and non-logical premises of M are true. The rules of inference receive truth, therefore, all theorems must be true. So the theory must be consistent, therefore the Gödel sentence must be unprovable and therefore true. Gödel sentence: is true only if unprovable; if provable, it is not true. Problem: this induction can of course not be formalized in M. But one often feels that it is somehow "informally valid". If that is true, only the truth of the Gödel theorem is proved, not its particular truth. Solution: we might be able to fill the gap by establishing a principle that if we can prove something informally, it must certainly be true. (Vs: That’s plausible, but not undisputed!). In any case, the arguments for the particular truth of the Gödel theorem are weaker than those for its simple truth. |
Field I H. Field Realism, Mathematics and Modality Oxford New York 1989 Field II H. Field Truth and the Absence of Fact Oxford New York 2001 Field III H. Field Science without numbers Princeton New Jersey 1980 Field IV Hartry Field "Realism and Relativism", The Journal of Philosophy, 76 (1982), pp. 553-67 In Theories of Truth, Paul Horwich Aldershot 1994 |
| Antirealism | Quine Vs Antirealism | Field I 64 Infinite/Anti-Realism/Field: the Anti-R can only assume an infinite number of entities if there is an infinite number of physical entities. (E.g. infinitely many parts of a light beam). But it would be inappropriate to want to test the adequacy of the number theory by assumptions about the physical world. Truth/Ontology/Field: sure, the truth of the number theory would require infinitely many objects for the quantifiers, but its conservatism would not! And conservatism is all we need! Physics: How about atypical applications such as differential equations, etc.? Here the existence of many entities such as real numbers, functions, differential operators, etc. seems to be called for. How should nominalistic inferences become easier here? Where ever are the nominalistic premises? We would only have them if we were able to somehow represent the theory of electromagnetism nominalistically, and that seems hardly possible. Indispensability: if it is true that mathematics does not only facilitate inferences, it would theoretically be indispensable. How can indispensability be represented in terms of conservatism? Quine Putnam Argument/VsAnti-Realism: (see above): only by truth! We must assume the truth of mathematics for its usefulness in the extra-mathematical realm. FieldVs: this is certainly an exaggeration. Parts of the usefulness may also be explained by conservatism (but not only). Field I 65 In the end, I try to show that thesis: mathematics is not just indispensable. |
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 Field IV Hartry Field "Realism and Relativism", The Journal of Philosophy, 76 (1982), pp. 553-67 In Theories of Truth, Paul Horwich Aldershot 1994 |
| Aristotle | Nozick Vs Aristotle | II 145 Relation/Law/Incident/Language/Interpretation/Nozick: Wittgenstein needed people to teach the language with its instances. Nozick: but it cannot be people who teach a natural law with its instances. Causal laws also apply for people, inter alia, and were valid before people existed. The consent of people to something depends on causality and cannot determine causality itself. (FN 22). Law/Nozick: therefore seems to have no own ontological status, because it cannot reach for incidents itself. Nevertheless, if a natural law only determines a pattern, it is merely descriptive. Without ontological status it cannot support counterfactual conditionals beyond actual events and how could laws then be used to explain something? Explanation/Nozick: how does a higher level pattern explain a lower level one? Is every explanation implicitly only a repetition? Explanation/Law/NozickVsAristotle: explanatory laws need not be necessary truths, but do they need to be anything at all? If events proceed according to laws, what is the connection between the event and the law? It can of course not be causal. ((s) recourse). But even any logical connection must be interpreted in turn. Can a lawlike statement interpret itself? I.e. can a law give instructions for the interpretation? Problem: these instructions would have to be interpreted again II 146 If the interpretation was to be fixed, the law would have to include something analogous to reflexive self-reference. This itself is mysterious. Hence, we must not treat laws as related to statements. Gödel: there is no formal system in which all the truths of number theory can be proved. Nozick: that is bad luck for a picture of all the facts from which the statements of fact can be completely derived. Determinism/Nozick: should therefore not rely on derivability from causal laws! (FN 23). NozickVsDeterminism: claims: if the initial state was repeated, the later states would also repeat themselves. Problem: in a re-collapsing universe other laws could apply for another big bang. I.e. the subjunctivist conditional, (subjunction = counterfactual conditional, unlike implication (metalinguistical)) on which determinism is based would be wrong. |
No I R. Nozick Philosophical Explanations Oxford 1981 No II R., Nozick The Nature of Rationality 1994 |
| Black, Max | Thomson Vs Black, Max | Horwich I 161 Material Equivalent/T-Schema/Thomson: "material equivalent" is itself defined in terms of truth! "Principle of Definition"/PdD/Black/Thomson: we must not confuse it with what we understand when we understand the T-scheme! Also not with a general T-Def, which cannot be formulated. A "principle of definition" probably). Tarski/Black: "we only seem to see that the assertion of a proposition is true, logically equivalent to the assertion of the proposition itself". (see above redundancy theory). ThomsonVsBlack: he makes a mistake when he says this is pointless. It is simply wrong! (Group: ThomsonVs Redundancy Theory? BlackVs Redundancy Theory?). For example, suppose we do not know anything about number theory and someone tells us: (5) 43 ε prime ⇔ any number that divides 43 is either 43 or 1 (ε: epsilon, is element of) Now we could think, either 43 is a prime number and... or 43 would not be a prime number and.... But we would definitely think that (5) is the consequence of some definition of "prime number" (whatever that would be). And then we would say that (5) shows an equivalence between "43 ε prime" and "any number...". And we would not say that this is a coincidence, that 43 and the class of prime numbers are mentioned on the left and 43 again on the right, but not the class of prime numbers. For if it were a coincidence, perhaps we could also get the following: (6) 43 ε Prime ⇔ Ramanujan is dead Thomson: but here we would not assume that there would be a general formula from which this is a consequence. Now we could assume that the rule that is fulfilled in the case of (5) and not in (6) is also fulfilled for (1) and