@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}
}