A Shared Framework for Consequence Operations and Abstract Model Theory
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In this paper we develop an abstract theory of adequacy. In the same way as the theory of consequence operations is a general theory of logic, this theory of adequacy is a general theory of the interactions and connections between consequence operations and its sound and complete semantics. Addition of axioms for the connectives of propositional logic to the basic axioms of consequence operations yields a unifying framework for different systems of classical propositional logic. We present an abstract model-theoretical semantics based on model mappings and theory mappings. Between the classes of models and theories, i.e., the set of sentences verified by a model, it obtains a connection that is well-known within algebra as Galois correspondence. Many basic semantical properties can be derived from this observation. A sentence A is a semantical consequence of T if every model of T is also a model of A. A model mapping is adequate for a consequence operation if its semantical inference operation is identical with the consequence operation. We study how properties of an adequate model mapping reflect the properties of the consequence operation and vice versa. In particular, we show how every concept of the theory of consequence operations can be formulated semantically.
Mathematics Subject Classification (2010)Primary 03C95 Secondary 03B22
KeywordsConsequence operation abstract model theory Galois correspondence completeness adequacy classical propositional logic
This work was supported by the Austrian Science Fund(FWF):I141-G15 (Project Leader: Gernot Kleiter) as part of the LogICCC programme of the European Science Foundation. Many thanks to three anonymous reviewers for their very detailed, and extremely useful comments. They helped a lot to improve a first version of the paper. I am indebted to Reinhard Kleinknecht for having first introduced me to consequence operations and the notion of adequate model mappings.
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