the other T-sentences. And that the T-sentences exemplify a relation of logical equivalence. (see below: but do not make explicit!). Thomson: but this is wrong! Example in (1) "London is a city" is true ⇔ London is a city are right: London and the class of cities (mentioned) left: one sentence and the class of true sentences (mentioned). Thomson: why do we accept (1) as true? Because we assume that "London is a city" is used as the name of the phrase "London is a city". (> name of sentence). And we assume this because we assume that the sentence mentioned to the left is the same sentence, Horwich I 162 that is needed on the right. And that is almost certainly what we are supposed to understand. Problem: whoever expresses (1) cannot tell us what we should assume here! In other words: We assume that (1) is a sentence of the meta-language for which there is a designation rule that states that "London is a city" is the name of the sentence "London is a city". N.B.: (1) itself does not tell us that this rule exists! ((s) because it needed a sentence in meta meta language). |
ThomsonJF I James F. Thomson "A Note on Truth", Analysis 9, (1949), pp. 67-72 In Theories of Truth, Paul Horwich Aldershot 1994 ThomsonJJ I Judith J. Thomson Goodness and Advice Princeton 2003 Horwich I P. Horwich (Ed.) Theories of Truth Aldershot 1994 |
| Field, H. | Gödel Vs Field, H. | Field I 66 Realism/Mathematics/Gödel: ("What is Cantor's Continuum Problem?", 1947) (Pro Quine Putnam Argument, VsField, VsAnti Realism): even with a very narrow definition of the term "mathematical data" (only equations of number theory) we can justify quite abstract parts by explanatory success: Gödel: even without having to accept the necessity of a new axiom, and even if it has no intrinsic necessity at all, a decision about its truth is possible by examining its "explanatory success" with induction. The fertility of its consequences, especially the "verifiable" ones, i.e. those which can be demonstrated without the new axiom, but whose proofs are made easier by the new axiom. Or if one can combine several proofs to one. For example the axioms about the real numbers, which are rejected by the intuitionists. I 67 FieldVsGödel: if no mathematical entities are indispensable, then one does not have to call the so-called "mathematical data" true. But at the beginning I had said that there can be no other goal of mathematics than truth. |
Göd II Kurt Gödel Collected Works: Volume II: Publications 1938-1974 Oxford 1990 Field IV Hartry Field "Realism and Relativism", The Journal of Philosophy, 76 (1982), pp. 553-67 In Theories of Truth, Paul Horwich Aldershot 1994 |
| Field, H. | Verschiedene Vs Field, H. | Field I 51 Infinity/Physics/Essay 4: even without "part of" relation we do not really need the finity operator for physics. VsField: many have accused me of needing every extension of 1st level logic. But this is not the case. I 52 I rather assume that the nominalization program has not yet been advanced far enough to be able to say what the best logical basis is. Ultimately, we are going to choose only a few natural means that go beyond the 1st level logic, preferably those that the Platonist would also need. But we can only experience this by trial and error. I 73 Indispensability Argument/Logic/VsField: if mE may be dispensable in science, they are not in logic! And we need logic in science. Logical Sequence Relation/Consequence/Field: is normally defined in terms of model theory: (Models are mE, semantic: a model is true or not true.) Even if one formulates them in a proven theoretical way ("there is a derivation", syntactically, or provable in a system) one needs mE or abstract objects: arbitrary sign sequences of symbol tokens and their arbitrary sequences. I 77 VsField: some have objected that only if we accept a Tarski Theory of truth do we need mE in mathematics. FieldVsVs: this led to the misunderstanding that without Tarskian truth mathematics would have no epistemic problems. Mathematics/Field: indeed implies mE itself, (only, we do not always need mathematics) without the help of the concept of truth, e.g. that there are prime numbers > 1000. I 138 Logic of Part-of-Relation/Field: has no complete evidence procedure. VsField: how can subsequent relations be useful then? Field: sure, the means by which we can know that something follows from something else are codifiable in an evidentiary procedure, and that seems to imply that no appeal to anything stronger than a proof can be of practical use. FieldVsVs: but you do not need to take any epistemic approach to more than a countable part of it. I 182 Field Theory/FT/Relationalism/Substantivalism/Some AuthorsVsField: justify the relevance of field theories for the dispute between S/R just the other way round: for them, FT make it easy to justify a relationalist view: (Putnam, 1981, Malament 1982): they postulate as a field with a single huge (because of the infinity of physical forces) and a corresponding part of it for each region. Variant: the field does not exist in all places! But all points in the field are not zero. FieldVsPutnam: I do not think you can do without regions. Field II 351 Indeterminacy/Undecidability/Set Theory/Number Theory/Field: Thesis: not only in the set theory but also in the number theory many undecidable sets do not have a certain truth value. Many VsField: 1. truth and reference are basically disquotational. Disquotational View/Field: is sometimes seen as eliminating indeterminacy for our present language. FieldVsVs: that is not the case :>Chapter 10 showed that. VsField: Even if there is indeterminacy in our current language also for disquotationalism, the arguments for it are less convincing from this perspective. For example, the question of the power of the continuum ((s)) is undecidable for us, but the answer could (from an objectivist point of view (FieldVs)) have a certain truth value. Uncertainty/Set Theory/Number Theory/Field: Recently some well-known philosophers have produced arguments for the impossibility of any kind of uncertainty in set theory and number theory that have nothing to do with disquotationalism: two variants: 1. Assuming that set theory and number theory are in full logic of the 2nd level (i.e. 2nd level logic, which is understood model theoretically, with the requirement that any legitimate interpretation) Def "full" in the sense that the 2nd level quantifiers go over all subsets of the 1st level quantifier range. 2. Let us assume that number theory and the set theory are formulated in a variant of the full logic of the 2nd level, which we could call "full schematic logic of level 1". II 354 Full schematic logic 1st Level/LavineVsField: denies that it is a partial theory of (non-schematic!) logic of the 2nd level. Field: we now better forget the 2nd level logic in favour of full schematic theories. We stay in the number theory to avoid complications. We assume that the certainty of the number theory is not in question, except for the use of full schemata. |
