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There are basically three kinds of time travel stories in terms of they handle paradoxes:

  1. Stories that obey the Novikov self-consistency principle, like the TV show LOST and the movie Interstellar, where you can’t change the past, just bring about past events.
  2. Stories like the TV show Dragon Ball Z, where time travel involves going into a parallel Universe
  3. Stories like the Back to the Future that don’t make logical sense

But the Indian movie 24 seems to be in a fourth category. Here’s how time travel works in 24:

  1. There is a watch that allows its wearer to travel through time.
  2. There can only be one copy of the watch at any given time.
  3. You can only input a date and time into the watch if you are alive at that date and time and the watch exists at that date and time.
  4. Once you input an (allowed) date and time, your consciousness will be sent back into the version of your body at the date and time indicated.
  5. Once your consciousness is sent to the destination date and time, the watch teleports from wherever it was at that destination date and time and appears around your wrist.
  6. From then on, a new course of events for your life plays out.

My question is, do these rules make logical sense, or do they allow for the possibility of time travel paradoxes of some kind?

If they do allow paradoxes, is there anything paradoxical that’s featured in the movie itself?

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    According with (6), it sounds more like a variation on the infinite multi-verse scenario where each decision creates a new 'path' or universe that the time traveller follows. – user62584 Dec 31 '19 at 6:58
  • @Jeeped But I’m wondering if this system of rules can ever lead to a paradox. In the movie the most prominent feature of the time travel is (5). Like one person may use the watch and go to a certain time, then another person may steal the watch and go to an earlier time than when the first person used the watch, thereby preventing the first person from ever using it. I’m just wondering there’s some sequence of such moves that might result in an inconsistency. – Keshav Srinivasan Dec 31 '19 at 7:03
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    Paradoxes (aka time incursions) occur because one event changes the conditions that a future event depends upon. If you are constantly spawning new universes on new events with one decision going on one timeline in one universe and the other decision going another timeline in another universe then there are no paradoxes. They simply follow new paths and do not intersect nor affect each other in any way. The only way a paradox could occur if somewhere in the future, the timelines were merged. – user62584 Dec 31 '19 at 7:08
  • Just FYI, you more or less described the rules of Quantum Leap (minus the watch bit) – Machavity Dec 31 '19 at 13:29
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The rules you have described do appear to lead to paradoxes.

If you travel into your past, there are two obvious problems. One is that, presumably, you can do things differently this time around, and that means changing the past—which requires a parallel worlds interpretation in order to make sense. Secondly, if the watch moves from where it was in the past to your past self, that once again appears to the change the course of history. How does the watch make it to your [present] self if it teleported away to be with your [past] self?

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As Jeeped said in the comments, this doesn't actually sound like a fourth distinct kind of time travel: it sounds like a variant of the second kind that you list.

Based on your description, "going back in time" in 24 could be explained as entering a new timeline at a point that looks like a past point in your own timeline. Nothing in the new timeline is causally prior to any of the time traveler's previous experiences (which occurred in the original timeline). The details numbered 2-5 in your question all seem consistent with this interpretation of what's happening when the watch is used: they just add some additional limitations and effects.

That mechanic would not give rise to any paradoxes, because causality only goes one direction, from the original timeline to the "newer" timelines. It does brings up the questions of what happens to the original timeline that the time traveler came from, and whether the "new" timelines are created each time the watch is used, or just selected from some kind of preexisting multiverse of timelines.

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When information travels through time, paradoxes are likely to arise.

At least in scenarios where no different timelines are created. In the case of 24, a time traveler does not travel physically but brings knowledge of the future (his present) to the past. From there, it is possible for a time traveler to go to the past and

  1. become rich by betting on sports events.
  2. kill Hitler, or at least any bad guys from his lifetime.
  3. get the watch before they did in their original timeline
  4. get the watch before some other users used it in the original timeline, preventing them to use it.
  5. die (accidently or not) before they were supposed to, including before they could get the watch.

and many other situations similar to well-known time travelers stories.

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Yes, what you describe makes lot of sense, and it may or not lead to paradox depending on the context.

As a rule of thumb, but keep in mind I may be wrong, time travel doesn't exist: it's a made up thing invented by humans being to tell stories. So whatever rules the writers came up with are perfectly fine and working in the context of the story which is being told, because that universe is self contained and self defined.

There are basically three kinds of time travel stories in terms of [sic] they handle paradoxes

No.

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