*4.1.* L*-Institutions*

The extension of the concept of institution from binary to many-valued truth may be achieved at several structural levels. The most primitive level is to consider a plain set of truth values, either in general or in some particular form. At higher levels, we may consider various order theoretic structures. Traditionally, the binary situation is treated as a Boolean algebra in order to support the common logical connectives such as ∧, ∨, ¬, etc. and their semantics. The many-valued approach treats the structure of truth values rather axiomatically, so we can consider order theoretic structures of various degrees of complexity. At the end, the most constrained such structure is in fact the binary Boolean algebra.

Given a set *L*, called the *space of the truth values*, an *L-institution*

$$\mathcal{T} = \left( \text{Sign}^{\mathcal{T}}, \text{Sen}^{\mathcal{T}}, \text{Mod}^{\mathcal{T}}, \models^{\mathcal{T}} \right)$$

is like an ordinary institution with the only difference that the *Satisfaction Relation* is an indexed family of *L-fuzzy relation*, i.e., |=<sup>I</sup> <sup>Σ</sup> : |*Mod*<sup>I</sup> (Σ)| × *Sen*<sup>I</sup> (Σ) → *L* for each Σ ∈ |*Sign*<sup>I</sup> |. Then, the *Satisfaction Condition* obtains the following form: for each morphism *ϕ* : Σ → Σ ∈ *Sign*<sup>I</sup> ,

$$(M' \mathop{=}^{\mathcal{T}}\_{\Sigma'} \operatorname{Sen}^{\mathcal{T}}(\boldsymbol{\varrho})\boldsymbol{\rho}) \ = \ (\operatorname{Mod}^{\mathcal{T}}(\boldsymbol{\varrho})M' \mathop{=}^{\mathcal{T}}\_{\Sigma} \boldsymbol{\varrho})\tag{9}$$

holds for each *M* ∈ |*Mod*<sup>I</sup> (Σ )| and *ρ* ∈ *Sen*<sup>I</sup> (Σ). The Satisfaction Condition says that the *truth degree is an invariant with respect to change of notation*.

For L = (*L*, ) partial order, an L*-institution* means just an *L*-institution. Evidently, the ordinary institutions are just L-institutions for which L is the binary Boolean algebra. For this reason, in the context of the theory of L-institutions, ordinary institutions may be refereed to as *binary institutions*. The step from classic binary institutions to many-valued institutions is hardly new; this idea had appeared already in the early age of institution theory in the form of the so-called 'galleries' of [93]. The 'generalised institutions' of [94] are very similar to L-institutions; however, they introduce an additional monadic structure on the sentence functor meant to model substitution systems. A fully abstract treatment of many-valued semantics appears very early in [50]; however, it differs form the approach of L-institutions in two quite important aspects. One is its single-signature feature. The other is the collapse of model theory modulo elementary equivalence, which makes it unusable for the development of a proper fully abstract many-valued model theory. In other words, Pavelka's approach in [50] would correspond to an L-institution that has only one signature <sup>Σ</sup> and also such that <sup>|</sup>*Mod*(Σ)| ⊆ *<sup>L</sup>Sen*(Σ).

Now, we present the following examples from [48,49,95] very briefly; for more details, the reader should study them from these publications.


In the rest of this section, we present the main developments that have happened in the area of L-institutions over the past decade or so. Our discussion includes the following aspects.