Field IV Hartry Field "Realism and Relativism", The Journal of Philosophy, 76 (1982), pp. 553-67 In Theories of Truth, Paul Horwich Aldershot 1994 |
| Hilbert | Frege Vs Hilbert | Berka I 294 Consistency/Geometry/Hilbert: Proof through analogous relations between numbers. Concepts: if properties contradict each other, the concept does not exist. FregeVsHilbert: there is just nothing that falls under it. Real Numbers/Hilbert: here, the proof of consistency for the axioms is also the proof of existence of the continuum.(1) 1. D. Hilbert, „Mathematische Probleme“ in: Ders. Gesammelte Abhandlungen (1935) Bd. III S. 290-329 (gekürzter Nachdruck v. S 299-301) Thiel I 279 Hilbert: Used concepts like point, line, plane, "between", etc. in his Foundations of Geometry in 1899, but understood their sense in a hitherto unfamiliar way. They should not only enable the derivation of the usual sentences, but rather, in its entirety, specify the meaning of the concepts used in it in the first place! Thiel I 280 Later this was called a "definition by postulates", "implicit definition" >Definition. The designations point, line, etc. were to be nothing more than a convenient aid for mathematical considerations. FregeVsHilbert: clarifies the letter correspondence that his axioms are not statements, but rather statement forms. >Statement Form. He denied that by their interaction the concepts occurring in them might be given a meaning. It was rather a (in Frege’s terminology) "second stage concept" that was defined, today we would say a "structure". HilbertVsFrege: the point of the Hilbert’s proceeding is just that the meaning of "point", "line", etc. is left open. Frege and Hilbert might well have been able to agree on this, but they did not. Frege: Axiom should be in the classical sense a simple, sense-wise completely clear statement at the beginning of a system. Hilbert: statement forms that combined define a discipline. From this the "sloppy" figure of speech developed E.g. "straight" in spherical geometry was then a great circle. Thiel I 343 Formalism: 1) "older" formalism: second half of the 19th century, creators Hankel, Heine, Thomae, Stolz. "Formal arithmetic", "formal algebra". "Object of arithmetic are the signs on the paper itself, so that the existence of these numbers is not in question" (naive). Def "Permanence Principle": it had become customary to introduce new signs for numbers that had been added and to postulate then that the rules that applied to the numbers of the original are should also be valid for the extended area. Vs: that would have to be regarded as illegitimate as long as the consistency is not shown. Otherwise, you could introduce a new number, and E.g. simply postulate § + 1 = 2 und § + 2 = 1. This contradiction would show that these "new numbers" did not really exist. This explains Heine’s formulation that "existence is not in question". (> "tonk"). Thiel I 343/344 Thomae treated the problem as "rules of the game" in a somewhat more differentiated way. FregeVsThomae: he had not even precisely specified the basic rules of his game, namely the correlation to the rules, pieces and positions. This criticism of Frege was already a precursor of Hilbert’S proof theory, in which also mere character strings are considered without regard their possible content for their production and transformation according to the given rules. Thiel I 345 HilbertVsVs: Hilbert critics often overlook that, at least for Hilbert himself, the "finite core" should remain content-wise interpreted and only the "ideal", not finitely interpretable parts have no directly provable content. This important argument is of a methodical, not a philosophical nature. "Formalism" is the most commonly used expression for Hilbert’s program. Beyond that, the conception of formalism is also possible in a third sense: i.e. the conception of mathematics and logic as a system of action schemes for dealing with figures that are free of any content. HilbertVsFrege and Dedekind: the objects of the number theory are the signs themselves. Motto: "In the beginning was the sign." Thiel I 346 The designation formalism did not come from Hilbert or his school. Brouwer had hyped up the contrasts between his intuitionism and the formalism of Hilbert’s school to a landmark decision. |
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 Berka I Karel Berka Lothar Kreiser Logik Texte Berlin 1983 T I Chr. Thiel Philosophie und Mathematik Darmstadt 1995 |
| McGee, V. | Field Vs McGee, V. | II 351 Second Order Number Theory/2nd Order Logic/HOL/2nd Order Theory/Field: Thesis (i) full 2nd stage N.TH. is - unlike 1st stage N.TH. - categorical. I.e. it has only one interpretation up to isomorphism. II 352 in which the N.TH. comes out as true. Def Categorical Theory/Field: has only one interpretation up to isomorphism in which it comes out as true. E.g. second order number theory. (ii) Thesis: This shows that there can be no indeterminacy for it. Set Theory/S.th.: This is a bit more complicated: full 2nd order set theory is not quite categorical (if there are unreachable cardinal numbers) but only quasi-categorical. That means, for all interpretations in which it is true, they are either isomorphic or isomorphic to a fragment of the other, which was obtained by restriction to a less unreachable cardinal number. Important argument: even the quasi-categorical 2nd order theory is still sufficient to give most questions on the cardinality of the continuum counterfactual conditional the same truth value in all interpretations, so that the assumptions of indeterminacy in ML are almost eliminated. McGee: (1997) shows that we can get a full second order set theory by adding an axiom. This axiom limits it to interpretations in which 1st order quantifiers go above absolutely everything. Then we get full categoricity. Problem: This does not work if the 2nd order quantifiers go above all subsets of the range of the 1st order quantifiers. (Paradoxes) But in McGee (as Boolos 1984) the 2nd order quantifiers do not literally go above classes as special entities, but as "plural quantifiers". (>plural quantification). Indeterminacy/2nd Order Logic/FieldVsMcGee: (see above chapter I): Vs the attempt to escape indeterminacy with 2nd order logic: it is questionable whether the indeterminacy argument is at all applicable to the determination of the 2nd order logic as it is applicable to the concept of quantity. If you say that sentences about the counterfactual conditional have no specific truth value, this leads to an argument that the concept "all subsets" is indeterminate, and therefore that it is indeterminate which counts as "full" interpretation. Plural Quantification: it can also be indeterminate: Question: over which multiplicities should plural quantifiers go?. "Full" Interpretation: is still (despite it being relative to a