A Shared Framework for Consequence Operations and Abstract Model Theory

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.


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., [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 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 Mod 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 . Mod 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 • Mod. 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 • Mod. After studying adequacy in its most general form, we investigate how properties of Mod reflect properties of Cn and vice versa. We treat the cases where Cn is a propositional consequence operation and where Mod is a propositional model mapping. Furthermore, we determine for every basic notion of the theory of consequence operations a semantical equivalent.

Completeness.
To show that every propositional consequence operation is complete with respect to classical logic, we employ semantics based on relatively maximal sets. The set of its relatively maximal sets is an adequate semantics for a finitary consequence operation (for the notion of adequacy, refer to Definition 4.1 and especially the subsequent example). This fact hinges essentially on the Lindenbaum Lemma. Consequently, by identifying a relatively maximal set with its characteristic function, every finitary consequence operation has a bivalent semantics. Furthermore, the set of relatively maximal sets forms a minimal semantics, i.e., every semantics that does not contain all relatively maximal sets, is not adequate. The framework of this section is very general and can be applied to prove the completeness of many logics. The results of this section can be found, e.g., in [3], [17, p. 25-28].

Definition 2.8.
A set E is a closure system iff it is closed under intersection, i.e., if D ⊆ E, then D ∈ E.
The following fact is well-known (see, e.g., [6]) and implies that the closure system of all Cn-theories forms an adequate semantics for Cn. Lemma 2.9. Every consequence operation Cn, i.e., closure operator, determines a closure system {T |Cn(T ) = T }. Conversely, every closure system E determines a closure operator Cn through the condition: Definition 2.10. G ⊆ E is a generator set of a closure system E iff for every E ∈ E: E = G for some G ⊆ G. A subset of E is a minimal generator set iff it is not generated by one of its proper subsets. T is called completely meet The next lemma shows that for a consequence operation Cn, it is sufficient to consider a generator set of the closure system of Cn. Hence, every generator set also forms an adequate semantics for Cn. Observe that every relatively maximal set is a theory. We denote the set of relatively Cn-maximal theories by RELM AX(Cn) and the set of relatively maximal extensions of T by RELEXT (T ).
No relatively maximal set T can be generated by a set of closures not containing T . This implies that if RELM AX(Cn) is a generator set, it is a minimal generator set.

Lemma 2.13. T is relatively Cn-maximal iff it is completely meet irreducible in the closure system given by Cn.
From the Lindenbaum Lemma, it follows that RELM AX(Cn) is a generator set.

Lemma 2.14 (Lindenbaum Lemma). Let Cn be finitary. For every
The following lemma states that the set of all relatively maximal sets forms a minimal adequate semantics for finitary consequence operations. As already pointed out, it can be obtained by Lemma 2.13. If Cn is finitary, then the Lindenbaum Lemma is valid. As a consequence, RELM AX(Cn) is a generator set, and we have the following tool to prove the completeness of finitary consequence operations.

Theorem 2.16. If Cn is finitary and RELM AX(Cn)
is a subset of the closure system generated by Cn , then Cn ≤ Cn.

Propositional Consequence Operations
The concept of propositional consequence operation (see, e.g., [15], [17, p. 110]) covers all systems of classical propositional logic. Since it describes the connectives of classical logic, the underlying language is the formal language generated by the connectives ¬, ∧, ∨, →. We denote this language by 'L AL ' and write 'Cn(T, A 1 , . . . , A n )' instead of 'Cn(T ∪{A 1 , . . . , A n })'. It is fundamental that every finitary, consistent, and structural consequence operation that satisfies the conditions (¬), (∧), (∨), and (→), is identical with classical logic. We sketch the proof in this section (for a detailed proof see, e.g., [17, pp. 123-127] Example. A is a classical consequence of T iff for every Boolean valuation α: If |= α T , then |= α A. We denote the classical consequence operation by Cn |= . It can easily be verified that the classical consequence operation is a propositional consequence operation.
In order to prove Theorems 4.8 and 4.14, we have to employ several properties of propositional consequence operations. Lemma 2.18. Let Cn be a propositional consequence operation. Then: The following closure properties of relatively maximal sets play a key role in proving that every propositional consequence operation is complete with respect to classical logic (for a proof see [17, p. 125-126]).

Lemma 2.19. Let
Cn be a propositional consequence operation and T be relatively Cn-maximal. Then: Consistency and completeness can be formulated in a familiar way within the theory of propositional consequence operations. Lemma 2.20. Let Cn be a propositional consequence operation. Then:

The Completeness of Finitary Propositional Consequence Operations.
If Cn is a finitary propositional consequence operation, then the set of all its maximally consistent sets is a generator set of {T |Cn(T ) = T }. Consequently, this set is an adequate semantics for Cn. By MAX(Cn) we denote the set of all maximally Cn-consistent sets, and by EXT M AX(T ), we denote the set of all maximally consistent extensions of T .

