From constants to consequence, and back

Dag Westerståhl

pp. 957-971

in: Sten Lindström, Erik Palmgren, Dag Westerståhl (eds), The philosophy of logical consequence and inference, Synthese 187 (3), 2012.

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).