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The Structure of Logical Consequence: Proof-Theoretic Conceptions View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by St Andrews Research Repository THE STRUCTURE OF LOGICAL CONSEQUENCE: PROOF- THEORETIC CONCEPTIONS Ole Thomassen Hjortland A Thesis Submitted for the Degree of PhD at the University of St. Andrews 2010 Full metadata for this item is available in the St Andrews Digital Research Repository at: https://research-repository.st-andrews.ac.uk/ Please use this identifier to cite or link to this item: http://hdl.handle.net/10023/892 This item is protected by original copyright This item is licensed under a Creative Commons License UNIVERSITY OF ST ANDREWS The Structure of Logical Consequence: Proof-Theoretic Conceptions by Ole Thomassen Hjortland A thesis submitted in partial fulfillment for the degree of Doctor of Philosophy in the School of Philosophical, Anthropological and Film Studies October 2009 Declaration of Authorship I, Ole Thomassen Hjortland, hereby certify that this thesis, which is approximately 70.000 words in length, has been written by me, that it is the record of work carried out by me, and that it has not been submitted in any previous application for a higher degree. I was admitted as a research student in September 2006 and as a candidate for the degree of Doctor of Philosophy in September 2006; the higher study for which this is a record was carried out in the University of St Andrews between September 2006 and September 2009. date signature of candidate I hereby certify that the candidate has fulfilled the conditions of the Resolution and Regulations appropriate for the degree of Doctor of Philosophy in the University of St Andrews and that the candidate is qualified to submit this thesis in application for that degree. date signature of supervisor In submitting this thesis to the University of St Andrews we understand that we are giving permission for it to be made available fo ruse in accordance with the regulations of the University Library for the time being in force, subject to any copyright vested in the work not being affected thereby. We also understand that the title and the abstract will be published, and that a copy of the work may be made and supplied to any bona fide library or research worker, that my thesis will be electronically accessible for personal or research use unless exempt by award of an embargo as requested below, and that the library has the right to migrate my thesis into new electronic forms as required to ensure continued access to the thesis. We have obtained any third-party copyright permissions that may be required in order to allow such access and migration, or have requested the appropriate embargo below. The following is an agreed request by candidate and supervisor regarding the electronic publication of this thesis: Access to Printed copy and electronic publication of thesis through the University of St Andrews. date signature of candidate date signature of supervisor iii So many and so various laws are giv’n; So many laws argue so many sins – Milton, Paradise Lost UNIVERSITY OF ST ANDREWS Abstract School of Philosophical, Anthropological and Film Studies Doctor of Philosophy by Ole Thomassen Hjortland The model-theoretic analysis of the concept of logical consequence has come under heavy criticism in the last couple of decades. The present work looks at an alter- native approach to logical consequence where the notion of inference takes center stage. Formally, the model-theoretic framework is exchanged for a proof-theoretic framework. It is argued that contrary to the traditional view, proof-theoretic se- mantics is not revisionary, and should rather be seen as a formal semantics that can supplement model-theory. Specifically, there are formal resources to provide a proof-theoretic semantics for both intuitionistic and classical logic. We develop a new perspective on proof-theoretic harmony for logical constants which incorporates elements from the substructural era of proof-theory. We show that there is a semantic lacuna in the traditional accounts of harmony. A new theory of how inference rules determine the semantic content of logical constants is developed. The theory weds proof-theoretic and model-theoretic semantics by showing how proof-theoretic rules can induce truth-conditional clauses in Boolean and many-valued settings. It is argued that such a new approach to how rules determine meaning will ultimately assist our understanding of the apriori nature of logic. Acknowledgements Philosophy is at its best when thought is extended into dialogue. The philosophical conversation denies an idea the suffocating intimacy of the author’s intellectual harbour, and launches it into the open sea of scorn and scrutiny. In Arché and St Andrews I learned that the hardships of Neurath’s boat are best faced