 Given these definitions, it is clear that knowledge implies both belief and evidence (i.e., both \(A\varphi \rightarrow B\varphi\) and \(A\varphi \rightarrow \oBox\varphi\) are valid). Still, it is interesting to note not only that the agent might believe a given \(\varphi\) without having a basic piece of evidence supporting it \((B \varphi \rightarrow \oBox\varphi\) is NOT valid, as beliefs are defined in terms of combined pieces of evidence), but also that she might have a basic piece of evidence supporting \(\varphi\) without believing \(\varphi\) \((\oBox\varphi \rightarrow B\varphi\) is NOT valid, as the basic evidence supporting \(\varphi\) not be part of all maximally consistent combinations).