  We study methods to obtain the consistency of forcing axioms, and
particularly higher forcing axioms. We first force over a model with a
supercompact cardinal $\theta>\kappa$ to get the consistency of the forcing
axiom for $\kappa$-strongly proper forcing notions which are also
$\kappa$-lattice, and then eliminate the need for large cardinals. The proof
goes through a natural reflection property for $\kappa$-strongly proper
forcings. We also produce a model of this forcing axiom with $2^\kappa$
arbitrarily large, and prove the inconsistency of certain natural
strengthenings of the axiom.
