Forcing axioms via ground model interpretations

We study principles of the form: if a name sigma is forced to have a certain property phi, then there is a ground model filter g such that sigma g satisfies phi. We prove a general correspondence connecting these name principles to forcing axioms. Special cases of the main theorem are: center dot Any forcing axiom can be expressed as a name principle. For instance, PFA is equivalent to: center dot A principle for rank 1 names (equivalently, nice names) for subsets of omega 1. center dot A principle for rank 2 names for sets of reals. center dot lambda-bounded forcing axioms are equivalent to name principles. Bagaria's characterisation of BFA via generic absoluteness is a corollary. We further systematically study name principles where phi is a notion of largeness for subsets of omega 1 (such as being unbounded, stationary or in the club filter) and corresponding forcing axioms.