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. Fields of science natural sciencesmathematicspure mathematicsdiscrete mathematicsmathematical logichumanitiesphilosophy, ethics and religionphilosophy Programme(s) FP7-IDEAS-ERC - Specific programme: "Ideas" implementing the Seventh Framework Programme of the European Community for research, technological development and demonstration activities (2007 to 2013) Topic(s) ERC-SG-ID1 - ERC Starting Grant Interdisciplinary Panel Call for proposal ERC-2009-StG See other projects for this call Funding Scheme ERC-SG - ERC Starting Grant Host institution UNIVERSITETET I OSLO EU contribution € 507 769,33 Address PROBLEMVEIEN 5-7 0313 Oslo Norway See on map Region Norge Oslo og Viken Oslo Activity type Higher or Secondary Education Establishments Administrative Contact Karen Haugland (Ms.) Principal investigator øystein Linnebo (Dr.) Links Contact the organisation Opens in new window Website Opens in new window Total cost No data Beneficiaries (2) Sort alphabetically Sort by EU Contribution Expand all Collapse all UNIVERSITETET I OSLO Norway EU contribution € 507 769,33 Address PROBLEMVEIEN 5-7 0313 Oslo See on map Region Norge Oslo og Viken Oslo Activity type Higher or Secondary Education Establishments Administrative Contact Karen Haugland (Ms.) Principal investigator øystein Linnebo (Dr.) Links Contact the organisation Opens in new window Website Opens in new window Total cost No data BIRKBECK COLLEGE - UNIVERSITY OF LONDON Participation ended United Kingdom EU contribution € 432 885,67 Address MALET STREET WC1E 7HX London See on map Region London Inner London — West Camden and City of London Activity type Higher or Secondary Education Establishments Administrative Contact Craig Bryce (Mr.) Links Contact the organisation Opens in new window Website Opens in new window Total cost No data