*1.3. Beyond Classical Institutional Model Theory*

The concept of institution is abstract enough to accommodate any logical system based on satisfaction between sentences and models of any kind, including non-classical logics. However, the developments discussed above, albeit highly abstract and axiomatic, may be considered "classical" in the sense that they reflect concepts, methods and results that have been originally worked out at a concrete level in first-order model theory. Classical institutional model theory may be effective to some extent in non-classical contexts but not entirely satisfactory. For instance, non-classical logical situations that are beyond the usual binary satisfaction relation between models and sentences, such as local satisfaction in modal logics or many-valued satisfaction, admit classical institution–theoretic formalisations but at the cost of flattening the satisfaction relation to the binary case [44,48], which is a process that alters the nature of the respective logics. Consequently, there is a loss of information, and important non-classical logic developments cannot be completed naturally or not at all. For example, when considering institutions for modal logics, this is completed on the basis of global satisfaction, which is much less relevant than the local satisfaction relation. In addition, in the flattening of many-valued satisfaction, the possibility of grading the consequence relation [49] is lost. Moreover, logic encodings that are based on *theoroidal comorphisms* are difficult to define because of the multifaceted nature of the concept of theory in many-valued logics [49,50]. The answers to these challenges is given

by the *stratified institutions* [51–53] and the L*-institutions* [49] that represent extensions of the ordinary concept of institution that accommodate properly models with states and local satisfaction, and many-valued semantic truth, respectively. Technically, these two new mathematical structures are generalisations of the ordinary concept of institution. This survey is about these two extensions of ordinary institution theory with emphasis on model theory-motivated developments rather than computing science. Regarding the technical level of this survey, while avoiding technical vagueness, we will also deliberately try to avoid intricate technicalities that pervade many institutional model theory works. In order to achieve such a balance in the presentation, we will employ more informal explanations while providing pointers to works where the respective technical details can be found.

Before surveying the theories of stratified and of L-institutions, respectively, we will review the ordinary concept of institution.

## **2. Institutions**

In this section, we will first discuss the role played by category theory in institution theory, we will review some basic notational conventions, and finally, we will recall the concept of institution.
