Multi context logics (MCL) is a formalism that allow to integrate multiple logical theories (contexts) in a more complex structure called multi context system. In the past 20 years multi context logics have been developed for contexts in propositional logics, rst order logics, description logics and temporal logic. The two principles MCL are locality and compatibility. The principle of locality states that a context axiomatizes in a logical theory a portion of the world, and that every statement entailed by such a theory is intended to hold within such a portion of the world. The principle of compatibility instead states that, since di erent contexts can describe overlapping portions of world, the theories they contain must be constrained so that they describe compatible situations. Following these two principles, the formal semantics of an MCL is the result of a suitable composition of the semantics associated to each single context. This takes the name of Local models semantics. The e ects of the two principles above on the inference engines that can be de ned on a multi contextual knowledge base are the following: Locality principle implies that inference rules applied to knowledge inside a context (aka, local inference rules) allow to infer local truths; Compatibility principle instead implies that certain facts in a context can be inferred on the base of other facts present on other compatible contexts. This information propagation is formalized via a special type of inter contextual inference rules called bridge rules
In general terms, a multi context logic is de ned on a family of logical languages fLigi2I where each Li is used to specify what holds in the i-th context. The set I of context indexes (aka context names) can be either a simple set, or a set equipped with some algebraic structure, like total or partial order, and operations on context indices. The relations and functions de ned on I can be used to specify the organization of contexts in terms of an algebraic structure. For instance a partial order no I, can be used to represent that a context is wider (more general) than another context, e.g., football sport, means that the context of sport is more speci c that the context of football. To represent what is true in a scene in di erent time stamps, we can enrich I with a total order so that CV Luciano 2010 CV Luciano 2011 represents the fact that the context describing Luciano's curriculum vit at 2010 precedes the context describing his CV at 2011.
A model for a multi context logic fLigi2I is a pair hMI ; Ci composed of a family of local models MI = fMigi2I , where each Mi is a model of Li, and a compatibility relation C among the local models. The formal structure of C can vary depending on the type of local models and the type of constraints it is necessary to impose on the local models of di erent contexts. For this reason we don't give a general de nition of C, which will be completely de ned for each speci c multi context logic. Satis ability of formulas in Li is de ned w.r.t1. the local models. Namely If is a formula of the language Li, then the a multi context model satis es i : i Mi j=i , where j=i is the satis ability relation associated to the local logic Li.
A Multi context theory in a multi context logic LI = fLigi2I is a family of theories fTigi2I , where each Ti is a set of statements in the logics Li, and a set BR of bridge rules. Intuitively each Ti axiomatizes the constraints on the local models Mi, while the bridge rules BR axiomatizes the compatibility relations. Bridge rules are cross logical axioms and their syntactic form depends on the local logics, so as in the case of the compatibility relation their syntactic depends on the syntax of each Li.
In my invited talk, I will go through the many possible examples of multi context logics, starting from the simplest one, the propositional multi context logics, going through hierarchical meta logics, multi context logics for beliefs and propositional attitudes, non-monotonic propositional context logics, distributed rst order logics, distributed description logics, and logics for semantic import and contextualized knowledge repository for the semantic web.