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Journal of Mathematics
Volume 2013 (2013), Article ID 126347, 8 pages
Categorical Abstract Algebraic Logic: Meet-Combination of Logical Systems
School of Mathematics and Computer Science, Lake Superior State University, Sault Sainte Marie, MI 49783, USA
Received 20 December 2012; Accepted 11 March 2013
Academic Editor: Abdul Hamid Kara
Copyright © 2013 George Voutsadakis. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
The widespread and rapid proliferation of logical systems in several areas of computer science has led to a resurgence of interest in various methods for combining logical systems and in investigations into the properties inherited by the resulting combinations. One of the oldest such methods is fibring. In fibring the shared connectives of the combined logics inherit properties from both component logical systems, and this leads often to inconsistencies. To deal with such undesired effects, Sernadas et al. (2011, 2012) have recently introduced a novel way of combining logics, called meet-combination, in which the combined connectives share only the common logical properties they enjoy in the component systems. In their investigations they provide a sound and concretely complete calculus for the meet-combination based on available sound and complete calculi for the component systems. In this work, an effort is made to abstract those results to a categorical level amenable to categorical abstract algebraic logic techniques.
The widespread and rapid proliferation of logical systems in several areas of computer science has led to a resurgence of interest in various methods for combining logical systems and in investigations into the properties inherited by the resulting combinations. One of the oldest methods for combining connectives is fibring . In fibring one combines two logical systems by possibly imposing some sharing of common connectives or identification of connectives from the constituent logical systems. When such interaction occurs, the combined connectives inherit all properties of the components from both logical systems, and this leads often to inconsistencies. A typical example of this strong interaction is the combination of an intuitionistic negation from one logical system with a classical negation from another. The combined connective behaves like a classical negation, and this outcome defeats any intended purpose for the combination. Fibring has been studied substantially since its original introduction, and both its virtues and its vices are relatively well understood. For instance in , fibring was presented as a categorical construction (see also ), in  fibred logical systems were investigated from the point of view of preserving completeness, in  some work was carried out on the effect of fibring in logics belonging to specific classes of the classical abstract algebraic logic hierarchy [6–8], and more recently, in  fibring was employed to obtain some modal logics, first considered in , in a structured way and to draw some conclusions regarding their algebraic character.
To avoid some of the drawbacks and undesired effects involved in the application of fibring, Sernadas et al. [11, 12] introduced, recently, another way of combining logical systems, called meet-combination, in which the combined connectives, instead of inheriting all properties they enjoy in the component logical systems, inherit only those properties that are common to both connectives. A very illuminating example of the difference that this entails as contrasted to the fibring method consists of the result of combining two logics and , one including a classical conjunction and one including a classical disjunction , with the intention of obtaining a combined connective “identifying" these two connectives from the component logics. Roughly speaking, if fibring is used, then, since in the combination the combined connective has all properties that are enjoyed by each of the connectives in either logic, the derivation shows that in the combined logic a single formula entails all other formulas; that is, there are only two possible theories, the empty theory and the entire set of formulas. On the other hand, this derivation would not be valid in the meet-combination of the two logics, since the afore-used Properties of Disjunction and Conjunction in and , respectively, are not shared by in and by in , respectively. Commutativity, however, is a shared property, whence the derived rule is a derived rule of the meet-combination.
In  Sernadas et al. start from a given logical system with a Hilbert style calculus and with a matrix semantics and define a new logic that incorporates all meet-combinations of connectives of of the same arity. Moreover, this system includes in a canonical way the connectives of the original logical system. Roughly speaking, the Hilbert calculus of the combination consists of all old Hilbert rules plus two new rules that ensure that the combined connectives inherit the common properties of the component connectives and only those properties. The matrix semantics consists, also roughly speaking, of the direct squares of the matrices in the original matrix semantics. In the main results, [11, Theorems 3.9 and 3.13], it is shown that soundness and a special form of completeness, called concrete completeness, are inherited in from . Moreover, Sernadas et al.  investigate in some detail the case of classical propositional logic, which constitutes the main motivation and paradigmatic example behind their work. Based on classical propositional calculus, they present several interesting examples, which, in addition, serve as illustrations for various sensitive points of the general theory.
