I’ve been trying to understand this concept on my own, but I’m unable to grasp it. Google keeps referring to a paper published in the Aristotelian Society, but that article is hidden behind a paywall. Any explanation of the concept would be most appreciated.
Facing the exact question
The question does not invite general reflections on the nature of moral inquiry. The text box makes that plain. It focuses on an article published by James Lenman. Answers that do not address the article do not confront the OP's query.
James Lenman's 'What is moral inquiry?' has a broad title but specific and not immediately obvious concerns.
Here's the wide panorama opened by the title:
Moral inquiry is an activity in which we all at some level engage. And it's natural to think that some of us some of the time do it quite well while others do it less well. And when we do it well it is natural to think of it as a way of finding things out about how we and others ought and ought not to act. Only it's not very easy to understand what this activity is and how it is meant to work. What exactly is it we are supposed to be doing when we try to determine what we ought morally to do? (James Lenman, 'What is Moral Inquiry?', Proceedings of the Aristotelian Society, Supplementary Volumes, Vol. 81 (2007), pp. 63-81: 63.
What moral inquiry isn't - the pursuit of individual reflective equilibrium
The article is chiefly gunning against a particular type of answer, namely that moral inquiry is best seen as the pursuit of individual reflective equilibrium:
Here is a natural and widely familiar story. We start out with a bunch of considered moral judgements, judgements we find attractive and plausible and of whose truth we are stably confident, judgements whose credibility is not compromised by the circumstances in which we formed them being circumstances where we were in fear or distress or stood to gain or to lose personally from forming certain judgements rather than others. (Sometimes considered judgements so understood are referred to as 'intuitions' and I'll follow that practice here though it's worth being clear that intuitions, so understood, are not 'gut reactions'.)
We then seek to impose theoretical unity and coherence on these judgements in ways that we hope will illuminate other moral questions where we are less confident what we should say. This we do by considering and comparing general principles and theoretical conceptions, themselves intuitively plausible, in whose light we seek to make sense of our more particular considered judgements. In the pursuit of coherence among these things, we 'work from both ends' (Rawls, 1972, p. 20; 1999, p. 18), looking for attractive general and theoretical ideas to which our considered judgements conform, while open to the possibility of revising the latter as our estimation of their credibility shifts in the light of modifications to our more general and theoretical understandings.
The standard name for this process is the pursuit of reflective equilibrium, a term most at home in ethics following its deployment there by John Rawls. It is widely accepted among moral philosophers that the story of reflective equilibrium is roughly right as an account of how moral inquiry must proceed. Thus Shelly Kagan, in the opening chapter of Normative Ethics, urges that what we are seeking in normative ethics is a theory that fits as well as possible with our firmest intuitions and that enjoys an intuitively plausible rationale (1998, pp. 11-17). Brad Hooker, in Ideal Code, Real World, urges that in doing moral philosophy we are looking for theories with the virtues of internal consistency, coherence with the convictions we have after careful reflection, unification by fundamental and independently attractive principles and helpfulness in settling questions where we are uncertain or in disagreement (2000, ch. 1, esp. p. 4). More recently still, Mark Timmons (2002, pp. 12 17) urges that we should evaluate moral theories with regard to the extent to which they are consistent, intuitively appealing, cohere with our considered moral judgements in particular by entailing and explaining them, as well as cohering with other non-moral beliefs and yielding determinate verdicts.
What moral inquiry is - a communal pursuit
Moral equilibrium doesn't entirely let us down. It does so only when taken in an individualistic way. Lenman's stress is on the inter-personal. The account of reflective equilibrium as presented so far is a story about how we each individually can and should conduct moral inquiry - almost as if we were alone in the world or answerable to no-one. This is the wrong perspective. Morality is a communal affair.
Moral inquiry is what we do when we try - as we constantly do - to constitute a moral community together. And in the light of that fact, we can make progress with the problem of what authority might attach to my moral intuitions, say, for definiteness, to the considered judgement I have that murder, torture and rape are morally wrong. What that signals, as I have suggested, is that, if you want to join me in constituting a moral community, that is some thing you have to deal with when we deliberate together about how that community should be morally ordered.
