Logical Frameworks for Truth and Abstraction: An Axiomatic Study
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...
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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. Booknews Cantini (philosophy, U. of Florence, Italy) explores logical systems that are usually termed type-free or self-referential. Though such systems emerge from the traditional discussion on logical and semantical paradoxes, he eschews both paradoxes and any grand logic inspired by a foundational program. Instead he argues that there are rich theories of self-application that can serve as foundations for property theory and formal semantics, that those theories can provide new perspectives on classical topics such as inductive definitions and predicative mathematics, and that they show promise of practical application. Revised, expanded, and updated as well as translated from the original 1988 edition in Italian. Annotation c. Book News, Inc., Portland, OR (booknews.com)
PrefaceContentsIntroduction1IIntroducing operations13IIExtending operations with reflective truth43IIIInductive models and definability theory85IVType-free abstraction with approximation operator103VType-free abstraction, choice and sets125VILevels of implication and intensional logical equivalence151VIIOn the global structure of models for reflective truth177VIIILevels of reflective truth215IXLevels of truth and predicative well-orderings257XReducing reflective truth with levels to finitely iterated reflective truth285XIProof-theoretic investigation of finitely iterated reflective truth311XIINon-reductive systems for type-free abstraction and truth351XIIIThe variety of non-reductive approaches379XIVEpilogue: applications and perspectives401Bibliography425Index441List of Symbols453
\ From the PublisherE. Martin\ ...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...\ Studia Logica\ \ \ \ \ \ BooknewsCantini (philosophy, U. of Florence, Italy) explores logical systems that are usually termed type-free or self-referential. Though such systems emerge from the traditional discussion on logical and semantical paradoxes, he eschews both paradoxes and any grand logic inspired by a foundational program. Instead he argues that there are rich theories of self-application that can serve as foundations for property theory and formal semantics, that those theories can provide new perspectives on classical topics such as inductive definitions and predicative mathematics, and that they show promise of practical application. Revised, expanded, and updated as well as translated from the original 1988 edition in Italian. Annotation c. Book News, Inc., Portland, OR (booknews.com)\ \
