DNA does change. There are "point mutations", for example, in which say a single nucleotide changes, say from guanine to cytosine. . . . CTG TCA . . . becomes . . . CTG GCA . . . If there is a strand of DNA that suffers such a change, is it then not the same strand of DNA? This is exactly like the question whether persons become different persons if they lose say half a finger. And now we have the problem of DNA identity. When are two descriptions sufficiently similar to count as descriptions of the same strand of DNA? Anthony Quinton has the general issue right, in a 1962 article in the Journal of Philosophy called "The Soul": 'No general account of the identity of a kind of individual thing can be given which finds that identity in the presence of another individual thing within it. For the question immediately arises, how is the identity through time of the identifier to be established? It, like the thing it is supposed to identify, can present itself at any one time only as it is at that time. However alike its temporally separate phases may be, they still require to be identified as parts of the same, continuing thing.' By the way, 6% of identical twins do not have identical DNA, so the members some pairs of identical twins would be metaphysically identical and some would not.
Before we make things complicated, let's try whether a simple approach might work. We can say that the "I" refers unproblematically to the agent as a human being and that the somewhat special meanings are those of "shoot" and "robber". These words have special meanings within the game just as "threatening your knight" has a special meaning in chess. To be sure, there's the difference that in video games -- unlike chess -- the player is "embodied" as some sort of virtual personality. So you can move "your" fist or cloak "your" body or lose "your" left arm. But how is this different from moving "your" rook and protecting "your" king or losing "your" queen in chess?
More interesting in your sense may be video games in which one creates a coherent personality. Perhaps you play a little girl and I play her grandmother. This is like collaborative fiction writing or improvization theater. In the end -- as with all good fiction -- one can debate about the psychology and motivations of such characters as well as about what they would have done if their situation had been different in some specific way. These are interesting issues that have been discussed for ages. But I don't think they have much to do with personal identity. However similar a fictional character may be to its author, the two remain distinct even if the fictional character is described in first-personal language. When Vladimir Nabokov writes, in his novel Lolita, "I want to be with her," the "I" refers to Humbert Humbert, not to Nabokov.
You mean the word "expect" in a normative sense, I take it. You are asking others to accept and respect your self-identification and suggesting to them that they ought to accept and respect it. So you are asking for more than a basic right of self-identification. Still, I think what you ask is reasonable and something we others ought to accept and respect much as we ought to accept and respect another person's (newly changed or old) religious identification, sexual preference or choice of lifestyle when such choices do not harm us or third parties.
Obviously, a choice like yours may be hard for some persons to accept -- a wife may find it hard to accept that the man she loved and married now asks to be treated as a female. But leaving a narrow class of such exceptions aside, I don't think you are asking too much. Many may find it difficult to express their acceptance and respect in an easy and natural way as any explicit expression may strike them as awkward for themselves and also for you. But I don't think you really ask for, or need, such an explicit verbal acceptance. In fact, it may be more accepting just to treat you as a female, as you say, as just another woman -- without making a big to-do about it. Yes, we can do this, and we should.
Yes, I think it is wrong to harm a benevolent and charitable person just because of what they did in the past when they held values that are different than what they currently hold. But we cannot run a legal system so as to avoid this wrong. Just imagine that juries, to convict, would have to find beyond a reasonable doubt that the accused has not had a change of values since he committed the crime in question. It would not be hard for many accused (and their lawyers and jury experts) to create such doubt. Many criminals would be acquitted and many of these would then commit further crimes. Others would be emboldened to commit crimes by the confident expectation that, if caught, they would find a way to plead a subsequent change of heart.
What I'm suggesting then is that our current practice of holding people responsible for their past conduct is the lesser evil. And we mitigate this evil in various ways: through statutes of limitation, through pardons, and through the occasional jury nullification (where a jury acquits even while its members understand that, on the basis of all the evidence, they ought to have convicted). Despite such mitigating measures, the wrong you are calling attention to is not entirely avoided or avoidable (much like the occasional conviction and punishment of innocent persons is practically unavoidable).
