Alex Oliver and Timothy Smiley
- Published in print:
- 2013
- Published Online:
- September 2013
- ISBN:
- 9780199570423
- eISBN:
- 9780191755866
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199570423.003.0014
- Subject:
- Philosophy, Logic/Philosophy of Mathematics, Philosophy of Language
This chapter applies plural logic to set theory. Set theory may be axiomatized with full plural logic as the underlying logic and with the function sign { }—read ‘set of’—as its only nonlogical ...
More
This chapter applies plural logic to set theory. Set theory may be axiomatized with full plural logic as the underlying logic and with the function sign { }—read ‘set of’—as its only nonlogical primitive. This style of axiomatization is inspired by Cantor, both with regard to the idea of a set as a collection of many members and the plural language used to express it. Cantor’s idea of sets as collections does not accommodate empty or singleton sets. So a system of Cantorian set theory is developed which excludes them. It is argued that this is no loss. But for those who prefer to retain the anomalous sets, the chapter concludes by explaining how a conventional iterative set theory can be based on full plural logic.Less
This chapter applies plural logic to set theory. Set theory may be axiomatized with full plural logic as the underlying logic and with the function sign { }—read ‘set of’—as its only nonlogical primitive. This style of axiomatization is inspired by Cantor, both with regard to the idea of a set as a collection of many members and the plural language used to express it. Cantor’s idea of sets as collections does not accommodate empty or singleton sets. So a system of Cantorian set theory is developed which excludes them. It is argued that this is no loss. But for those who prefer to retain the anomalous sets, the chapter concludes by explaining how a conventional iterative set theory can be based on full plural logic.
Alex Oliver and Timothy Smiley
- Published in print:
- 2013
- Published Online:
- September 2013
- ISBN:
- 9780199570423
- eISBN:
- 9780191755866
- Item type:
- book
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199570423.001.0001
- Subject:
- Philosophy, Logic/Philosophy of Mathematics, Philosophy of Language
This book aims to be the natural point of entry to what will be a new subject for most readers. Technicalities have been kept to a minimum, and anyone who is familiar with the classical predicate ...
More
This book aims to be the natural point of entry to what will be a new subject for most readers. Technicalities have been kept to a minimum, and anyone who is familiar with the classical predicate calculus should be able to follow it. The book tackles the logic of plural terms (‘Whitehead and Russell’, ‘the men who wrote Principia Mathematica’, ‘Henry VIII’s wives’, ‘the real numbers’, ‘√−1’, ‘they’); plural predicates (‘surrounded the fort’, ‘are prime’, ‘are consistent’, ‘imply’); and plural quantification (‘some things’, ‘any things’). Current logic is singularist: it only allows terms to stand for at most one thing. By contrast, the foundational thesis of this book is that a particular term may legitimately stand for several things at once, in other words, there is such a thing as genuinely plural denotation. Plural logic is logic based on plural denotation. The book begins by making the case for taking plural phenomena seriously, and argues, by eliminating rival singularist strategies, that the only viable response is to adopt a plural logic. The subsequent development of the conceptual ground includes the distinction between distributive and collective predicates, the theory of plural descriptions, multivalued functions, and lists. A formal system of plural logic is then presented in three stages, before being applied to Cantorian set theory as an illustration.Less
This book aims to be the natural point of entry to what will be a new subject for most readers. Technicalities have been kept to a minimum, and anyone who is familiar with the classical predicate calculus should be able to follow it. The book tackles the logic of plural terms (‘Whitehead and Russell’, ‘the men who wrote Principia Mathematica’, ‘Henry VIII’s wives’, ‘the real numbers’, ‘√−1’, ‘they’); plural predicates (‘surrounded the fort’, ‘are prime’, ‘are consistent’, ‘imply’); and plural quantification (‘some things’, ‘any things’). Current logic is singularist: it only allows terms to stand for at most one thing. By contrast, the foundational thesis of this book is that a particular term may legitimately stand for several things at once, in other words, there is such a thing as genuinely plural denotation. Plural logic is logic based on plural denotation. The book begins by making the case for taking plural phenomena seriously, and argues, by eliminating rival singularist strategies, that the only viable response is to adopt a plural logic. The subsequent development of the conceptual ground includes the distinction between distributive and collective predicates, the theory of plural descriptions, multivalued functions, and lists. A formal system of plural logic is then presented in three stages, before being applied to Cantorian set theory as an illustration.
Alex Oliver and Timothy Smiley
- Published in print:
- 2016
- Published Online:
- February 2017
- ISBN:
- 9780198744382
- eISBN:
- 9780191843877
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198744382.003.0014
- Subject:
- Philosophy, Logic/Philosophy of Mathematics, Philosophy of Language
This chapter applies plural logic to set theory. Set theory may be axiomatized with full plural logic as the underlying logic and with the function sign { }—read ‘set of’—as its only nonlogical ...