concept of "fullness") quasi-unambiguous. But that does not diminish the indeterminacy. McGeeVsField: (1997): he asserts that this criticism is based on the fact that 2nd order logic is not considered part of the real logic, but rather a set theory in disguise. FieldVsMcGee: this is wrong: whether 2nd order logic is part of the logic, is a question of terminology. Even if it is a part of logic, the 2nd order quantifiers could be indeterminate, and that undermines that 2nd order categoricity implies determinacy. "Absolutely Everything"/Quantification/FieldVsMcGee: that one is only interested in those models where the 1st. order quantifiers go over absolutely everything, only manages then to eliminate the indeterminacy of the 1st order quantification if the use of "absolutely everything" is determined!. Important argument: this demand will only work when it is superfluous: that is, only when quantification over absolutely everything is possible without this requirement!. All-Quantification/(s): "on everything": undetermined, because no predicate specified, (as usual E.g. (x)Fx). "Everything" is not a predicate. Inflationism/Field: representatives of inflationist semantics must explain how it happened that properties of our practice (usage) determine that our quantifiers go above absolutely everything. II 353 McGee: (2000) tries to do just that: (*) We have to exclude the hypothesis that the apparently unrestricted quantifiers of a person go only above entities of type F, if the person has an idea of F. ((s) i.e. you should be able to quantify over something indeterminate or unknown). Field: McGee says that this precludes the normal attempts to demonstrate the vagueness of all-quantification. FieldVsMcGee: does not succeed. E.g. Suppose we assume that our own quantifiers determinedly run above everything. Then it seems natural to assume that the quantifiers of another person are governed by the same rules and therefore also determinedly run above everything. Then they could only have a more limited area if the person has a more restricted concept. FieldVs: the real question is whether the quantifiers have a determinate range at all, even our own! And if so, how is it that our use (practices) define this area ? In this context it is not even clear what it means to have the concept of a restricted area! Because if all-quantification is indeterminate, then surely also the concepts that are needed for a restriction of the range. Range/Quantification/Field: for every candidate X for the range of unrestricted quantifiers, we automatically have a concept of at least one candidate for the picking out of objects in X: namely, the concept of self-identity! ((s) I.e. all-quantification. Everything is identical with itself). FieldVsMcGee: Even thoguh (*) is acceptable in the case where our own quantifiers can be indeterminate, it has no teeth here. FieldVsSemantic Change or VsInduction!!!. II 355 Schematic 1st Stage Arithmetic/McGee: (1997, p.57): seems to argue that it is much stronger than normal 1st stage arithmetic. G. is a Godel sentence PA: "Primitive Arithmetic". Based on the normal basic concepts. McGee: seems to assert that G is provable in schematic PA ((s) so it is not true). We just have to add the T predicate and apply inductions about it. FieldVsMcGee: that’s wrong. We get stronger results if we also add a certain compositional T Theory (McGee also says that at the end). Problem: This goes beyond schematic arithmetics. McGee: his approach is, however, more model theoretical: i.e. schematic 1st stage N.TH. fixes the extensions of number theory concepts clearly. Def Indeterminacy: "having non-standard models". McGee: Suppose our arithmetic language is indeterminate, i.e. It allows for unintended models. But there is a possible extension of the language with a new predicate "standard natural number". Solution: induction on this new predicate will exclude non-standard models. FieldVsMcGee: I believe that this is cheating (although some recognized logicians represent it). Suppose we only have Peano arithmetic here, with Scheme/Field: here understood as having instances only in the current language. Suppose that we have not managed to pick out a uniform structure up to isomorphism. (Field: this assumption is wrong). FieldVsMcGee: if that’s the case, then the mere addition of new vocabulary will not help, and additional new axioms for the new vocabulary would help no better than if we introduce new axioms simply without the new vocabulary! Especially for E.g. "standard natural number". Scheme/FieldVsMcGee: how can his rich perspective of schemes help to secure determinacy? It only allows to add a new instance of induction if I introduce new vocabulary. For McGee, the required relevant concept does not seem to be "standard natural number", and we have already seen that this does not help. Predicate/Determinacy/Indeterminacy/Field: sure if I had a new predicate with a certain "magical" ability to determine its extension. II 356 Then we would have singled out genuine natural numbers. But this is a tautology and has nothing to do with whether I extend the induction scheme on this magical predicate. FieldVsMysticism/VsMysticism/Magic: Problem: If you think that you might have magical aids available in the future, then you might also think that you already have it now and this in turn would not depend on the schematic induction. Then the only possible relevance of the induction according to the scheme is to allow the transfer of the postulated future magical abilities to the present. And future magic is no less mysterious than contemporary magic. FieldVsMcGee: it is cheating to describe the expansion of the language in terms of its extensions. The cheating consists in assuming that the new predicates in the expansion have certain extensions. And they do not have them if the indeterminist is right regarding the N.Th. (Field: I do not believe that indeterminism is right in terms of N.Th.; but we assume it here). Expansion/Extenstion/Language/Theory/FieldVsMcGee: 2)Vs: he thinks that the necessary new predicates could be such for which it is psychological impossible to add them at all, because of their complexity. Nevertheless, our language rules would not forbid her addition. FieldVsMcGee: In this case, can it really be determined that the language rules allow us something that is psychologically impossible? That seems to be rather a good example of indeterminacy. FieldVsMcGee: the most important thing is, however, that we do not simply add new predicates with certain extensions. |
Field IV Hartry Field "Realism and Relativism", The Journal of Philosophy, 76 (1982), pp. 553-67 In Theories of Truth, Paul Horwich Aldershot 1994 |
