, Volume 187, Issue 3, pp 957-971
Date: 22 Mar 2011

From constants to consequence, and back

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Abstract

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).