More
This chapter applies plural logic to set theory. Set theory may be axiomatized with full plural logic as the underlying logic and with the function sign { }—read ‘set of’—as its only nonlogical primitive. This style of axiomatization is inspired by Cantor, both with regard to the idea of a set as a collection of many members and the plural language used to express it. Cantor's idea of sets as collections does not accommodate empty or singleton sets. So a system of Cantorian set theory is developed which excludes them. It is argued that this is no loss. The detailed development of the theory is put in the appendix. It is the first actually worked-out application of plural logic.Less
This chapter applies plural logic to set theory. Set theory may be axiomatized with full plural logic as the underlying logic and with the function sign { }—read ‘set of’—as its only nonlogical primitive. This style of axiomatization is inspired by Cantor, both with regard to the idea of a set as a collection of many members and the plural language used to express it. Cantor's idea of sets as collections does not accommodate empty or singleton sets. So a system of Cantorian set theory is developed which excludes them. It is argued that this is no loss. The detailed development of the theory is put in the appendix. It is the first actually worked-out application of plural logic.
Alex Oliver and Timothy Smiley
- Published in print:
- 2016
- Published Online:
- February 2017
- ISBN:
- 9780198744382
- eISBN:
- 9780191843877
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198744382.003.0015
- Subject:
- Philosophy, Logic/Philosophy of Mathematics, Philosophy of Language
This chapter is conceived as meeting the challenge left by the abandoned second volume of Russell's Principles of Mathematics. An outline of higher-level plural logic is presented making use of ...
More
This chapter is conceived as meeting the challenge left by the abandoned second volume of Russell's Principles of Mathematics. An outline of higher-level plural logic is presented making use of pseudo-singular terms, which are syntactically singular but semantically plural. The pseudo-singular ‘multitude’ is used to express iterated plurality (multitudes of multitudes, multitude of multitudes of multitudes etc). The logical primitives of first-level plural logic are reconfigured to obtain the higher-level system. Singular and plural variables are replaced by a single stock of all-level variables, and inclusion is factored into its lateral and vertical components. Vertical inclusion as well as identity are taken as primitives, while lateral inclusion (submultitude) can be defined. The striking resemblance between higher-level plural logic and Cantorian set theory suggests that the conventional conception of sets as collections rests on a mistake about logical form.Less
This chapter is conceived as meeting the challenge left by the abandoned second volume of Russell's Principles of Mathematics. An outline of higher-level plural logic is presented making use of pseudo-singular terms, which are syntactically singular but semantically plural. The pseudo-singular ‘multitude’ is used to express iterated plurality (multitudes of multitudes, multitude of multitudes of multitudes etc). The logical primitives of first-level plural logic are reconfigured to obtain the higher-level system. Singular and plural variables are replaced by a single stock of all-level variables, and inclusion is factored into its lateral and vertical components. Vertical inclusion as well as identity are taken as primitives, while lateral inclusion (submultitude) can be defined. The striking resemblance between higher-level plural logic and Cantorian set theory suggests that the conventional conception of sets as collections rests on a mistake about logical form.
Alex Oliver and Timothy Smiley
- Published in print:
- 2016
- Published Online:
- February 2017
- ISBN:
- 9780198744382
- eISBN:
- 9780191843877
- Item type:
- book
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198744382.001.0001
- Subject:
- Philosophy, Logic/Philosophy of Mathematics, Philosophy of Language
This book tackles the logic of plural terms (‘Whitehead and Russell’, ‘the men who wrote Principia Mathematica’, ‘Henry VIII's wives’, ‘the real numbers’, ‘√—1’, ‘they’); plural predicates ...
More
This book tackles the logic of plural terms (‘Whitehead and Russell’, ‘the men who wrote Principia Mathematica’, ‘Henry VIII's wives’, ‘the real numbers’, ‘√—1’, ‘they’); plural predicates (‘surrounded the fort’, ‘are prime’, ‘are consistent’, ‘imply’); and plural quantification (‘some things’, ‘any things’). Current logic is singularist: it only allows terms to stand for at most one thing. By contrast, the foundational thesis of this book is that a particular term may legitimately stand for several things at once, in other words, there is such a thing as genuinely plural denotation. Plural logic is logic based on plural denotation. The book begins by making the case for taking plural phenomena seriously, and argues, by eliminating rival singularist strategies, that the only viable response is to adopt a plural logic. The subsequent development of the conceptual ground includes the distinction between distributive and collective predicates, the theory of plural descriptions, multivalued functions, and lists. A formal system of plural logic is then presented in three stages, before being applied to Cantorian set theory as an illustration. A system of higher-level plural logic is then outlined. It bears a striking similarlty to the set theory.Less
This book tackles the logic of plural terms (‘Whitehead and Russell’, ‘the men who wrote Principia Mathematica’, ‘Henry VIII's wives’, ‘the real numbers’, ‘√—1’, ‘they’); plural predicates (‘surrounded the fort’, ‘are prime’, ‘are consistent’, ‘imply’); and plural quantification (‘some things’, ‘any things’). Current logic is singularist: it only allows terms to stand for at most one thing. By contrast, the foundational thesis of this book is that a particular term may legitimately stand for several things at once, in other words, there is such a thing as genuinely plural denotation. Plural logic is logic based on plural denotation. The book begins by making the case for taking plural phenomena seriously, and argues, by eliminating rival singularist strategies, that the only viable response is to adopt a plural logic. The subsequent development of the conceptual ground includes the distinction between distributive and collective predicates, the theory of plural descriptions, multivalued functions, and lists. A formal system of plural logic is then presented in three stages, before being applied to Cantorian set theory as an illustration. A system of higher-level plural logic is then outlined. It bears a striking similarlty to the set theory.
