Fibered universal algebra for first-order logics

We extend Lawvere-Pitts prop-categories (aka. hyperdoctrines) to develop a general framework for providing “algebraic” semantics nonclassical first-order logics. This includes natural notion of substitution, which allows logics be considered as structural closure operators just propositional are in abstract algebraic logic. then establish an extension the homomorphism theorem from universal algebra generalized and characterize two on prop-categorical semantics. The first closes class structures (which interpreted morphisms prop-categories) under satisfaction their common theory second associated consequence. It turns out, these have characterizations that closely mirror Birkhoff's characterization algebras equational Blok Jónsson's consequence, respectively. These unique They do not analogs, example, Tarskian classical we consider much more than traditional intuitionistic or triposes (i.e., topos representing indexed partially ordered sets). Nonetheless, our knowledge, results still new, even when restricted special classes prop-categories.