*4.5. The Logic of* L*-Institutions*

Many-valued logic in the institution theoretic framework can be approached at two different levels, namely that of consequence (L-entailment systems) and that of semantics (L-institutions). The former is of course more abstract than the latter, but the relationship between them is non-trivial. All these have been addressed in [49] as follows.

4.5.1. Entailment Theoretic Connectives

In an L-entailment system (*Sign*, *Sen*, "), a Σ-sentence *ρ* is

• A *conjunction* of sentences *ρ*<sup>1</sup> and *ρ*<sup>2</sup> when for any set of sentences *E*,

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

• A *residual conjunction* of sentences *ρ*<sup>1</sup> and *ρ*<sup>2</sup> when for any set of sentences *E*,

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

• An *implication* of sentences *ρ*<sup>1</sup> and *ρ*<sup>2</sup> when for any set of sentences *E*,

$$E \vdash \rho = E \cup \{\rho\_1\} \vdash \rho\_2 \mathbf{y}$$

• A *disjunction* of sentences *ρ*<sup>1</sup> and *ρ*<sup>2</sup> when L has joins and for any set of sentences *E*,

$$E \vdash \rho = (E \vdash \rho\_1) \lor (E \vdash \rho\_2);$$

• A *negation* of the sentence *ρ* when for any sentence *e*,

$$\{\rho, \rho'\} \vdash e = 1;$$

• A *universal χ-quantification* of a Σ -sentence *ρ* for *χ* : Σ → Σ signature morphism when for any set of Σ-sentences *E*

$$E \vdash\_{\Sigma} \rho \;=\; \chi(E) \vdash\_{\Sigma'} \rho';$$

• An *existential χ-quantification* of a Σ -sentence *ρ* for *χ* : Σ → Σ signature morphism when for any Σ-sentence *e*

$$
\rho \vdash\_{\Sigma} e \;=\; \rho' \vdash\_{\Sigma'} \chi(e).
$$

These definitions can be extended at the level of the L-entailment system. For instance, we say that the L-entailment system *has conjunctions* when *any* two Σ-sentences have a conjunction and similarly for the other connectives.

When L is the binary Boolean algebra, the above definitions yield the usual entailment theoretic connectives from the institution theory literature (e.g., [108]). In binary logic, the inequalities that are implicit in the equation defining the entailment theoretic implication are known as *Modus Ponens* (≤) and the *Deduction Theorem* (≥). This terminology can be extended to L-entailment systems.

As in the binary situation, we can consider the *least entailment system* that "contains" a given entailment system and that has some of the connectives defined above. This is supported by the following result from [49]: any intersection of entailment systems (that share the same sentence functor) is an entailment system. Moreover, the property of having a certain connective is invariant with respect to such intersections.
