This paper provides a thorough theoretical investigation on the logical characterizations of the Common Prior Assumption (CPA). CPA has played a prominent role in economic theory, its essential meaning being that differences in beliefs among agents can be completely explained by differences in information they received. Using classical tools from modal logic, Joseph Halpern gives two different characterizations of CPA and compares them to those existing in the economics literature. The first CPA characterization is based on frame distinguishability, a concept closer to the notions typically considered in the economics literature, the results obtained for CPA frame distinguishability being similar to those already existing ones. In particular, the paper shows that finite frames (actually, finite spaces) that satisfy the CPA can be distinguished from those that do not, in terms of disagreements in expectation. However, it is also proved that there exist no formulas in the defined language which can distinguish infinite spaces satisfying the CPA, from those spaces that do not. A second approach to CPA characterization is based on finding a sound and complete axiomatization. In the proposed axiom system, the key axiom involves disagreements in expectation. Since the axiomatization is proved to be sound and complete both for the finite and infinite case, this logical language is too weak to distinguish finite and infinite spaces. This phenomenon is frequently met in various aspects of (fundamental) mathematics. Essentially, the result obtained says that in economics there exist ``nonstandard'' models that satisfy all the properties of the CPA expressible in the considered axiomatized language, despite the fact that these models do not satisfy the CPA (similarly to the nonstandard models of the real numbers that satisfy all the properties of the reals). Thus the two approaches characterizing the CPA behave quite differently in the case of infinite spaces. The answer to the question asking which of the two proposed CPA characterizations is more appropriate depends on each specific application. Frame distinguishability seems to be a more adequate notion when a frame is given, while axiomatizations could be more useful when only some facts about the frame are provided (e.g. implying no common knowledge of disagreement in terms of CPA).