| Objectivism | Putnam Vs Objectivism | Field II 325 Universe/Standard-Platonism/Field: (thesis: "there is only one universe"). Problem/PutnamVsPlatonism: how do we manage at all to pick out the "full" (comprehensive) universe and to oppose this to a partial universe, and in accordance with this the standard element relationship as opposed to a non-member standard relationship? (Putnam 1980). (Here from the perspective of "one universe"). Putnam: thesis we cannot do that. That is, that the "incomplete content" of the terms "set" and "element of" is not sufficient to determine the truth value of all theoretical conclusions. PutnamVsObjectivism: concluded the same anti-objectivist methodological consequences such as the Platonism of perfection: although the standard Platonism has incorporated this idea that we have a set-theoretical universe, this is not really part of Platonism per se. PutnamVsPlatonism/Field: if he's right, this standard Platonism cannot be maintained. Field: Putnam is right: the "anti-objectivist" methodology is the right conclusion, whatever the ontological consequences are. The "Platonism of perfection" shows us this. Field II 338 PutnamVsObjectivism/Set Theory/Field: (Putnam 1980 first half): thesis: even if we assume that part of the standard Platonism, who says that there is only one universe, Problem: then there is nothing in our conclusion practices, which could determine the specific truth value of typical undecidable sentences. Field II 339 This can be easily extended to the relation of semantic consequence in logic 2nd order. In short: (ia) Nothing in our practices determines that the term "set" picks out the entire set-theoretical universe V and not any suitably closed part of V. (ib), even if the entire set-theoretical universe V could be picked out, there would be nothing in our practice that could determine that our term "e" picks out, the element relation E on V unlike any other relation on V that obeys our axioms. (ii) The indeterminacy in (ia) and (ib) is sufficient to allow the truth value of typical undecidable sentences of the set theory to be undetermined. ((ib) alone would also be sufficient, often also (ia) alone), Undecidable sentences/Field: which are covered by this scheme? What are the semantic facts that are determined by our conclusion practices? How is the semantics of "quantity" and "e" defined by our practice far enough to allow the quantifier "only a finite number of" to be sufficiently defined. And with that, how the truth will be determined by F-decidable, but otherwise by undecidable sentences, also for the number theory. |
Putnam I Hilary Putnam Von einem Realistischen Standpunkt In Von einem realistischen Standpunkt, Vincent C. Müller Frankfurt 1993 Putnam I (a) Hilary Putnam Explanation and Reference, In: Glenn Pearce & Patrick Maynard (eds.), Conceptual Change. D. Reidel. pp. 196--214 (1973) In Von einem realistischen Standpunkt, Vincent C. Müller Reinbek 1993 Putnam I (b) Hilary Putnam Language and Reality, in: Mind, Language and Reality: Philosophical Papers, Volume 2. Cambridge University Press. pp. 272-90 (1995 In Von einem realistischen Standpunkt, Vincent C. Müller Reinbek 1993 Putnam I (c) Hilary Putnam What is Realism? in: Proceedings of the Aristotelian Society 76 (1975):pp. 177 - 194. In Von einem realistischen Standpunkt, Vincent C. Müller Reinbek 1993 Putnam I (d) Hilary Putnam Models and Reality, Journal of Symbolic Logic 45 (3), 1980:pp. 464-482. In Von einem realistischen Standpunkt, Vincent C. Müller Reinbek 1993 Putnam I (e) Hilary Putnam Reference and Truth In Von einem realistischen Standpunkt, Vincent C. Müller Reinbek 1993 Putnam I (f) Hilary Putnam How to Be an Internal Realist and a Transcendental Idealist (at the Same Time) in: R. Haller/W. Grassl (eds): Sprache, Logik und Philosophie, Akten des 4. Internationalen Wittgenstein-Symposiums, 1979 In Von einem realistischen Standpunkt, Vincent C. Müller Reinbek 1993 Putnam I (g) Hilary Putnam Why there isn’t a ready-made world, Synthese 51 (2):205--228 (1982) In Von einem realistischen Standpunkt, Vincent C. Müller Reinbek 1993 Putnam I (h) Hilary Putnam Pourqui les Philosophes? in: A: Jacob (ed.) L’Encyclopédie PHilosophieque Universelle, Paris 1986 In Von einem realistischen Standpunkt, Vincent C. Müller Reinbek 1993 Putnam I (i) Hilary Putnam Realism with a Human Face, Cambridge/MA 1990 In Von einem realistischen Standpunkt, Vincent C. Müller Reinbek 1993 Putnam I (k) Hilary Putnam "Irrealism and Deconstruction", 6. Giford Lecture, St. Andrews 1990, in: H. Putnam, Renewing Philosophy (The Gifford Lectures), Cambridge/MA 1992, pp. 108-133 In Von einem realistischen Standpunkt, Vincent C. Müller Reinbek 1993 Putnam II Hilary Putnam Representation and Reality, Cambridge/MA 1988 German Edition: Repräsentation und Realität Frankfurt 1999 Putnam III Hilary Putnam Renewing Philosophy (The Gifford Lectures), Cambridge/MA 1992 German Edition: Für eine Erneuerung der Philosophie Stuttgart 1997 Putnam IV Hilary Putnam "Minds and Machines", in: Sidney Hook (ed.) Dimensions of Mind, New York 1960, pp. 138-164 In Künstliche Intelligenz, Walther Ch. Zimmerli/Stefan Wolf Stuttgart 1994 Putnam V Hilary Putnam Reason, Truth and History, Cambridge/MA 1981 German Edition: Vernunft, Wahrheit und Geschichte Frankfurt 1990 Putnam VI Hilary Putnam "Realism and Reason", Proceedings of the American Philosophical Association (1976) pp. 483-98 In Truth and Meaning, Paul Horwich Aldershot 1994 Putnam VII Hilary Putnam "A Defense of Internal Realism" in: James Conant (ed.)Realism with a Human Face, Cambridge/MA 1990 pp. 30-43 In Theories of Truth, Paul Horwich Aldershot 1994 SocPut I Robert D. Putnam Bowling Alone: The Collapse and Revival of American Community New York 2000 Field IV Hartry Field "Realism and Relativism", The Journal of Philosophy, 76 (1982), pp. 553-67 In Theories of Truth, Paul Horwich Aldershot 1994 |
| Quine, W.V.O. | Verschiedene Vs Quine, W.V.O. | Davidson I 55 CreswellVsQuine: he had a realm of reified experiences or phenomena facing an unexplored reality. Davidson pro - - QuineVsCresswell >Quine III) Kanitscheider II 23 Ontology/language/human/Kanitschneider: the linguistic products of the organism are in no way separated from its producer by an ontological gap. Ideas are certain neuronal patterns in the organism. KanitscheiderVsQuine: Weak point: his empiricism. One must therefore view his epistemology more as a research programme. Quine VI 36 VsQuine: I've been told that the question "What is there?" is always a question of fact and not just a linguistic problem. That is correct. QuineVsVs: but saying or assuming what there is remains a linguistic matter and here the bound variables are in place. VI 51 Meaning/Quine: the search for it should start with the whole sentences. VsQuine: the thesis of the indeterminacy of translation leads directly to behaviorism. Others: it leads to a reductio ad absurdum of Quine's own behaviorism. VI 52 Translation Indeterminacy/Quine: it actually leads to behaviorism, which there is no way around. Behaviorism/Quine: in psychology one still has the choice whether one wants to be a behaviorist, in