Objectif 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. Champ scientifique 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) Thème(s) ERC-SG-ID1 - ERC Starting Grant Interdisciplinary Panel Appel à propositions ERC-2009-StG Voir d’autres projets de cet appel Régime de financement ERC-SG - ERC Starting Grant Institution d’accueil UNIVERSITETET I OSLO Contribution de l’UE € 507 769,33 Adresse PROBLEMVEIEN 5-7 0313 Oslo Norvège Voir sur la carte Région Norge Oslo og Viken Oslo Type d’activité Higher or Secondary Education Establishments Contact administratif Karen Haugland (Ms.) Chercheur principal øystein Linnebo (Dr.) Liens Contacter l’organisation Opens in new window Site web Opens in new window Coût total Aucune donnée Bénéficiaires (2) Trier par ordre alphabétique Trier par contribution de l’UE Tout développer Tout réduire UNIVERSITETET I OSLO Norvège Contribution de l’UE € 507 769,33 Adresse PROBLEMVEIEN 5-7 0313 Oslo Voir sur la carte Région Norge Oslo og Viken Oslo Type d’activité Higher or Secondary Education Establishments Contact administratif Karen Haugland (Ms.) Chercheur principal øystein Linnebo (Dr.) Liens Contacter l’organisation Opens in new window Site web Opens in new window Coût total Aucune donnée BIRKBECK COLLEGE - UNIVERSITY OF LONDON Participation terminée Royaume-Uni Contribution de l’UE € 432 885,67 Adresse MALET STREET WC1E 7HX London Voir sur la carte Région London Inner London — West Camden and City of London Type d’activité Higher or Secondary Education Establishments Contact administratif Craig Bryce (Mr.) Liens Contacter l’organisation Opens in new window Site web Opens in new window Coût total Aucune donnée