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.
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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