Dictionary of Arguments

Philosophical and Scientific Issues in Dispute

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The author or concept searched is found in the following 8 entries.

Disputed term/author/ism	Author	Entry	Reference
Church-Turing Thesis	Lorenzen	Berka I 266 Church thesis/Lorenzen: the thesis is an equation of "constructive" with "recursive". >Constructivism, >Recursion, >Recursivity. LorenzenVsChurch: this is a too narrow view: thus it no longer permits the free use of the quantification over the natural numbers. >Quantification, >Numbers, >Infinity. I 267 Decision-making problem/ChurchVsLorenzen: (according to Lorenzen): Advantage: greater clarity: when limiting to recursive statements, there can never be a dispute as to whether one of the admitted statements is true or false. The definition of recursiveness guarantees precisely the decision-definition, that is, the existence of a decision-making process. >Decidability, >decision problem.⁽¹⁾ 1. P. Lorenzen, Ein dialogisches Konstruktivitätskriterium, in: Infinitistic Methods, (1961), 193-200	Lorn I P. Lorenzen Constructive Philosophy Cambridge 1987 Berka I Karel Berka Lothar Kreiser Logik Texte Berlin 1983
Cybernetics	Foerster	Brockman I 209 Cybernectics/circles/circularity/Foerster: The substance of what we have learned from cybernetics is to think in circles: A leads to B, B to C, but C can return to A. >Circularity, >Recursion, >Thinking. Such kinds of arguments are not linear but circular. The significant contribution of cybernetics to our thinking is to accept circular arguments. This means that we have to look at circular processes and understand under which circumstances an equilibrium, and thus a stable structure, emerges. >Recursivity, >Processes, >Replication, cf. >Artificial Intelligence, >Artificial consciousness, >Robots, >Machine Learning. Obrist, H.U. “Making the Invisible Visible: Art Meets AI”, in: Brockman, John (ed.) 2019. Twenty-Five Ways of Looking at AI. New York: Penguin Press.	Förster I H. von Foerster Wissen und Gewissen Frankfurt 1997 Brockman I John Brockman Possible Minds: Twenty-Five Ways of Looking at AI New York 2019
Decidability	Quine	II 112 Decidability/Proof-Theoretical Analogy/Quine: the Concept of mechanical procedure is recursivity (e.g. to prove or even formulate Goedel's theorem or Church's theorem of undecidability.) But to prove the decidability of the theory we do not need a definition of the mechanical process, we simply present a method that everyone would call mechanical. >Proofs, >Provability. II 191ff, Undecidable Logics: is a general theory for a single symmetrical two-digit predicate. II 198 Also undecidable: is a general theory of two-digit formulas that have no quantifiers except (Ex)(y)(Ez).	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
Excluded Middle	Lorenzen	Berka I 271 Sentence of the excluded middle/Dialogical Logic/intuitionistic/logical constants/Lorenzen: If the particle is given its dialogical meaning also in the meta-language, then one can no longer generally prove the only classical valid A v i A. >Dialogical logic, >Provability, >Metalanguage, >Logical particles, >Intuitionism. Solution/Gentzen: one considers the sequences with additional infinite rules: (n)A > B(n) v C > A > (x)B(x) v C (n)A u B(n) > C > A u (Ex)B(x) > C which are allowed for derivation. Axiom: all sequences are allowed as axioms A u p > q v B for false or true constant prime formulas p or q. >G. Gentzen. LorenzenVsRecursiveness/LorenzenVsFormalism: this is no longer a formalism in the sense of a definition of a recursive enumeration, but a "semi-formalism" (concept by Schütte). >Recursion, >Recursivity. Trivially, this is consistent. Any formula that can be derived from Peano's arithmetic is it also here. >Consistency. This is a "constructive" consistency proof, if the dialogical procedure is recognized as constructive. >Constructivism. I 272 Infinity/premisses/dialogical logic/Lorenzen: one can state a step number l < e0 to each formula that can be derived in the Peano formalism with the following: e0 = ω to the power of ω to the power of ω to the power of ... P can thus first calculate an ordinal number e The calculation process is recursive, so even in the narrowest sense constructive. >Recursion. The statements that are used in the consistency proof are generally not recursive.⁽¹⁾ 1. P. Lorenzen, Ein dialogisches Konstruktivitätskriterium, in: Infinitistic Methods, (1961), 193-200	Lorn I P. Lorenzen Constructive Philosophy Cambridge 1987 Berka I Karel Berka Lothar Kreiser Logik Texte Berlin 1983
Finiteness	Hilbert	Thiel I 245 Finite/Hilbert: in the sense of Hilbert, it is only a question of how statements about infinite objects can be justified by means of "finite" methods. >Infinity, >Circularity, cf. >Recursion, >Recursivity. Hilbert found the finiteness in the "operational" method, especially of the combinatorics, arithmetics, and elemental algebra already exemplarily realized. They were "genetically" (constructively) built up into the second third of the 19th century, while the construction of geometry was a prime example for the axiomatic structure of a discipline. >Constructivism, >Geometry, >Number theory, >Arithmetics, >Axioms, >Axiom systems. I 246 Each finite operation is an area that is manageable for the person who is acting. This area can change during the process. I 247 The fact that the arithmetic functions required for Goedel's proof are even primitively recursive is remarkable in that not all effectively computable functions are primitively recursive, and the primitive recursive functions are a true subclass of the computable functions. >K. Gödel, >Completeness/Gödel, >Incompleteness/Gödel. I 248 An effectively computable, but not primitive, recursive function is e.g. explained by the following scheme for the calculation of their values (not proved) (x 'is the