*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/9780199570423.003.0015
- Subject:
- Philosophy, Logic/Philosophy of Mathematics, Philosophy of Language

The book concludes with a Postscript describing some items of unfinished business. It begins with three tricky topics: first, pseudo-singular terms—syntactically singular but semantically plural noun ...
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The book concludes with a Postscript describing some items of unfinished business. It begins with three tricky topics: first, pseudo-singular terms—syntactically singular but semantically plural noun phrases—and then two extensions of full plural logic in very different directions—higher-level plural logic and higher-order plural logic. It adds a bunch of other items the authors have hardly exploredLess

The book concludes with a Postscript describing some items of unfinished business. It begins with three tricky topics: first, pseudo-singular terms—syntactically singular but semantically plural noun phrases—and then two extensions of full plural logic in very different directions—higher-level plural logic and higher-order plural logic. It adds a bunch of other items the authors have hardly explored

*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 ...
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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 ...
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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.