This is an interesting and complicated question. Personally I'd say that a "god" is anything that people worship or consider worthy of worship. So I'd define it in relational terms. Of course that raises the question: what does "worship" mean? And perhaps that can be answered anthropologically rather than theologically.
To be honest, though, I'm not sure that any particular quality can be identified. I don't really see what's wrong with saying that a "god" is whatever people want to call a god. It's simply a matter of convenience. As in Wittgenstein's example of "games": there is no one quality which all games have and which nothing else has; we simply use the word in a way which is useful to us and which varies according to situation. I'd say the same is true of "god". It's certainly true of "religion".
However, we do need to distinguish between "gods" as a species, as it were, and "God" as the name of a character, as it were. That is, "God" is a name which picks out a certain entity (existent or non-existent as the case may be). And we can sensibly ask questions of that entity, including whether it has whatever quality we think is essential to divinity. Thus, if we accept that divinity is connected to being worthy of worship, one can sensibly ask if "God" is worthy of worship; I suppose that if you accept my definition then what you're asking there is if the character called God is a god.
frob2900 said:
What kind of definition are you looking for? By talking about necessary and sufficient conditions you are placing the concept in a logical system. What are the axioms, the obvious assumptions we can make about divinity? Well I submit that these lie in the realm of faith and so a cultural/subjective perspective is the only honest one.
You don't need any axioms or indeed propositions at all if you're only seeking to define something. If I lay down the necessary and sufficient conditions for something's being a triangle, I merely offer a definition; I don't lay down any other axioms or derive any propositions from them. Similarly, I can (if I want) define "god" as "that than which no greater can be conceived", or whatever, and I'm not making any claim at all, except to say that when I talk about a "god" I mean something which falls under that description.
Oh yes, I'll get to the other questions as soon as I can...