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Bolzano’s definition of consequence in effect associates with each set X of symbols (in a given interpreted language) a consequence relation ({Rightarrow_X}) . We present this in a precise and abstract form, in particular studying minimal sets of symbols generating ({Rightarrow_X}) . Then we present a method for going in the other direction: extracting from an arbitrary consequence relation ({Rightarrow}) its associated set ({C_Rightarrow}) of constants. We show that this returns the expected logical constants from familiar consequence relations, and that, restricting attention to sets of symbols satisfying a strong minimality condition, there is an isomorphism between the set of strongly minimal sets of symbols and the set of corresponding consequence relations (both ordered under inclusion).

DOI: 10.1007/s11229-011-9902-z

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