* I call it 'philosophy of religions' rather than 'philosophy of religion' to distinguish it from the many things that get lumped together under the latter label. 'Religions' or 'a religion' gives us something distinct from 'religion' qua system or attitude or virtue, even though obviously there will be relations between the two.
* One always runs into the difficulty of defining religion, but part of the reason for the difficulty of this is that religion is inherently an orientation to something other than itself, and people regularly run into problems trying to define 'a religion' independently of that to which a given religion is oriented. One can think of an analogy here, as if one tried to define 'thought' without taking into account the fact that thought has objects, and, indeed, extremely different and widely variable objects; this becomes perplexing because intentionality is the very heart of what thought is, so trying to abstract from it gets us strange results. Likewise, we should simply avoid trying to define everything religious as if it were all univocally so; we should instead pick something particular and discuss it.
* It is more useful, then, to go around a different way and consider the structure of the orientation as an orientation. A crude, intuitive starting point can be the idea of liminality from anthropology. Van Gennep in studying rites and rituals, particularly rites of passage, saw them as having a threefold structure: preliminal, liminal, postliminal. The pre- and post- make sense of rites of passage, which involve a transition, but they presuppose a this-side-of-the-threshold and a that-side-of-the-threshold first. The essential idea is that human beings experience human life as involving fundamental borders -- again, this is a crude, intuitive place to begin, but every given religion has, as a genus, orientation across a fundamental border of human life. The 'across' is important. Everyone recognizes life-borders; but for the general kind of orientation that a religion has to be, the life-border has to be a crossable threshold. And in particular, a religion, as an orientation across a life-border, is a particular kind of orientation to a power across the border, to which either we can cross, or which can cross to our side. For instance, human beings recognize death as a limit. Merely recognizing death as a limit does not get one to a religion. Recognizing it as a crossable limit (there is something beyond death) gets closer, and taking there to be a power across the border of some kind is much closer, and organizing one's life in part around the power across the border of death is the sort of thing I here call 'a religion'. The primary qualification of all this crude groundwork is that it generally seems more convenient not to call 'a religion' every particular cross-liminal orientation to a power beyond the threshold of human life, since virtually all people (perhaps indeed all) have a tangle of such orientations, and it seems reasonable to say that 'a religion' is a given kind of orientation-tangle.
* This is, incidentally, why 'grave goods' are significant for the history of religion. It is impossible to make sense of a consistent custom of placing goods in graves without taking there to be some actual point to doing so. Graves indicate a recognition of death as a life-border; grave goods seem only to have a point if they are taken to cross that life-border as a threshold, even if only symbolically; while we have no direct information that far back, the simplest hypothesis is recognizing that the dead across the border have some kind of agency and power, even if only on their side; and the practice is orienting our life to that power. Death is not the only life-border, by any means, but it is the first for which we have definite evidence that it was probably taken as a crossable threshold with an other-side significant for this-side, even though we don't have any direct evidence for how people thirteen thousand years ago conceived the crossability, or the other side, or the significance.
* All this is, again, crude, but it puts us into position to make a more fundamental move. Rather than talk directly about such powers, which is the temptation, we should talk about the kinds of attributions under which something is classified as a beyond-threshold power. This has several advantages, one of which is that it gives us much more flexibility than other approaches, because it takes into account that human beings relate different kinds of attributions in different ways, and mix them up in various ways.
* So what are the kinds of attributions that we find? There are at least three, and probably only three. First, we can attribute a distinct personal or quasi-personal power on the basis of a local causal inference. Call these tutelar attributions. Second, we can attribute an abstract power on the basis of a global or general causal inference. Call these preternatural attributions. Third, we can attribute an ultimate power on the basis of a global or general causal inference. Call these deity attributions. And, as noted, these can be mixed. For instance, if we stick with death as our example of a threshold, the power across the way may be attributed as
tutelar, like Hades or Osiris, distinct 'caretakers' for matters concerned with the other side of the death-threshold;
preternatural, like Death as a powerful abstract entity;
tutelar-preternatural, like the Angel of Death or Grim Reaper as a personation of abstract Death;
deity-preternatural, like Death worshipped as an ultimate divinity in a death cult;
deity-tutelar, like ultimate Judge of the dead;
and so forth. It's important to keep in mind that we are talking attributions, not things; the same thing may have multiple different attributions. Thus, to take a very different example, Christians use all different kinds of attributions to talk about God. 'God' usually indicates a deity attribution (unsurprisingly), that is an ultimacy of power; but God is also referred to as 'Heaven' or as 'Deity' or 'Divinity', which is a preternatural attribution; and God is referred to by attributions indicating provident caretaking, like 'Father', and such attributions are tutelar attributions.
