Subjectivity and logic

A friend of mine challenged the system of formal logic claiming that since logic has a subjective nature, trying to formalize it with hard and fast rules will ultimately be problematic. There certainly is some truth to the premise, since logicians will still disagree with one another, which I suppose is evidence of logic's non-mathematical nature. The question I wish to pose asks what element in logical thinking will allow two intelligent and informed individuals to come to two different conclusions?
While some aspects of logic work with mathematical precision, I will contend that the translation of normal sentences into a more purely logical forms add a degree of interpretation. In addition, I think that one's individual values can play a role in determining what constitutes an okay premise or not.