Modal logic and model theory

Abstract

We propose a first order modal logic, theQS4E-logic, obtained by adding to the well-known first order modal logicQS4 arigidity axiom schemas:A → □A, whereA denotes a basic formula. In this logic, thepossibility entails the possibility of extending a given classical first order model. This allows us to express some important concepts of classical model theory, such as existential completeness and the state of being infinitely generic, that are not expressibile in classical first order logic. Since they can be expressed in$L_{\omega _1 \omega }$-logic, we are also induced to compare the expressive powers ofQS4E and$L_{\omega _1 \omega }$. Some questions concerning the power of rigidity axiom are also examined.