4.7.2. Graded Definability by Graded Interpolation

Both in the concrete classical case and in the institution theoretic context, interpolation constitutes a principal cause for the definability property, i.e., that implicitly implies explicit definability. In fact, in [44], it has been revealed that interpolation in the Craig–Robinson form is what is needed in order to establish definability. In this way, we can dispense with implications, and while implications plus Ci obtain Craig–Robinson interpolation, there are important situations when we have the latter in the absence of implications, such as in many-sorted Horn clause logics (cf. [44]).

In [109], we have extended both the implicit and the explicit definabilities from the their binary version of [40,44] to many-valued truth as follows.

• In any L-entailment system, for any *κ* ∈ *L*, a signature morphism *ϕ* : Σ → Σ is *defined κ-implicitly* by a set *E* ⊆ *Sen*(Σ ) when for any diagram of pushout squares such as below

(14)

and for any Σ <sup>1</sup>-sentence *ρ*, we have that

$$
\mu(\theta'E') \cup \upsilon(\theta'E') \cup \mathfrak{u}\rho \vdash \upsilon\rho \ge \kappa.
$$

• In any L-entailment system, for each *κ* ∈ *L*, a signature morphism *ϕ* : Σ → Σ is *κ-explicitly defined* by a set of sentences *E* ⊆ *Sen*(Σ ) when for each pushout square of signature morphisms such as

$$\begin{array}{c} \Sigma \xrightarrow{\begin{smallmatrix} \varphi\\ \varphi \end{smallmatrix}} \Sigma'\\ \varphi \xrightarrow{\begin{smallmatrix} \varphi\\ \varphi \end{smallmatrix}} \Sigma'\_1 \end{array} \tag{15}$$

and each *ρ* ∈ *Sen*(Σ <sup>1</sup>), there exists a finite set of sentences *E<sup>ρ</sup>* ⊆ *Sen*(Σ1) such that

$$(\theta'E' \cup \rho \vdash \mathcal{q}\_1E\_\rho) \* (\theta'E' \cup \mathcal{q}\_1E\_\rho \vdash \rho) \ge \kappa.$$

The main result of this development is a theorem that generalises the binary truth result of [40]. It says that in any L-institution with a form of model amalgamation and which enjoys Craig–Robinson interpolation (with respect to designated classes of signature morphisms), a signature morphism is defined *κ*-explicitly when it is defined -implicitly provided the truth values *κ* and are related by a condition similar to one of the conditions underlying the implication of Rc from Ci.
