  This paper proposes first-order modal $\xi$-calculus as well as genealogical
Kripke models. Inspired by modal $\mu$-calculus, first-order modal
$\xi$-calculus takes a quite similar form and extends its inductive
expressivity onto a different dimension. We elaborate on several vivid examples
that demonstrate this logic's profound utility, especially for depicting
genealogy of concurrent computer processes. Bisimulation notion for the logic
has also been thoroughly examined.
