  The Maximality Principle MP is a scheme which states that if a sentence of
the language of ZFC is true in some forcing extension V^P, and remains true in
any further forcing extension of V^P, then it is true in all forcing extensions
of V. A modified maximality principle MP_Gamma arises when considering forcing
with a particular class Gamma of forcing notions. A parametrized form of such a
principle, MP_Gamma(X), considers formulas taking parameters; to avoid
inconsistency such parameters must be restricted to a specific set X which
depends on the forcing class Gamma being considered. A stronger necessary form
of such a principle, Box MP_Gamma(X), occurs when it continues to be true in
all Gamma forcing extensions.
  This study uses iterated forcing, modal logic, and other techniques to
establish consistency strengths for various modified maximality principles
restricted to various forcing classes, including CCC, COHEN, COLL (the forcing
notions that collapse ordinals to omega), <kappa directed closed forcing
notions, etc., both with and without parameter sets. Necessary forms of these
principles are also considered.
