Plurals, Predicates and Paradox (completed)
The aim of this project was to transform our understanding of the logical paradoxes, their solution and significance for mathematics, philosophy and semantics.
About the project
Plurals, Predicates and Paradox set out 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 denied that an acceptable resolution had been found and aimed 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.
The project opened 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 developed a new challenge to a great variety of philosophical applications of HOL.
Set out to reveal greater similarities between HOL and set theory than generally appreciated.
Explored 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 the project showed that a properly circumscribed form of collapse is a valuable source of mathematical and semantic insight.
Examined controlled forms of collapse using notions of modality and groundedness. This enabled us to motivate ZF set theory and valuable semantic theories, explain the nature of cognition about sets and properties, and show that mathematics can not be fully extensionalized.
Applied these insights to solve the paradoxes and criticize influential uses of HOL.
European Research Council (ERC Starting Grant)
- Department of Philosophy, Classics, History of Art and Ideas.
From January 2010 until December 2013.
In August 2012, the project moved from Birkbeck, University of London to the University of Oslo.
We cooperated 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.
- “Untyped Pluralism”, forthcoming in Mind
- “Set Theory, Type Theory, and Absolute Generality” (with Stewart Shapiro), forthcoming in Mind
Selected publications most relevant to the PPP project:
- “Response to Florio and Shapiro” (with Agustín Rayo), forthcoming in Mind
- “The Potential Hierarchy of Sets”, Review of Symbolic Logic 6:2 (2013), 205-28
- “Hierarchies Ontological and Ideological” (with Agustín Rayo), Mind 121 (2012): 269-308
- “Reference by Abstraction”, Proceedings of the Aristotelian Society 112:1 (2012), 45-71
- “Identity and Discernibility in Philosophy and Logic” (with James Ladyman and Richard Pettigrew), Review of Symbolic Logic 5:1 (2012), 162-86
- “Metaontological Minimalism”, Philosophy Compass 7:2 (2012), 139-51
- “Some Criteria for Acceptable Abstraction”, Notre Dame Journal of Formal Logic 52:3 (2011), 331-338
- “Pluralities and Sets,” Journal of Philosophy 107:3 (2010), 144-164
- “The Context Principle in Frege’s Grundgesetze”, forthcoming in Philip Ebert and Marcus Rossberg (eds.), Essays on Frege's Basic Laws of Arithmetic (Oxford: Oxford University Press)
- “Truth in Mathematics,” forthcoming in M. Glanzberg (ed.), Oxford Companion to Truth (Oxford University Press)
- “Impredicativity in the Neo-Fregean Programme,” forthcoming in Philip Ebert and Marcus Rossberg (eds.), Abstractionism in Mathematics: Status Belli (Oxford: Oxford University Press)
- “Higher-Order Logic,” in L. Horsten and R. Pettigrew (eds.), Continuum Companion to Philosophical Logic (London: Continuum, 2011), 105-127
- “Predicativity and Impredicative Definitions”, in James Fieser and Bradley Dowden (eds.), Internet Encyclopedia of Philosophy (2010), URL=<http://www.iep.utm.edu/predicat/>
Jon Erling Litland
- "On Some Counterexamples to the Transitivity of Grounding," Essays in Philosophy, 14:1 (2013)
- “The Groundedness Approach to Class Theory”, forthcoming in Inquiry
- “Bernard Bolzano”, forthcoming in: A. Malpass (ed.) An Introduction to the History of Philosophical and Formal Logic: From Aristotle to Tarski, contracted with Continuum Press
- “Comparing Peano Arithmetic, Basic Law V, and Hume’s Principle”, Annals of Pure and Applied Logic, 163 (2012):1679-1709
- Invited submission for a survey article on set-theoretic reflection principles. Philosophia Mathematica.