The Blueprint for Constructing Institutions of Statements

In the following subsections, we define Institutions of Statements (IoS). Each Institution of Statements is characterized by seven parameters that we will introduce step by step. The reader can keep track of the development consulting the scheme in Figure 1.

**Figure 1.** Stepwise construction of an Institution of Statements (IoS).

#### *3.2. Base Category*

To define a certain Institution of Statements, we first have to choose a base category comprising as well the basic syntactic entities as the semantic domains. The base category fixes, somehow, the linguistic and conceptual universe we intend to work within.

**Definition 1** (First parameter: Base Category)**.** *The first parameter of an Institution of Statements is a chosen base category* Base*.*

**Remark 2** (Uniformity)**.** *All components of a logic are related with each other; thus, it seems to be natural to require that they all live in the same category (universe). It turns out, however, that this uniformity requirement is not as trivial, as it looks like at a first glance. Some effort may be needed to present known logic's and formalisms in such a uniform way. One underlying cause for this kind of additional effort is that we follow the tradition in logic and work here with the semantics-as-interpretation paradigm (Tarskian semantics) (see Section 3.3.2).*

*One problem could be that we become obliged to reflect a distinction between different syntactic entities, like predicate symbols and sort symbols, for example, already on the semantic level. In the more practical relational data model example, this is quite appropriate while it needs getting used to in the mFOL-example.*

*In other words: We take the chance to present the mFOL-example in a more unconventional way also with the intention to illustrate the flexibility of our framework.*

**Example 1** (FOL: Base category)**.** *There is a particular dependency in many-sorted first-order signatures. We have first to establish a set of sort symbols before we can define the arity of predicates (compare Chapter 12 in [2]). We fix a set S* ∈ Set*Obj of sort symbols and consider S as a graph (without edges) in* Graph*. Relying on the coercion* Graph GRAPH *(see Remark 1), we choose the interpretation category* Set*<sup>S</sup>* = [*<sup>S</sup>* <sup>→</sup> Set] *in* CAT *as base category* Base*FOL. We call the objects in* [*S* → Set] *S-sets and the morphisms S-maps, respectively.*

**Example 2** (ALC: Base category)**.** *The prototypical description logic* Attributive Concept Language with Complements (ALC) *can be seen as a fragment of unsorted FOL without functions [36], thus the category* Base*ALC* := Set *in* CAT *is our base category of choice.*

**Example 3** (mFOL: Base category)**.** *In this example, we describe the traditional formalism* many-sorted first-order logic without functions *as such.*

*In this formalism,* sorts *and* predicates *are the only concepts. Therefore, we use the set MmF* := {*S*,*P*} ∈ Set*Obj of* concept symbols*. Relying on coercion* Set SET*, we define the base category as a slice category* Base*mF* := SET/*MmF. (For any category* C *and any object T in* C *we can define the slice category* C/*T with objects all pairs* (*A*, *ϕ*) *of an object A and a morphism ϕ* : *A* → *T in* C *and morphisms f* : (*A*, *ϕ*) → (*B*, *ψ*) *given by morphisms f* : *A* → *B in* C *satisfying the condition ϕ* = *f* ; *ψ.) We use the term MmF*-typed set *for the objects in* SET/*MmF, i.e., for pairs* (*A*, *τA*) *of a set A and a* typing map *τ<sup>A</sup>* : *A* → *MmF.*

**Example 4** (CT: Base category)**.** *Located on the same abstraction (modeling) level as the examples FOL and ALC and reflecting the viewpoint that a category is a graph equipped with composition and identities, we outline a diagrammatic version of the theory of small categories; thus, the category* Graph *of small graphs is chosen as the base category* Base*CT. Note that* Graph *is isomorphic to the*

*interpretation category* [*MCT* → Set] *with MCT the graph E sc tg <sup>V</sup> thus we follow somehow the same pattern as in example FOL.*

**Example 5** (RM: Base category)**.** *The relational data model relies on the concepts* table*,* column

*and* datatype*. Analogously to example mFOL, we use a graph MRM* := ( *<sup>T</sup> <sup>c</sup> <sup>D</sup>* ) <sup>∈</sup> Graph*Obj of concept symbols and define the base category as a slice category* Base*RM* := GRAPH/*MRM relying on the coercion* Graph GRAPH*. We use the term MRM*-typed graph *for the objects in* GRAPH/*MRM, i.e., for pairs* (*G*, *τG*) *of a graph G and a* typing morphism *τ<sup>G</sup>* : *G* → *MRM.*
