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Philosophical and Scientific Issues in Dispute

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The author or concept searched is found in the following 6 entries.

Disputed term/author/ism	Author	Entry	Reference
Calculus	Mates	I 63 artificial language/formal language/counterpart/Mates: the statements of the natural language correspond the artificial formulas, as a counterpart, not as abbreviations. >Symbols, >Equivalence, >Propositional forms, >Propositional functions, >Formal language, >Natural language. If symbols are associated with no sense, then it is an uninterpreted calculus. >Interpretation/Mates. I 115 Propositional calculus: the propositional calculus has no quantifiers. >Propositional calculus, >Quantifiers, >Quantification.	Mate I B. Mates Elementare Logik Göttingen 1969 Mate II B. Mates Skeptical Essays Chicago 1981
Correctness	Logic Texts	II 109 Def correct/correctness/propositional logic/Hoyningen-Huene: be A and B statement logical formulas. The conclusion from A to B is called propositionally correct, exactly when A > B is propositionally true. II 110 The trick is that in [the above] definition the required propositional truth of A > B means different things, depending on whether A > B is a statement or a propositional formula. >Formula, >Statement, >Proposition, >Truth, >Logical truth.	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
Formal Language	Mates	I 63 Artificial language/formal/counterpart/Mates: the statement forms of the natural language comply with formulas of the artificial, namely as a counterpart, not as abbreviations. >Propositional forms, >Propositional functions, >Natural language, >Equivalence. If symbols are not assigned to meaning, then "uninterpreted calculus". >Interpretation, >Sense, >Symbols. I 74 artificial language L/Mates: E.g. statement j: always true in relation to an interpretation I - values of "j": statements of the language L - values of I: interpretations of L. Cf. >Value progression/Frege, >Ideal language, >Universal language.	Mate I B. Mates Elementare Logik Göttingen 1969 Mate II B. Mates Skeptical Essays Chicago 1981
Implication	Logic Texts	II 109 Implication: Instead of a logically correct conclusion, one also speaks of a valid or deductive conclusion, instead of conclusion one also speaks of implication. The premises imply the conclusion. Def correct/correctness/statement logic/Hoyningen-Huene: be A and B statement logical formulas. The conclusion from A to B is called propositionally correct, exactly when A > B is propositionally true. >Correctness. II 110 The trick is that in [the above] definition the required propositional truth of A > B means different things, depending on whether A > B is a statement or a propositional formula. >Statement, >Formula.	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
Incompleteness	Gödel	Thiel I 227 ff Incompleteness Theorem/Goedel/Thiel: ... this metamathematical statement corresponds in F to a one-digit statement form G(x) which then must occur somewhere in the counting sequence. If G(x) takes the h'th place, it is therefore identical with the propositional form called Ah(x) there. Goedel's result will be, that in F neither the proposition G(h) arising from G(x) by the insertion of h nor its negative ~G(h) is derivable. "Undecidable in F". Suppose G(h) is derivable in F, then only the derivation of true statements would be allowed, so G(h) would also be true. Thus, since G(x) was introduced as an image of $Ax(x) in F, $Ah(h) would be valid. But that would mean, since Ah(x) is identical with G(x), $G(h). G(h) would therefore be non-derivable in F - this is a contradiction. >Derivation, >Derivability. This derivation first only proves the validity of the "if-then-statement" S G(h)>$ G(h). This must now be inserted: (S G(h)>$ G(h))> $ G(h). This follows from the general scheme (A>~A)>~A. On the other hand, if we then assume that the negative ~G(h) is derivable, then ~G(h) would also be true. This would be equivalent to the validity of ~$ Ah(h) thus with S Ah(h). Thiel I 228 This in turn agrees with S G(h), so that both assertion and negative would be derivable, and we would have a formal contradiction. If F is contradiction-free at all, our second assumption S ~G(h) is not valid either. This is an undecidable assertion. Cf. >Decidability, >Indecidability. Thiel I 228 This proof sketch establishes a program. Important roles in the execution of this program are played by the "Goedelization" and the so-called "negative representability" of certain relations in F. Def Goedelization: Goedelization is first of all only a reversibly definite assignment of basic numbers to character sequences. We want to put the expressions of F into bracket-free form. >Goedel numbers. For this we write the logical connective signs not between, but in front of the expressions. We write the logical operators as "indices" to the order functor G. Terminology order functor G. Quantifiers: we treat quantifiers as two-digit functors whose first argument is the index, the second the quantified propositional form. >Quantifiers, >Quantification. Thiel I 229 Then the statement (x)(y)(z) ((x=y)>(zx = zy) gets the form (x)(y)(z)G > G = xyG = G times zxG times zy. We can represent the members of the infinite variable sequences in each case by a