*4.4. Many-Valued Theories, Consistency and Compactness*

In binary institution theory, a Σ-theory is a set of Σ-sentences. (However, in many works, including [18,44], etc., this is called 'presentation', the word 'theory' being used for 'presentations' that are closed under semantic consequence. This owes to the algebraic specification tradition which considers theories that are 'presented' by (finite) sets of sentences, these being in fact specification modules.) Any theory may be represented by its characteristic function *Sen*(Σ) → 2, which for each sentence gives a truth value for its membership to the respective theory. This new perspective on theories is the basis for the generalisation of the concept of theory to many-valued truth. For any fixed set *L* and for any functor *Sen* : *Sign* → **Set**, a Σ*-theory* is just a function *X* : *Sen*(Σ) → *L*. When L = (*L*, ≤, ∧) is a complete meet-semilattice, for any Σ-theory *X* : *Sen*(Σ) → *L* and for any *E* ⊆ *Sen*(Σ), we denote

$$X(E) = \bigwedge \{ X(e) \mid e \in E \}. \tag{12}$$

Note that a theory in an L-institution I corresponds exactly to a theory in its binary flattening <sup>I</sup> by representing any function *<sup>X</sup>* : *Sen*(Σ) <sup>→</sup> *<sup>L</sup>* as the set {(*ρ*, *<sup>X</sup>*(*ρ*)) <sup>|</sup> *<sup>ρ</sup>* <sup>∈</sup> *Sen*(Σ), *X*(*ρ*) = 0} (0 denotes the bottom element of L).

The concept of Galois connection between syntax and semantics in binary institution theory admits a natural extension to many-valued truth. Let L be a complete meetsemilattice. In any L-institution:


For each signature Σ, the mappings (\_)∗ defined above represent a Galois connection between (P|*Mod*(Σ)|, <sup>⊇</sup>) and (*LSen*(Σ), <sup>≤</sup>).
