4.4.1. Closure Systems

Concepts of closures of theories can be regarded as axiomatic treatments of consequence relations. This approach originates from Tarski's work [106] and later on was applied by Pavelka [50] to many-valued theories. The following definition from [49] extends the latter to the multi-signature framework. Given a partial order L = (*L*, ≤), an L*-closure system* is a tuple (*Sign*, *Sen*, C) where



In the binary framework, there is a straightforward equivalence between the concepts of entailment system and closure system: *E* "<sup>Σ</sup> *e* if and only if *e* ∈ CΣ*E*. However, in the many-valued framework, the relationship between the two concepts is much more interesting. Let us present two of them from [49].

• Provided some conditions on L are fulfilled, the following closure applies to any graded entailment system. Let L = (*L*, ≤, ∗) be a complete meet-semilattice with a binary operation ∗ and let (*Sign*, *Sen* ") be an L-entailment system. The following definition draws inspiration from Goguen's many-valued interpretation of Modus Ponens [107]. A theory *X* : *Sen*(Σ) → *L* is *weakly closed* with respect to the entailment system when for each entailment *E* "<sup>Σ</sup> *ρ*,

$$X(E) \* (E \vdash \rho) \leq X(\rho).$$

If ∗ is increasing monotone, then in [49], we have proved that the weakly closed theories are closed under arbitrary meets. This allows for the following definition: for any theory *X*, let *X*◦, called the *weak closure* of *X*, denote the least weakly closed theory greater than *X*. In [49], we have also proved that the weak closure (\_)◦ defines an L-closure system.

• The second closure system on many-valued theories has a semantic nature, so its basic framework is now stronger than in the case of the previous closure system. Note that in any <sup>L</sup>-institution, the Galois connection between (P|*Mod*(Σ)|, <sup>⊇</sup>) and (*LSen*(Σ), <sup>≤</sup>) determines an L-closure system (*Sign*, *Sen*,(\_)∗∗). This allows for the following definition. In any L-institution, a Σ-theory is *strongly closed* when *X* = *X*∗∗. Moreover, *X*∗∗ is called the *strong closure* of *X*. The relationship between the two closure systems has been established in [49] as follows. When L is a complete residuated lattice, in any L-institution and for any Σ-theory *X*, if *X*◦ denotes its weak closure with respect to the semantic L-entailment system, then *X*◦ ≤ *X*∗∗.
