  We introduce and study a family of axioms that closely follows the pattern of
parametrized diamonds, studied by Moore, Hru\v{s}\'ak, and D\v{z}amonja in
[13]. However, our approach appeals to model theoretic / forcing theoretic
notions, rather than pure combinatorics. The main goal of the paper is to
exhibit a surprising, close connection between seemingly very distinct
principles. As an application, we show that forcing with a measure algebra
preserves (a variant of) $\diamondsuit(\mathfrak{d})$, improving an old result
of M. Hru\v{s}\'ak.

Logic