To give the gist of Warburton's argument we need to distinguish between moral motivation and moral obligation. Warburton suggests a 'threefold cord' of moral motivation -- three types of incentives for acting morally.
(1) Moral Sense: We have a natural taste, capable of refinement, for moral action, and a natural distaste, also capable of refinement, for immoral action. We are charmed by morality, repulsed by immorality, at least to some extent, even setting aside any reasoning about consequences.
(2) Perception of Essential Differences: We have the capacity to reason about the differences between, say, malevolent actions and benevolent actions, and can see that one is as a rule more appropriate to a given situation; we are also able to see that, at least in the long run, for the most part, all other things being equal, etc., virtue has good consequences, vice has bad consequences.
(3) The Will of a Superior: We are obligated by moral authority.
Warburton thinks all of these are necessary for a complete moral life. The moral sense serves as the most general basis; it is something that both theists and atheists have, since it is a sort of human instinct. By reasoning upon the differences in the natures of acts, we can improve upon the moral sense by showing that it is not capricious but, at least roughly, captures the seeds of general moral rules. And this far, too, an atheist who reasons well might go. However, Warburton argues that reason shows another step to be required for a properly moral life, namely, obligation; and this requires a moral governor. It's obligation that Warburton thinks is morality in the strict and proper sense; we can call the having of any of the threads of the threefold cord 'morality in a broad sense'.
Thus it's important to be clear about what Warburton is claiming when he says that atheists cannot be moral. Obviously, atheists can be moral in a broad sense: they have the moral sense, and they can reason about consequences. Thus, Warburton concedes a rather robust moral life to the reasonable atheist. What he denies is that the atheist can consistently hold that there is any such thing as moral obligation. The atheist, just like the theist, can volunteer for virtue's army; but he can't say he was drafted, nor can he say that there's anything more to his volunteering than taste and calculation. The primary argument for this is what I have called the Obliger Argument. Obligation implies an obliger; we cannot, properly speaking, obligate ourselves; the atheist, as atheist, recognizes no moral obliger; so he can recognize no moral obligation.
Clearly, an atheist wanting to reject this argument would need to do one of three possible things:
(1) Argue that obligation is not a genuinely legitimate part of morality;
(2) Argue that there are obligations without obligers;
(3) Argue that we can, in fact, obligate ourselves.
Cockburn is not interested in the atheist's response as such. She does, however, want to reject the claim that morality properly consists in the will of a superior authority rather than in the perception of moral differences, and so is interested in the problem for this reason.
Her opening gambit is to deny the basic premise of the Obliger Argument, that obligation implies an obliger, at least in the sense the argument requires:
To this I answer, that in the common acceptation of the word, obligation implies only a perception of some ground or reason, upon which it is founded, but not necessarily a superior will. When we say a man is under an obligation to be grateful to a benefactor, we mean, that the relation interceding between them requires it of him; and so that he is obliged to do to others, as he would have them do to him, implies an equity in the thing, that brings him under such obligation.
In other words, Warburton's phenomenology of obligation is not quite right: we say that we are obligated to do something when we have identified a reason that requires it of us.
Her second move is to look at the sense in which we can say that reason is our obliger. Warburton had argued that if we say that reason obligates us, we are in effect making a person a self-obliger. To this Cockburn replies:
Very true, but it is just the same, whatever principle we suppose morality to be originally founded on; a free-agent must be always the immediate obliger of himself: Whether he judges, that the will of the superior is to be the only rule of his actions; or that he ought to act conformably to the necessary relations, and essential differences of things, or to his consciousness of right and wrong; or that a prospect of rewards and punishments should solely influence his actions; in either case it is equally the perception and judgment of the mind, or his reason, that obliges him to act accordingly; and this is so far from being an absurdity, that it is essential to moral choice and free agency.
We can, in other words, distinguish the obligation as such, from the immediate application of the obligation to oneself; and in the latter sense we are all, indeed, the immediate obligers of ourselves. This is true on any account of obligation whatsoever. But does our being self-obligers mean that we are able to dissolve at will any 'obligation' we choose to impose on ourselves? If that's the case, Warburton is right: this is not a real obligation at all. So Cockburn's third step is not surprising, since she argues that the sense in which we obligate ourselves does not mean that we can relinquish the obligation. While there is a sense in which we are self-obligers, the ground of obligation is our rational recognition of the essential differences between (say) murder and benevolence; it's only in voluntary contracts that the obliger and obligated can dissolve the obligation at will.
This, then, is Cockburn's basic response to the Obliger Argument. The weak point in all this, I think, is the first step: obligation is, indeed, in common acceptation a requirement following from a reason, but it's also equally clear that not all requirements following from reason are moral obligations, and that not even all very good reasons impose requirements. Warburton, remember, effectively distinguishes between moral motivations and moral obligations. This is a legitimate distinction, and the strength of Warburton's position is the difficulty in moving from 'good reasons to be moral' to 'obligated to be moral'. As Cockburn suggests, the distance between the two is not so great as Warburton likes to think, but that's not enough to show that it is traversable without appeal to the will of a superior. That would require, I think, a more systematic account of obligation than Cockburn generally gives. However, setting aside this weakness, the basic line of argument seems to me to be a powerful one. Warburton too easily conflates original obligation and immediate obligation, and this already takes him halfway to his moral positivism.
All quotations from Cockburn are from Catherine Trotter Cockburn, Philosophical Writings. Patricia Sheridan, ed. Broadview Press [Petersborough, ON: 2006] 140-141.