Why did Dumbledore have a say over what books were allowed when Dippet was headmaster?

Question

Armando Dippet was the headmaster of Hogwarts when Tom Riddle was attended the school, at least from when Riddle was in his fifth year until after he graduated. When Dumbledore is teaching Harry about Voldemort's past in The Half-Blood Prince, he states that Dippet denied Riddle a job as a teacher because he believed he was too young.

"Hardly anyone knew of it at the time — I was one of the few in whom the then headmaster confided — but Voldemort first approached Professor Dippet and asked whether he could remain at Hogwarts as a teacher."

...

"But he didn't get the job, sir?"

"No, he did not. Professor Dippet told him that he was too young at eighteen, but invited him to reapply in a few years, if he still wished to teach."

• The Half-Blood Prince Chapter 20 (Lord Voldemort's Request)

However, later in the book Harry is in Slughorn's memory of Tom Riddle. When talking about Horcruxes, Slughorn warns Riddle that Dumbledore is strict about the subject of horcruxes.

"Of course," he muttered, "this is all hypothetical, what we're discussing, isn't it? All academic . . ."

"Yes, sir, of course," said Riddle quickly.

"But all the same, Tom . . . keep it quiet, what I've told — that's to say what we've discussed. People wouldn't like to think we've been chatting about Horcruxes. It's a banned subject at Hogwarts, you know. . . . Dumbledore's particularly fierce about it. ..."

• The Half-Blood Prince Chapter 23 (Horcruxes)

Dippet is clearly headmaster at the time of this memory. In that case, why would Slughorn say that Dumbledore is particularly fierce about it, right after saying how it's banned? That seems to imply that Dumbledore is in charge of the banning and subsequent "punishment" if the rule is broken. Unless Dumbledore was deputy headmaster, but it still seems odd that he would say Dumbledore instead of Dippet.

Answer 1

All he's saying is that Professor Dumbledore is particularly fierce about keeping the Dark Arts out of school. That doesn't necessarily mean that he's the one making the decision to keep it out, just that he feels strongly about it.

Let's update it as an example:

"People wouldn't like to think we've been chatting about Cruciatius Curse. It's a banned subject at Hogwarts, you know... Professor Longbottom's particularly fierce about it..."

The implication isn't that Neville is making the decision to ban Crucio, but that among the staff he's particularly outspoken. And like Neville, Dumbledore was outspokenly against the Dark Arts and for deeply personal reasons.

In any case, Dippet appeared to trust Dumbledore and rely on his advice. From the same scene in Half-Blood Prince:

"How did you feel about that, sir?" asked Harry hesitantly. "Deeply uneasy," said Dumbledore. "I had advised Armando against the appointment — I did not give the reasons I have given you, for Professor Dippet was very fond of Voldemort and convinced of his honesty. But I did not want Lord Voldemort back at this school, and especially not in a position of power."

Dumbledore was also influential enough to talk Dippet into hiring a groundskeeper he thought had just opened the Chamber of Secrets and killed someone. He probably was deputy headmaster by the way, given that Professor McGonnagal was shown to take over by default when Dumbledore died. Keep in mind, that by this time he was already considered one of the most respected wizards in the world. Even if he didn't hold the power in Hogwarts, his opinion would be highly-respected.

Answer 2

I believe Dumbledore proposed the ban on the subject, and Dippet merely signed off on his suggestion.

We know that horcrux magic is exceptionally rare and unknown¹. We also know that Dumbledore’s knowledge is exceptional, especially when it comes to fighting the Dark Arts. I don’t think Armando Dippet would have known about horcruxes unless Dumbledore told him.

No, I think Dumbledore proposed the ban to Dippet, and when the topic was explained, Dippet agreed to ban the subject from the library. This means that if you were flouting this ban, it was Dumbledore who would be truly angry, not Dippet. Hence it’s Dumbledore who gets described as “fierce”, as his anger derives from a genuine distaste for the subject, not just following school rules.

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¹ Although this is in part due to the Hogwarts ban on the subject, they’re such a dark and dangerous branch of magic that I believe this would be true whether or not they were banned.