|Derivability: question which statements can be obtained according to the rules of a calculus.|
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|Berka I 18/19
Derivability/Bolzano: exists, if certain ideas i, j, which make the premises A, B, C, true, also make the conclusions M, N, O... true.
Namely the epitome (the totality) of the ideas is intended to make the entire conclusions and the whole premises true. (> To make it true, truthmaker).
Comprise/include/Bolzano: Premises: are here the included,
Conclusions: the comprehensive/including sentences.
Derivability/Bolzano: Problem: sentences obtained through an arbitrary exchange of ideas from given true must not always be true.
(s) Exchange of ideas: insert for variables.
Bolzano: thus the relationship of derivability can also exist under false theorems.
E.g. Follow-up relationship/Bolzano/(s): (content-related): if it is warmer in one place, a higher temperature is displayed in this place. In reality, higher temperature is displayed because it is warmer. The thermometer does not generate the temperature. That is, the follow-up relationship consists only in one direction: Heat > Temperature: (s) reformulated, but somewhat correspondingly in Bolzano. Different in the derivability:
E.g. derivability/Bolzano/(s): if the sentence "... higher temperature" is true, the sentence "it is warmer" is also true and vice versa. Reversible ratio of two true sentences. Content is not decisive.
Follow-up relationshiop/Bolzano: is not already present when the corresponding sentences are all true.
Follow-up relationship/implication/truth/(s): 1. In order to avoid empirical problems (confirmation, etc.) one can say: For example, if the sentence "the temperature is higher ..." is true, it is warmer there. Then problem of intensity, the display by the thermometer does not exist. (Truth instead of intension).
K. Berka/L. Kreiser
Logik Texte Berlin 1983