Quantifiers, Propositions and Identity: Admissible Semantics for Quantified Modal and Substructural Logics

Description

Many systems of quantified modal logic cannot be characterised by Kripke's well-known possible worlds semantic analysis. This book shows how they can be characterised by a more general 'admissible semantics', using models in which there is a restriction on which sets of worlds count as propositions. This requires a new interpretation of quantifiers that takes into account the admissibility of propositions. The author sheds new light on the celebrated Barcan Formula, whose role becomes that of legitimising the Kripkean interpretation of quantification. The theory is worked out for systems with quantifiers ranging over actual objects, and over all possibilia, and for logics with existence and identity predicates and definite descriptions. The final chapter develops a new admissible 'cover semantics' for propositional and quantified relevant logic, adapting ideas from the Kripke-Joyal semantics for intuitionistic logic in topos theory. This book is for mathematical or philosophical logicians, computer scientists and linguists Content: ""Cover"" ""Title"" ""Copyright"" ""Contents"" ""Introduction and Overview "" ""Chapter 1. Logics with Actualist Quantifiers "" ""1.1. Syntax "" ""1.2. Logics "" ""1.3. Incompleteness and Admissibility "" ""1.4. Some History of the Quantifiers. "" ""1.5. Model Structures "" ""1.6. Premodels and Models "" ""1.7. Soundness """"1.8. Infinitely Many Constants "" ""1.9. Canonical Models and Completeness "" ""1.10. Completeness and Canonicity for QS "" ""1.11. Kinds of Incompleteness "" ""1.11.1. Incompleteness for Kripkean Models "" ""1.11.2. Kripkean S-frame Incompleteness "" ""1.11.3. Non-Canonical S-frame Incompleteness """"Chapter 2. The Barcan Formulas "" ""2.1. Logics with CBF "" ""2.2. Contracting Domains for All "" ""2.3. Constant Domains for CBF "" ""2.4. One Universal Domain "" ""2.5. The Deductive Role of Commuting Quantifiers "" ""2.6. Completeness with CBF and BF """"2.7. Completeness with UI and BF. "" ""2.8. S-frame Incompleteness Revisited "" ""Chapter 3. The Existence Predicate "" ""3.1. Axiomatising Existence "" ""3.2. Completeness for Kripkean E-Models "" ""3.3. Necessity of (Non)Existence "" ""3.4. Independence of BF from NNE """"3.5. What is the Role of the Barcan Formula? "" ""Chapter 4. Propositional Functions and Predicate Substitution "" ""4.1. Functional Model Structures "" ""4.2. Predicate Substitution: Notation and Terminology ""