Theres' a great book by Harry Harrison called One Step From Earh, where he went through pretty much every possible philosophical interpretation of how matter transporter's could/might work including that they might just copy a person rather than "move" them...
There's also at least one short story by Philip k. Dick where every time someone time "travels", they actually sending a copy of themselves to the next time - the story ends in havoc as the population of the world ends up being dominated by millions of different age copies of the first time traveller...
The problem with this is that there's a point where philosophy meets physics or
chemistry or biology
As in the play, of course, your cells are replaced over time, so that you are not the same set of components that you were (say) 7 years ago - but this is also true at the level of particles, only far faster....
And also at the level of cognition (given brain plasticity, and how it is believed memory really works, we are continually modifying who we "are" as we experience new things, even just recalling old things)...
On a more interesting take about paradox, Behold the Man, by Michael Moorcock was pretty ingenious...and for paradox busting par excellence, Robert Heinlein's classic By His Bootstraps is the business...
So taking the memory model as a template for the philosophical challenge of continuous identity, it seems to me that there's really several problems
- Fidelity
if something is a high fidelity copy of the previous instance of a person, and the previous version is replaced, then from everyone elses point of view, this is the same
person.
- Memory
if the next instance of a person has the same (or very similar) memories to the previous instance, then they can delude themselves that they are the same person, as they will have the illusion of continuous existence - this is actually no different than how vision works, where your visual cortex as to make an apparently coherent and spatial and temporal continuous visual space out of what your eyes/retina detect, despite that that is intermittent and imperfect...
- Consciousness
this is very tricky, since the locus of attention moves ahead (anticipation/ etc) as well as behind (memory) the current moment...however, if the brain is just a machine, then it is reasonable for the model it runs of the world to include prediction, and that model itself is copied from instance to instance of the person, providing the illusion of continued consciousness....
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