@misc{Lexicon of Arguments, title = {Quotation from: Lexicon of Arguments – Concepts - Ed. Martin Schulz, 29 Mar 2024}, author = {Lorenzen,Paul}, subject = {Excluded Middle}, note = {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}, note = { Lorn I P. Lorenzen Constructive Philosophy Cambridge 1987 Berka I Karel Berka Lothar Kreiser Logik Texte Berlin 1983 }, file = {http://philosophy-science-humanities-controversies.com/listview-details.php?id=796508} url = {http://philosophy-science-humanities-controversies.com/listview-details.php?id=796508} }