(1) Propositional Core: These constitute the primary constitutive point of the teaching, and anything entailed by it.
(2) Pragmatic Core: These constitute the primary regulative point of the teaching (i.e., how it affects practice). The organization of this core is dependent on (1), which gives the practices governed by the Pragmatic Core a point.
(3) Core Symbols: These are all the standard material vehicles for the doctrines: parables, vivid phrases, rituals, artworks, etc. Thus, some are propositional, some are pragmatic, some are pictorial, etc.
(4) Supplementary Elements: These may be propositional, pragmatic, or symbolic; if propositional, they might consist of (a) corollaries derived from the combination of this doctrine's propositions with other doctrinal propositions; (b) facts independently discovered (by reason, by experience, or by both) that are suggestive or supportive of the Core Elements in some way; (c) speculative guesses, hypotheses, and the like attempting to give more specificity, clarity, or utility to the Core Elements; if pragmatic, they consist of (a) indepedently developed practices that are assimilated; (b) programs for reform. (I could continue this, but I won't here).
It is important to recognize that a doctrine, although analyzable, is also highly integrated; the Core Symbols, for instance, are the primary means of conveying the Propositional and Pragmatic Core; the Pragmatic Core, or practical aspect, is given its point by what is primarily intended to be conveyed by way of the Core Symbols (namely, the Propositional Core); the Propositional Core makes a practical difference only because it is closely associated with a Pragmatic Core; the Propositional Core, while consisting of propositions with definite truth value, usually consists of vague propositions (in the sense of requiring further elaboration) that require some sort of Supplementary Elements to play any role in thought at all. And so forth.
I will divide my discussion of this proposal into two parts:
Part I: An Example
Part II: Further Thoughts
Let's take the doctrine of hell as an example. Naturally, this will be a rather crude characterization of the doctrine of hell, or, if you prefer, the characterization of a rather crude doctrine of hell; my point is not to explain the doctrine of hell, but to give a relatively simple and straightforward example of this analysis of doctrine.
(1) Propositional Core: Without salvation, sin is punished.
(2) Pragmatic Core: guidelines and standards of practice for the working out of one's salvation
(3) Core Symbols: These are manifold, e.g., "the fire that shall never be quenched" (Mk 9), "where the worm does not die" (Mk 9), "the second death" (Rv 20:14), the parable of Lazarus and the rich man (Lk 16), "chains of darkness" (2 Pt 2), etc.
(4) Supplementary Elements: Also manifold; a good recent example would be C. S. Lewis's The Great Divorce.
It's unfortunate that most people know him for nothing else, but a good way to see this religious doctrine (as teaching) in action is to read through Jonathan Edwards's Sinners in the Hands of an Angry God. It hits every aspect of the above analysis of the doctrine of hell. If you want a look at a rather different approach to building up the Supplementary Elements to the Core Symbols than is found in Edwards's sermon, I recommend Aquinas's question on damnation (keep in mind, though, that it's in the Supplement; i.e., it's part of the Summa he never finished -- the current question is just a summary of the very early Sentence Commentary question on the subject, and so shouldn't necessarily be taken as Aquinas's mature view).
This example brings up another issue, namely, that doctrine can be embedded within doctrine. There is good reason to regard the doctrine of hell as (in effect) a side issue in the doctrine of heaven. I won't go into the whole argument for this, but there are actually a large number of reasons for this (ranging from the context of the Core Symbols, to what seems to be implied by the Propositional and Pragmatic Cores of both doctrines, to the fact that most "scandal of hell" objections to Christianity turn out to be on closer inspection scandal of heaven objections). So the doctrine is just a chunk of teaching whose point (as expressed in its Propositional Core) might well be subordinate to another chunk of teaching.
All of this just gives the basic idea. This post was originally intended to be a relatively short one, but it keeps expanding just due to the fact that I'm attempting to communicate more carefully than I usually do in posts (and so can't just jot down what comes to mind). So it may be a bit before I manage to post the second part. It should include a further discussion of some of the details of the analysis, and a look at how it (or something it approximates) might make more sense than Atran's quasi-propositions for at least some of the things he seems to want to say.