You're correct, it is indeed contradictory. Taken purely logically, then, if neither can live while the other survives, then if Harry is alive Voldemort must be dead and if Voldemort is alive Harry must be dead. Since we know both Harry and Voldemort are alive, the statement is clearly false. Since the statement is part of a piece that refers to them as alive, and one killing the other, then the passage as a whole is clearly illogical.
However, there are a few sensible ways to read it.
One is to suggest that their life under this doom (in both the common sense, and the original sense of being fated to something) is incomplete. Neither truly live until after it comes to pass.
The other, is that as a prophesy it is speaking about the future, so even though the present tense is used, it should be thought about in terms of the future. This is an unusual use of tenses for most cases, but reasonable if we consider that we're talking about a magical trance that leads people to speak dialogue in a completely different register and rhythm than that the author normally uses! And this also makes it make perfect sense; some point in the future will come, in which one of them must be dead in order for the other one to survive.
Spoiler: Hover over the text to read it if you have read the end of the last book, or don't care about the plot being given away:
It also has a different interpretation, that becomes clear later. Voldemort is in a state that is neither life nor death, and for that reason cannot be killed. This state can only be ended when certain objects are destroyed, and Harry is one of those objects. This means that while Harry survives, Voldemort cannot truly live, and cannot truly die. Harry dies, and comes back to life. Harry's death is the destruction of the last object that keeps Voldemort in his non-life/non-death state, so Voldemort truly lives when Harry stops surviving. Then they can kill him.
The strangeness is justified as fitting with the general strangeness and cryptic opacity of the prophesy as a whole (again, especially in taking how it doesn't fit with the normal style of the books). It's justified further in light of the plot twist above.
It's also deliberately strange in its phrasing, because you're meant to be wondering about it until you've read on through the series.
OP's "reference book of English grammar" is misleading. No native speaker would ever say/write "it is a grand rule to [do something]. As regards that supposed "rule", it's demolished by...
There is hope [for us yet.]
There is the matter of [actual usage to be considered].
Whilst it's true we normally say "There is a first time for everything" (not "...the first time..."), this has nothing to do with "rules of grammar" - it's just the established idiomatic usage in that precise construction. In a slightly different construction, we normally use the...
There is always the next [example to be considered].
Turning to the specific question as asked, there's nothing at all wrong with OP's example...
There was the following article in the New York Times.
...providing the article itself is either quoted or referenced in the text following. But it would be "non-standard" if the were simply replaced by a with no other changes. Note that if we remove the word following, it would be valid with either the or a - it largely depends on whether or not the speaker wants to emphasise/single out "the" specific article he's talking about.
Best Answer
Certainly not, a quick glance at the dictionary demonstrates that the use of and is not limited to a boolean operation:
And is a conjunction connecting two related units, and the connection is not necessarily boolean:
The larger context informs of the relationship between encouraging immigration and increasing the trade:
Recognizing the legislative source, and the productive analysis of parallelism, the exhibition mentioned in the larger context seems to have a three-fold purpose: immigration and trade or tourism. The conjunction and seems to be "boolean" in reference to the purpose--both are included together.
It is not necessarily "boolean" in reference to outcome. At some expositions, the the exhibit may effectively encourage immigration. At some expositions, the the exhibit may effectively increase the trade. At some expositions, the the exhibit may accomplish both.
Interestingly, the or presses the and toward a boolean interpretation in reference to the purpose. By contrast with the conjunction and, the conjunction or sets the advertising part of the there-fold purpose apart as an alternative to the pair: encouraging immigration and increasing the trade.
Still and would not necessarily have a strictly boolean function in reference to the outcomes. At any exposition, the the exhibit may fulfill any combination of the listed purposes--which were linguistically presented as a single three-fold purpose.
Conclusion:
The semantic complexity of purpose, rooted in the complexity of human reasoning and interaction, transcends a simple boolean arrangement. Unless they anticipate litigation, both the average writer and the average reader can manage this complexity without the need for precise boolean logic. Though this legislative language likely anticipated litigation, it remains a matter of subjective judgement to determine the "purpose" of an exhibit.