Semantics
The basic framework of this section is that of abstract model theory (see, e.g., [7,9]). As observed, for instance, by Cohn [6, p. 205], abstract model theory can be based on a Galois correspondence between models and its theories. To be as general as possible, the set of structures Str is not specified. The following relation-based construction of Galois correspondences is attributed to Birkhoff (first edition of [5], 1940). The starting point of the construction is a satisfaction relation |= Str on Str × L. The corresponding polarities [5] are the model mapping Mod and the theory mapping T h. Between Mod and T h, it obtains a Galois correspondence. All theorems of the present section derive from the theory of Galois correspondences. The mapping T h • Mod, for instance, is a consequence operation. It is the semantical consequence operation.
Propositional model mappings are obtained by adding axioms for the connectives of propositional logic. As we shall see in Sect. 4.3, they induce propositional consequence operations. At the end of this section, we review an alternative approach towards abstract semantics that is based on theories instead of models [10].

Model Mappings and Semantical Inference Operations
Let Str be an arbitrary non-empty set and L be a formal language.
The class of models Mod(T ) of T is the set of all models that satisfy T . The semantical theory T h(M ) of M is the set of all sentences that are satisfied by M . As already pointed out, if two mappings are defined by a binary relation in the way described above, then they form a Galois correspondence [5]. Mappings that satisfy (3.1) are called antitone. The following consequences are well-known within the theory of Galois correspondences.

Lemma 3.3. 1. M ⊆ Mod(T ) iff T ⊆ T h(M ). 2. T h(Mod(T h(M ))) = T h(M ). 3. Mod(T h(Mod(T ))) = Mod(T ).
The mapping T h • Mod : 2 L → 2 L is the semantical inference operation. T h • Mod is a consequence operation. Mod • T h is a closure operator. This is an immediate consequence of the fact that Mod and T h form a Galois correspondence. The operation Mod is a bijection from T HE to AXC. The inverse of Mod is T h. Moreover, Mod is a dual isomorphism 2 between these two sets (see, e.g., [5]). Mod inverts the order of T HE, i.e., larger theories correspond to smaller classes of models and vice versa.

Theorem 3.10. The mappings Mod and T h determine a dual isomorphism between T HE and AXC.
We conclude this section with a general remark concerning an alternative approach towards semantics that is based on theories of models, rather than models [10].
Since Moreover, since the set of all relatively maximal theories (or completely meet irreducible theories, or totally prime theories) is a minimal generator set, if a logic is minimally generated, it is sufficient to consider relatively maximal theories. Abstract connectives are then interpreted as closure conditions on relatively maximal sets (see also Remark 1 below). This approach leads to interesting results concerning intuitionistic and classical propositional logic. One can give, for instance, elegant semantical characterizations of intuitionistic and propositional consequence operations [10, Theorem 3.2].
However, this approach is not feasible if we move to first-order logic. If we identify models with their theories, we cannot distinguish between elementary equivalent models, i.e., models that have the same first-order theory. As a consequence, many important concepts and results of first-order logic become inaccessible. Isomorphy of models and categoricity of theories 3 are excluded from this framework. Moreover, the theorem of Löwenheim-Skolem which permits a model-theoretical characterization of first-order logic [7] cannot be obtained.

Propositional Model Mappings
We have seen that T h • Mod is a consequence operation. It is natural to ask which properties of Mod are sufficient for T h • Mod being a propositional consequence operation. This question is dealt with in Theorem 4.9. We call these model mappings 'propositional model mappings'. However, the fact that T h • Mod is a propositional consequence operation does not imply that Mod is a propositional model mapping.
Consequently, for propositional model mappings the approach taken here and in [10] are equivalent.

Adequacy
A semantical system is adequate with respect to a consequence operation, if and only if both yield the same theorems 5 . This leads to a concept of adequacy that does not dependent upon the kind of logic considered. We study the interaction between a consequence operation and its adequate semantics in its most general form. We show how all concepts defined in the framework of consequence operations (Definition 2.5) can be expressed semantically. We then investigate adequacy for propositional consequence operations. However, only if we require Mod to be a propositional model mapping, we obtain familiar connections between semantics and syntax of classical logic. As a main result, we shall see that every consequence operation that has an adequate propositional model mapping is a propositional consequence operation but not vice versa.
Satisfiability and logical truth are semantical equivalents for consistency and tautology.
Each notion in Definition 2.5 can be expressed semantically if the concept of adequate model mapping is used as a link between syntax and semantics. Since T HE and AXC are dually isomorphic (Theorem 3.10), we have the following picture (see also [13]). Cn(L) is the largest set in T HE and corresponds therefore to the smallest set of AXC. Because Mod is antitone, this set is Mod(L). Since the correspondence is one to one, all consistent sets correspond to model classes different from Mod(L). Maximally consistent sets are next in the ordering within T HE and correspond consequently to sets of the form M ∪Mod(L), such that all members of M are elementary equivalent. The converse need not be true. Observe further that since Mod(Cn(T )) = Mod(T ), maximally consistent sets and consistent and complete sets have the same models. Each theory that is not maximal, corresponds to a subset of the remaining axiomatic classes inversely ordered. On the lower end of T HE, Cn(∅) corresponds to Str. We obtain Let M ⊆ Str and N, N ∈ Str. With the help of the following notion, a semantical formulation for completeness can be given. The following result states that a set is complete, if and only if any two "consistent" models of this set satisfy exactly the same formulas. As a corollary, we obtain The converse is not true because there may be structures N such that T h(N ) is not maximal.

Adequacy for Propositional Consequence Operations
The definition of propositional consequence operations concerns the connectives of L AL . Hence, if Cn is a propositional consequence operation, we can establish relations between semantics and syntax that involve these connectives.