. Several mechanisms for combining logics have appeared in the literature. Synchronization is one of the simplest: the language of the combined logic is the disjoint union of the given languages, but the class of models of the resulting logic is a subset of the cartesian product of the given classes of models (the interaction between the two logics is imposed by constraining the class of pairs of models). Herein, we give both a model-theoretic and a proof-theoretic account of synchronization as a categorial construction (using coproducts and cocartesian liftings) . We also prove that soundness is preserved by possibly constrained synchronization and state sufficient conditions for preservation of model existence and strong completeness. We provide an application to the combination of dynamic logic and linear temporal logic. Keywords: combination of logics, synchronization of logics, model existence, completeness, dynamic logic, temporal logic. 1 Introduction There has been a recent g...

wing the ideas in [4], to cope with possible non{truth{functionality of constructors. In the spirit of the theory of institutions and general logics [8, 9], we consider a logic to consist of an indexing functor to a suitable category of logic systems. In our case, the logic systems of interest are non{truth{functional (ntf) rooms . For simplicity, we shall only work at this level of abstraction. As shown in [3], everything can be smoothly lifted to the fully edged indexed case. In the sequel, AlgSig' denotes the category of algebraic many sorted signatures with a distinguished sort ' (for formulae) and morphisms preserving it. Given one such signature , we denote by Alg() the category of {algebras and {algebra homomorphisms, and by cAlg() the class of all pairs hA; i with A a<

generalizes the common situation when truth-values are ordered, we require a whole Tarskian closure operation as in [2]. In the sequel, AlgSig denotes the category of algebraic many sorted signatures with a distinguished sort (for formulae) and morphisms preserving it. Given such a signature hS; Oi, we denote by Alg(hS; Oi) the category of hS; Oi- algebras, and by cAlg(hS; Oi) the class of all pairs hA; i with A 2 jAlg(hS; Oi)j and a closure operation on A . Denition 1. A layered parchment is a tuple P = hSig; L; Mi where: { Sig is a category (of abstract<F13

We show that the category proposed in [5] of logic system presentations equipped with cryptomorphisms gives rise to a category of parchments that is both complete and translatable to the category of institutions, improving on previous work [15]. We argue that limits in this category of parchments constitute a very powerful mechanism for combining logics.

We propose in this paper a generalization of bring of propositional logic systems using the language of category theory. We generalize the categorial construction of bring using arbitrary colimits and limits, instead of the usual simple kind of colimit. We prove that limits and colimits preserve completeness (under reasonable conditions) in the category of propositional Hilbert calculi endowed with general algebraic semantics. 1

In [6] it was shown that fibring could be used to combine institutions presented as c-parchments, and several completeness preservation results were established. However, their scope of applicability was limited to propositional-based logics. Herein, we extend these results to a broader class of logics, possibly including variables, terms and quantifiers.

Two Hilbert calculi for higher-order logic (or theory of types) are introduced.

In this paper we address the question of recovering a logic system by combining two or more fragments of it. We show that, in general, by fibring two or more fragments of a given logic the resulting logic is weaker than the original one, because some meta-properties of the connectives are lost after the combination process. In order to overcome this problem, the categories Mcon and Seq of multiple-conclusion consequence relations and sequent calculi, respectively, are introduced. The main feature of these categories is the preservation, by morphisms, of meta-properties of the consequence relations, which allows, in several cases, to recover a logic by fibring of its fragments. The fibring in this categories is called meta-fibring. Several examples of well-known logics which can be recovered by metafibring its fragments (in opposition to fibring in the usual categories) are given. Finally, a general semantics for objects in Seq (and, in particular, for objects in Mcon) is proposed, obtaining a category of logic systems

The problem of bring paraconsistent logics is addressed. Such logics raise new problems in the semantics of bring since previous work assumed verum-functional models. The solution is found in a general notion of interpretation system presentation that \species" the intended valuations in some appropriate meta language. Fibring appears as a universal construction in the category of interpretation system presentations, generalizing the results for systems with verum-functional semantics. As an illustration, the bring of paraconsistent system C 1 and modal system K, while sharing propositional symbols, conjunction, disjunction and implication, is obtained. The bring of the whole hierarchy fCn gn2N leads to the limit paraconsistent logic C lim . Fibring is shown to be a promising technique for generating new paraconsistent logics. 1 What is bring? In recent years, the problem of combining logics has deserved the attention of many researchers in mathematical logic. Beside...

This text aims at providing a bird’s eye view of possible-translations semantics ([10, 24]), defined, developed and illustrated as a very comprehensive formalism for obtaining or for representing semantics for all sorts of logics. With that tool, a wide class of complex logics will very naturally turn out to be (de)composable by way of some suitable combination of simpler logics. Several examples will be mentioned, and some related special cases of possible-translations semantics, among which are society semantics and non-deterministic semantics, will also be surveyed. 1 Logics, translations, possible-translations Let a logic L be a structure of the form 〈S, �〉, where S denotes its language (its set of formulas) and � ⊆ Pow(S)×Pow(S) represents its associated consequence relation (cr), somehow defined so as to embed some formal model of reasoning. Call any subset of S a theory. As usual, capital Greek letters will denote theories, and lowercase Greek will denote formulas; a sequence such as Γ, α, Γ ′ � ∆ ′ , β, ∆ should be read as asserting that Γ ∪ {α} ∪ Γ ′ � ∆ ′ ∪ {β} ∪ ∆. Morphisms between any two of the above structures will be called translations. So, given any two logics, L1=〈S1, �1 〉 and L2=〈S2, �2〉, a mapping t: S1 → S2 will constitute a translation from L1 into L2 just in case the following holds: