  The object of this paper is twofold: In the first part, we unify and extend
the recent developments on honesty theory of perturbed substochastic semigroups
(on $L^{1}(\mu)$-spaces or noncommutative $L^{1}$ spaces) to general state
spaces; this allows us to capture for instance a honesty theory in preduals of
abstract von Neumann algebras or subspaces of duals of abstract $C^{\ast
}$-algebras. In the second part of the paper, we provide another honesty theory
(a semigroup-perturbation approach) independent of the previous
resolvent-perturbation approach and show the equivalence of the two approaches.
This second viewpoint on honesty is new even in $L^{1}(\mu)$ spaces. Several
fine properties of Dyson-Phillips expansions are given and a classical
generation theorem by T. Kato is revisited.
