, Volume 187, Issue 3, pp 957-971

First online:

From constants to consequence, and back

  • Dag WesterståhlAffiliated withDepartment of Philosophy, Linguistics and Theory of Science, University of Gothenburg Email author 

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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).


Logical constants Consequence relations Bolzano consequence