|Proof in logic, mathematics: finite string of symbols, which derives a statement in a system from the axioms of the system together with already proven statements.|
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|Proofs||P. Lorenzen Ein dialogisches Konstruktivitätskriterium (1959) in Karel Berka/L. Kreiser Logik Texte Berlin, 1983
Berka I 267
Definition Proof/Dialogical Logic/Lorenzen: E.g. the assertion of such statements as
(1) (Ex) R (x, n)
should not trigger a meaningless dispute!
It is obvious, then, to agree on that the person who claims (1) is also obliged to give a number m, so that
(2) R (m,n)
is true. If he cannot do this, he has "lost" his claim.
(1) is "proved" by (2) if this latter statement is true. Since (2) is decisive-definite, it is here defined how to decide whether the statement presented as "proof" is really a "proof".
Definition proof-definite/Lorenzen: proof-definite are statements whose use is defined in the dialogue so that it is clear how to decide whether a statement is a proof. To this belongs in mathematics, for example, assertions about the derivability of a figure in a calculus.
Thiel I 256 ff
Proof/Lorenzen: Dialogical proof procedure according to Lorenzen (Proponent/Opponent) ... + ... in a certain situation the proponent wins quite independently of whether the opponent can justify his position or not.
It wins against any opponent regardless of the truth value, but not independently of the knowledge of the truth value! (> Intuitionist).
Because he needs this knowledge to make the right choice.
This separates the effectively universal theses from the just classic generally valid ones. The classical ones are the ones that he can find on the basis of the knowledge of the truth values through the winning strategy.
The effective ones are those through which a dialogue can be won without knowledge of the truth values.
Constructive Philosophy Cambridge 1987
K. Berka/L. Kreiser
Logik Texte Berlin 1983
Philosophie und Mathematik Darmstadt 1995