Compactness of first-order fuzzy logics

One of the nice properties of the first-order logic is the compactness of satisfiability. It state that a finitely satisfiable theory is satisfiable. However, different degrees of satisfiability in many-valued logics, poses various kind of the compactness in these logics. One of this issues is the compactness of $K$-satisfiability. Here, after an overview on the results around the compactness of satisfiability and compactness of $K$-satisfiability in many-valued logic based on continuous t-norms (basic logic), we extend the results around this topic. To this end, we consider a reverse semantical meaning for basic logic. Then we introduce a topology on $[0,1]$ and $[0,1]^2$ that the interpretation of all logical connectives are continuous with respect to these topologies. Finally using this fact we extend the results around the compactness of satisfiability in basic ogic

The Craig Interpolation Property in First-order G\"odel Logic

In this article, a model-theoretic approach is proposed to prove that the first-order G\"odel logic, $\mathbf{G}$, as well as its extension $\mathbf{G}^\Delta$ associated with first-order relational languages enjoy the Craig interpolation property. These results partially provide an affirmative answer to a question posed in [Aguilera, Baaz, 2017, Ten problems in G\"odel logic]