An algebraic study of exactness in partial contexts

DMFDMFʼs are the natural algebraic tool for modelling reasoning with Körnerʼs partial predicates. We provide two representation theorems for DMFDMFʼs which give rise to two adjunctions, the first between DMFDMF and the category of sets and the second between DMFDMF and the category of distributive lattices with minimum. Then we propose a logic L{1}L{1} for dealing with exactness in partial contexts, which belongs neither to the Leibniz, nor to the Frege hierarchies, and carry on its study with techniques of abstract algebraic logic. Finally a fully adequate and algebraizable Gentzen system for L{1}L{1} is given.