Two interesting dilemmas are the Ship of Theseus paradox and the Loki's Wager fallacy.
Basically, the ship of Theseus was an important ship in ancient Greece (for reasons that aren't important). This ship was so important that the Athenians decided it needed to be preserved. So, through the years, they would find old, decaying boards and replace them with new boards. Of course, that meant that eventually none of the material that was in the original ship was left. The controversial question among philosophers was deceptively simple: Is it the same ship?
What does this have to do with abortion? Many times the other side will pose a question about cyborgs. Yes, cyborgs. They ask where along the spectrum one draws the line between human and non-human.
It is a question designed to put the abortion opponent in a corner as to what organ or body system is necessary to humanity. If we say "Once the brain is replaced, you've got a robot.", they can say "Aha! So abortion is okay until the brain has formed.".
But this rests upon an unfounded assumption. Consider the following gradient.
At the left border, we have someone who is 100% organic. This entity is a human. On the far right border, we have the 0% organic "robot" category. Regardless of where in between one draws the line between man and machine, the human embryo falls on the leftern border.
This is related to the fallacy of Loki's Wager.
The story goes like this: In Norse mythology, the trickster Loki loses a bet to some dwarves, and, per the terms of the contract, the dwarves get to take his head. Loki tells them that, fine, they could have his head, but insisted that they had absolutely no right to any part of his neck. Though some places were clearly part of his head and some part of his neck, the squabbled endlessly as to where precisely to draw the line, so he kept his head indefinitely.
"Loki's wager" is now used to refer to the logical fallacy that anything that cannot be defined precisely cannot be discussed.
EDIT: Also, I now have administrator privileges, and have bookmarked this blog's spam queue. Hopefully I can check it regularly, and legitimate posts don't get "censored" for too long.