4.7.1. Graded Interpolation versus Many-Valued Robinson Consistency

Let us first have a look at the binary institution theoretic version of Robinson consistency (abbr. *Rc*). In an institution, a commutative square of signature morphisms such as below

is a *Robinson consistency (Rc) square* when any finite sets *Ei* of Σ*i*-sentences, *i* = 1, 2, with 'inter-consistent reducts' (i.e., {*ρ* ∈ *Sen*(Σ) | *E*<sup>1</sup> |= *ϕ*1*ρ*}∪{*ρ* ∈ *Sen*(Σ) | *E*<sup>2</sup> |= *ϕ*2*ρ*} has a model) has 'inter-consistent Σ -translations' (i.e., *θ*1*E*<sup>1</sup> ∪ *θ*2*E*<sup>2</sup> has a model).

The many-valued version of this is based on a many-valued concept of 'inter-consistency' which is relative to arbitrary truth values and, very importantly, the two truth values of the inter-consistency of the reducts and of the translations, respectively, are in general not necessarily equal. Then, we obtain the expected bi-directional causality between Rc and a somehow stronger version of Ci. There are many aspects underlying this result that deserve mention.