* A good example of why it is useful to think in terms of attributions can be seen in terms of tutelar attributions. Tutelar attributions largely arise because we think of the world socially; we are very able to relate to things to the extent we see them as intelligent powers, or at least like such. Thus it can be argued that we take personality (in at least a broad, loose sense) to be the default; we take things to be at least candidates for at least quasi-personal agency, until we have reason to think they are not. Thus we find, throughout the world, people applying tutelar attributions to natural things. But if we look at many of these, there is not necessarily any sharpness to the attribution; people apply the attribution -- the tree has a numen, a hidden (cross-threshold) quasi-personal agency, and who knows it has related to everything else. You could have people who hold that all trees have the same numen, or that each tree has a distinct numen; you could have people who hold that the numen of a tree is very different from the numen of a river, and people who don't. You can have one thing (like a river) with multiple tutelar attributions for various different reasons, or you could take the attributions each to be identifying something separate. The primary driver for separating tutelar attributions is difficulty in seeing how they are related (cf. Greer's argument for polytheism); arguments from evil perhaps also play a role, by suggesting that certain tutelar attributions cannot be combined. But nothing about tutelar attributions as such prevents people from uniting them together, either, and people have done so throughout history. Think of the interpretatio graeca and the interpretatio romana. 'Sulis' and 'Minerva' are distinct tutelar attributions; but nothing prevents combination into 'Sulis Minerva', if you can find some commonality. This is why we have the peculiarity in almost every polytheism that apparently distinct gods are sometimes also apparently conflated. Plutarch has a work in which a character argues that the God of the Jews must be Dionysus, because their primary holy day is the Sabbath, which obviously indicates a feast of Sabazios, who is Dionysus or Zeus, just depending. This is excellent reasoning for someone who primarily thinks in terms of tutelar attributions. For Jews, who subordinate tutelar attributions to the deity attribution of the Ineffable Name, which they mostly only allow to be united to tutelar attributions connected to their history (God of Abraham, Isaac, and Jacob), it inevitably sounds like gibberish; the Lord who is One is equated with a god who is also another god or two. This is because the function of tutelar attributions is very different from the function of a deity attribution; the former indicate caretakings, the latter an ultimacy. But caretakings can flow in and out of each other.
* Preternatural attributions arise from abstraction; abstracting, we posit an abstract entity with a reality of its own. This may be separate from any tutelar appellation -- the Forms are not gods, but more divine than the gods -- but they may also be combinable, since it's fairly natural for us to personify abstractions, and indeed, personifying abstractions involves combining with tutelar attributions. Thus we get the goddess Virtue, which is a preternatural attribution that has begun to be combined with specific tutelar attributions (e.g., the power that has patronage of this temple). The reverse can happen as well. We can start with a tutelar attribution and allegorize it into abstraction, as Lucretius does with Venus.
* Deity attributions are a puzzle. They are much more rare, and while they do get associated with tutelar and preternatural attributions, it's very hard to see how deity attributions arise psychologically. The Lockean account takes us to get to God as an ultimate attribution by removing limits from our own mind; but this just seems to give us a tutelar attribution. The same happens with accounts that try to use a primary tutelar attribution. More plausible are cases where we start with the preternatural abstraction of Divinity, but it's still unclear how we get beyond this. Many accounts of religion really require the assumption that there are only tutelar or preternatural divine titles; but the Ineffable Name, for instance, or 'God' as it is usually used by Christians, Jews, Muslims, and Sikhs, is neither. There seem three possible paths: the Cartesian path (the capacity for deity attributions is always innate), the traditionary path (the capacity for deity attributions is taught, ultimately from something for which it is innate), and we-know-not-how.
* A pecularity -- but it is an advantage -- of the attribution approach is that one can attribute without belief. The obvious example of this is Santa Claus, who is characterized as a power across a life-border under tutelar attributions. Super-liminal powers are attributed to him, and we reorganize our lives around him. Atheists who try to lump together gods and Santa Claus are not wholly wrong; they both involve tutelar attributions. But attributions can be in different modes. We can attribute in a proper mode, as with the devout or as with the superstitious. This involves real belief. We can attribute in a mode of disenchantment. The Hopi believe the kachinas existed, maybe still do, but young Hopi have to learn eventually that the things that they call kachinas in feasts and the like are not, because the real kachinas are gone, who knows where, and we don't know what happened to them. There are many agnostics likewise who still use tutelar attributions, just in disenchantment rather than proper mode. The attributions stand, but we use them specifically as being not what we thought. We can attribute in an as-if mode, playing along, as with the Renaissance literary use of Greek and Roman mythology, swearing to Jove and the like. And we can attribute in a fictional mode, either as symbolically representing something known under other attributions in other modes (Aslan) or as purely fictional. The tutelar attributions of Santa Claus are attributed (by adults) in disenchantment, as-if, and fictional modes. It would be a mistake, though, to assume that the tutelar attributions of gods in whom one believes are always in proper mode; religious believers of all types often use every mode of attribution of their divinities, depending on context, and in fact, in many religions, it can be very difficult sometimes to tell which is being used because all the modes are used so promiscuously. Greek polytheism is a good example; it was not a belief-based religion, so there was no penalty for not prioritizing the proper mode if you didn't want to do so, and the Ancient Greek religion was practiced in part by literally making up and acting out stories about the gods known by tradition. Most of what we know about the Greek gods is from fictional stories about local ritual traditions, which we know were sometimes believed to be at least plausible but sometimes not believed at all, and, allowing for cultural differences, the same is true of many other polytheisms. Divine Caesar received his tutelar attributions, and you practiced part of the Roman religion by merely attributing them (whatever the mode) in relevant ritual and ceremony; the Christians got into trouble because they refused to do so even in as-if mode. Similarly, all agnostics and atheists still use tutelar and preternatural attributions at least occasionally; they just stop using the proper mode. When discussing things with religious believers, they often use disenchantment or as-if modes; they may still use religious attributions as symbols or in fictional stories. They personify abstractions like everyone else; they allegorize concretes like everyone else; they interact with their world as if there were persons on the other side like everyone else. The attributions can still stand even if you don't believe them. This is why there is sometimes still a clear distinction between a 'Catholic atheist' and a 'Muslim atheist'; they still carry around them, although unbelieved, a religion, a tangle of cross-liminal attributions of power.