, Volume 187, Issue 3, pp 957–971

From constants to consequence, and back


DOI: 10.1007/s11229-011-9902-z

Cite this article as:
Westerståhl, D. Synthese (2012) 187: 957. doi:10.1007/s11229-011-9902-z


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 constantsConsequence relationsBolzano consequence

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© Springer Science+Business Media B.V. 2011

Authors and Affiliations

  1. 1.Department of Philosophy, Linguistics and Theory of ScienceUniversity of GothenburgGothenburgSweden