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HUMAN BEING - Definition

How should the term "human being" be defined?  Biped primate?  Yeah, but if killing one is often punishable by death, we need to know where the line is between human being and anything that isn't one.  The conservatives would say the only non-arbitrary dividing line is conception.  The liberals disagree on the dividing line, which at least illustrates the conservatives' point, if it doesn't outright prove it.

But what if the liberals all agreed to the first heartbeat as the dividing line?  The conservatives would, of course, say it doesn't matter if they all agree; it's still arbitrary.  The liberals might say, "That's interesting.  Then the conservatives' agreement doesn't make their dividing line non-arbitrary."

Conservatives:  "It's not agreement that makes conception non-arbitrary.  It's the fact that before that line, the cells are definitely not a human being, and after that line the fetus has the potential to become anything a human being can become.

Liberals:  "Only if it acquires a heartbeat."

Conservatives:  "And blood, and bones, and a brain, and a spine, and its first breath, and any number of other things you might arbitrarily choose."

Liberals:  "Then why is conception any less arbitrary?"

Conservatives:  "Because before then, it's in two pieces!"

Liberals:  "Being in one piece isn't a bit more necessary than any of the other things you just mentioned.  And if all of them are equally necessary, they're all equally arbitrary."

Conservatives:  "The first one is not arbitrary because it's first."

Liberals:  "Formation of sperms and eggs happens before that.  You might as well say killing sperms or eggs is murder."

I would call the above argument a stalemate.  What am I missing?