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A contradiction of causality within the timeline brought about by Time Travel. Theorized to be dangerous to the fabric of reality, and known to be dangerous to the brains of anyone who tries to get their head around them. So don't. It's usually what The Professor worries about during a Time Travel story.
Punishments for violating this rule may vary. You might instantly vanish from history or cause your time-travelling self to be erased; you might be immune but find the world around you different; you might destroy reality itself; heck, you might even accidentally unleash killer flying time monkeys.
This is all fictional, of course. In reality, a confirmed paradox would disallow time travel to work at all. You see, a logical paradox is not a thing. It is a sign in a human-created model that either you have attempted something impossible or that you have incomplete understanding of how something works. Here's the thing: all our notions of causality are based on the fact that time only moves in the one direction and a paradox exploits this cause and effect relationship. Once you throw Time Travel into the equation and have time to move in a different way, it's really anyone's guess what will happen but it's not good.
There are 3 types of famous cases of this:Grandfather Paradox
This paradox gets its name for a very simple question: "what would happen to you if you went back in time your grandfather accidently died before he had offspring?"
More generally, this means doing something that makes your time travel impossible or unnecessary. For instance, if success in the time travel endeavor means that the condition you set out to change never happens, then you won't ever have had any reason to come back and try to change it. Thus, without your intervention, it will happen after all, meaning you then come back to change it, meaning you don't have to, meaning you have to, and so on, and so on, etc.
While it can destroy the universe if the author is cruel, nowadays it's more likely to fall into the domain of the multiverse theory: According to this theory there are an infinite number of universes, differentiated by things ranging from a single misplaced atom to inexistence of life. According to this theory if you travel back in time what you really do is travel to the present day of an Alternate Universe just like your own, but which got started later. In this scenario, you can make anything happen and not trigger a paradox.
It's Very Confusing.Ontological Paradox
The kind of paradox that occurs in a Stable Time Loop if you're not very careful about what you're doing, involving events that are their own causes. While not a paradox in the strictest sense - events remain self-consistent - it does violate normal expectations in surprising waysThe Object Loop
When an object from the future is sent into the past, takes The Slow Path back to the future, and then gets sent back into the past again, in the same way, for the same purpose. For example you travel to the past and sell a pair of antique glasses you got from a friend, who it turns out got them from his grandfather, who bought them from the you the time traveler. Where do the glasses come from in the first place? The glasses have literally come from nothing. This is sometimes called a "closed timelike curve" in hard SF. You might ask how the object escapes erosion or other damage too? Since its origin point is also its end point the object cannot (relative to itself) loop endlessly as it would experience infinite decay (relative to itself) and thus would not exist to be sent to the past. Preventing the loop in the first place. Thus to exist the object must (improbably) escape all forms of damage/erosion/entropy between its arrival and departure. Which is technically not impossible but only deepens the weirdness of such an object existing. In one variation, information rather than a physical object comes out of nowhere (e.g. an engineer from the future giving the formula for transparent aluminum to the historical "inventor"). This version avoids the "erosion" issue, provided that the information is correctly transmitted and recordedThe Reverse Grandfather Paradox
When time travel is involved, cause and effect tend to get muddled. Say you remember being involved in an accident as a child, and would have died if not for the intervention of a mysterious stranger who showed up, saved your life and then vanished without a trace. Then you become a time traveler and find yourself at the scene of the accident, and there's a little kid who needs saving. That's right: you happen to be the mysterious rescuer. Instead of accidentally making your time travel unnecessary or impossible by meddling with the past, your meddling somehow made it possible in the first place. But then the question becomes how you "originally" (technically meaningless in this context) survived to time-travel and save yourself— and thus the "paradox", which is not actually a paradox in the logical sense, but a confusing and counter-intuitive result of time travel. This also precludes a Multiverse explanation, since both child and rescuer-adult occupy the same timeline and universe, if the child has a childhood-memory of being rescued by the adult-self.
Sorry for Confusing you.