The analysis of the "agreeing to disagree" type results is unified by considering functions which assign to each set of states of nature the value "True" or "False". We identify properties of such functions, being preserved under union, under disjoint union and under difference. The property of being preserved by disjoint union is used to generalize Aumann's, Milgrom and Stokey's and other results. Another proposition refers to all of these properties and implies Samet's generalization of Aumann's result to non-partitional information structures. The two generalizations are used for proving some additional "agreeing to disagree" type results.