In this paper formal properties of CI in different frameworks are studied. The first part is devoted to the comparison of three different frameworks for study CI: probability theory, theory of relational databases and Spohn's theory of ordinal conditional functions. Although CI-models arising in these frameworks are very similar (they satisfy semigraphoid axioms) we give examples showing that their formal properties still differ (each other). On the other hand, we find that (within each of these frameworks) there exists no finite complete axiomatic characterization of CI-models by finding an infinite set of sound inference rules (the same in all three frameworks). In the second part further frameworks for CI are discussed: Dempster-Shafer theory, possibility theory and (general) Shenoy's theory of valuation-based systems.