Explain This Passage

[rlevatter]

No: HPFGUIDX 180429

Well, I'm new to this group and obviously the archive is huge,
so my apologies if this has been asked/answered already.

But there's a passage in book 5, OoP, that bothers me. If it's
a mistake, it seems such a basic error I can't believe JKR made
it. So I think I must be misinterpreting, but what's your take.

Defining terms:

Mudblood: a witch/wizard with two Muggle parents
Half-blood: a witch/wizard with one Muggle and one Magic parent
Pure-blood: a witch/wizard with two Magic parents

>From bottom of page 842 (paperback edition), OoP [Dumbledore
explaining the prophesy to Harry]:

"He [Voldemort] chose the boy he thought most likely to be a
danger to him...and notice this, Harry. He chose, not the
pureblood [Neville Longbottom] (which, according to his
creed, is the only kind of wizard worth being or knowing),
but the half-blood, like himself."

Voldemort IS a half-blood, witch mother and Muggle father.
And Neville is a pureblood, whose parents were both magical.
But Harry is also a pureblood, is he not? Both James and
Lily were magical. Lily was a Mudblood, but James' magical
geneology goes back many generations. Surely if being a
pureblood meant not merely that one's parents were magical
but that each line was magical for many generations, only a
small number of inbreeding families could still be purebloods.
Voldemort and Snape were both half-bloods, both with Muggle
fathers, but it makes no sense to call Harry a half-blood.

Or does it? What am I missing?

RL