In this paper we investigate strong logics of first and
second order that have certain absoluteness properties. We begin
with an investigation of first order logic and the strong logics
ω-logic and β-logic, isolating two facets of
absoluteness, namely, generic invariance and faithfulness. It turns
out that absoluteness is relative in the sense that stronger
background assumptions secure greater degrees of absoluteness. Our
aim is to investigate the hierarchies of strong logics of first and
second order that are generically invariant and faithful against the
backdrop of the strongest large cardinal hypotheses. We show that
there is a close correspondence between the two hierarchies and we
characterize the strongest logic in each hierarchy. On the
first-order side, this leads to a new presentation of Woodin's
Ω-logic. On the second-order side, we compare the strongest
logic with full second-order logic and argue that the comparison
lends support to Quine's claim that second-order logic is really set
theory in sheep's clothing.