linguistics one is forced to be one. One acquires language through the behavior of others, which is evaluated in the light of a common situation. It literally does not matter what other kind psychological life is! Semantics/Quine: therefore no more will be able to enter into the semantic meaning than what can also be inferred from perceptible behaviour in observable situations Quine XI 146 Deputy function/Quine/Lauener: does not have to be unambiguous at all. E.g. characterisation of persons on the basis of their income: here different values are assigned to an argument. For this we need a background theory: We map the universe U in V so that both the objects of U and their substitutes are included in V. If V forms a subset of U, U itself can be represented as background theory within which their own ontological reduction is described. XI 147 VsQuine: this is no reduction at all, because then the objects must exist. QuineVsVs: this is comparable to a reductio ad absurdum: if we want to show that a part of U is superfluous, we can assume U for the duration of the argument. (>Ontology/Reduction). Lauener: this brings us to ontological relativity. Löwenheim/Ontology/Reduction/Quine/Lauener: if a theory of its own requires an overcountable range, we can no longer present a proxy function that would allow a reduction to a countable range. For this one needed a much stronger frame theory, which then could no longer be discussed away as reductio ad absurdum according to Quine's proposal. Quine X 83 Logical Truth/Validity/Quine: our insertion definitions (sentences instead of sets) use a concept of truth and fulfillment that goes beyond the framework of object language. This dependence on the concept of ((s) simple) truth, by the way, would also concern the model definition of validity and logical truth. Therefore we have reason to look at a 3rd possibility of the definition of validity and logical truth: it gets by without the concepts of truth and fulfillment: we need the completeness theorem ((s) >provability). Solution: we can simply define the steps that form a complete method of proof and then: Def Valid Schema/Quine: is one that can be proven with such steps. Def Logically True/Quine: as before: a sentence resulting from a valid schema by inserting it instead of its simple sentences. Proof Procedure/Evidence Method/Quine: some complete ones do not necessarily refer to schemata, but can also be applied directly to the propositions, X 84 namely those that emerge from the scheme by insertion. Such methods generate true sentences directly from other true sentences. Then we can leave aside schemata and validity and define logical truth as the sentence generated by these proofs. 1st VsQuine: this tends to trigger protest: the property "to be provable by a certain method of evidence" is uninteresting in itself. It is interesting only because of the completeness theorem, which allows to equate provability with logical truth! 2. VsQuine: if one defines logical truth indirectly by referring to a suitable method of proof, one deprives the completeness theorem of its ground. It becomes empty of content. QuineVsVs: the danger does not exist at all: The sentence of completeness in the formulation (B) does not depend on how we define logical truth, because it is not mentioned at all! Part of its meaning, however, is that it shows that we can define logical truth by merely describing the method of proof, without losing anything of what makes logical truth interesting in the first place. Equivalence/Quine: important are theorems, which state an equivalence between quite different formulations of a concept - here the logical truth. Which formulation is then called the official definition is less important. But even mere terms can be better or worse. Validity/logical truth/definition/Quine: the elementary definition has the advantage that it is relevant for more neighboring problems. 3. VsQuine: with the great arbitrariness of the choice of the evidence procedure it cannot be excluded that the essence of the logical truth is not grasped. QuineVsVs: how arbitrary is the choice actually? It describes the procedure and talks about strings of characters. In this respect it corresponds to the sentence. Insertion definition: it moves effectively at the level of the elementary number theory. And it stays at the level, while the other definition uses the concept of truth. That is a big difference. |
Davidson I D. Davidson Der Mythos des Subjektiven Stuttgart 1993 Davidson I (a) Donald Davidson "Tho Conditions of Thoughts", in: Le Cahier du Collège de Philosophie, Paris 1989, pp. 163-171 In Der Mythos des Subjektiven, Stuttgart 1993 Davidson I (b) Donald Davidson "What is Present to the Mind?" in: J. Brandl/W. Gombocz (eds) The MInd of Donald Davidson, Amsterdam 1989, pp. 3-18 In Der Mythos des Subjektiven, Stuttgart 1993 Davidson I (c) Donald Davidson "Meaning, Truth and Evidence", in: R. Barrett/R. Gibson (eds.) Perspectives on Quine, Cambridge/MA 1990, pp. 68-79 In Der Mythos des Subjektiven, Stuttgart 1993 Davidson I (d) Donald Davidson "Epistemology Externalized", Ms 1989 In Der Mythos des Subjektiven, Stuttgart 1993 Davidson I (e) Donald Davidson "The Myth of the Subjective", in: M. Benedikt/R. Burger (eds.) Bewußtsein, Sprache und die Kunst, Wien 1988, pp. 45-54 In Der Mythos des Subjektiven, Stuttgart 1993 Davidson II Donald Davidson "Reply to Foster" In Truth and Meaning, G. Evans/J. McDowell Oxford 1976 Davidson III D. Davidson Essays on Actions and Events, Oxford 1980 German Edition: Handlung und Ereignis Frankfurt 1990 Davidson IV D. Davidson Inquiries into Truth and Interpretation, Oxford 1984 German Edition: Wahrheit und Interpretation Frankfurt 1990 Davidson V Donald Davidson "Rational Animals", in: D. Davidson, Subjective, Intersubjective, Objective, Oxford 2001, pp. 95-105 In Der Geist der Tiere, D Perler/M. Wild Frankfurt/M. 2005 Kanitsch I B. Kanitscheider Kosmologie Stuttgart 1991 Kanitsch II B. Kanitscheider Im Innern der Natur Darmstadt 1996 Quine I W.V.O. Quine Word and Object, Cambridge/MA 1960 German Edition: Wort und Gegenstand Stuttgart 1980 Quine XIII Willard Van Orman Quine Quiddities Cambridge/London 1987 |
