This book is about the possible-worlds semantic analysis of systems of logic that have quantifiers binding individual variables. Our approach is based on a notion of “admissible” model that places a restriction on which sets of worlds can serve as propositions. We show that admissible models provide semantic characterisations of a wide range of logical systems, including many for which the well-known model theory of Kripke [1963b] is incomplete. The key to this is an interpretation of quantification that takes into account the admissibility of propositions.

This is a subject that bristles with choices and challenges. Should terms be treated as rigid designators, or should their denotations vary from world to world? Should individual constants and variables be treated the same in this respect, or differently? Should each world have its own domain of existing individuals over which the quantifiable variables range, or should there be just a single domain of individuals? If there are varying domains, how should they relate to each other? Can any function from worlds to individuals be regarded as the “meaning” of some individual concept? Should an arbitrary mapping from individuals to propositions be admissible as a propositional function? Can we deductively axiomatise the class of valid formulas determined by each answer to these questions?