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Plurals Predicates and Paradox Towards a Type Free Account

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 unc... ver más

31/12/2013

Innovasjon Norge

941K€

Presupuesto del proyecto: 941K€

Líder del proyecto

Innovasjon Norge No se ha especificado una descripción o un objeto social para esta compañía.

Fecha límite participación Sin fecha límite de participación.

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Financiación concedida El organismo FP7 notifico la concesión del proyecto el día 2013-12-31 No tenemos la información de la convocatoria

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2 Participantes

Innovasjon Norge

0.00€ | Lider

BIRKBECK COLLEGE UNIVERSITY...

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Líder del proyecto

Innovasjon Norge No se ha especificado una descripción o un objeto social para esta compañía.

Presupuesto del proyecto 941K€

Fecha límite de participación Sin fecha límite de participación.

Descripción del proyecto 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.

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