According to the standard argument from targetless higher-order representations, the possibility of such representations presents a dilemma for higher-order theorists. In this article I argue that there are two theoretically well-motivated replies to the standard argument. Consequently, the standard argument against higher-order theories fails. I then go on to argue that while certain versions of higher-order theory can adequately respond to the standard argument, they both, nevertheless, fail to explain the fineness-of-grain that phenomenally conscious experience appears to have.