Between a priori and analytical contradiction

Rabbi Shalom,
In the book 'Two Carts and a Balloon' you distinguish between two types of contradictions: analytical contradiction and a priori contradiction. You then argue that belief in an analytical contradiction is meaningless, whereas an a priori contradiction can have meaning. In this way you allow belief in knowledge and choice without falling into a meaningless statement or having to give up one of the premises. 
I don't understand, how can one even imagine an a priori contradiction, and say that it has any meaning at all? Why is it 'okay' to believe in such contradictions? 
I would love a clearer explanation of the difference between the types of contradictions. 
thanks