From constants to consequence, and back

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).