TWO PHD STUDENTSHIPS

Birkbeck College, University of London

Department of Philosophy

 

Two three-year, full-time studentship are available with funding from the European Research Council (ERC). The studentships are associated with a research project in philosophical logic and the philosophy of mathematics, concerned with higher-order logic, the logical paradoxes, modality and the foundation of set theory.

 

Overview of the project – “Plurals, predicates, and paradox”

This project aims to transform our understanding of the logical paradoxes, their solution, and their significance for mathematics, philosophy, and semantics. It seeks to show that some of the key inferences which give rise to the paradoxes should not uncritically be blocked, as is customary, but rather can 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. But many logicians and philosophers now believe the crisis has been resolved by the iterative conception of sets and formal theories of truth due to Tarski and Kripke. This project denies that an acceptable resolution has been found and aims to do better. Although existing theories are extremely valuable, strong pressure remains from mathematics and semantics to accept certain problematic principles. This pressure arises from the widespread use of (and apparent need for) higher-order logics (HOL), which allow quantification into the positions of predicates or plural noun phrases.

 

The project 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. We examine controlled forms of collapse using notions of modality and groundedness. This enables us to provide a novel motivation for ZF set theory, to formulate valuable semantic theories, to explain the nature of cognition about sets and properties, and to show that mathematics cannot be fully extensionalized. We end by applying 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.

 

Further information about the project is available at http://www.oysteinlinnebo.org/ppp/project.pdf.

 

The research team

The Principal Investigator is Dr Øystein Linnebo. There will also be two three-year postdoctoral researchers and two PhD students.

 

The PhD studentships

  • Applicants must have a MA or equivalent in Philosophy or some related field by the commencement date.
  • A strong background in logic and the foundations of mathematics will be an advantage.
  • Applicants must be willing work on issues related to the project, as well as contribute actively to weekly seminars and regular workshops associated with the project.
  • Ideally, it is hoped that one student will work on philosophical issues concerning higher-order logic, set theory and their history; and the other, on mathematical cognition, especially thought about sets, properties and other mathematical objects and the role of nominalization in such thought. However, some flexibility will be offered in order to recruit the best students and maximize their potential.

 

Application process

Applicants must do two things: firstly, apply to be a PhD student at Birkbeck in the normal way; and secondly, submit the project-specific material listed below.

 

Applying to be a PhD student at Birkbeck:

 

Project specific material:

  • A writing sample
  • A statement of no more than 1000 words explaining how your research proposal contributes to the research project

 

The application should be emailed to Ms Simmi Pahwa at S.Pahwa@bbk.ac.uk by 15 February 2010.

 

Informal enquiries about the project may be made to Dr Øystein Linnebo at o.linnebo@bbk.ac.uk. Shortlisted candidates may be interviewed in person or over the phone in late February or early March.