Modal Logic via Global Consequence

In modal logic, semantic consequence is usually defined locally by truth preservation at all worlds in all models (with respect to a class of frames). It can also be defined globally by truth preservation in all models (with respect to a class of frames). The latter is called global consequence, which is much less studied than the standard local one. In this paper we first study the relationship between local and global consequence. Then we give some correspondence results for global consequence. Finally, we illustrate two applications of global consequence, connecting it with informational consequence and update consequence proposed in formal semantics. Some results in the paper are already known, which are collected in the paper for the sake of completeness. The others appear to be new. We suggest that global consequence is not only interesting theoretically, but also useful for application.