If we're going to include computer science, the big kahuna is of course P vs NP, first proposed by Gödel in a now-famous letter to von Neumann in the 1950s. A fair amount of the existing progress on it has been made by logicians.
Furthermore, the problem can be stated in a particularly logical way: "Is existential second-order logic able to describe languages (of finite linearly ordered structures with nontrivial signature) that first-order logic with least fixed point cannot?"
