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Say we have a model $M$ of a theory $T$ of some core fuzzy logic.

When dealing with compactness, we run in to a situation where the new model being built (by the use of compactness over $M$), will necessarily need a newer BL-algebra for the truth in it to be evaluated (for example, see here: Compactness and completeness in Gödel logic).

It is known that certain logics do not have this problem (the easiest being standard first order logic).

My question is, is there a classification on BL-algebras that says the BL-algebras having this property will allow for compactness (in the classical sense) to go through?

Edit: Also I know that there are studies of this question from the viewpoint; Which logics have (classical) compactness. I would be very thankful if someone could point me in the direction of a comprehensive survey of these results.

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    $\begingroup$ Let me point out a quite "trivial" statement: for every class K of BL chains which is first-order definable (in the setting of classical model theory) it holds that the "many-valued first-order logic given by using safe models over algebras in K" satifies compactness. [In particular, this says that the so called general semantics has compactness] The reason why this is trivial is the fact that such semantics can be modelled using many-sorted classical first-order logic (one sort for the elements of the model and other sort for the truth values), which has compactness. $\endgroup$
    – boumol
    Commented Jun 4, 2015 at 0:18
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    $\begingroup$ The details about how to "model using many-sorted classical first-order logic" are straightforward, but very tedious. They can be found, for instance, in Section 5.2 of the paper dx.doi.org/10.1016/j.apal.2009.01.012 $\endgroup$
    – boumol
    Commented Jun 4, 2015 at 0:19

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