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|Berka I 8
Conditional/ relationship/Conclusion/Bolzano: for Bolzano follows M, N, O..from A, B, C... only if
(1) each (semantic) model of A, B, C ... is also a model of M, N, O .... That is, if each of the final sentences M, N, O, can be deduced separately from the assumptions A, B, C.
(2) the premises are the (s) content-related) reason for the conclusion.
Berka: that is a very strong conclusion concept.
TarskiVsBolzano: it is enough for him if the first condition is fulfilled.
GentzenVsBolzano: for Gentzen it is sufficient if at least one of the final sentences is derivable from the set of premisses.
Special case: if the claim quantity contains only one final sentence, the Bolzano's and the Gentzen's inference system are identical.
Conclusion/Bolzano: additional condition: one must be able to decide which concepts are logical concepts.
Berka I 20 f
E.g. Entailment/Bolzano: (content-wise): if it is warmer in a place, then there are higher temperatures at the location - in reality, higher temperatures are shown, because it's warmer - the thermometer does not generate the temperatures. I.e. the entailment only exists in one direction: heat > temperature: - different with Deducibility/Bolzano/(s): if the sentence "... higher temperatures" is true, the sentence "it's warmer" is true, and vice versa. Reversible relation of two true sentences. Content is not decisive here - entailment: only in one direction - derivability: goes in both directions, regardless of truth - Entailment/Bolzano: for reason - Derivability/>Mates: formal.
K. Berka/L. Kreiser
Logik Texte Berlin 1983