Algebraic LogicAlgebraic Logic is the discipline that studies "Bridge Theorems" that allow to cross the mirror between Logic and Algebra by associating a purely semantic interpretation (such as the amalgamation property) with a given metalogical property (such as the interpolation property). This allows to study metalogical phenomena through the lenses of their semantic counterparts, which are typically amenable to the powerful methods of Universal Algebra, Lattice Theory and Category Theory. This perspective proved to be very fruitful both in the study of concrete logical systems such as Fuzzy, Modal and Intuitionistic Logics as well as in its most general formulation known as Abstract Algebraic Logic.
Mathematical Fuzzy LogicMathematical Fuzzy Logic is the part of Mathematical Logic that studies logical systems which extend the two-valued semantics of classical logic by allowing formulas to take value in the real unit interval. These formalisms provide a more faithful representation of those properties and predicates which, for their own nature, are perceived as graded. Several points of contact with other areas of mathematics such as functional analysis, probability theory, universal algebra, real convex geometry, have been explored and are a fruitful ongoing line of research.
Modal and Intuitionistic LogicModal and Intuitionistic Logics are some of the main non-classical logics. Modal Logic is known for extending the expressive power of classical (and many-valued) logic, while preserving many of its desirable computational features. On the other hand, Intuitionistic Logic is the deductive system that governs the constructive aspects of mathematics. Both these logics have a natural semantics consisting of topological Kripke frames, which proved to be a versatile tool in the modellization of problems ranging from Theoretical Computer Science and Artificial Intelligence to Philosophy and Metaphysics. In view of Duality Theory, this topological semantics is dually equivalent to a purely algebraic one which, moreover, is amenable to the methods of Algebraic Logic.