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In the Doctor Who episode Deep Breath, the Doctor and the Half-Face Man are standing in a doorway hundreds of feet above London and the following exchange takes place:

HALF-FACE MAN: Self-destruction is against my basic programme.
DOCTOR: And murder is against mine.
[...]
DOCTOR: You realise, of course, one of us is lying about our basic programming.
HALF-FACE MAN: Yes.
DOCTOR: And I think we both know who that is.

The next we see, the Half-Face Man is impaled on the top of Big Ben. It's never stated explicitly whether he jumped out or the Doctor pushed him, but do we know what happened? I'd say the events towards the end of Death in Heaven (spoilers!) suggest he jumped out, but is there any more solid evidence or word-of-Moffat proof?

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    He made it to Heaven and was interviewed by Missy.
    – Elliott Frisch
    Commented May 15, 2015 at 3:21

3 Answers 3

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The Doctor has canonically murdered both his own people and more than one Dalek. Also, he is regularly ready to kill Daleks, but due to some external intervention is prevented. Since he states that [exactly] one of the two is lying about their basic programming, we can conclude that the half-face person is incapable of self-destruction and the Doctor, indeed, killed him.

Additionally, this is a hint to Rule Number One: The Doctor lies. He also lies in this case.

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    The Doctor also has stated repeatedly throughout the series that he does not like robots. So it's totally within his nature to have pushed him.
    – Mr Lister
    Commented May 13, 2015 at 12:38
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    "Since he states that [exactly] one of the two is lying about their basic programming" Of course...he could be lying about that. :-)
    – T.J. Crowder
    Commented May 13, 2015 at 13:02
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    @T.J.Crowder: Yes, I considered that. But Half-Face agrees, so although it might have been a lie. Half-Face took it at face value, I think.
    – bitmask
    Commented May 13, 2015 at 13:17
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    Also, is it murder if it's a machine? If I smash my computer with a hammer it's not murder, just destruction of property.
    – AndyD273
    Commented May 13, 2015 at 17:58
  • @AndyD273 Half-Face does seem to have a personality. The whole droid/robot thing in scifi is a bit odd on this issue really...
    – Rand al'Thor ♦
    Commented May 13, 2015 at 19:08
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This is intentionally left open to interpretation by the viewers. I have the feeling it will never be fully explained.

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The script is ambiguous about it and we, the audience, are left to wonder what happened in those final moments.

THE DOCTOR: Those people down there. They are never small to me. Don't make assumptions about how far I will go to protect them, because I've already come a very long way. And unlike you I don't expect to reach the promised land.

The Half-faced man extinguishes his fire-arm.

THE DOCTOR: (cont'd) You realise of course, one of us is lying about his basic programming.

HALF-FACE MAN: Yes.

THE DOCTOR: And I think we both know who that is.

[snip]

EXT. STREET BIG BEN - DAY Craning up Big Ben to see - the Half-Face Man lies on the roof, speared by the spire, clearly dead.

DISSOLVE TO: Close on the Doctor's face as he stares down, sombre. He lifts his eyes direct to camera.

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