The aim of this book is to present the fundamental theoretical results concerning inference rules in deductive formal systems. Primary attention is focused on:
• admissible or permissible inference rules
• the derivability of the admissible inference rules
• the structural completeness of logics
• the bases for admissible and valid inference rules.
There is particular emphasis on propositional non-standard logics (primary, superintuitionistic and modal logics) but general logical consequence relations and classical first-order theories are also considered.
The book is basically self-contained and special attention has been made to present the material in a convenient manner for the reader. Proofs of results, many of which are not readily available elsewhere, are also included.
The book is written at a level appropriate for first-year graduate students in mathematics or computer science. Although some knowledge of elementary logic and universal algebra are necessary, the first chapter includes all the results from universal algebra and logic that the reader needs. For graduate students in mathematics and computer science the book is an excellent textbook.
Admissibility of Logical Inference Rules, 1st Edition
Preface and acknowledgments. Introduction. Syntaxes and Semantics.
Syntax of formal logic systems. First-order semantics and universal algebra. Algebraic semantics for propositional logics. Admissible rules in algebraic logics. Logical consequence relations. Algebraizable consequence relations. Admissibility for consequence relations. Lattices of logical consequences. Semantics for Non-Standard Logics.
Algebraic semantics for intuitionistic logic. Algebraic semantics, modal and tense logics. Kripke semantics for modal and temporal logic. Kripke semantics for intuitionistic logic. Stone's theory and Kripke semantics. The finite model property. Relation of intuitionistic and modal logics. Advanced tools for the finite model property. Kripke incomplete logics. Advanced tools for Kripke completeness.Criteria for Admissibility.
Reduced forms. T
–translation of inference rules. Semantic criteria for admissibility. Some technical lemmas. Criteria for determining admissibility. Elementary theories of free algebras. Scheme–logics of first–order theories. Some counterexamples. Admissibility through reduced forms. Bases for Inference Rules.
Initial auxiliary results. The absence of finite bases. Bases for some strong logics. Bases for valid rules of tabular logics. Tabular logics without independent bases. Structural Completeness.
General properties, descriptions. Quasi–characteristic inference rules. Some preliminary technical results. Hereditary structural completeness. Structurally complete fragments. Related Questions.
Rules with meta–variables. The preservation of admissible for S4 rules. The preservation of admissibility for H
. Non-compact modal logics. Decidable Kripke non–compact logics. Non–compact superintuitionistic logics. Index. Bibliography.