This is the fallacy of asking to be given an exemption to a rule that others are held to. It's typically used as an excuse for special treatment others don't receive, or to win arguments by claiming to have special insights others don't have.
This is fallacious because even if someone has certain expertise or is part of a specific group, they still have to provide evidence and cogent reasons for their position.
A fairly well-known example is the common argument that the universe must logically have a creator. This goes:
The universe has a beginning
Therefore the universe has a cause.
The special pleading here is that it's insisted that an ultimate cause exists; to avoid the infinite regress, it's claimed that this cause is the sole exception and does not require a beginning, and therefore does not itself require a cause. The proponents naturally try to justify this (for instance, arguing infinite regress is impossible, and thus the "uncaused cause" necessarily must exist) though this simply creates another case of special pleading (as there is no reason to believe an uncaused cause is more able to exist than an infinite regress): there's a reason this gets called the "existence paradox."
Looks like this fallacy, but isn't:
- When there really is a reason why someone should be given special treatment. For example, an exemption for murder is often granted for someone acting in self-defense. Or, to link to an example above, "I'm an ambulance driver on duty, so I shouldn't have to stop at red lights".
- Mitigating circumstances not admitted in trial may be considered in sentencing; after the court has established guilt, it seeks to determine what penalty the particular case warrants.
- The philosophy of Søren Kierkegaard strikes some—especially atheists—as a kind of special pleading, since it is based on the idea of the "leap of faith", used as a justification for believing in things like miracles (which are special pleadings to the laws of nature). However, Kierkegaard's philosophy—and most sane theology since then—is essentially irrational: it explicitly argues that the rules of logic simply do not work when exploring the deeper questions of human experience. Whether or not you agree, it is highly important to understand this about that area of philosophy, and attempting to make an attack on it on the grounds of this particular logical fallacy is rather like critiquing pop music for ignoring the rules of polyphony. Of course, many people also reject such arguments precisely on the basis that they are irrational.