*2.2. The Concept of Institution*

The original standard reference for institution theory is [18]. An *institution*

$$\mathcal{T} = (\text{Sign}^{\mathcal{L}}, \text{Sen}^{\mathcal{L}}, \text{Mod}^{\mathcal{L}}, \text{|}^{\perp})$$

consists of


$$M' \left| = \stackrel{\mathcal{T}}{=}\_{\varrho \sqsupset \square} \operatorname{Sen}^{\mathcal{T}}(\varrho)\rho \text{ if and only if } \operatorname{Mod}^{\mathcal{T}}(\varrho)M' \right| = ^{\mathcal{T}}\_{\square \varrho} \rho \tag{1}$$

holds for each *M* ∈ |*Mod*<sup>I</sup> (*ϕ*✷)| and *ρ* ∈ *Sen*<sup>I</sup> (✷*ϕ*). This can be expressed as the satisfaction relation |= being a natural transformation:

$$\begin{array}{ccc} \Box \mathcal{q} & \operatorname{Sen}^{\mathcal{T}}(\Box \mathcal{q}) \xrightarrow{\mid \vdash^{\mathcal{Q}}\_{\mathcal{Q}\rho}} [|\operatorname{Mod}^{\mathcal{X}}(\Box \mathcal{q})| \to 2] \\ \varphi & \operatorname{Sen}^{\mathcal{X}}(\varphi) \\ \varphi \sqcap & \operatorname{Sen}^{\mathcal{T}}(\mathcal{q} \sqcap) \xrightarrow[|\rnot^{\mathcal{Z}}\_{\mathcal{q}\complement}] \end{array} \Big| \begin{array}{ccc} \operatorname{Mod}^{\mathcal{X}}(\Box \mathcal{q}) \mid \to 2 \mid \\ \bigwedge^{\mathcal{X}}(\varphi) & \\ \hline \end{array}$$

([|*Mod*(Σ)| → 2] represents the 'set' of the 'subsets' of |*Mod*(Σ)|).

We may omit the superscripts or subscripts from the notations of the components of institutions when there is no risk of ambiguity. For example, if the considered institution and signature are clear, we may denote |=<sup>I</sup> <sup>Σ</sup> just by |=. For *M* = *Mod*(*ϕ*)*M* , we say that *M* is the *ϕ-reduct* of *M* .

The literature shows myriads of logical systems from computing or from mathematical logic captured as institutions. Many of these are collected in [25,44]. In fact, an informal thesis underlying institution theory is that any 'logic' may be captured by the above definition. While this should be taken with a grain of salt, it certainly applies to any logical system based on satisfaction between sentences and models of any kind. In [44], one can read how propositional logic PL, (many-sorted) first order logic FOL together with many of its fragments, partial algebra, various flavours of modal logic, intuitionistic logics, preordered algebra, multialgebras, membership algebra, higher-order logics with various semantics, many-valued logics, etc. can be captured as institutions. In all these cases, the effort to capture the (model theory of the) respective logical system as an institution implies a conceptual adjustment of some of its aspects in the direction of a higher mathematical rigour, an emblematic case being that of the variables (see for example the relevant discussion in [55]). In many cases, some important concepts have been extended, most notably concepts of signature morphisms. In order to fully understand these conceptual developments, it is worth looking in the literature at detailed examples of mainstream concrete institutions.
