I revisit this topic now and then, and I don't think my ideas have changed much, but maybe it's worth a re-run, maybe I have some new words I can use.
In my previous remarks, I made reference to the idea of a larger "story" that a photograph, or group of pictures, could be part of. I also have been known to remark that a picture should "say something", and there are probably a half dozen other similar turns of phrase.
In all cases I mean something that's not really a "story" at all, and I don't mean "say something" in any necessarily verbal way. My intention is something very broad, which is perhaps story-like in a sense. In intention is to convey the idea that a picture should in some sense act like a good short story, a poem, a novel, in the way it evokes something in the right sort of mind. The viewer, ideally, is left with some complex of emotional response, of new ideas, a sensation (not necessarily correct) of some insight. While it might be possible to express this happy result in words, it's not a requirement. I suppose one ought to be able to "talk around" it a bit, if you're one of those people (like me) who enjoys wrestling verbally with the non-verbal.
So it's not a "story", really, it's more like a result of a story.
I can't think of a good word for this thing that Art should do, it seems to encompass a lot. "say something", "tell a story", "challenge the audience" and so on, but there is more, and these aren't really right anyways. There's a little bit of that lame cop-out "I know it when I see it" going on here, to boot.
Perhaps there's a great word or phrase in another language? That would certainly be nice.