While the wrong is not, I think, reasonably avoidable in our criminal justice system, we can do better in our personal lives by being ready to forgive wholeheartedly in cases where a former offender really did have a change of heart.
The answer is NO.
Whatever incoherence there might be in wishing that *I* were just like David Beckham, this does not render it incoherent or irrational to desire or view as beneficial things which would, in effect, make one a different person.
Thus suppose that I wish that the person sitting in this chair one minute from now (and from then on) shall not be subject to any of the worries and temptations that distract me from what's important and that he shall otherwise be committed to the same ends as I am. Now would this person be me? That's an irrelevant question, because nothing about this topic was contained or implied in my wish. So my wish is perfectly coherent -- and also rational, I think, for my ends would be better promoted if my wish came true.
Now, of course, if your real end is that *you* should experience the positive reactions that many visit upon Beckham, then you better not wish for the person sitting in your chair a minute from now to be just like Beckham. For suppose you get what you wish for. Then whatever adulation would be bestowed upon that just-like-Beckham creature sitting in your chair would not be experienced by *you*, and so you sadly would not attain your real end.
Morale. A desire for a change that would make you a radically different person is incoherent only if (necessary condition) it also contains or implies the desire that this different person be you.
Even if this necessary condition is fulfilled, there may still be nothing incoherent about the desire -- for that different person may be you! Your wife's husband may be a different person a year from now not only by you being replaced by someone else, but also by your changing. For example, a year ago today, I resolved to lose my bad temper. More precisely, I desired that *I* would not get angry at people any more. I managed to live up to my resolution. So I've become a different person as far as my temper is concerned. But, on any non-eccentric account of personal identity, the person I am now is the person who made that resolution a year ago. I changed over the course of this past year -- I wasn't replaced by someone else. Since what actually happened is exactly what I desired, my desire wasn't incoherent. I desired that *I* should become a different (non-angry) person. I achieved what I desired, because the different (non-angry) person I've become is not so radically different as to be someone other than the person who made the resolution.
The problem you are raising here is actually very nicely discussed in Derek Parfit's famous book Reasons and Persons (Part III). Parfit asks you to imagine tele-transportation, where your body is carefully scanned (and destroyed in the process), the data e-mailed to some destination, and a human being constructed at this destination who is an exact replica of you, including your memories and whim for hazelnut chocolate. You'd be scared to travel this way, but seeing that others do it safely all the time, you too do it and get used to it.
Now one can ask whether the person getting out of the machine at the destination really is the same person as the one who walked into another machine at the departure point. As Allen Stairs wrote back then, something can be said for either answer. But there's a third thing one might say: once the story's been told, there is not further question to be answered. You can say what you like about sameness, the important thing is that you really have no serious reason to avoid using this technology -- provided it works, of course!
One day you travel by e-mail, the technology actually works a little better than usual. The scan does not actually destroy the person at the departure point. So now we have two people, just as you imagined in your question. As you write: if they both were identical with the pre-departure person, then they'd be identical with each other -- and this they surely are not, seeing that they are miles apart from one another and having a heated conversation with each other on the telephone (about who gets to be with hubby and the kids).
When you used the technology in the old days, when the scanner destroyed the person at the departure point, you thought of e-mail travel as being just as good as taking the train. You may be a different person each time, strictly speaking, but why mind? Parfit examines the plausibility of this attitude for the branchline case, where you are the person who survived the scanning. He imagines that the scan did damage after all and that you are going to die rather soon. Can you be as cheerful about this, in light of the person at the destination point, as you used to be pre-scan in your previous e-mail travels? It would be hard to be cheerful like this in the face of imminent death, but Parfit makes a good case that you have reason to be. So have a look and see what you think.
