  We recently formulated a new large-cardinal axiom of strength intermediate
between a totally indescribable cardinal and an $\omega$-Erd\H{o}s cardinal,
positing the existence of what we called an "extremely reflective cardinal",
and we showed that the property of being extremely reflective was in fact
equivalent to the property of being remarkable, and we sought to argue that
this axiom should be seen as intrinsically justified. This built on related
earlier work in which the notion of an $\alpha$-reflective cardinal was
formulated. Then Welch and Roberts put forward a family of reflection
principles, Welch's principle implying the existence of a proper class of
Shelah cardinals and provably consistent relative to a superstrong cardinal,
and Roberts' principle implying the existence of a proper class of 1-extendible
cardinals and provably consistent relative to a 2-extendible cardinal. Roberts
tentatively argued that his principle should be seen as intrinsically justified
(at least on the assumption that a weaker form of reflection involving
reflection of second-order formulas with a second-order parameter should be
seen as intrinsically justified). This work overlapped with previous work of
Victoria Marshall's on reflection principles. We analyze the relationship
between reflection principles equivalent to those studied in my earlier work
and stronger but similar reflection principles which are natural extensions of
those of Welch and Roberts. We also show how a natural strengthening of
Roberts' reflection principle yields the existence of supercompact cardinals,
and in the process solve a question which Marshall left open, of whether her
theory $B_0(V_0)$ is strong enough to imply the existence of supercompact
cardinals. We also manage to resolve negatively her question of whether her
theory $B_1(V_0)$ implies the existence of a huge cardinal.