standard letter signaling the sort and e.g. prefixed points: thus for instance x,y,z,...by x,°x,°°x,...As counting character we take instead of \|,\|\|,\|\|\|,... zeros with a corresponding number of preceding dashes 0,'0,''0,... >Sequences. With this convention, each character in F is either a 0 or one of the one-digit functors G1 (the first order functor!), ', ~. Two-digit is G2, three-digit is G4, etc. Thiel I 229 E.g. Goedelization, Goedel number, Goedel number: Prime numbers are assigned in each case:.... Primes. Thiel I 230 In this way, each character string of F can be uniquely assigned a Goedel number and told how to compute it. Since every basic number has a unique representation as a product of prime numbers, it can be said of any given number whether it is a Goedel number of a character string of F at all. Metamathematical and arithmetical relations correspond to each other: example: Thiel I 230 We replace the x by 0 in ~G=x'x and obtain ~G = 0'0. The Goedel number of the first row is: 2²³ x 3¹³ x 5³⁷ x 7²⁹ x 11³⁷, the Goedel number of the second row of characters is: 2²³ x 3¹³ x 5³¹ x 7²⁹ x 11³¹. The transition from the Goedel number of the first row to that of the second row is made by division by 5⁶ x 11⁶ and this relation (of product and factor) is the arithmetic relation between their Goedel numbers corresponding to the metamathematical relation of the character rows. Thiel I 231 These relations are even effective, since one can effectively (Goedel says "recursively") compute the Goedel number of each member of the relation from those of its remaining members. >Recursion. The most important case is of course the relation Bxy between the Goedel number x, a proof figure Gz1...zk and the Goedel number y of its final sequence... Thiel I 233 "Negation-faithful representability": Goedel shows that for every recursive k-digit relation R there exists a k-digit propositional form A in F of the kind that A is derivable if R is valid, and ~A if R does not (..+..). We say that the propositional form A represents the relation R in F negation-faithfully. Thiel I 234 After all this, it follows that if F is ω-contradiction-free, then neither G nor ~G is derivable in F. G is an "undecidable statement in F". The occurrence of undecidable statements in this sense is not the same as the undecidability of F in the sense that there is no, as it were, mechanical procedure. >Decidability. Thiel I 236 It is true that there is no such decision procedure for F, but this is not the same as the shown "incompleteness", which can be seen from the fact that in 1930 Goedel had proved the classical quantifier logic as complete, but there is no decision procedure here, too. Def Incomplete/Thiel: a theory would only be incomplete if a true proposition about objects of the theory could be stated, which demonstrably could not be derived from the axiom system underlying the theory. ((s) Then the system would not be maximally consistent.) Whether this was done in the case of arithmetic by the construction of Goedel's statement G was for a long time answered in the negative, on the grounds that G was not a "true" arithmetic statement. This was settled about 20 years ago by the fact that combinatorial propositions were found, which are also not derivable in the full formalism. Goedel/Thiel: thus incompleteness can no longer be doubted. This is not a proof of the limits of human cognition, but only a proof of an intrinsic limit of the axiomatic method. Thiel I 238 ff One of the points of the proof of Goedel's "Underivability Theorem" was that the effectiveness of the metamathematical derivability relation corresponding to the self-evident effectiveness of all proofs in the full formalism F, has its exact counterpart in the recursivity of the arithmetic relations between the Goedel numbers of the proof figures and final formulas, and that this parallelism can be secured for all effectively decidable metamathematical relations and their arithmetic counterparts at all. >Derivation, >Derivability.	Göd II Kurt Gödel Collected Works: Volume II: Publications 1938-1974 Oxford 1990 T I Chr. Thiel Philosophie und Mathematik Darmstadt 1995
Inserting	Logic Texts	II 133 Insertion/substitution/identity/truth preservation: Logical equivalence is (...) a weakening of the identity of statements. Logically equivalent statements are not the same in all properties, but only in logical terms. If one statement is logically true, so is the other and vice versa. If a certain statement follows logically from one, then it follows logically from the other and vice versa. >Substitution, >Substitution (Insertion), >Equivalence, >Logical truth. Insertion Theorem: Let FA be a propositional logical formula which contains a partial form A. Let FB be a formula which results from FA when A is replaced by a propositional formula B, (not necessarily everywhere). If A is now ≡ B, then FA ≡ FB applies. II 134 Logically equivalent formulas have the same inference sets. Logically equivalent formulas can be inferred from the same prerequisites. Redundancy Theory/Hoyningen-Huene: therefore, in propositional logic one does not really have to distinguish between "A" and "It is true that A". (In propositional logic such properties are abstracted from). >Redundancy theory, >Propositional logic.	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

