*4.2. Semantic Connectives*

Institution theory has developed its own general approach to logical connectives [3,24,25]. This was refined in [6] to stratified institution theory. With 3/2-institutions, there are two ways to approach this issue.


We argue that the straightforward approach does not work, which means that in order to have sound semantic connectives, we have to rely on the representation result. Our argument is based on an important property of the semantic connectives, namely that they should be preserved by the translations along signature morphisms. For instance, for a signature morphism *ϕ*, if *ρ* is a semantic disjunction of *ρ*<sup>1</sup> and *ρ*<sup>2</sup> in the signature ✷*ϕ*, then *Sen*(*ϕ*)*ρ* should be a semantic disjunction of *Sen*(*ϕ*)*ρ*<sup>1</sup> and *Sen*(*ϕ*)*ρ*<sup>2</sup> in *ϕ*✷. This holds naturally in ordinary institution theory as well as in stratified institution theory, the proof of this relying on the satisfaction condition. In fact, in the stratified institutions case, this property can be established from the corresponding ordinary institution theory property via the <sup>S</sup> representation, since as noticed in [6], the common propositional connectives and the quantification connectives do coincide in <sup>S</sup> and in <sup>S</sup> .

In order to understand what is wrong with the straightforward approach to the semantic connectives in 3/2-institutions, let us attempt to establish the preservation property for the semantic disjunction. Let *ϕ* be a signature morphism and assume that *ρ* is a semantic disjunction of *ρ*<sup>1</sup> and *ρ*<sup>2</sup> in ✷*ϕ*, which means that for each ✷*ϕ*-model *M*, *M* |= *ρ* if and only if *M* |= *ρ<sup>k</sup>* for some *k* ∈ {1, 2}. We have to establish the same property for *Sen*(*ϕ*)*ρ*, *Sen*(*ϕ*)*ρ*1, and *Sen*(*ϕ*)*ρ*2. A first issue with this is the existence of these translations. We can overcome this by requiring that *Sen*(*ϕ*)*ρ* is a semantic disjunction of *Sen*(*ϕ*)*ρ*<sup>1</sup> and *Sen*(*ϕ*)*ρ*<sup>2</sup> when *all three translations do exist*. Let us attempt to prove the property under this new formulation. We have to prove that for any *ϕ*✷-model *M* ,

*M* |= *Sen*(*ϕ*)*ρ* if and only if *M* |= *Sen*(*ϕ*)*ρ<sup>k</sup>* for some *k* ∈ {1, 2}.

However, *M* |= *Sen*(*ϕ*)*ρ* means *M* |= *ρ* for all *M* ∈ *Mod*(*ϕ*)*M* . Since *ρ* is the semantic disjunction of *ρ*<sup>1</sup> and *ρ*2, this further means that for each *M* ∈ *Mod*(*ϕ*)*M* , there exists *k* ∈ {1, 2}, such that *M* |= *ρk*. At this point, we have to get back to *ϕ*✷, i.e., to establish that there exists *k* ∈ {1, 2} such that *M* |= *Sen*(*ϕ*)*ρk*, which means that for all *M* ∈ *Mod*(*ϕ*)*M* , *M* |= *ρ<sup>k</sup>* for the *same k*. This is a gap because for one *M* we may have *M* |= *ρ*<sup>1</sup> and for another *M* we may have *M* |= *ρ*2. So, the property cannot be established.

This failure to prove the preservation of semantic disjunctions along signature morphisms also tells us about the crucial role played by the reducts; that in fact the satisfaction in 3/2-institutions has the reducts as an implicit parameter. This perspective provides a solution to our problem. Additionally, here we are; this situation calls for a stratified institution approach. In the particular case of the semantic disjunctions, this means that, under the representation given by corollary 1, we should define *ρ* as a semantic disjunction of *ρ*<sup>1</sup> and *ρ*2, when for each *χ*-model *M*:

$$\{N \in \left[M\right] \mathbb{I}\_{\mathcal{X}} \mid N \vdash \rho\} = \{N \in \left[M\right] \mathbb{I}\_{\mathcal{X}} \mid N \vdash \rho\_1\} \cup \{N \in \left[M\right] \mathbb{I}\_{\mathcal{X}} \mid N \vdash \rho\_2\}.$$

Similar definitions can be derived in the case of the other semantic connectives by following the stratified institutions approach [6].
