|Law of the Excluded Middle: an assertion is either true or false. "There is no third possibility."See also bivalence, anti-realism, multivalued logic.|
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|P. Lorenzen Ein dialogisches Konstruktivitätskriterium (1959) in Karel Berka/L. Kreiser Logik Texte Berlin, 1983
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.
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.
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).
Trivially, this is consistent. Any formula that can be derived from Peano's arithmetic is it also here.
This is a "constructive" consistency proof, if the dialogical procedure is recognized as constructive.
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 = w to the power of w to the power of w to the power of ...
P can thus first calculate an ordinal number e
The statements that are used in the consistency proof are generally not recursive.
Constructive Philosophy Cambridge 1987
K. Berka/L. Kreiser
Logik Texte Berlin 1983