We will now recast the above results in terms of a strong logic. We shall make full use of large cardinal axioms and in this setting we are interested in logics that are “well-behaved” in the sense that the question of what implies what is not radically independent. For example, it is well known that CH is expressible in full second-order logic. It follows that in the presence of large cardinals one can always use set forcing to flip the truth-value of a purported logical validity of full second-order logic. However, there are strong logics—like ω-logic and β-logic—that do not have this feature—they are well-behaved in the sense that in the presence of large cardinal axioms the question of what implies what cannot be altered by set forcing. We shall introduce a very strong logic that has this feature—Ω-logic. In fact, the logic we shall introduce can be characterized as the strongest logic with this feature (see Koellner (2010) for further discussion of strong logics and for a precise statement of this result). 