In his (1971) Theory of Justice, Rawls explicitly follows Quine's Word and Object on the nature of explication; he treats explication as a form of elimination: "we start with a concept the expression for which is somehow troublesome; but it serves certain ends that cannot be given up. An explication achieves these ends in other ways that are relatively free of difficulty." (95 citing Word and Object, in the original edition pp. 257-262).
Quine (1960), who treats such explication as synonymous with 'analysis,' attributes this version to Wittgenstein, and explicitly contrasts it with Carnap. Quine understands explication as a means toward dissolving problems (hence the nod to Wittgenstein); for him explication is a form of elimination. Along the way, he (Quine) makes fun of ordinary language philosophers for failing to see that the introduction of technical vocabulary is a means of circumventing the problematic parts of ordinary language and show that apparent problems are merely verbal ones.
By contrast, the Carnapian notion of explication was first articulated in his 1945 "The Two Concepts of Probability" (with a nod to Langford 1942), and then elaborated in Logical Foundations of Probability (1950). I very much like (recall) Hannes Leitgeb's summary: "For Carnap, an explication is the construction of a new concept C′ that replaces a given, often pre-theoretic concept C of philosophical interest, so that the extension of C′ coincides with that of C in the clear-cut and uncontroversial cases but C′ is permitted to, and indeed ought to, improve upon C in terms of exactness, fruitfulness, and simplicity in all the unclear or fuzzy cases." My only addition to this is that for Carnap the new concept C' is constructed in an artificial language. And for Carnap the utility of this new concept C' is (at Leitgeb notes) within science (and not ordinary life; I have discussed some of the tricky issues with this here).*
It turns out, however, that early Rawls used a notion of explication that was distinct from both the Carnapian and Quine-ian version.** In fact, early Rawls explicitly contrasted his notion of explication with analysis (see 3.4 below). I quote from Rawls'(1951) "Outline of a Decision Procedure for Ethics."
3.2 The term "explication" is given meaning somewhat graphically as follows: Consider a group of competent judges making considered judgments in review of a set of cases which would be likely to arise in ordinary life. Then an explication of these judgments is defined to be a set of principles, such that, if any competent man were to apply them intelligently and consistently to the same cases under review, his judgments, made systematically nonintuitive by the explicit and conscious use of the principles, would be, nevertheless, identical, case by case, with the considered judgments of the group of competent judges. The range of an explication is specified by stating precisely those judgments which it is designed to explicate, and any given explication which successfully explicates its specified range is satisfactory.
3.3 The next objective, then, in the development of the present method is to discover and formulate an explication which is satisfactory, by and large, over the total range of the considered judgments of competent moral judges as they are made from day to day in ordinary life, and as they are found embodied in the many dictates of common-sense morality, in various aspects of legal procedure, and so on. If reasonable principles exist for deciding moral questions, there is a presumption that the principles of a satisfactory explication of the total range of the considered judgments of competent judges will at least approximate them. On the basis of this presumption the explication of these judgments is designed to be a heuristic device for discovering reasonable principles. Therefore, while explication is an empirical inquiry, it is felt that it is likely to be a way of finding reasonable and justifiable principles in view of the nature of the class of judgments which make up its range.
3.4 Since the concept of an explication may not be clear, I shall try to clarify it by stating some of the things that an explication is not. First, an explication is not an analysis of the meaning of the ethical terms used in the judgments constituting its range. An explication attempts to do nothing more than that explicitly stated above, and in no way concerns itself with the sense of ethical expressions or with their linguistic meaning. Second, an explication is not concerned with what people intend to assert when they use ethical expressions or make moral judgments in particular cases....Finally, there is only one way of showing an explication to be unsatisfactory, and that is to show that there exist considered judgments of competent judges on specifiable cases for which it either fails to yield any judgments at all or leads one to make judgments inconsistent with them. Conversely, the only way to show that an explication is satisfactory is to evidence that its explicit and conscious application can be, or could be, a cause of the judgments in its range.
3.5 Having noted some of the things that an explication is not, I consider some positive features thereof. First, an explication must be such that it can be applied intelligently by a competent judge; and since a competent judge is not required to have a special training in logic and mathematics, an explication either must be formulated or formulatable in ordinary language and its principles must be capable of an interpretation which the average competent man can grasp. Second, an explication must be stated in the form of principles, the reason for this demand lying in the use of explication as a heuristic device. The typical form of a considered judgment is as follows: since A, B, C,..., and M, N, 0,...., are the facts of the case and the interests in conflict, M is to be given preference over N., O . .. A considered judgment does not provide any reasons for the decision. It simply states the felt preference in view of the facts of the case and the interests competing therein. The principles of an explication must be general directives, expressible in ordinary language, such that, when applied to specific cases, they yield the preferences expressed in considered judgments. Finally, an explication, to be completely successful, must be comprehensive; that is, it must explicate, in view of the explication itself... all considered judgments; and it is expected to do this with the greatest possible simplicity and elegance. The requirement of simplicity means that, other things being equal, an explication is more or less satisfactory according to the number of principles which it uses; and although this demand is difficult to state precisely, it is clear that nothing is gained if we require a separate principle for each case or for each class of cases. Rawls, 1951, 184-186.
In context (3.1; see also 3.3), Rawls is clear that explication is a "heuristic device" to generating "reasonable and justifiable principles" from considered judgments of competent judges. (This is different from the 1971 procedure; see here for some inchoate reflections on it.) Crucially, explication itself is an "empirical enquiry" (3.3). Another striking contrast of the early Rawlsian form of explication with the Carnapian and Quine-ean notions of explication is that it must be "formulated or fumulatable in ordinary language;" it is not a species of language engineering at all. And, in fact, Rawls adds for good measure that its principles "must be capable of an interpretation which the average competent man can grasp" (3.5)
Explication is, thus, the generation of reasonable principles which deliver judgments of cases alike to those of the intuitive judgments of competent judges making considered judgments. The point, to put it informally, is to create an objective realm grounded in principles that can deliver judgments and, thereby, eliminate a subjective element in ethics.
This is not the place to explore the uses of such a heuristic and Rawls's decision procedure and how it was transformed in his later work in slightly different decision procedures. (It is interesting though -- see 3.6*** -- already in 1951 Rawls is inching his way toward (recall) a representative agent approach that instantiates at least an ideal of consensus.) It should be clear, however, that early Rawlsian explication is a different beast than the competing Carnapian and Quine-ian (and TJ Rawlsian) notions of explication. Even though, and this is the main point, the early Rawlsian version of explication shares with its rivals the aim of displacing and eliminating an element of ordinary experience that prevents as it were automated or algorithmic judgments
For, early Rawls' explication is part of a piecemeal approach (and so shares some sensibility with Carnap's): "We should expect satisfactory explications of but delimited areas of the considered judgments. Ethics must, like any other discipline, work its way piece by piece." (4.4) And, in fact, Rawls suggests his method of finding reasonable ethical principles "is" (as Delaney emphasized) analogous to the method used to establish the reasonableness of the criteria of inductive logic." The comparison is not accidental because for early Rawls "succesful explication" can be thought of as "representing the invariant in the considered judgments of competent judges."+