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Plurals, Predicates, and Paradox: Towards a Type-Free Account

Objective

This project aims to transform our understanding of the logical paradoxes, their solution and significance for mathematics, philosophy and semantics. It seeks to show that some of the key inferences in the paradoxes should not uncritically be blocked, as is customary, but rather be tamed and put to valuable mathematical, philosophical and semantic use. By adopting a richer logical framework than usual, the paradoxes can be transformed from threats to valuable sources of insight. When discovered at the turn of the previous century, the paradoxes caused a foundational crisis in mathematics. Many logicians and philosophers now believe the crisis has been resolved. This project denies that an acceptable resolution has been found and aims to do better. A strong push remains towards paradox. This push arises from the widespread use of (and need for) higher-order logics (HOL), which allow quantification into the positions of predicates or plural noun phrases. Phase I seeks to reveal greater similarities between HOL and set theory than generally appreciated. Phase II explores four arguments that HOL collapses to first-order logic, i.e. that every higher-order entity defines a corresponding first-order entity. These arguments are generally ignored as they threaten to reintroduce the paradoxes. But we show that a properly circumscribed form of collapse is a valuable source of mathematical and semantic insight. Phase III examines controlled forms of collapse using notions of modality and groundedness. This enables us to motivate ZF set theory and valuable semantic theories, explain the nature of cognition about sets and properties, and show that mathematics cannot be fully extensionalized. Phase IV applies these insights to solve the paradoxes and criticize influential uses of HOL.

Field of science

  • /natural sciences/mathematics
  • /humanities/philosophy, ethics and religion/philosophy

Call for proposal

ERC-2009-StG
See other projects for this call

Funding Scheme

ERC-SG - ERC Starting Grant

Host institution

UNIVERSITETET I OSLO
Address
Problemveien 5-7
0313 Oslo
Norway
Activity type
Higher or Secondary Education Establishments
EU contribution
€ 507 769,33
Principal investigator
øystein Linnebo (Dr.)
Administrative Contact
Karen Haugland (Ms.)

Beneficiaries (2)

UNIVERSITETET I OSLO
Norway
EU contribution
€ 507 769,33
Address
Problemveien 5-7
0313 Oslo
Activity type
Higher or Secondary Education Establishments
Principal investigator
øystein Linnebo (Dr.)
Administrative Contact
Karen Haugland (Ms.)
BIRKBECK COLLEGE - UNIVERSITY OF LONDON

Participation ended

United Kingdom
EU contribution
€ 432 885,67
Address
Malet Street
WC1E 7HX London
Activity type
Higher or Secondary Education Establishments
Administrative Contact
Craig Bryce (Mr.)