The author or concept searched is found in the following controversies.

Disputed term/author/ism	Author Vs Author	Entry	Reference
Hume, D.	Sellars Vs Hume, D.	I 9 Imagination/Hume: the accumulation of ideas is imagination, it is not a fortune, but a reservoir, a collection without Scrapbook, a theater without a stage, a "river of perceptions". (SellarsVsHume). The place is not different from the action in which it takes place, the idea is not given in a subject. I 10 The imagination is not a factor, not a mediator, no decisive determination. Nothing happens by, everything happens in the imagination. Sellars II 327 Hume/Sellars: our "perceptions" are "images" of facts in a general spatiotemporal world. The uniformities of natural events tend to be reflected in our "ideas" as uniformities. The difference between acts of thinking and lightning obliterates. SellarsVsHume: he has difficulties in explaining the reference relation between a present idea and an earlier event. II 328 He notes the propositional form of his "ideas" not sufficiently and spoils himself thus the way to an explicit explanation of the difference between the conclusions Lightning now, so soon thunder and: Yesterday at 10:00 lightning, so yesterday at 10:01 thunder. SellarsVsHume: because it does not clearly distinguish between thoughts and impressions, it can be assumed that a natural derivative corresponds not only a logical but also a temporal sequence. His theory must be extended so that it also includes cases such as the above or backwards: Now thunder, therefore a moment ago lightning. Perception/complex/SellarsVsHume: he does not explicitly state that the perception of a configuration itself is a configuration of perceptions! Although this is true in the core, in principle, the previously troubles if you understand "perceptions" in the sense of "sensation or impression".	Sellars I Wilfrid Sellars The Myth of the Given: Three Lectures on the Philosophy of Mind, University of London 1956 in: H. Feigl/M. Scriven (eds.) Minnesota Studies in the Philosophy of Science 1956 German Edition: Der Empirismus und die Philosophie des Geistes Paderborn 1999 Sellars II Wilfred Sellars Science, Perception, and Reality, London 1963 In Wahrheitstheorien, Gunnar Skirbekk Frankfurt/M. 1977

The author or concept searched is found in the following disputes of scientific camps.

Disputed term/author/ism	Pro/Versus	Entry	Reference
Propositional Knowledge	Versus	Frank I 325 Proposition/Lewis/Chisholm: VsPropositional Knowledge: instead: self-ascription of (non-propositional) features - Castaneda: late: "Guise": objects are objective guises (the ego) - this also in propositional form. Hector-Neri Castaneda (1987b): Self-Consciousness, Demonstrative Reference, and the Self-Ascription View of Believing, in: James E. Tomberlin (ed) (1987a): Critical Review of Myles Brand's "Intending and Acting", in: Nous 21 (1987), 45-55 James E. Tomberlin (ed.) (1986): Hector-Neri.Castaneda, (Profiles: An International Series on Contemporary Philosophers and Logicians, Vol. 6), Dordrecht 1986	Fra I M. Frank (Hrsg.) Analytische Theorien des Selbstbewusstseins Frankfurt 1994

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
Propos. Attitude	Chisholm, R.	I 49 intentional attitudess / propositional att./ Chisholm: 1st each of these settings has a primary form that does not have a propositional object. 2nd we can characterize these familiar forms de re and de dicto by primary non-propositional forms.	Link to abbreviations/authors