1.2.1. The Institution-Theoretic Trend

The definition in the late 1970s by Goguen and Burstall of the concept of *institution* as a formal definition of the intuitive notion of logic [16–18] achieved the *full axiomatic approach to model theory*. In institution theory, all three components of logical systems—namely, the syntax, the semantics, and the satisfaction relation between them—are treated fully abstractly by relying heavily on *category theory* [19]. Very briefly, the above-mentioned formalization is a category–theoretic structure (*Sign*, *Sen*, *Mod*, |=), called *institution*, consisting of a category *Sign* (of so-called signatures), two functors (*Sen* for the syntax and *Mod* for the semantics), and a family |= of binary relations, which are all bound to satisfy certain consistency axioms. We will clarify precisely this definition below in the paper. In our survey, we will follow this trend of axiomatic model theory known as *institution-independent model theory*, or *institutional model theory*, or *institution-theoretic model theory*. The first in this list of synonymous terminologies may be actually the most informative, as the word 'independent' suggests a model theory that is not confined to any particular logical system.
