The Bootstrap Paradox, first popularized by science fiction writer Robert A. Heinlein in his book By His Bootstraps, gets its name from the impossible idea of "pulling yourself up by the bootstraps". In the scenario, a time-traveler is pulled up by information that seemingly came from his future self, or nowhere.
In the book, the protagonist Bob Wilson is working on a thesis on time travel, before time travel has been invented. One day, someone who [spoiler alert] will turn out to be his future self sits on his bed, and tells him his thesis is incorrect, and to come through a time portal to the future.
Bob does so, and meets a man called Diktor 3,000 years in the future, who has traveled to the time to take advantage of a now docile humanity, and install himself as king. Importantly for one of the book's paradoxes, Diktor has a notebook which translates the language of his slaves into much more convenient 20th century English.
Returning to his time, Bob (relatably) soon grows tired of his thesis, and decides to travel to the future. He finds Diktor's notebook, conveniently placed near the time portal, and decides to utilize it for his own advantage, traveling to ten years before he met Diktor and setting himself up as king before Diktor has the chance.
Of course, he does this before realizing that he was in fact Diktor all along, and it simply means "king" in the language of the future slaves. While this is of itself a type of Grandfather Paradox or causal loop, it's the notebook which is important to the Bootstrap Paradox.
Bob, now Diktor, continues to use the translation as well as other useful notes on how to become king of the locals in this time period, and as he does so the book becomes degraded. He copies the information over to a new notebook and destroys the old one, before realizing that in fact the new notebook in its current condition is the notebook he received while younger. He ponders where the information the book contained actually came from, as all he did was learn from it and then copy it to a new book for himself to learn from.
A more recent example of the Bootstrap Paradox is in the Terminator movies, where the artificial intelligence Skynet learns how to create Terminators after one comes back to the past and dies, meaning it can be studied.
Unlike the Grandfather Paradox – where you go back in time and kill your own grandfather, making it impossible for you to have been born to grow up and kill your own grandfather – the Bootstrap Paradox is self-consistent, and can't be undone with its own internal logic alone.
Nevertheless, there are a few problems in "closed causal loops" as they are sometimes termed, in that they appear to show something in the future affecting the past. The law of entropy also poses a problem for versions of the loop, such as when a physical object is involved.
Say Bob had not copied the information across to a new book, but had sent the same physical book back in time. The book would, obligingly, follow the law that systems flow from a state of order to one of disorder, and eventually crumble as it made its way forwards and backwards in time. Eventually, as it crumbled, we'd have ourselves a paradox again, with nowhere for young Bob to get the information in the first place. Some have suggested this must degradation must happen with the information itself too, rather than just the pages of the book.
Bizarre as the whole idea is, scientists have proposed theoretical ways of taking the paradox for a spin.
Seth Lloyd, professor of mechanical engineering at the Massachusetts Institute of Technology, suggested taking a bit of information, "Bit A", and copying it onto a second bit, "Bit B".
“Bit B is sent back into the past, where it turns out that B in the past is in fact the same bit that becomes A in the future. Rather than engendering a self-inconsistency, however, the bootstrap/unproved theorem experiment is entirely self-consistent," Lloyd explained to Popular Mechanics.
“The bootstrap paradox isn’t always paradoxical. In our proposed experiment, for example, we predict that the [...] bit will turn out to be entirely random,” he says. “This makes sense, as at no point in the future or the past was there any bias introduced to make it anything other than random.”
This would demonstrate that paradoxes of this type were self-consistent and not paradoxes that require a resolution (be it the creation of parallel universes every time you create a logical inconsistency, or some mechanism that prevents you from creating the paradox in the first place). However, to demonstrate this you would need a time machine, which let's face it, is probably not going to happen unless someone pops out of a time machine and tells us precisely how to get it done.