4.5.2. Model Theoretic Connectives

The many-valued semantic connectives mimic those defined for binary institutions [30,34,44,108], etc., but now, their interpretation is in a many-valued truth context. A Σ-sentence *ρ* is an L-institution that is

• A *semantic conjunction* of sentences *ρ*<sup>1</sup> and *ρ*<sup>2</sup> when L has meets and for each Σmodel *M*,

$$(M \vdash \rho) = (M \vdash \rho\_1) \land (M \vdash \rho\_2);$$

• A *semantic residual conjunction* of sentences *ρ*<sup>1</sup> and *ρ*<sup>2</sup> when L is a residuated lattice and for each Σ-model *M*,

$$(M \vdash \rho) = (M \vdash \rho\_1) \* (M \vdash \rho\_2);$$

• An *semantic implication* of sentences *ρ*<sup>1</sup> and *ρ*<sup>2</sup> when L is a residuated lattice and for each Σ-model *M*,

$$(M \mid = \rho) = (M \mid = \rho\_1) \Rightarrow (M \mid = \rho\_2);$$

• A *semantic disjunction* of sentences *ρ*<sup>1</sup> and *ρ*<sup>2</sup> when L has joins and for each Σ-model *M*,

$$(M \vdash \rho) = (M \vdash \rho\_1) \vee (M \vdash \rho\_2);$$

• A *semantic negation* of a sentence *ρ* when L is a residuated lattice for each Σ-model *M*,

$$(M \mid = \rho') = (M \mid = \rho) \Rightarrow 0\varphi$$

• A *semantic universal χ-quantification* of a Σ -sentence *ρ* for *χ* : Σ → Σ signature morphism when L is a complete meet-semilattice and for each Σ-model *M*

$$(M \mid =\_{\Sigma} \rho) = \bigwedge \{ M' \mid =\_{\Sigma'} \rho' \mid \text{Mod}(\chi)M' = M \};$$

• An *semantic existential χ-quantification* of a Σ -sentence *ρ* for *χ* : Σ → Σ signature morphism when L is a complete join-semilattice and for each Σ-model *M*

$$(M \mid =\_{\Sigma} \rho) = \bigvee \{ M' \mid =\_{\Sigma'} \rho' \mid Mod(\chi)M' = M \}.$$

These definitions can be extended at the level of the respective L-institution. For instance, we say that the L-institution *has conjunctions* when any two Σ-sentences have a conjunction, etc.

The semantic connectives represent yet another situation when the binary flattening diverges from the respective L-institution. In general, it is not possible to establish a general causality relationship between the semantic connectives in the L-institution and in its binary flattening.