| Smart, J. C. | Quine Vs Smart, J. C. | II 118 ff The Oxford trained philosopher today turns one ear to common sense and the other one to science. Historians who do not want to be outflanked claim that the real driving force behind development was fashion. Even quantum theorists are heard to say that they do not attribute reality so much to the tiny objects of their theory as primarily to their experimental apparatuses, i.e. to ordinary things. In refreshing contrast to that is the Australian philosopher Smart: he represents a shamelessly realistic conception of physical elementary particles. The worldview of the physicist is not only ontologically respectable, but his language gives us a truer picture of the world than common sense. (Smart mainly studies physics). There have also been materialists who believe that living beings are indeed material, but subject to biological and psychological laws, which cannot be reduced to physical laws in principle. This was the emergence materialism. Smart's materialism is more robust than that. II 119 Smart Thesis: He denies that there are any laws in the strict sense in psychology and biology at all. The statements there are site-specific generalizations about some terrestrial plants of our acquaintance. SmartVsEmergence. They are at the same level as geography or reports on consumer behavior. That even applies to statements about cell division. They will most likely be falsified at least elsewhere in outer space, if not even here with us. (Law: explanatory force) Smart admits that statements about the small processes in biology tend to have more explanatory force. (Precisely, they come indeed closer to physicochemistry.) Biology describes a site-specific outgrowth, while physics describes the nature of the world. Psychology then describes an outgrowth on this outgrowth. II 120 Colors: Smart on the color concept: Color dominates our sensory experience, with its help we distinguish objects. But, that's the point of Smart's explanations: color differences rarely have an interesting connection to the laws of physics: a mixed color can appear to us as a pure one depending on contingent mechanisms inside us. It can be assumed that extraterrestrial beings have similar concepts of distance and electric charge, but hardly similar concepts of color. To view the world sub specie aeternitatis we have to avoid the concept of color and other secondary qualities. Primary: length, weight, hardness, shape, etc. are those that are easiest to incorporate in physical laws. For Smart, physicalism wins. On the subject of "humans as machines", today's opponents of mechanistic thought refer to Godel's theorem, which states that no formal proof method can cover the entire number theory. II 121 Smart, who represents the mechanistic view, argues against this rather gloomy application of the great Gödel theorem. The place where man defies the barriers of formal proof theory is that of the informal and largely resultless maneuvers of scientific method. Determinism: Smart agrees with Hobbes that >determinism and freedom are not antithetic to one another: deterministic action is considered free if it is in a certain way mediated by the agent. Ethics: The differentiation of activities for which one can be responsible, and those for which this is not true, follows the social apparatus of rewarding and punishing. Responsibility is assigned a place where reward and punishment tended to work. Disposition/Smart: This corresponds to an important element in the use of "he could have done." Smart continues to infer on "it could have" (e.g. broken). He brings this into context with the incompleteness of information relating to causal circumstances. Quine: I welcome this thesis for modalities. These modalities are not based on the nature of the world, but on the fact that we ourselves, e.g. because of ignorance, disregard details. There is a conception mocked by Smart, according to which the present moment moves forward through time at a velocity of sixty seconds per minute. Furthermore, there is the idea that sentences about the future are neither true nor false. Otherwise fatalism would get the the reins in his hand. Such thoughts are widespread and confused and partially go back to Aristotle. They have been put right with great clarity by Donald Williams et al. As Smart puts them right again, distinctive details are added. II 122 Incredible contrast between probability and truth. Smart: "probably" is an indicator; such as "I", "you" "now" "then" "here", "there". A word that depends on the use situation. For a specific statement of fact is, if at all, true at all times, whether we know it or not, but even then it can be more or less probable, depending on the situation. So modality concept of probability finally ends in subjective ambiguity, like the modalities. Quine: Smart is an honest writer. Smart overcomes all moral dilemmas; the materialist takes the bull by the horns and effortlessly wins over the moralists! |
Quine I W.V.O. Quine Word and Object, Cambridge/MA 1960 German Edition: Wort und Gegenstand Stuttgart 1980 |
| 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 XIII Willard Van Orman Quine Quiddities Cambridge/London 1987 |
| Tarski, A. | Field Vs Tarski, A. | Brendel I 68 T-Def/FieldVsTarski: does not do justice to physicalistic intuitions. (Field 1972). Semantic concepts and especially the W concept should be traceable to physical or logical-mathematical concepts. Tarski/Brendel: advocates for a metalinguistic definition himself that is based only on logical terms, no axiomatic characterization of "truth". (Tarski, "The Establishment of Scientific Semantics"). Bre I 69 FieldVsTarski: E.g. designation: Def Designation/Field: Saying that the name N denotes an object a is the same thing as stipulating that either a is France and N is "France" or a is Germany and N is "Germany"... etc. Problem: here only an extensional equivalence is given, no explanation of what designation (or satisfiability) is. Bre I 70 Explanation/FieldVsTarski/Field: should indicate because of which properties a name refers to a subject. Therefore, Tarski’s theory of truth is not physicalistic. T-Def/FieldVsTarski/Field/Brendel: does not do justice to physicalistic intuitions - extensional equivalence is no explanation of what designation or satisfiability is. Field I 33 Implication/Field: is also in simpler contexts sensibly a primitive basic concept: E.g. Someone asserts the two sentences. a) "Snow is white" does not imply logically "grass is green". b) There are no mathematical entities such as quantities. That does not look as contradictory as Fie I 34 John is a bachelor/John is married FieldVsTarski: according to him, a) and b) together would be a contradiction, because he defines implication with quantities. Tarski does not give the normal meaning of those terms. VsField: you could say, however, that the Tarskian concepts give similar access as the definition of "light is electromagnetic radiation". FieldVsVs: but for implication we do not need such a theoretical approach. This is because it is a logical concept like negation and conjunction. Field II 141 T-Theory/Tarski: Thesis: we do not get an adequate probability theory if we just take all instances of the schema as axioms. This does not give us the generalizations that we need, for example, so that the modus ponens receives the truth. FieldVsTarski: see above Section 3. 