*4.2. Flattening* L*-Institutions to Binary Institutions*

The general reduction of many-valued truth to binary truth advocated by the skeptics of many-valued truth can also be applied to L-institutions. It works as follows. Given any Linstitution I = (*Sign*, *Sen*, *Mod*, |=), we define the binary institution <sup>I</sup> = (*Sign* , *Sen* , *Mod* , <sup>|</sup><sup>=</sup> ):


This flattening idea has been present in several places in the fuzzy logic literature. For instance, in [99], our pairs (*ρ*, *κ*) are called 'signed formulas' and given the same interpretation as here.

The flattening of L-institutions to binary institutions has the advantage of reducing things to a well-studied and matured framework and functions well in some aspects, but it falls short in several areas that involve some fine-grained aspects of multiple truth values. Thus, while the flattening <sup>S</sup> of stratified institutions does not pose many limitations, the situation is different with the flattenings of L-institutions.

## *4.3. The Graded Semantic Consequence*

Given an L institution such that L is a complete meet-semilattice, for each Σ-model *M* and each set *E* of Σ-sentences, we define

$$(M \mid \!=\_{\Sigma} E) = \bigwedge \{ M \mid \!=\_{\Sigma} \rho \mid \rho \in E \}. \tag{10}$$

Given an L-institution, there are two ways to extend the satisfaction relation to a semantic consequence relation between sets of sentences and single sentences, both of them generalising the semantic consequence relation of binary institution theory.


(*<sup>E</sup>* <sup>|</sup>=<sup>Σ</sup> *<sup>e</sup>*) = {(*<sup>M</sup>* <sup>|</sup>=<sup>Σ</sup> *<sup>E</sup>*) <sup>⇒</sup> (*<sup>M</sup>* <sup>|</sup>=<sup>Σ</sup> *<sup>e</sup>*) <sup>|</sup> *<sup>M</sup>* ∈ |*Mod*(Σ)|}. (11)

The graded semantic consequence is more subtle and more in the spirit of many-valued truth than the crisp one, although the definition of the latter requires more infrastructure on the space of the truth values, namely that L *is a residuated lattice* [100,101]. Hence, "⇒" of (11) represents the residuated implication operation. This difference in subtlety may be traced to the fact that while the crisp semantic consequence corresponds to the semantic consequence of the binary flattening <sup>I</sup> of the <sup>L</sup>-institution <sup>I</sup> (that *<sup>E</sup>* <sup>|</sup><sup>=</sup> *<sup>e</sup>* holds in <sup>I</sup> means {(*ρ*, 1) <sup>|</sup> *<sup>ρ</sup>* <sup>∈</sup> *<sup>E</sup>*} |= (*e*, 1) in <sup>I</sup> ), the graded semantic consequence is a concept beyond <sup>I</sup> . The graded semantic consequence appears in a disguised form in [50] within the context of Pavelka's theory of fuzzy consequence operators and in a form that is more explicitly similar to ours in [102] within the framework of 'graded consequence relations'. However, both these semantic frameworks are less general than ours, in both of them models being in fact fuzzy theories.

One of the important properties of the semantic consequence in binary institution theory is that it satisfies the axioms of entailment systems. The graded semantic consequence enjoys the same property but for the following refined many-valued concept of entailment. This has been proved in a full form in [49]. In a restricted single signature framework, this has also been proved in [102].