In the present paper, we adapt the framework of  to a categorical level, using notions and techniques of categorical abstract algebraic logic [13, 14]. Our main goal is providing a framework in which, starting from a -institution whose closure system is axiomatized by a set of rules of inference, we may construct a new -institution that includes, in a precise technical sense, natural transformations corresponding to meet-combinations of operations available in the original -institution. The closure system of this new -institution is created by essentially mimicking the process of  to create a new set of rules of inference, suitable for the new sentence functor, and by using this new set of rules to define the inferences in the newly created structure. Under conditions analogous to those imposed by Sernadas et al. in , we are also able to establish a form of soundness and a form of restricted completeness for the new system, with respect to a suitably constructed matrix system semantics, under the proviso that these properties are satisfied by the original system.
We close this section by providing an outline of the contents of the paper. In Section 2, we introduce the basic notions underlying the framework in which our work will be carried out. The inspiration comes from categorical abstract algebraic logic [13, 14] and, more specifically, uses the notion of a category of natural transformations on a given sentence functor and, implicitly, many aspects of the theory of -rule based -institutions, where is a category of natural transformations on the sentence functor of the -institution under consideration. A recent reference on this material is . The reader should be aware that basic categorical notions are used rather heavily, but the elementary references to the subject [16–18] should be enough for necessary terminology and notation.
In Section 3 the basic constructions that take after corresponding constructions in  are presented. Here the meet-combination of logical systems refers to logical systems based on sentence functors, whose “signatures" are categories of natural transformations on the sentence functors and whose rules of inference and model classes are all categorical in nature. The goal is to work in a framework that would be amenable to categorical abstract algebraic logic methods and techniques so as to be able to consider aspects drawing from both theories.
In Sections 4 and 5, we show that a form of soundness and a form of restricted completeness are inherited by the meet-combination, subject to the condition that it is present in the components being combined. These results yield also results on conservativeness and on consistency, which are presented in Section 6.
Finally, based on the thorough work of , we present in Section 7 some examples showcasing various aspects of the general theory. These examples are relevant to both the theory developed in  and to its extension elaborated on in the present paper and, whenever appropriate, we draw attention to points where the two theories overlap and points where some differences occur.
2. Basic Framework
In the sequel we consider an arbitrary but fixed category Sign, called the category of signatures, and an arbitrary but fixed Set-valued functor , called the sentence functor. Also into the picture in a critical way will be an arbitrary but fixed category of natural transformations on , which we view as the clone of all algebraic operations on . We remind the reader here of the precise definition of such a category, as presented, for example, in . The clone of all natural transformations on is defined to be the locally small category with collection of objects and collection of morphisms -sequences of natural transformations . Composition is defined by A subcategory of this category containing all objects of the form for , and all projection morphisms , , , with given by and such that, for every family of natural transformations in , the sequence is also in , is referred to as a category of natural transformations on .
A natural transformation in is called a constant if, for all and all , If is a constant, then we set , to denote the value of the constant in , which is independent of .
An -rule of inference or simply an -rule is a pair of the form , sometimes written more legibly , where , are natural transformations in . The elements , , are called the premises and the conclusion of the rule.
An -Hilbert calculus is a set of -rules. Using the -rules in , one may define derivations of a natural transformation in from a set of natural transformations in . Such a derivation is denoted by . If the calculus is fixed and clear in a particular context, we might simply write .
Given two functors and , with categories of natural transformations on , respectively, a pair , where is a functor and is a natural transformation, is called a translation from to . Moreover, it is said to be -epimorphic if there exists a correspondence between the natural transformations in and that preserves projections (and, thus, also arities), such that, for all , all and all , An -epimorphic translation from to will be denoted by , with the relevant categories of natural transformations on , respectively, understood from context.
An -algebraic system consists of (i)a functor , with a category of natural transformations on ; (ii)an -epimorphic translation . An -matrix system or, simply, -matrix is a pair consisting of (i)an -algebraic system ; (ii)an axiom family on , that is, a collection of subsets . We perceive of the elements of as truth values for evaluating the natural transformations in and those of as being the designated ones. An -matrix semantics is a class of -matrices. Given a natural transformation in , we set where and . The matrix satisfies at under , written , if . An -rule is a rule of an -matrix semantics , written if , for all , implies , for every -matrix , all , all -assignments in , and all . If the semantics is clear from context, we simply write .
