4.4.2. Consistency

The following is a generalisation of the concept of consistent theory from binary institution theory to L-institutions. According to [49], in any L-institution, a Σ-theory *T* is *consistent* when there exists a Σ-model *M* such that *T M*∗. *E* is *consistent* when there exists *κ* > 0 such that *E* is *κ*-consistent; otherwise, it is *inconsistent*. Note that the concept of *κ*-consistency can be derived from the corresponding consistency concept from binary institution theory by considering the binary flattening of the respective L-institution.

Now, we introduce another concept of consistency that is relative to a fixed truth value. First, we prepare some notations. For any truth value *κ* ∈ *L*, let *T<sup>κ</sup>* denote the *constant theory* defined by *Tκρ* = *κ* for each sentence *ρ*. For any Σ-theory *T* and Γ ⊆ *Sen*(Σ), the theory *T*|Γ is defined for each *ρ* ∈ *Sen*(Σ) by

$$(T|\Gamma)\rho = \begin{cases} T\rho\_{\prime} & \rho \in \Gamma\\ 0, & \text{otherwise}. \end{cases}$$

In any L-institution, for any truth value *κ*, a set *E* of Σ-sentences is *κ-consistent* when *Tκ*|*E* is consistent. Note that this concept can also be reduced to binary consistency since *E* is *κ*-consistent if and only if (*E*, *κ*) = {(*e*, *κ*) | *e* ∈ *E*} is consistent in the binary flattening of the respective L-institution. Note also that in the binary case, both concepts of consistency defined above collapse to the same concept.
