On the translation from quantified modal logic into the counterpart theory revisited

Abstract

The counterpart theory which was introduced by David Lewis is an alternative semantics to the possible worlds semantics for quantified modal logic. Lewis interprets modal claims by using a translation from quantified modal logic into the counterpart theory. Due to the flexibility of semantics of the counterpart theory, Lewis's translation may translate an unsatisfiable formula to a satisfiable one. In this paper, two properties are defined to describe Lewis's translation, called the faithfulness and fullness. The former implies a translation which preserves the satisfiability of formulas, whereas the latter implies the preservation of the unsatisfiability. We show that Lewis's translation is faithful but not full. To make Lewis's translation full, two auxiliary axioms are added to restrain the counterpart relation such that every possible object has exactly one counterpart in every possible world. Under the circumstances, we show that Lewis's translation is a faithful and full translation. © 2011 Springer-Verlag.