Binary predication from binary variables
This is really a job for Logistic Regression. Input variables can be categorical/boolean and the prediction can be categorical/boolean as well.

However, if your target variable contains multiple categories, you should use multinomial logistic regression or Ordinal Logistic Regression (if there is an order in your categories). It also changes the way you interpret the output.

The another approach would be to take any nonparametric method, Decision Tree for example. Where input and target can be boolean/categorical/continuous. Note that in case of a categorical variable, it is often recommended to transform it into dummy variables.