To understand why the question even arises, you have to go back to a text that isn't often read these days, William Warburton's Divine Legation of Moses. Even though it is scarcely read today, it exercised a powerful influence on its time. In the course of the work (Cockburn has in mind the second edition in particular) Warburton advances a particular view of moral obligation. It is with this that Catharine Cockburn (1679-1749) disagrees; and atheists are significant test case.
Warburton argues that morality is not based on perception of the fitness of things (such as Clarke, or Cockburn, argue for). An atheist could recognize the 'natural essential differences' between things; but this is not sufficient for moral knowledge. Knowledge of one does not ground knowledge of the other. (This is Warburton's version of 'Hume's fork'.) What the atheist is missing is that (in Cockburn's summary) "nothing but the will, or the law of a superior, can constitute the morality of actions" (138). As Cockburn argues, this seems to get things backwards. But she also argues against the claim that an atheist could only know the bare 'natural essential differences' rather than moral fitnesses as well. Her argument for this is worth quoting in full.
In order to judge of this point, let us suppose of a society of Atheists, one fallen into a pit, where he must inevitably perish if unassisted; and another of them happening to travel that way, who could with great ease relieve him. Will these two persons perceive nothing, but the natural essential difference between leaving a man to perish in a pit, and helping him out of it? Would not the distressed consider one of these as inhumanity to be detested, and the other as a good action deserving grateful return? Might not the traveler be too conscious, that one of these actions would be better than the other, have a goodness in it more to be approved? Yet we will supose some business or pleasure he is intent upon, stifles this consciousness and prevails with him, to leave the distressed to his miserable fate; and that he afterwards relates to the rest of society, how he had hurried from the melancholy object, in pursuit of his inclinations. Can it be imagined, that they would coldly consider this action, only as not agreeable to reason? Or would they not rather judge it to be wrong, inhuman, and worthy of detestation. It cannot, I think, be doubted, that such a society might be capable of these sentiments. And what is this but to perceive the moral difference of things, tho' they have not discovered a superior will to enforce the observance of them? Or tho' they may think the guilty secure from that punishment, which they must be conscious so great an immorality deserves. (138n)
Thus Cockburn's view is that there is good reason to think that moral differences aren't so wholly independent of the facts as Warburton claims; and since atheists can clearly distinguish 'natural essential differences' -- for instance, the difference between leaving a man to die and helping him to live -- there is no reason to deny that they could see the moral difference between the two -- that one of these is better than the other. She notes elsewhere that this is not a conflation of natural differences and moral differences, but only a recognition that when we recognize natural differences, it creates a fitness for certain types of action, and a lack of fitness for other types of action. And this is sufficient for identifying grounds of obligation.
It is interesting, as a side issue, to think of how Hume is to be placed with regard to this argument. Cockburn's argument is formulated in the terms of what Hume calls 'the abstract theory of morals', which Hume sharply rejects and criticizes at great length at the beginning of Book III of the Treatise and elsewhere. And Hume can at least plausibly be read as accepting some version of Hume's fork (although, as I've noted, there are hermeneutical difficulties here). But Hume, while not a moral rationalist, is a moral sense theorist, and it seems that a moral sense theorist would have to accept some closely similar version of this argument.
So it would appear that we can see a sort of double divide in the ethical disputes of the period. On the one hand, we have what we can call moral positivism, like that espoused by Warburton, found in those who locate morality purely in convention or else in the 'will of a superior'; on the other hand, we have the moral naturalists, and they fall roughly into two camps, the moral rationalists, like Malebranche, Clarke, Price, and Cockburn herself; and the moral sentimentalists like Hume. The latter two are similar in that they both hold that in some sense we naturally perceive the moral status of things; where they differ sharply is in what we should actually mean by 'perceive', and what exactly this moral status consists in. On the other hand, the moral sentimentalists and the moral positivists will both agree that the moral rationalist is failing to observe the is/ought distinction properly; but moral positivists will insists that moral sentimentalists are also failing in this regard, and the moral sentimentalists will insist the same of the moral positivists. Of course, this is very crude and idealized, and one wonders how far it can actually be taken, but it's an interesting line of thought worth further investigation.
Page references are to Catharine Trotter Cockburn, Philosophical Writings, Patricia Sheridan, ed. [Broadview Press (Peterborough, Ontario: 2006)].