4.4.3. Compactness

Compactness can be thought both in semantic and consequence theoretic terms. This is what happens in every logic, and it extends also to many-valued truth.


$$E \vdash \gamma = \bigvee \{ E\_0 \vdash \gamma \mid E\_0 \text{ finite} \subseteq E \} $$

The following characterisation from [49] brings closer to something that sounds more familiar. In any compact L-entailment system (*Sign*, *Sen*, ") such that the meet operation ∧ is join-continuous, for any finite *κ* ∈ *L*, if *κ* ≤ (*E* " *γ*), then there exists finite *E*<sup>0</sup> ⊆ *E* such that *κ* ≤ (*E*<sup>0</sup> " *γ*).
