I have training data that classifies articles (article title and a summary) to one of two classes, let's say class A and class B

I want to be able to classify new articles. The problem is, the new articles that I want to classify can either be in class A or class B or they can not belong to either.

Should my training data include samples that do not belong to both classes? In other words, should I treat "none" as a separate class. Or is there another way (probably generative learning algorithms) where I can use class probabilities and makes decisions whether an article belong to a class based on the probability? I was using Multinomial Naive-Bayes but the results are very bad for articles that do not belong to either class