Question: There are many movies and TV episodes in which a character "crosses his own timeline", sometimes even meeting his past or future self (e.g. *The Day of the Doctor*). Technically, I should say meeting his past self *and* meeting his future self, I suppose, since this is actually what happens.

When they supposedly alter their own future resulting in their never having travelled in time (as in *Back to The Future* or *Looper*), or narrowly avoid doing so, this is known as the "grandfather paradox".

How are we, the audience, supposed to make sense of this, if we actually think it through logically?

If you perform an action which has an effect that you never survived or were never ever born, then how did you travel in time in the first place? (Although it's not really "in the first place", if you travelled backwards in time...)

I know that there is some quantum mechanics theory relevant to this but the average cinema-goer doesn't know about all that stuff, surely...

Answer: 
There are two logical ways of dealing with time travel paradoxes:

* [**Stable time loop**](http://tvtropes.org/pmwiki/pmwiki.php/Main/StableTimeLoop). When you travel to the past, whatever you do, you won't actually change anything (which pretty much implies there is no free will).

In fiction, the way this is often done is that the time traveller attempts to change the past, but doesn't succeed, because it was him who caused the final outcome in the first place. Or something like that.
* [**Alternate timelines**](http://tvtropes.org/pmwiki/pmwiki.php/Main/AlternateTimeline). When you travel back in time, you cause the creation of an alternate timeline. So there are now two timelines: the one where you left and the one where you arrived. This means that you can do anything and it won't change the original timeline, so no paradoxes are possible.

Though many films and TV shows (including BTTF and Doctor Who) don't use any of the two approaches and instead use approach that doesn't make sense (people slowly vanishing, monsters killing people because change occured, but only sometimes).
My belief is that the audience is guided towards the nonsensical belief that there are two dimensions of time - ordinary time, and "meta-time" - that is, that before "now" in "meta-time", the universe had such-and-such a history, but one second later in "meta-time", history was "changed", and the past became a different past.

This doesn't actually make sense from a physics point of view, and they are usually careful not to be explicit about it.

I am not talking about "personal time" - how time subjectively appears to a character - although it is easy to get confused between "personal time" and this idea of meta-time, because they are usually coterminous from the point of view of a particular protagonist.

The only way to make it really consistent is if the protagonist who "changes the past" is really God, and everything else is effectively just a figment of his imagination, or a dream in other words. Otherwise, as soon as you start thinking about what would happen if someone else also "changed the past", it all falls apart, because the "personal time" theory doesn't fit.