In the present article, we construct the theory of near Boolean families of valuation rings which extends the corresponding theory for Boolean families. We establish the fact that the local-global principle (LGP) for such families is effectively elementary. We indicate sufficient conditions for 1-embeddability of holomorphy rings of near Boolean families that possess the LGP property. By way of application, we prove that the elementary theory of the ring of integrably totally $p$-adic integers and the elementary theory of the class of all closed subrings of almost all algebraic numbers are decidable.