Fallacy One: A poem/story can mean anything you want it to mean

No. It can't. It's that simple.

However, there's also not one exact meaning for most creative writing, and a person with different insight or knowledge may extract a different layer of meaning from a piece than I would based on my knowledge and background.

For instance, when Alice follows the White Rabbit down the hole, the White Rabbit could symbolize many different things: white, purity and Rabbit, close in spelling to Rabbi--could this be heralding a religious experience? Maybe, but what other evidence supports that theory. Or, white rabbit could symbolize nature, she's leaving civilization to follow the wild, untamed side of her. Can that be supported?

Or White like cocaine? Or some other popular drug when Lewis Carroll was writing? Could the White Rabbit symbolize a pusher?

And so on...

But suggesting that the White Rabbit is Aaron Burr and the story is an allegory about the founding of America, no. (Not that anyone's tried that, as far as I know). Or that Alice is projecting her Electra complex onto the White rabbit, again, no. Literature can't mean just anything.

Similarly, I thought I understood Hunter Thompson's writings, then I was talking about them with someone who had a much more....extensive, I'll say...knowledge of street drugs. All the sudden, imagery was explained so that I found out what was really happening in a scene. I thought I'd understood it before, and I had, but only on one level. The deeper meaning added a great deal to the text.

If I wrote a story and named a character "Paul," there are many possible hidden meanings:

• St. Paul. If I used it in this context, I'd probably creating a preachy, self-righteous character who has firm values (that I find misguided!) He might be the 2nd wave of a situation, or someone who had a blinding realization. Those types of evidence would prove the character was named after St. Paul.
  
• Paul McCartney. A character with soulful eyes, glib, sentimental, musical, creative, all those could be traits proving I was going for a McCartney subtext. Or it could be a character who is the lonely last of a realm, to take a different tact (yea, I know Ringo's alive. Still. Creative license)
  
• Paul Wellstone. One of my all time favorite politicians. I'd be making a point about the character's motives and ethics, probably.
• Paul my 10th grade boyfriend. If he was the genesis of the character, it would be an anti-intellectual character, one who doesn't think girls need to go to college. And related ideas. Rural, traditional. Lots of flannel shirts!
  

I think I put those in order of how likely a reader would be to pick up on the meaning. If a St. Paul subtext is intended, it's reasonable to think readers--especially well read ones--would at least catch a hint of it. My ex-boyfriend, though,...you'd have to be a serious student of JJ biography to catch that. And that's assuming I didn't do something cute like call him Pablo or make him charming (that would throw people off!) or use his middle name (Michael) instead.

But if I named a character Paul, and the name seemed symbolically important, there are limits to what it might mean. It wouldn't mean "Don't eat tuna because they kill dolphins."

With literature, showing the causal link between text and meaning is crucial, just like in math. A word or action can't mean what you want it to mean just because you say it does. Connecting the ideas through a reasonably logical explanation of causation is important. And fun!