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I've read somewhere that there is this writing rule stating that, for some superpower, it would be less believable if two completely different settings are present in order to obtain the same superpower. In the article, I remember it mentioned in the movie Spiderman (which I've not watched), the protagonist and another character gain the power through different means.

What's this rule and does it have a name?

First, welcome to StackExchange!

Now onto your question: there aren't any enforced rules when it comes to superpowers or even fiction. The closest thing would be a consistency guideline. Consistency, while not a rule, is usually something a reader will be quick to call out if they perceive it to be broken.

When you hear readers complain about a story having many plot-holes, a lack of consistency is sometimes the cause. But what the reader expects to be a constant changes from story to story. As the writer, you set what is believable or unbelievable in your setting.

Take My Hero Academia (Boku no Hīrō Akademia) for example: the story establishes early on that people in that universe are born with superpowers. Their genetics determine what 'quirk' they are born with and from there they can train their power and become stronger, but never change the ability itself. Without getting into any spoilers, the plot deals with a rare example of where that isn't the case. While that may break the consistency established at the beginning of the story, the plot goes on to explain how this happens and what it means in that universe.

The bottom line is: As an author, you determine everything that is a possibility in your world. As long as you give the reader enough information as to how anomalies can occur (or even foreshadow that they may occur), the consistency isn't broken.

I have been researching comics history for a few decades and I have never heard of such a rule. Others in the thread have given examples.

It's true that there are a few stories in which all superpowers have a common source, typically an alien contaminant into the Earth biosphere (J. Michael Straczynski's series Rising Stars and Supreme Power both explore this option, as does the Wild Cards anthology), however these are the exception rather than the rule.

Keep in mind that the most established superhero universes (Marvel and DC), the characters were created in tandem and only after the fact organised into a unified 'universe' ... so you might have the Human Torch (who flies because, I suppose, heat rises) battle the Sub-Mariner (who flies because he has little wings on his feet) ... and there was never any sense of contradiction there.

For something to genuinely be considered a "rule" of writing, which will delineate bad writing from the rest, it generally has to denote something that is hard to do successfully. Given the fact that most Superhero universes violate this rule all the time, and they're currently quite popular, evidence for this rule being able to separate good writing from bad is pretty minimal.

So odds are good this was just a statement from someone who prefers that sort of thing, aggrandizing their own preferences by declaring them to be a "rule".

That having been said, "believe-ability" is an interesting domain. The thing is, in the general Superhero milieu, pretty much anything which does not directly contradict previously established rules is more or less acceptable. To the extent that Superheros are science fiction, they are very clearly on the softer side of it. The softer the sci-fi, the less "believe-able" it needs to be; the audience is pretty willing to swallow a lot of oddball concepts if they are interesting and you're giving them what they're interested in (cool action/etc).

Sci-fi hardness is a useful way to examine this "rule". It is effectively promotes a kind of "One Big Lie (TV Tropes)" style of superpowers and world-building. That is, you pick one, or maybe a few, mechanisms for how superpowers work, and you stick with just that. Different powersets are just a matter of how someone chooses to use the special rules and physical laws you create.

Is that more "believe-able"? Well, it is easier to swallow a single change to reality than the anything-goes style of many superhero universes. Some people may take a work more seriously if it is like reality except for one thing, particularly those who like hard Sci-Fi. And having this kind of focus can lead to some interesting worldbuilding scenarios, which can lead to unique superheroics and the like.

I have never heard of such a rule in writing, but there is a principle of logic that sort of goes along with this: Occam's razor. For those unfamiliar, Occam's Razor is a way of determining the most likely answer to a question by eliminating answers that require the introduction of more assumptions.

It makes more sense if a known and understood cause or principle underlies something as fantastic as super power. There's certainly a benefit to this kind of logic when developing a story.

In comics, for example, while it may be that heroes have radically different backstories on how they gain their powers, there are "acceptable" methods which make sense to readers: Magic, advanced technology, radiation, ancient beings/artifacts of power, and intense mental/spiritual training are all general sources.

Notably, in the case of Spiderman, both Peter Parker and Norman Osborne gain their strength, durability, and quickness from science. In Peter's case a radioactive (genetically modified) spider. In Osborne's case from an experiment attempting to create a super-soldier. Not too different, actually.

However, once a story has established specific instances of a power, near-identical versions are more believable if the source is the same. For example, Marvel's She-Hulk and Red Hulk both gained their powers through the original Hulk. DC's speedsters are all connected to the Speed Force.

Considering the medium, there's considerable latitude, but the principle does hold a certain truth to it.

I've never heard of that rule but I think it is actually more of mnemonic for not messing up than an actual rule.

The problem it would solve is that you might be tempted to forget that the source of the power is actually part of the definition of the super power as much as the abilities it grants. A super power that has a different source is by definition a different super power even if it grants the exact same abilities. And that should be reflected in the flavour you give to the abilities it grants.

I think your confusion stems from not making a clear distinction between the super power and the abilities it gives. The distinction is not really needed to read or write about super heroes but it is pretty important if you try to make your own supers with unique powers.

I'd suggest reading the rule book for some RPG that has rules for creating super powered characters with custom powers. For example GURPS has expansions for both Supers genre and Powers in general, and the rules explain much of the basic "genre logic".

EDIT:

I probably should give an example since this is not really obvious without one.

Lets have three characters with ability to cling to the walls using three different sources. One was bitten by a radioactive spider and got "spider abilities" because of that. One has telekinetic abilities that allow clinging to the wall. And one has super-science suction pads in his suit.

The first would be a "natural" ability. It would be highly reliable and the character could use it instinctively. He'd be able to move very rapidly and accurately on all kinds of surfaces. And the ability would work unless he is sick, poisoned, or the surface is coated with some exotic substance he cannot stick to.

The second would be a psionic ability. The character would have lots of control over it and be able to use it in flexible fashion. He would be able to move fast and with high precision but would lack the kind of fluency the first character would have. He just would not be quite as good at using it, especially under pressure since he would need some level of mental focus to use it. It also could be disabled by anything that stops him from concentrating or that blocks or disrupts psionic powers. On the positive side the ability would not really care about the specific surface and the character would be able to control the specifics of how it works. They might for example be able to repel surfaces to slide frictionlessly or even use it to throw objects.

The third just has some weird and unrealistic gizmos. It would probably break if handled carelessly or if maintenance is skipped. Fixing it might require expensive or otherwise hard to get parts. And the character would not be able to move with very high speed or precision since the gizmo would have strict limits on how it works. Moving too fast or trying to do a fancy move would probably see the gizmo simply not work and the character falling in an embarrassing fashion. It would also fail on all sorts of normal surfaces because it is just a machine. And an EMP or even dust in the air might make the gizmo stop working.

I hope this example makes my meaning (which I hope is related to the rule in the question) clear. The source of the power is an intrinsic part of the power itself and it should be clear to the reader that the "same" ability from different sources would be entirely different thing. The reader should never think that the above three characters with the "same" power actually have the same power.