*3.4. The Logic of Stratified Institutions*

The development of an in-depth model theory in the axiomatic style relies also on the possibility to 'internalise' important logical concepts such as propositional connectives and quantifiers. In ordinary institution theory, this has been achieved very early in [30] (for a more comprehensive treatment, see also [44]). The axiomatic semantic definitions of the common propositional connectives and of quantifiers have been extended to stratified institutions in [52]. Although presented in a different form closer to [53], the definitions below are equivalent to those of [52]. The following notation is useful for what follows. For any Σ-model *M* and any Σ-sentence *ρ*, we let

$$[[\mathcal{M}, \rho]] = \{ w \in [M]\_{\Sigma} \mid M \doteq^w \rho \}.$$

3.4.1. Propositional Connectives

Given a signature Σ in a stratified institution, a Σ-sentence *ρ* is a *semantic*


• etc.

for each Σ-model *M*. A stratified institution *has (semantic) negation* when each sentence of the institution has a negation. It has *(semantic) conjunctions* when each two sentences (of the same signature) have a conjunction. Similar definitions can be formulated for disjunctions, implications, and equivalences. As in ordinary institution theory, distinguished negations are usually denoted by ¬\_ , distinguished conjunctions are usually denoted by \_ ∧ \_ , distinguished disjunctions are usually denoted by \_ ∨ \_ distinguished implications are usually denoted by \_ ⇒ \_ distinguished equivalences are usually denoted by \_ ⇔ \_ , etc. Note that MFOL, MPL together with their hybrid extensions HFOL, HPL, as well as OFOL have all these semantics propositional connectives. SAUT has conjunctions only.

When they exist, the semantic propositional connectives are inter-definable. Moroever, when they exist, the negations, conjunctions, disjunctions, implications, and negations coincide in <sup>S</sup> and <sup>S</sup> .

## 3.4.2. Quantifiers

Given a morphism of signatures *χ* : Σ → Σ , a Σ-sentence *ρ* is a *semantic*

• *Universal χ-quantification* of a Σ -sentence *ρ* when

$$\left[\left[M,\rho\right]\right] = \bigcap\_{\mathrm{Mod}(\chi)\,\mathrm{M}'=\mathrm{M}} \left\{ w \in \left[\left[M\right]\right]\_{\Sigma} \mid \left[\left[M'\right]\right]\_{\chi}^{-1} w \subseteq \left[\left[M',\rho'\right]\right]\right\}, \text{ and}$$

• *Existential χ-quantification* of a Σ -sentence *ρ* when

$$\left[M,\rho\right] = \bigcup\_{\mathrm{Mod}(\chi)\,\mathcal{M}'=\mathcal{M}} \left[\![\mathcal{M}']\!\_{\chi}(\left[\![\mathcal{M}',\rho']\right])\!\_{\chi}\right]$$

for any Σ-model *M*.

A stratified institution *has (semantic) universal* D*-quantification* for a class D of signature morphisms when for each (*χ* : Σ → Σ ) ∈ D, each Σ -sentence has a universal *χ*-quantification. A similar definition applies to existential quantification. Distinguished universal/existential quantifications are denoted by (∀*χ*)*ρ* /(∃*χ*)*ρ* .

When they exist, the universal and the existential *χ*-quantifications, respectively, coincide in <sup>S</sup> and <sup>S</sup> . So, on the one hand, the concepts of semantic propositional connectives and quantifications in ordinary institutions arise as an instance of those of stratified institutions when the underlying set of each [[*M*]]<sup>Σ</sup> is a singleton set. On the other hand, we have seen that the stratified institution concepts of propositional connectives and quantifications are in substance no more general than their ordinary institution theoretic correspondents. Therefore, an alternative equivalent way to introduce the stratified institution semantics of propositional connectives is to define them on the basis of <sup>S</sup> and then infer the above definitions as properties at the level of S.
