This paper considers the class of meaning that Grice (1975) referred to as conventional implicature (CI). There has been a great deal of discussion on this topic since the work of Potts (2005), and much new data has been brought to bear on the subject. However, in the midst of this new work, we have got away from Grice's original conception, and the fact that CIs as defined by Potts are different than what Grice originally intended is often overlooked. The dislocation construction known as a double-subject sentence (DSS), makes a nice test case for exploring the two conceptions of conventional implicature. I show that the DSS meets three of Potts' four CI criteria quite readily. Yet, Potts' criterion that CIs be speaker-oriented — which distinguishes them from Gricean CIs — is difficult to illustrate for the DSS. I then show that this criterion is empirically unsustainable and so should be abandoned. This leaves us with a system that is essentially Gricean, in which the DSS is given a straightforward account.