Logical Frameworks for Truth and Abstraction
An Axiomatic StudyBy
- A. Cantini, University of Florence, Department of Philosophy, Italy
This English translation of the author's original work has been thoroughly revised, expanded and updated.
The book covers logical systems known as type-free or self-referential. These traditionally arise from any discussion on logical and semantical paradoxes. This particular volume, however, is not concerned with paradoxes but with the investigation of type-free sytems to show that: (i) there are rich theories of self-application, involving both operations and truth which can serve as foundations for property theory and formal semantics; (ii) these theories provide a new outlook on classical topics, such as inductive definitions and predicative mathematics; (iii) they are particularly promising with regard to applications.
Research arising from paradoxes has moved progressively closer to the mainstream of mathematical logic and has become much more prominent in the last twenty years. A number of significant developments, techniques and results have been discovered.Academics, students and researchers will find that the book contains a thorough overview of all relevant research in this field.
Studies in Logic and the Foundations of Mathematics
Published: March 1996
...This is an appealing book, (relatively) easy to read, and attractive in its unified treatment of a range of issues...I found this to be a well-constructed book. The bibliography is thorough, and the indexes are well constructed and accurate...
E. Martin, Studia Logica
- Preface. Contents. Introduction. PART A: COMBINATORS AND TRUTH. I. Introducing operations. II. Extending operations with reflective truth. PART B: TRUTH AND RECURSION THEORY. III. Inductive models and definability theory. IV. Type-free abstraction with approximation operator. V. Type-free abstraction, choice and sets. PART C: SELECTED TOPICS. VI. Levels of implication and intentional logical equivalence. VII. On the global structure of models for reflective truth. PART D: LEVELS OF TRUTH AND PROOF THEORY. VIII. Levels of reflective truth. IX. Levels of truth and predicative well-orderings. X. Reducing reflective truth with levels to finitely iterated reflective truth. XI. Proof-theoretic investigation of finitely iterated reflective truth. PART E: ALTERNATIVE VIEWS. XII. Non-reductive systems for type-free abstraction and truth. XIII. The variety of non-reductive approaches. XIV. Epilogue: applications and perspectives. Bibliography. Index. List of Symbols.