Forcing axioms via ground model interpretations

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We study principles of the form: if a name σ is forced to have a certain property φ, then there is a ground model filter g such that σg satisfies φ. We prove a general correspondence connecting these name principles to forcing axioms. Special cases of the main theorem are:•Any forcing axiom can be expressed as a name principle. For instance, PFA is equivalent to:▪A principle for rank 1 names (equivalently, nice names) for subsets of ω1.▪A principle for rank 2 names for sets of reals.•λ-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 φ is a notion of largeness for subsets of ω1 (such as being unbounded, stationary or in the club filter) and corresponding forcing axioms.