Plurals, Predicates and Paradox - ERC Starting Grants Research Project (completed)
As part of a European Research Council Starting Grant, Professor Øystein Linnebo is leading a four-year research project entitled 'Plurals, Predicates, and Paradox: Towards a Type-Free Account', to run from January 2010 until December 2013. In August 2012, the project moved from Birkbeck, University of London to the University of Oslo.
About the project
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 and mathematical 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, arising 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.
The project will open up new approaches to the logical paradoxes and the foundations of mathematics, shed new light on the semantics of nominalization and the psychology of mathematics, and develop a new challenge to a great variety of philosophical applications of HOL.
The project is funded by the European Research Council (ERC Starting Grant) and the Department of Philosophy, Classics, History of Art and Ideas.
We cooperate with a number of international researchers and have co-organized conferences with the following groups and institutions: Bristol; Foundations of Logical Consequence Arché/St Andrews; Inexpressibility and Reflection in the Formal Sciences, Oxford; Institute of Philosophy, London; Munich Center for Mathematical Philosophy; Nominalizations: Philosophical and Linguistic Perspectives, Hamburg.