Given an order-enriched category, it is known that all its KZ-monadic subcategories can be described by Kan-injectivity with respect to a collection of morphisms. We prove the analogous result for Kan-injectivity with respect to a collection H of commutative squares. A square is called a Kan-injective consequence of H if by adding it to H Kan-injectivity is not changed. We present a sound logic for Kan-injectivity consequences and prove that in ``reasonable" categories (such as $\Pos$ or $\Top_0$) it is also complete for every set H of squares.
Keywords: order-enriched category, Kan-injectivity, KZ-monad, Kan-injectivity logic, locally ranked category
2010 MSC: 18C20, 18B35, 18D20, 54B30, 06B35,06D22, 18A15
Theory and Applications of Categories, Vol. 32, 2017, No. 10, pp 338-379.