From constants to consequence, and back

• Published in: Synthese (Dordrecht) Vol. 187; no. 3; pp. 957 - 971
• Main Author: Westerståhl, Dag
• Format: Journal Article
• Language: English
• Published: Dordrecht Springer 01.08.2012; Springer Netherlands; Springer Nature B.V
• Subjects: Bolzano consequence; consequence relations; Education; Epistemology; Filosofi; Filosofiämnen; Historisk-filosofiska ämnen; History and philosophy subjects; HUMANIORA och RELIGIONSVETENSKAP; HUMANITIES and RELIGION; Inference; Logic; Logical constants; Logical theorems; Mathematical constants; Mathematical functions; Mathematical theorems; Mathematicians; Mathematics; Metaphysics; Philosophy; Philosophy of Language; Philosophy of Science; Philosophy subjects; Predicate logic; Propositional logic; Tennis; Teoretisk filosofi; Theoretical philosophy; Theory; Truth; Truth functions; Consequence relations; Logical constants; Bolzano consequence
• ISSN: 0039-7857; 1573-0964; 1573-0964
• 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 ⇒x. We present this in a precise and abstract form, in particular studying minimal sets of symbols generating ⇒x. Then we present a method for going in the other direction: extracting from an arbitrary consequence relation ⇒ its associated set C⇒ 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).