Abstract
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.
Highlights
There exists a variety of different logics and each of these logics can be defined through different axioms, rules or semantics
We have shown that M od is a propositional model mapping is stronger than T h ◦ M od is a propositional consequence operation
Within our framework of adequacy, we have studied the interactions and connections between syntax and semantics
Summary
There exists a variety of different logics and each of these logics can be defined through different axioms, rules or semantics. There has been a great effort to provide a general framework for all these logics [4]. The syntactical theory of consequence operations (see, e.g., [15,16,17]) is such a framework. A similar degree of generality has been achieved in semantics by the development of abstract model theory [1,2]. Surma [14] was the first to study the
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