A deductive system for PC(ID)

TL;DR: A formal proof system based on the sequent calculus (Gentzen-style deductive system) for the logic FO(ID), its propositional fragment, and it is proved the soundness and completeness of this deductedive system for a slightly restricted fragment of PC(ID).

Abstract: The logic FO(ID) uses ideas from the field of logic programming to extend first order logic with non-monotone inductive definitions. This paper studies a deductive inference method for PC(ID), its propositional fragment.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 deductive system integrates inference rules for propositional calculus and definitions.We prove the soundness and completeness of this deductive system for a slightly restricted fragment of PC(ID). We also give a counter-example to show that cut-elimination does not hold in this proof system.