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Not a bureaucracy
Notability is not mentioned in our deletion policy. However, this is for a large part an issue of definition. If an editor calls an article not notable, he may in fact mean that it's original research, unverifiable, or a vanity page – all of which are plausible reasons for deletion.
An article is occasionally speedied because it is a "piece of crap", which is not a criterion for speedy deletion. However, it can in some cases be a synonym for "patent nonsense", which IS such a criterion.
In other words, we should judge articles and deletion nominations on their merit, not on their wording. Wikipedia is not a bureaucracy, and just because someone didn't use the most well-accepted terms doesn't mean that he is wrong.
Meaning of the term
Notability is frequently used as a synonym for Verifiability. One reason why people do this, is that "notability" is a clearly defined term (if obviously a subjective one), whereas "verifiability" has a different meaning in Wikipedia than it does in the outside world.
When used colloquially, something is "verifiable" if it can be proven to be true. On the other hand, Wikipedia policy calls something "verifiable" if it can be shown to exist in a reputable secondary source. Most users are unaware of this dichotomy, and while the second meaning is considered obvious by some oldbies and die-hard editors, most editors generally use the first meaning.
For instance, it is verifiable that I have a goldfish – all you have to do is drop by and look at the fishbowl. However, my goldfish is not Verifiable since no reputable source has ever written about it.
There are instances when a subject can be verifiable, but still not suitable for an encyclopedia. While not official policy, this is considered common sense by many users. This is generally the case if a group of subjects is either so large that it is entirely impractical for Wikipedia to have a good article on each member of that group, or so homogenous that it's hard to tell individual members apart. Of course, where exactly to draw the line is subject to much debate.
As an extreme example, let us consider natural numbers. Every natural number is verifiable, and it is quite easy to write a factual and NPOV article about it. However, as any mathematician can tell you, there is an infinite number of them, so by definition it is impossible to have an article on each of them. Thus, it is common practice to write only on those numbers with interesting mathematical properties, and delete the permastub on 57489574385943754 (number). The same reasoning applies to chemical compounds, for instance, or to years in the far past or future. We simply write about the notable ones, not about all of them.
A slightly less extreme example, and the most common one, is people. There are several billion people in the world, and few people disagree that it's entirely impractical to write about all of them. People tend to be verifiable from census records, but generally not Verifiable from any reliable source. We have extensive guidelines for dealing with people, e.g. WP:BIO, WP:VAIN, WP:NOT a genealogy database, and WP:CSD #A7. We also have the 9/11 memorial wiki to store articles on people that died in the terrorist attack, but are deemed not encyclopedic. This means that generally, Wikipedia writes about notable people, rather than all people. Which people exactly are notable is a matter of opinion, but it is generally agreed that for instance U.S. Presidents are notable, and most college students are not. Whether the word 'notable' is used, or some other term is applied, is irrelevant.
Roughly the same argument can be made for websites. There are several billion sites out there, and it has been established that Wikipedia is not a web directory. WP:VAIN may also apply here, and guidelines at WP:WEB are under discussion. Once more, websites are verifiable simply by surfing there, but generally not Verifiable from any reliable source, as the plethora of "cool website awards" simply don't cut it. Consensus generally agrees to not have articles on each and every 100-member internet forum, or random flash animation. They are often called "not notable" and this is backed by other guidelines.
Then, notability may apply to subjects that are largely original research. For instance, a neologistic catchphrase that is only in use on a certain high school, or a top-secret society that nobody knows about, or even a garage band you started with a bunch of friends that has never had any releases, gigs, or media presence. Almost by definition, all of these are impossible to Verify; there's no way to tell if the author is speaking the truth or is simply exaggerating or being WP:VAIN. The term "not notable" is often applied to these.
Finally, there are some cases where notability is heavily controversial. This generally applies to physical objects of some kind, the most obvious being schools. Some people hold that, given the large number of schools, it is entirely unfeasible to write an article about all of them, and would prefer to only have articles on notable schools. Other people disagree and do want to write about every school. It is only a matter of semantics whether they should argue that "all schools are notable", or that "non-notability is not a reason for deletion". The possibility of complete coverage in Wikipedia is a tempting goal in itself.
But still, there is some consensus here. It is generally accepted that all villages should have an article, regardless of size, and it is generally accepted that all individual Starbucks stores should not have individual articles, because it's hard to tell them apart and the information would be more easily organized in a list.
So the bottom line is that notability is a heavily used term and it's not going to go away any time soon. It is obviously a subjective term, but that's hardly a reason not to work with it. Subjects that are called non-notable usually fall short of some other policy or guideline, such as Verifiability or WP:VAIN. And it seems generally a good idea to establish consensual guidelines on broad areas, such as WP:BIO, WP:MUSIC, WP:WEB and WP:FICT. The system isn't perfect, but it works fine most of the time. Thus, cooperation is more important than arguing semantics.
- An Guideline proposal with a contrasting viewpoint
- Another Notability essay (somewhat conflicting)
- A failed Proposal for defining "notability" in Wikipedia context
- A second failed Proposal of this sort
- Wikipedia:Deletion policy
- Wikipedia:List of ways to verify notability of articles