Dictionary of Arguments

Philosophical and Scientific Issues in Dispute

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The author or concept searched is found in the following 2 entries.

Disputed term/author/ism	Author	Entry	Reference
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 3 controversies.

Disputed term/author/ism	Author Vs Author	Entry	Reference
Mereology	Wessel Vs Mereology	Wessel I 360 Mereology/LesniewskiVsSet Theory: Lesniewski rejected set theory and built up his mereology as a competitor, as a theory of concrete object groups or as part of whole theory. In this theory there are no empty object groups. (>empty set). I 361 WesselVsMereology: there is no reason to discard class logic when distinguishing between sets and accumulations. (Frege already pointed out this difference).	Wessel I H. Wessel Logik Berlin 1999
Various Authors	Lorenzen Vs Various Authors	Berka I 266 Overcountable/infinite/LorenzenVsSet Theory: fable realm of the "overcountable". ((s) not constructible at all, >constructivism). LorenzenVsHerbrand/ConstructivismVs "impredicative": fable realm of the "impredicative".⁽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
Various Authors	Lesniewski Vs Various Authors	Simons I 102 LesniewskiVsClasses/LesniewskiVsSet Theory/MereologyVsSet Theory: can be eliminated by reformulation. Instead of "a is an element of the class of bs": simply "a is a b", ontologically: "a ε b". If the term e.g. "the class of teaspoons" is not so eliminable, then it can only stand for a simple sum. But as a sum it can be described differently, especially as the sum of all objects, which consists only of teaspoons and non-detached teaspoon parts. (> Gavagai). N.B.: because it consists exclusively of teaspoons and their parts, (it is a fusion of the teaspoons) it contains itself as an element.	Simons I P. Simons Parts. A Study in Ontology Oxford New York 1987