Moral inquiry is, in large measure, the attempt by the members of a community - or of what aspires to be one - to arrive by co-deliberation at agreement on what might be an acceptable set of moral standards for the conduct of that community. It is an attempt to determine what moral norms we might, at our best, stably agree in endorsing as a basis for governing our lives together, but where by 'determine' I mean not so much discover as settle. To put it a little provocatively, moral inquiry is politics. As such, it is a project properly and inevitably shaped by those confidently held commitments and aspirations that we the parties to it bring to the co-deliberative table. Or, to put it another way, it is shaped by our moral intuitions.
Of course, it would be wrong to say that all moral judgements ex press our desires about what standards we would wish to inform a moral community. For moral community is always already (albeit more or less imperfectly) in place. And where moral community is in place, moral standards are in place, and many of the moral judgements we make simply serve to affirm and to apply these. Moral argument and moral inquiry can thus be addressed to two issues: firstly, how our own moral standards should be interpreted and applied ...
and secondly, what these standards ought to be. And in practice, of course, these things too are frequently deeply entangled, the more so as our engagement with the latter issue is never ex nihilo, always a matter of rebuilding a ship on which we remain afloat.
Moral inquiry is the attempt to reach and sustain certain kinds of agreement, shared understandings of what should be the moral terms of our life in community together. It is the business of seeking to build and sustain the moral commonalities that make human community and human relations possible. And that in large measure is the business of getting clear about which moral commitments and aspirations we share and which we can be brought to share. That is the currency in which our talk of moral intuition trades.
I have drawn out what I take to be the main lines of Lenman's article. I have not covered its full detail or followed up on what I consider its more peripheral but subtle concerns.
Endnote - in defence of Rawls
On Rawls' account in A Theory of Justice, the two principles of justice embody the reflective equilibrium of the hypothetical contractors behind 'the veil of ignorance'. 'Contractors', plural : Lenham concentrates on what I assume he takes to be the individualistic turn which the pursuit of reflective equilibrium has taken in more recent writings.
DePaul, M. R. 1993: Balance and Refinement: Beyond Coherence Methods of Moral Inquiry (London: Routledge).
DePaul, M. R. 1993: 'Why Bother With Reflective Equilibrium?', in M. R. DePaul and W. Ramsey (eds.), Rethinking Intuition: The Psychology of Intuition and Its Role in Philosophical Inquiry (Lanham, MD: Rowman and Littlefield), pp. 293-309.
Hooker, B. 2000: Ideal Code, Real World (Oxford: Clarendon Press).
Kagan, S. 1998: Normative Ethics (Boulder, CO: Westview Press).
Kagan, S. 2001: 'Thinking About Cases', in Social Philosophy and Policy, 18, pp. 44-63
Lenman, J. 'What is Moral Inquiry?', Proceedings of the Aristotelian Society, Supplementary Volumes, Vol. 81 (2007), pp. 63-81.
Rawls, J. 1951: 'Outline of a Decision Procedure for Ethics', Philosophical Review, 60, pp. 177-97; reprinted in his Collected Papers, ed. S. Freeman (Cambridge, MA: Harvard University Press, 1999), pp. 1-19.
Rawls, J. 1972: A Theory of Justice (Oxford: Clarendon Press; rev. edn. Oxford: Oxford University Press, 1999).
Timmons, M. 2002: Moral Theory: An Introduction (Lanham, MD: Rowman & Littlefield).
Moral Inquiry is just sharing moral knowledge with other people.
"Moral inquiry", within the present context, means inquiry into practical matters (as opposed to mere speculation or scientific inquiry). Hans-Georg Gadamer uses "moral" in this very sense in Truth and Method (p. 314); Albert R. Jonsen and Stephen Toulmin write that "moral knowledge is essentially particular" (1988, p. 330).
Understood with precision then, rhetorical reason guides and φρόνησις (phronesis) drives moral inquiry. The aim of moral inquiry is sound moral judgment, but judgment in hard cases is frustrated because the crux of the matter is hedged in by a potentially limitless parade of particulars. (Wikipedia: Rhetorical Reasoning)
You may also refer to this: