A Shared Framework for Consequence Operations and Abstract Model Theory

Christian Wallmann 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. 1. Introduction Currently 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., [1517]) 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 interaction between consequence operations and semantics using an axiomatic approach. However, apart from the work of Surma, there has been little effort to study what is common to all relations of adequacy between syntax and semantics. In the present contribution we develop such a theory of adequacy for consequence operations on the syntactical side and abstract model theory. We study how certain properties of consequence operations determine properties of its adequate semantics and vice versa. Another main aim of the paper is to show how syntactical concepts can be formulated semantically. Such a formulation clearly depends on the class of consequence operations and semantics considered. In particular, we treat adequacy for classical propositional logic. After presenting the basic axioms for consequence operations and some basic notions of logic, we state axioms for the connectives of classical propositional logic. The resulting concept of propositional consequence operation covers all systems of classical logic. By employing semantics based on relatively maximal sets, we can prove the completeness of many logics in a much more simplified way [3, 17]. We review some of these results and show that every finitary propositional consequence operation is complete with respect to classical logic [17]. In Sect. 3, we present an abstract semantics [7, 9] which has the same degree of generality as consequence operations. The set of structures on which the semantics is based on is not specified. It could be any non-empty set. As a consequence, our semantical framework covers many different systems, such as valuation semantics, semantics based on maximally consistent sets, and probability semantics. Roughly speaking, a model mapping M od assigns to every formula the set of structures that verify it. The theory T h(N ) of a model N is the set of all sentences verified by N . M od and T h form a Galois correspondence [6]a relation that is well-established within algebra [5, 6]. This observation is of main importance because many semantical facts derive immediately from the theory of Galois correspondences. The semantical consequence operation is given by the mapping T h M od. It turns out that a sentence A is a semantical consequence of a set of sentences T , if and only if every model of T is a model of A. We study a special class of model mappings for propositional consequence operations. This class, propositional model mappings, has the Negation property and the Conjunction property. Finally, we review an alternative approach towards abstract semantics that is based on deductively closed sets instead of models [10]. In the last section, we develop a theory of adequacy that can be applied to many different kinds of logic. A semantics is adequate for Cn iff Cn is identical with the semantical inference operation T h M od. After studying adequacy in its most general form, we investigate how properties of M od reflect properties of Cn and vice versa. We treat the cases where Cn is a propositional consequence operation and where M od is a propositional model mapping. Furthermore, we determine for every basic notion of the theory of consequence operations a semantical equivalent. 2. Syntax In this section, we outline the theory of consequence operations and the theory of propositional consequence operations. 2.1. Consequence Operations 2.1.1. Basic Framework. Let Av be a countable infinite set of propositional variables and f1, . . . , fn be a set of propositional connectives. A formal language is the smallest set closed under f1, . . . , fn and containing Av. Suppose that L is a formal language, that T, T L, and that A, B L. The axioms given in [15] are equivalent to those given in Definition 2.1. A mapping Cn : 2L 2L is a consequence operation or closure operator iff for every T, T : 1. T Cn(T ) (Reflexivity) and 2. if T Cn(T ), then Cn(T ) Cn(T ) (Transitivity). Corollary 2.2. Cn is a consequence operation iff for every T, T : 1. T Cn(T ), 2. Cn((Cn(T )) Cn(T ) (Idempotency), and 3. if T T , then Cn(T ) Cn(T ) (Monotonicity). For the remainder of this article, suppose that Cn is a consequence oper A consequence operation is structural [11] if it is closed with respect to substitution. Uniform and simultaneous substitution of an arbitrary formula for a propositional variable within a valid inference, yields also a valid inference. The validity of an inference consequently does not depend on the content, but only on the form, i.e., on the kind and order of the occurring connectives. A mapping e : Av L is called substitution. Every substitution can be extended to an endomorphism he : L L, i.e., to a uniform and simultaneous substitution. Definition 2.4. Cn is stronger than Cn (Cn Cn) iff for all T : Cn (T ) Cn(T ). Cn is properly stronger than Cn (Cn < Cn) iff Cn is stronger than Cn and there is some T such that Cn (T ) Cn(T ). Basic concepts of logic can be defined within the theory of consequence operations. Definition 2.5. 1. Cn is consistent iff Cn() = L. 2. Cn is finitary iff for each T L: Cn(T ) = {Cn(T ) : T T and T is finite}. 3. Cn is compact iff for each T L: If Cn(T ) = L, then there exists a finite T T such that Cn(T ) = L. 4. T is a Cn-theory iff T = Cn(T ). 5. T is Cn-consistent iff Cn(T ) = L. 6. T is Cn-complete iff for all A: If T {A} is consistent, then A Cn(T ). 7. T is Cn-maximally consistent iff T is consistent, and there does not exist a consistent T T . 8. T is a Cn-axiom system for T iff Cn(T ) = Cn(T ). 9. A is in T Cn-independent iff A T and A Cn(T \{A}). 10. A is a Cn-tautology iff A Cn(). Although finitariness and compactness are in many (...truncated)