How many possible hypotheses The training set contains $p$ papers. Each paper is annotated as research or non-research. To develop the research paper filter, we consider the $W$ most frequent phrases in a paper. The research paper filter will use the presence/absence of these $W$ phrases to decide if the paper is indeed a research paper or not.

How many possible hypothesis are there?
I have $2^W$ possible hypothesis, but if we are including conjunctive hypothesis symbols, then I would say $4^W$. Is my logic correct?

How many of them are consistent?
For this question, I have no idea where to begin. Any insight?
I'm not seeing where the relationship between hypotheses and research paper is. I know what the relationship is in reality, but your question is a bit unorganized. Perhaps we could think about it in a different way... 

I would first create a definition of possible hypotheses (i.e., $H_{0}$ and $H_{1}$ are my two possible hypotheses, and every hypothesis will have both a null and an alternative), and then associate a $H_{0}$ with a $H_{1}$ such that a combination of both null and alternative hypotheses equate to one hypothesis. 

With this, it's a matter of iteratively counting.


With $p \in P$, where $p$ is a paper in the set of papers $P$,
With $w \in W$, where $w$ is a word in a set of words $W$ that indicate whether $p$ is a research paper or not,
With $i$ as the number of times a $w$ occurred in a $p$, 
count the number of times a $w$ occurs in each $p$ and iterate $i$ accordingly.