In the sentence:
My philosophy professor demonstrated not only how to reason persuasively but also how to avoid logical fallacies.
What part of speech is the word how?
In the sentence:
My philosophy professor demonstrated not only how to reason persuasively but also how to avoid logical fallacies.
What part of speech is the word how?
Both the hows there are adverbs that premodify the respective to-infinitive clauses. The first how premodifies to reason persuasively, and the second to avoid logical fallacies.
The fact that both how to reason persuasively and how to avoid logical fallacies are complements of the verb demonstrated doesn't change the part of speech of how at all, because how itself is not a complement of the verb. Only those clauses are.
It is Adverb and the mood is interrogative