1. Here I showed a solution, but should have explained more. Feferman/Field: Solution: (Feferman 1991) incorporates schema letters together with a rule for substitution. Then the domain expands automatically as the language expands. Feferman: needs this for number theory and set theory. Problem: expanding it to the T-theory, because here we need scheme letters inside and outside of quotation marks. Field: my solution was to introduce an additional rule that allows to go from a scheme with all the letters in quotation marks to a generalization for all sentences. Problem: we also need that for the syntax,... here, an interlinking functor is introduced in (TF) and (TFG). (see above). II 142 TarskiVsField: his variant, however, is purely axiomatic. FieldVsTarski/FefermanVsTarski: Approach with scheme letters instead of pure axioms: Advantages: 1) We have the same advantage as Feferman for the schematic number theory and the schematic set theory: expansions of the language are automatically considered. 2) the use of ""p" is true iff. p" (now as a scheme formula as part of the language rather than as an axiom) seems to grasp the concept of truth better. 3) (most important) is not dependent on a compositional approach to the functioning of the other parts of language. While this is important, it is also not ignored by my approach. FieldVsTarski: an axiomatic theory is hard to come by for belief sentences. Putnam I 91 Correspondence Theory/FieldVsTarski: Tarski’s theory is not suited for the reconstruction of the correspondence theory, because fulfillment (of simple predicates of language) is explained through a list. This list has the form "Electron" refers to electrons "DNS" refers to DNS "Gene" refers to genes. etc. this is similar to (w) "Snow is white" is true iff.... (s)> meaning postulates) Putnam: this similarity is no coincidence, because: Def "True"/Tarski/Putnam: "true" is the zero digit case of fulfillment (i.e. a formula is true if it has no free variables and the zero sequence fulfills it). Def Zero Sequence: converges to 0: E.g. 1; 1/4; 1/9; 1/16: ... Criterion W/Putnam: can be generalized to the criterion F as follows: (F for fulfillment): Def Criterion F/Putnam: (F) an adequate definition of fulfilled in S must generate all instances of the following scheme as theorems: "P(x1...xn) is fulfilled by the sequence y1...yn and only if P(y1...yn). Then we reformulate: "Electron (x)" is fulfilled by y1 iff. y1 is an electron. PutnamVsField: it would have been formulated like this in Tarskian from the start. But that shows that the list Field complained about is determined in its structure by criterion F. This as well as the criterion W are now determined by the formal properties we desired of the concepts of truth and reference, so we would even preserve the criterion F if we interpreted the connectives intuitionistically or quasi intuitionistically. Field’s objection fails. It is right for the realist to define "true" à la Tarski. |
Field IV Hartry Field "Realism and Relativism", The Journal of Philosophy, 76 (1982), pp. 553-67 In Theories of Truth, Paul Horwich Aldershot 1994 Bre I E. Brendel Wahrheit und Wissen Paderborn 1999 Putnam I Hilary Putnam Von einem Realistischen Standpunkt In Von einem realistischen Standpunkt, Vincent C. Müller Frankfurt 1993 SocPut I Robert D. Putnam Bowling Alone: The Collapse and Revival of American Community New York 2000 |
| Various Authors | Quine Vs Various Authors | II 111ff QuineVsSemantic Theory: there is a lack of a general definition of meaning QuineVsUse Theory of Meaning: definition of meaning through use too vague! (Demarcation of what is detectable under the "circumstances") (QuineVsWittgenstein). III 272 Singular Term/QuineVsSingular Terms: the whole category of singular terms is logically superfluous and should be abolished! ((s) Instead: variable). V 58 Language Learning/Language Acquisition/Quine: E.g. the child learns that "red" is applied to blood, tomatoes, ripe apples, etc. The idea associated with that may be whatever it likes! Language bypasses the idea and focuses on the object. ((s) reference/(s): goes to the object, not an idea, which is in this case unnecessary.) Stimulus/Quine: has nothing mysterious in language learning. V 60 Problem: in progressive learning sentences are formed which have less to do with stimuli. E.g. about past and future. Quine: philosophers have great difficulty to specify accurately and in detail which connections it is about. QuineVsSupranaturalism. V 61 We only need orientation by external circumstances. Internal mechanisms are only insofar positive as we can hope that they will be clarified by neurophysiology. IX 199 Individuals/QuineVsFraenkel: we cannot follow him to simply waive individuals, because under TT this would exclude infinite classes and also the classical number theory. (Chapter 39). Solution: (from Chapter 4): the identification of individuals with their One classes. IX 199/200 But then we would have to make an exception in the interpretation: if x is an individual, then "x ε x" should count as true. (Above, "x ε y" became false if neither were objects of sequential type). Now (1) and (2) reduce to: (4) Ey∀x(x ε y (Tnx u Fx)), (5) (∀w(w ε x w ε y) u x ε z) > y ε z. Moreover, the definition of "Tnx" needs to be revised to make it match the new idea of the individual: " x VT y" by way of merging we can define (6) "T0x" stands for "∀y (y ε x y = x)" ((S) "all parts of individuals are identical with this one".) "T n + 1 x" stands for "∀y(y ε x > Tny)" ((s) "The set x is always one type higher than its elements y".) IX 237 Set Theory/QuineVsAckermann: (like ML and NB) but unlike ZF: does not fully guarantee the existence of finite classes. Additional concept "M". II 129 QuineVsZettsky: Zettsky: properties are identical if the classes to which they belong are the identical... but when are such classes identical? II 130 We cannot rely on the identity of the elements here (as with physical objects), as we simply have no antecedent principle of individuation for the properties (as elements of classes) here. |
Quine I W.V.O. Quine Word and Object, Cambridge/MA 1960 German Edition: Wort und Gegenstand Stuttgart 1980 Quine XIII Willard Van Orman Quine Quiddities Cambridge/London 1987 |
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The author or concept searched is found in the following theses of the more related field of specialization.
| Disputed term/author/ism | Author |
Entry |
Reference |
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| Decidability | Field, Hartry | II X Field: Thesis: I am very reluctant to say that undecidable questions of number theory have no specific truth value. II 349 Goedel-Theorem/Undecidability/Truth Value/Field: Thesis: We have seen that the Goedel-Theorem gives no reason to think that some undecidable propositions have certain truth values. ((s) That would be an objectivist view >objectivism). II 351 Uncertainty/undecidability/set theory/number theory/Field: Thesis: not only in the set theory but also in the number theory many undecidable propositions do not have a certain truth value. |
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