in the company of friends and colleagues. In the years leading up to my PhD in St Andrews, I benefited from the encour- agement of a number of excellent teachers. To Thorleif Lyng I owe that first inspiration to pursue philosophy, and, above all, to perceive the critical nerve of the dissident as a virtue. Paul Johansen taught me about self-discipline and hu- mility the Kyokushin way—by example. Every day is a struggle to follow their advice. I served my first years in the philosophical trenches at the University of Bergen. Not lacking in enthusiasm, but short of direction, I found my calling in the solemn but authoritative logic lectures by Kåre Johnsen. Eivind Kolflaath drew the line between sound argument and African drumming in a place where straight lines are hard to come by. I am thankful to both of them for all their support, and for having left me with fond memories of the days in the department. Finally, thanks to Michaeł Walicki for pausing his logic lecture to point out that “it’s not semantics—it’s the real world”. Since arriving in St Andrews I have been fortunate enough to work in an outstand- ing research environment. I am especially grateful for having had the privilege of working with people who share my passion for logic and formal philosophy. My deepest thanks goes to my supervisors. In a fast-paced atmosphere, Stephen Read has been a perfect mentor with his patience, depth and intellectual integrity. Both his unfailing advice and his keen wit have made my supervision most rewarding and enjoyable. I feel like I have only now started the apprenticeship. Crispin Wright has built a research centre that is any graduate student’s dream. The intensity with which philosophy is practiced at Arché is unparalleled. It is a reflection of a philosopher whose modus operandi is as uncompromising as his work is profound. Stewart Shapiro, Arché regular and author of the ‘the Bible’, happily took the role as devil’s advocate and has challenged my views since our first meeting. Finally, Graham Priest’s supervision, both in St Andrews and in Melbourne during my ix research visit, is a constant reminder of unwavering intellectual courage. I have the deepest respect for all of them, and I am grateful for all of their support and stimulation in the past three years. Arché would not have been the same without all the fantastic members—both local and associates—who contribute to its unique atmosphere. Although the Archean family is too numerous to mention, I want to thank the students and research fel- lows I have spent my time with: Derek Ball, Yuri Cath, Josh Clarkson, Paul Dim- mock, Dylan Dodd, Philib Ebert, Thomas Hodgson, Torfinn Huvenes, Jonathan Ichikawa, Dirk Kindermann, Ira Kiourti, Julia Langkau, Federico Luzzi, Paul Mc- Callion, Darren McDonald, Dilip Ninan, Marcus Rossberg, Anders J. Schoubye, Daniele Sgaravatti, Rachel Sterken, Paula Sweeney, Chiara Tabet and Elia Zar- dini. Special thanks to the Scafia, my Nordic compatriots: Andreas Stokke, under the auspices of his intellect and charisma, the small town life has become bearable; and Guðmundur Andri Hjálmarsson and Ralf Bader, who could carry the world on their serious shoulders. And to Björn Brodowski and Patrick Greenough, who between them have the age and wisdom to keep me off the narrow path. I also want to thank members of and visitors to the FLC project for all the hours spent discussing my work in progress. Roy Dychoff has offered invaluable mathematical guidance for years. Michael De and Walter Pedriali both deserve special mention for combining knowledgeable advice with their respective brands of humour. Pål Fjeldvig Antonsen, Julien Murzi, and Catarina Dutilh Novaes bravely took on the thankless job of reading and commenting on early drafts of the thesis. Their eye for detail has greatly improved the final version, and I can only regretfully say that I haven’t been able to do justice to all of their insightful criticisms. In June to October of 2008 I spent a semester at the University of Melbourne, Australia. I want to thank Allen Hazen, Andreas Pietz, Graham Priest, Greg Restall, Dave Ripley, and everyone else for the excellent discussions we had during my visit. My warmest thanks goes to Ricki Bliss whose company made my time in Australia a memory for life. My research was funded by ORS and the Philosophy Department. I am very grateful to our former Head of School, Peter Clark, and the Department for facili- tating my work. I also need to thank my friends at Le Rendezvous in St Andrews, Mustafa "Aldo" Chebbah and Rachid Mosseddaq, for their food and companion- ship over the last three years. It’s worth breaking the rules for my friends and family. It’s worth breaking the rules with my friends and family. My parents, Trond Hjortland and Inger-Johanne Thomassen, enabled me to choose any life I wanted.
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