Both were classmates on Gallifrey so shouldn't the Master know the Doctor's name?
If so, why was the Doctor willing to trust River just because she knew his name?
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Sign up to join this communityWell, if you consider the fact that the Master has never used the Doctor's name (apparently granting the Doctor the same respect the Doctor grants him) it seems unlikely that the Master has told anyone else the Doctor's name, though he would absolutely know it.
Everyone else who ever knew the Doctor's name was a Time Lord.
Every Time Lord except the Doctor and the Master was erased from time (unless that's changed sometime since Tennant's run).
Ergo, if River knows the Doctor's name, she MUST have learned it from the Doctor himself or from the Master. And the Doctor has a habit of walking directly into traps, just to see what will happen. Thus, someone who knows his name is someone one of his future states trusted, or someone he'll pretend to trust to see what the trap is.
Edit: This answer assumes the answer in the Title is 'Yes'. This is because the Doctor's original or 'real' name seemed to be widely known by the Gallifreyans he regularly interacted with. Because of their long history, it is therefore easy to decide that the Master does, indeed, know the original name of the Doctor.
The rest of this answer seeks to answer the question asked in the text above, namely, why the Doctor would trust someone simply because they knew his name given that at least one antagonist of his also knows his name.
Only the nickname is known from the bbc archive, and I guess the master used this name as well:
In the Fourth Doctor serial "The Armageddon Factor", the Doctor runs into a former class mate of his named Drax. Drax refers to the Doctor several times as "Thete", to which the Doctor replies "It's the Doctor." "Thete" is then revealed to be short for Theta Sigma, the Doctor's nickname at Prydon Academy on Gallifrey, as also referenced in The Happiness Patrol and "The Pandorica Opens".
Although I have no definite proof, I think yes, for the following reasons.
Neither "The Doctor" nor "The Master" are names, but titles they call themselves.
OHILA: Because you are the good man, as you call yourself?
DOCTOR: I call myself the Doctor.
OHILA: It's the same thing in your mind.
DOCTOR: I'd like to think so.
- Night of the DoctorMASTER: Doctor.
DOCTOR: Master.
MASTER: I like it when you use my name.
DOCTOR: You chose it. Psychiatrist's field day.
- The Sound of Drums
Other renegade Timelords seem to have done the same (The Monk, The Rani, The Corsair, The War Chief). But the acquisition of titles seems to be a Gallifreyan, or Timelord trait anyway - for example, Rassilon, which we understand to be his name, also had many titles including "Conquerer of Yssgaroth", the "Overpriest of Dronid", the "First Earl of Prydon", the "Patris of the Vortex", and the "Ravager of the Void".
Similarly, in Twice Upon A Time, the weird soul-gathering aliens say:
"The Doctor has many names. The Destroyer of Worlds. The Imp of the Pandorica. The Shadow of the Valeyard. The Beast of Trenzalore. The Butcher of Skull Moon. The Last Tree of Garsennon. The Destroyer of Skaro."
All of these "titles", along with "The Doctor" are referred to as his "names".
On several occasions, The Master has demanded and gained pleasure over people, including The Doctor, addressing him as "The Master", as in the quote above. This suggests that The Doctor does know The Master's real name and would have used it in the past, but calling him "Master" strokes his ego because he chose it for himself believing that that he is worthy of being everyone's master. It is logical then that they would know each others names as they knew each other before either of them took on these titles.
If we accept then that it is normal for Timelords to go by a title they choose for themselves rather than a name, this would explain why every Gallifreyan and Timelord that The Doctor has encountered has called him by his chosen title without question or reference to his "real" name. Any occasion where another Timelord has expressed surprise or disdain for the Doctor's "name" could therefore be taken as a reaction to the meaning behind the title he chose, not that he chose it. We all know that The Doctor's chosen life-course, as personified by the name "The Doctor", is at odds with the rest of the Timelords.
It may also be an unspoken rule among Timelords not to use their "dead" names (for want of a better expression) if they have chosen a title that represents their life's path. This may be the reason that the woman the Twelfth Doctor met in the barn on Gallifrey in Hell Bent does not say his name yet he silently nods to confirm who he is. This would also make it remarkable that he told Riversong because she is not (technically) a Timelord and because he used a name neither he, nor his people use anymore.
Could be yes and no. It possible that both know each other's given names but not the respective true names. I've seen the idea used in some fantasy stories, I think the concept has its origin in folklore. A true name would be given to only select few, as a sign of supreme trust.
Yes, Missy "The Master" calls the Twelfth Doctor "Doctor Who" because that is the Doctor's name.
BILL: So, why do you keep calling yourself Doctor Who?
MISSY: Because I'm pretending to be him. Because that's the whole point of this ridiculous exercise.
[...]
BILL: Yeah, but he's called the Doctor, so
MISSY: He says, I'm the Doctor, and they say, Doctor who? See, I'm cutting to the chase, baby. I'm streamlining. I'm saving us actual minutes.
BILL: Yeah, okay, whatever.
MISSY: Also it's his real name.
[...]
BILL: Sorry, what do you mean, it's his real name? Nobody knows the Doctor's real name.
MISSY: I do, because I grew up with him, and his real name is Doctor Who.
DOCTOR: Bill, she's just trying to wind you up.
MISSY: Chose it himself, you know, trying to sound mysterious. And then he dropped the Who when he realised it was a tiny bit on the nose.-- S10E11 (World Enough and Time)
I think the Master knows the Doctors name, but and the Doctor knows the Masters name, but they don't use it, because Doctor and Master are their chosen names, and they respect each other by not using their real name.