Some of your difficulty -- very reminiscient of Leibniz, by the way -- may be caused by the word "different." Take a very simple case, two water molecules perhaps. Are they different? In one sense, they are exactly the same. Yet in another sense they are different or (perhaps better) distinct. You can tell that they are not the same in this second sense by counting: there are two, not one. And you can tell this, in turn, by attending to their space-time locations.
Similarly with your more complicated example. At any given time, there are two distinct locations at which a human being with this DNA is located: you at one place and your clone at the other. If he is living on earth, he's likely to be a bit different from you due to what the two of you have eaten and experienced. But he may be living on a planet that is an exact replica of this one, and his life may then mirror yours exactly with him thinking and doing exactly what you think and do, perhaps even simultaneously. He would still be distinct from you by virtue of his location. You are here and he is there.
This clearly depends on how severe the damage is. If there's a slight crack in the glass, people would still call it a window. If glass and frame are lying about, smashed into a few thousand pieces, no one (except an eccentric philosopher) would say that what's left is a window.
So how severe -- you will ask -- may the damage be exactly? This depends on standard use of the words of our language. Interestingly, things may cease under one description and continue under another. After some creative modification, what was a Hercules statue is still a statue, but not one of Hercules. After further modification it's no longer a statue at all, but still a piece of bronze. Similarly, when a caterpillar turns into a butterfly, a caterpillar ceases to exist, a butterfly begins to exist, yet an animal continues to exist throughout.
Philosophers have argued a great deal over whether what we say about such transformations is merely conventional (having to do with the words we have in our language and how we use them) or whether what we say can track (or fail to track) some truths about how matters really are (as your question seems to suggest).
Philosophers have analogously argued about the deeper issue that we tend to think of every such transformation as involving some underlying thing or substance that persists: When water turns into ice, H2O persists. When H2O disintegrates, its chemical elements (hydrogen and oxygen) persist. When these elements get transformed through a nuclear reaction, elementary particles (electrons and protons) persists. Etc. Is this just the way we think, or is this the way things turn out to be -- or perhaps the way things turn out not to be once our physics is sufficiently advanced? (Think of Einstein's formula about how matter and energy can be transformed into each other: E = mc2.)
You need here the distinction between a thing and its name(s) or representation(s).
When someone says or writes "Mozart = Mozart," the two sound tokens or ink tokens are indeed subtly different. And even if somehow they were not, they would still be different in that one spoken "Mozart" precedes the other in time and one written "Mozart" is to the Southwest of the other.
But the law of identity is not supposed to be about these name tokens, but about what they refer to. That referent, the entity being named by each of these tokens, is exactly the same: the man Wolfgang Amadeus Mozart.
That it's the referents, rather than the names one has in mind here is more easily appreciated by considering an "equation" with different signs on each side. Thus consider:
Wolfgang Amadeus = Mozart
Someone asserting this equation is not making the grotesque error of equating the two sound or ink tokens. Rather, she's claiming that these tokens refer to the very same thing.
One point that's implicit in Professor Pogge's answer above, but that it might be useful to make explicit, is that philosophers often use the notion of "person" in such a way that it contrasts with the genetic notion of "human". Whether or not you are human is a matter of your DNA. But whether or not you are a person cannot be settled by genetic testing. Rather, as Professor Pogge notes, it is usually taken to be a matter of having certain capacities. Thus, there may well be humans who are non-persons (for example, some philosophers have suggested that humans in persistent vegetative states fall into this category -- another controversial example concerns fetuses) and there may be persons who are non-humans (some advanced mammals, for example -- or if you want to get into the realm of science fiction, Data or Spock from Star Trek).
Specifying which capacities are necessary and/or sufficient for personhood turns out to be quite difficult. Some of the essays in Matters of Life and Death (edited by Tom Regan) give a good introduction to the issue of personhood. See in particular the essays on abortion and animal rights.
On the question of personal identity over time, John Perry's A Dialogue on Personal Identity and Immortality provides a useful introduction to the relevant issues.