Find counter arguments by entering NameVs… or …VsName.
The author or concept searched is found in the following 4 entries.
| Disputed term/author/ism | Author |
Entry |
Reference |
|---|---|---|---|
| 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 >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 |
| 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 P can thus first calculate an ordinal number I 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 |
| Truth | Lorenzen | Berka I 270 Truth/Dialogical Logic/Lorenzen: with the infinite inductive definitions, one can transform, e.g. the semantic concept of truth into a dialogically definite concept. >Dialogical Logic, >Semantic Truth. There are two sets, the set T of the true formulas and the set F of the wrong formulas. Cf. >Truth/Kripke. I 271 If the formulas with the logical particles are constructed from decision-definite prime formulas, then T (true) and F (false) are defined infinitely inductively as follows: A e T u B e T > A u B e T A e F > A u B e F B e F > A u B e F (correcpondingly for v) A e F > i A e T A e T > i A e F (n)A(n) e T > (x)A(x) e T A(n) e F > (x)A(x) e F (correspondingly for (Ex)). Foundation/Lorenzen: for this definition one does not need ordinal numbers as step numbers, because the definition scheme is "sound". That is, one gets after a finite number of steps to a prime formula.(1) >Foundation, >Step number. 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 |
| |||
No results. Please choose an author or concept or try a different keyword-search.