successor of x): ψ(0,n) = n' ψ(m',0) = ψ(m,1) ψ(m',n')= ψ(m,ψ(m',n)). (I 247) If one wants to approach the general concept of comprehensibility, one has to accept the so-called μ operator as a new means of expression. Thiel I 249 Computability/Church/Thiel: how close is this to a concept of "general computability"? There is the concept of "Turing computability", the concept of the "l definability" in Church and the "canonical systems" in Post. >Calculability, >A. Turing, >E. Post. Each function, which is in one of these classes, is also demonstrable in the others. Church has then uttered the presumption that with this an adequate clarification of the general concept of computability is achieved. >Church Thesis. But it means that this is a "non-mathematical" presumption, and is not capable of any mathematical proof. It is an intuitive term: whether such a specification is "adequate" cannot be answered with mathematical means. >Proofs, >Provability, >Adequacy. I 250 Apart from finiteness and constructivity, there remain other questions: none of the definitions for the offered functional classes is finite: e.g. μ-recursive functions. The attempt to describe effective executability with classical means remains questionable, but if we interpret the existence quantifier constructively, we have already assumed the concept of constructivity. >Quantification, >Quantifiers, >Existential quantification.	T I Chr. Thiel Philosophie und Mathematik Darmstadt 1995
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
Infinity	Lorenzen	Berka I 266 "Over-countable"/infinite/LorenzenVsSet theory: fable realm of the "Over-Countable". ((s) is not constructible). >constructivism, >Set theory. Berka I 272 Infinite/premisses/dialogical logic/Lorenzen: one can state a step number l e0 = ω exp ω exp ω exp... P can thus first calculate an ordinal number I >Derivation, >Derivability. The calculation process is recursive, so even in the narrowest sense constructive. >Constructivism, >Recursion, >Recursivity, >Calculability. The statement forms that are used in the consistency proof are generally not recursive.⁽¹⁾ >Consistency, >Proofs, >Provability. 1. P. Lorenzen, Ein dialogisches Konstruktivitätskriterium, in: Infinitistic Methods, (1961), 193-200	Lorn I P. Lorenzen Constructive Philosophy Cambridge 1987 Berka I Karel Berka Lothar Kreiser Logik Texte Berlin 1983
Set Theory	Lorenzen	Berka I 269 Inductive Definition/Set Theory/LorenzenVsSet Theory: For example, an inductive definition of a set M by a(y) › y ε M, x ε M u b(x,y) › y ε M whereby a (x) and b(x,y) are already defined formulas in which M does not occur, is "explained" set theoretically that M should be the average of all sets N satisfying these implications with N instead of M. Lorenzen: whoever wants to defend a claim n ε M (sic) will hardly attempt all these sets N. As P, he will rather defend O against either directly a(n), or he will first give an m which he will defend b (m, n) and m ∈ M. >Dialogical logic. Step number/Lorenzen: in order to determine this procedure as the the dialogical sense of the inductive definition of M, we must also require of P to indicate the number of steps required for complete proof for each assertion of the form x ∈ M. >Step number. E.g. suppose, for example, he traces n ε M back to the assertion m ∈ M and has stated the step number v for n ε M... I 270 ...so he must specify a step number μ ‹ v for m ε M. Without such information, P could assert "smaller" ‹ for the integers in the following inductive definition 0 ‹ y for positive numbers y x ‹ y _› x +/ 1 ‹ y +/ 1 e.g. 1 ‹ 0, and begin a "proof" with the aid of 0 ‹ 1, 1 ‹ 2, 2 ‹ 3 .... Of course, the proof could not be finished, but O could not prove this. Dialogical logic/Lorenzen: in these dialogues, it is never permitted to intervene suddenly in the "free speech" of the opponent. If, on the other hand, P has to specify a step number v, he will have lost his assertion at the latest after v steps. Step number: the steps are, of course, natural numbers. If one wants to give infinite inductive definitions, i.e. such with an infinite number of premisses, a dialogical meaning, one must allow transfinite ordinal numbers as the step numbers. Inductive Definition/LorenzenVsHerbrand: For example, a function sequence f1, f2 ... is already defined and the induction scheme a(y) › y ε M (x)fx(y) ε M › y ε M is adressed. This definition is by no means "impredicative". >Imprecativeness. But it is also not really constructive either. We have infinitely many premises here f1 (y) ∈ M, f2 (y) ε M ... which are necessary to prove y ∈ M. Infinite: in dialogue one cannot defend every premise, one will therefore allow O to select an fm(y) e M. This must then be claimed and defended by P. In addition, P must specify a generally transfinite ordinal number as the step number. Step number: the step number of a premise must always be specified as less than the step number of the conclusion. Winning strategy: of P: must provide the step numbers for all opponent's elections. II. Number-class/second/Lorenzen: set-theoretically one can prove easily the existence of suitable ordinal numbers of the II. number class. One can define transfinite recursion through this: y ε M0 ‹› a(y) y ε Mλ ‹› (x)fx(y) ε Ux x ‹ λ Mx. . Then M = Ul l › μ Ml for a suitable μ and if M is to be a set of natural numbers, μ can be taken from the II. number class. Constructively, if the inductive definition is to be constructive, the ordinal numbers used must also be "constructive". Here it is obvious to limit oneself to the recursive ordinal numbers of Church and Kleene.⁽¹⁾ >Constructivism, >Intuitionism, >Recursion, >Recursivity. 1. P. Lorenzen, Ein dialogisches Konstruktivitätskriterium, in: Infinitistic Methods, (1961), 193-200	Lorn I P. Lorenzen Constructive Philosophy Cambridge 1987 Berka I Karel Berka Lothar Kreiser Logik Texte Berlin 1983