In the remainder of this paper, by a logical system, or simply a logic, we understand a pentuple , where (i) is a category; (ii) is a sentence functor; (iii) is a category of natural transformations on ; (iv) is an -Hilbert calculus; (v) is a -matrix semantics.
Let be a logical system. Define the product logical system or, simply, product logic as follows:
the logic has the same signature category as .
The sentence functor is defined by setting for all , and, similarly, for morphisms.
The category of natural transformations on has the same objects as and its morphisms into are pairs of natural transformations in . We call the members of the combined natural transformations or combined operations or, following , but rather apologetic for abusing terminology, combined connectives.
Given in , we set in and, accordingly, given in , we set Every -rule gives rise to an -rule The calculus is an “enrichment" of in the sense that it contains all rules of the form , for , and some additional -rules devised for dealing with the combined operations: (i)for each in , the lifting rule (LFT) is included in to enforce inheritance by in of all the common properties of and in ; (ii)for each constant in , the special colifting rules (cLFT) are included in to enforce that should enjoy in only those properties that are common properties of and in . The reason for allowing only the special co-lifting rules (i.e., ones that admit only constants), rather than the (general) co-lifting rules, is that, unless this restriction is imposed, the rules are not in general sound. This will become apparent in the analysis to follow.
Before introducing the semantics of , we show, following , that given constant natural transformations in , the two combined constructors and are closely related.
Theorem 1 (Sernadas, Sernadas, and Rasga). Let be a logical system. Consider a constant natural transformation in and set . Then and are interderivable in .
Proof. Apply first cLFT twice and then LFT, in each direction. One gets the following proof:
Let and be -algebraic systems with the same underlying sentence functors and the same signature functor component . Let be defined, for all , by and similarly for morphisms, and let be given, for all , by Denote by the -algebraic system Moreover, given two -matrix systems and , let where , such that, for all ,
The semantics is the class consisting of all -matrix systems of the form , for , having underlying -algebraic systems , respectively, with the same underlying sentence functors and the same signature functor components. The semantics will be called the product semantics, taking after .
Finally, we let and stand for satisfaction and entailment in the product logic .
Recall that, given a natural transformation in , we use the notation to denote the natural transformation in .
Proposition 2. Let be a logical system and consider the product system . Suppose that in , and , where the th component of is , for all . Then Moreover, for all , , all and all ,
Proof. We have the following equivalences: iff iff and , iff and .This proves the Proposition.
Proposition 3. Let be a logical system. If the -rule is sound in , then the -rule , is sound in .
Proof. Suppose that and are in so that , , , and , such that , for all . Then, by Proposition 2, for all . Thus, by soundness of in , we get that and . Therefore, again by Proposition 2, and, hence, is sound for .
Let be in and suppose that , and . Then, by the definition of ,
Proposition 4. Let be a logical system. The lifting rule LFT is sound in .
Proof. Suppose that in , , and , such that, for some and , This implies that and . These imply that , whence, by (24), This proves the soundness of lifting.
Let be a category and a sentence functor with a category of natural transformations on . Recall that a natural transformation is called a constant if, for all , all and that we use the notation , for this value, which is independent of .
A class of -matrix systems is said to be a -semantics if, for all and in , every constant in and all , Intuitively, a semantics is a -semantics if and only if every constant is consistently interpreted as true or false under all matrix systems in the semantics, that is, under all combinations of interpretations and designated truth values included in the semantics.
Proposition 5. Let be a logical system, where is a -semantics. For all constants in , the special co-lifting rules are sound in .
Proof. Let , a constant in , , and , such that, for some and , Then (recalling the notation for constants) , whence Since is a -semantics, we get that the four following relations hold: Therefore, we obtain that which show that the special co-lifting rules are sound in .
Theorem 6 (soundness). Let be a logical system, where is a -semantics. If is sound, then the product logic is also sound.
Proof. We have shown in Proposition 3 that all rules inherited by are sound in . By Proposition 4, the lifting rule is sound in and, since is assumed to be a -semantics, by Proposition 5, the special co-lifting rules are sound in . Therefore the product logic is also sound.
