  The logic FO(ID) uses ideas from the field of logic programming to extend
first order logic with non-monotone inductive definitions. Such logic formally
extends logic programming, abductive logic programming and datalog, and thus
formalizes the view on these formalisms as logics of (generalized) inductive
definitions. The goal of this paper is to study a deductive inference method
for PC(ID), which is the propositional fragment of FO(ID). We introduce a
formal proof system based on the sequent calculus (Gentzen-style deductive
system) for this logic. As PC(ID) is an integration of classical propositional
logic and propositional inductive definitions, our sequent calculus proof
system integrates inference rules for propositional calculus and definitions.
We present the soundness and completeness of this proof system with respect to
a slightly restricted fragment of PC(ID). We also provide some complexity
results for PC(ID). By developing the proof system for PC(ID), it helps us to
enhance the understanding of proof-theoretic foundations of FO(ID), and
therefore to investigate useful proof systems for FO(ID).