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

Disputed term/author/ism	Author Vs Author	Entry	Reference
Church, A.	Lorenzen Vs Church, A.	Berka I 266 Church thesis/Lorenzen: the thesis is an equating of "constructive" with "recursive". (S) so all structures are recursively possible? Or: there is only one recursive structure. (Slightly different meaning). LorenzenVsChurch: view to narrow: it allows no longer the free use of the quantification of the natural numbers. I 267 Decision Problem/ChurchVsLorenzen: (according to Lorenzen): Advantage: greater clarity: when limited to recursive statement forms there can never arise dispute whether one of the approved statements is true or false. The definition of recursivity guarantees precisely the decision definiteness, that means the existence of a decision process.⁽¹⁾ 1. P. Lorenzen, Ein dialogisches Konstruktivitätskriterium, in: Infinitistic Methods, (1961), 193-200	Lorn I P. Lorenzen Constructive Philosophy Cambridge 1987 Berka I Karel Berka Lothar Kreiser Logik Texte Berlin 1983

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

Disputed term/author/ism	Pro/Versus	Entry	Reference
Constructivism	Pro	Berka I 266 Constructivism: Lorenzen per - LorenzenVsHerbrand - LorenzenVsChurch (too narrow conception of constructiveness as recursivity) - LorenzenVsImpredicativity	Berka I Karel Berka Lothar Kreiser Logik Texte Berlin 1983