A logic is -complete if it is complete with respect to constant natural transformations. More precisely, for all sets of constants in , we have that
Proposition 7. If a logic is -complete, then, for all sets of constants in ,
Proof. Suppose that . Then, since includes all -rules of the form , for all , we get that . Therefore, by the -completeness of , we get that . Thus, there exists a model , together with , and , such that and . Hence, the model is such that and . Therefore , showing that is also -complete.
Proposition 8. Let be a logic and suppose that, for some in , Then it is also the case that
Proof. Suppose that . By the lifting rule, we must have or . Therefore, by hypothesis, or . Suppose, without loss of generality, that the first holds. Thus, there exists a model , and , such that Thus, we must have or This implies that either or bears witness to and concludes the proof.
To formulate the following proposition we introduce a convenient notation: given a set of natural transformations in , we write
Proposition 9. Let be a logic and suppose for some set of constants in Then it is also the case that
Proof. If , then, by the special co-lifting property, . Thus, by hypothesis, . Hence, there exists , , and , such that while, at the same time, These relations imply that whence .
Theorem 10 (-completeness). If the logic is -complete, then the product logic is -complete also.
Proof. If is -complete, then, by Proposition 7, we get that, for all sets of constants in , Thus, by Proposition 8, for all sets of constants in , Finally, by Proposition 9, we get that, for all sets of constants in , This proves that is -complete.
6. Conservativeness and Consistency
Theorem 11 (conservativeness). Let be a logic. For every set of natural transformations in ,
Proof. Suppose . If is such that, for some , , , , then, we get that , whence, by the hypothesis, , and, therefore, . This shows that .
Theorem 12 (consistency). If the logic is consistent, then so is the product logic .
Proof. This follows directly from conservativeness.
7. Examples from Classical Propositional Logic
We present a simple example, essentially borrowed from , with the twofold goal of, first, seeing how the theory of  can be easily accommodated in the categorical framework (becoming actually a trivial case) and, second, showcasing the difference between the soundness of special co-lifting and the lack of soundness obtained by allowing the full power of the general co-lifting rule.
Suppose, first, that is a logic, such that contains two binary natural transformations and two constants that obey the usual laws of conjunction, disjunction, truth, and falsity of classical propositional logic. Then, if , we have that This can be shown by observing that the hypothesis yields, by special co-lifting, and . These, by following usual derivations in , yield and , whence, by lifting, we finally obtain the conclusion. In fact, if we arrange for to consist, essentially, of Boolean algebras and evaluations together with Boolean filters, it is the case that where are the two projection natural transformations; that is, “commutativity" is valid in general, not just for constants. However, the derivation (50) cannot be inferred directly from this using -completeness, since there are nonconstant natural transformations involved.
To illustrate, using the same example, that the general co-lifting rule fails, we may employ Boolean models to show that In fact, note that whereas the first belonging to the product filter of 2-element Boolean algebras, the second failing to do so.
Note, next, that A straightforward computation shows that in the direct product of 2-element Boolean algebras, the left-hand side evaluates to , whereas the right-hand side to . Even though this serves as a counterexample for an analog of Theorem 1 concerning the exchangeability of components in the context of , this problem does not arise in our context. In fact, our reformulation of [11, Theorem 2.1] in the form of Theorem 1 would only ensure that
Suppose now that in , one has the, possibly derived, rule , where are both constants in . Then it can be shown that In fact, follows from the special co-lifting, whereas lifting helps establish the opposite direction Finally, if one has available in a disjunction and an implication , both behaving classically, then, since both derived rules are rules of , one obtains the rule in by an application of lifting.
We close with a generally phrased (rather informally formulated) problem that would be of interest in the context developed in the present work from the point of view of abstract algebraic logic. For more details on the motivations and the state of the art in that theory, as well as the precise definitions and more insights on the notions employed in the phrasing of this problem, the reader is referred to [13–15] and further references therein.
Problem for Investigation. Suppose that we have some knowledge about the algebraic classification of the -institution , where is a logic in the sense of the present paper, possibly satisfying some additional conditions. The closure system is the system induced by the set of -rules, as detailed in, for example, . What corresponding information may then be drawn about the -institution , that corresponds, in a similar manner, to the product logic ?
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