CHAPTER 1 - THE ROLE OF LOGICAL SYSTEMS

Abstract

This chapter discusses the role of logical systems. The logical systems provide the basic inference mechanisms for theorem provers. As such they fit within and greatly influence the form of the search representation component of the theorem provers. First-order logic provides a natural environment in which to develop the mechanisms because of its expressive power and well-understood semantics and its familiarity. The chapter highlights the manner of problem representation within first-order logic. In practice, one is limited to confirming that finite sets of formulas are valid or unsatisfiable. With care in formulation, most real-world problems and mathematical problems are representable to a large extent. The test actually is for unsatisfiability using a Skolem conjunctive form, a quantifier-free formula in conjunctive normal form with existential quantifiers replaced by Skolem functions. The keystone to all the formats or procedures considered is the Herbrand Theorem, which is painlessly obtainable from the Completeness theorem via the Compactness theorem. A formula in Skolem co-junctive form is unsatisfiable iff a related finite set of ground clauses is unsatisfiable, where each ground clause is obtained from the given formula by replacement of variables by suitable Herbrand terms. The ground clause set is testable by truth tables.