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The Epistemology of Natural Deduction Seyed Hassan Masoud

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The Epistemology of Natural Deduction Seyed Hassan Masoud The Epistemology of Natural Deduction by Seyed Hassan Masoud A thesis submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy Department of Philosophy University of Alberta © Seyed Hassan Masoud, 2015 Abstract Natural deduction and the axiom method, as two derivation methods of formal deductive logic, are extensionally equivalent in the sense that they prove the same theorems and they derive the same premise-conclusion arguments: one has a derivation of a given conclusion from a given premise set if and only if the other does. They differ significantly, however, from an epistemo- logical point of view. Being based on the notion of hypothetical reasoning, providing the most logically detailed steps within a derivation, and being tied closely to the conception of logic as attempting to model mathematical reasoning are some features that give natural deduction systems an epistemic flavor the axiom-based systems, to a large extent, lack. This work investigates these features through a comparison of the two sorts of deduction systems. Analyzing the historical background of the advent and flourishing of the two methods, looking at the real examples of mathematical reasoning, and taking into account the way a rational agent carries out a logical derivation, give historical, experiential, and phenomenological dimensions to the work. All these features turn around the epistemic insight that deductive reasoning is a way of coming to know whether and how a given conclusion logically follows from a given set of premises. The general results include finding the main characteristic feature of natural deduction as having a method of making a hypothesis and deriving some conclusions from it in combination with previous premises if any. Also the notion of epistemic rigor is defined which proves to be useful in investigating the significant advantages of natural deduction. Epistemic rigor can be measured by looking at the simplicity and purity of the rules of inference. Another distinctive feature of natural deduction is the two-dimensionality of its derivations which is due to the fact ii that the formulas that occur within the scope of an assumption are not at the same level as other formulas outside the scope of that assumption. Furthermore, it is shown that, to a large extent, the best way of formalizing mathematical proofs is using natural deduction. The whole work is done within the framework of an epistemic approach to logic which characterizes logic as the science of reasoning. iii I dedicate this work to: My wife, Zahra and My son, Soroush iv Acknowledgments What I have learned―in philosophy in general and logic in particular―as well as the research I have conducted which yielded this dissertation would be impossible, if not for the intellectual and financial support by the Department of Philosophy at the University of Alberta. For more than five years, I felt at home in the department, surrounded by unexpectedly warm- hearted and amiable people―professors, students, and staff. If it was possible, I would have liked to mention all the members of the department one by one. First of all, I would like to express my deepest gratitude towards Professor Bernard Linsky, who was my interim advisor from the beginning and then my thesis supervisor. He is known as a great person to be around, and I have been lucky enough to experience innumerable moments when he surprised me by his heartening words and insightful suggestions. Having read this dissertation several times, he spent an ample amount of time going through the chapters and giving me invaluable comments. Professor Linsky has been an inspiration for me throughout the course of this long journey. Also, I owe many thanks to Professor Jeffry Pelletier, a great teacher from whom I learnt a lot in several courses and in our influential discussions. Professor Pelletier shared his wisdom with me through his many precious comments and meticulous ideas for this dissertation. Next, I have to thank Professor Bruce Hunter, another member of the committee, who made me more sensitive about epistemological aspects of logic, and gave me great ideas about my research. I feel fortunate that during my time here at the University of Alberta, Professor Allen Hazen moved to the Department of Philosophy and ran the weekly seminar of the Logic Group in which I could meet logic-oriented professors and students on a regular basis. I greatly benefited from participating in the thriving discussions of various topics in v mathematical and philosophical logic enriched by his vast knowledge. In addition to the Logic Group, Professor Hazen had a substantial impact on my research with his comments. I’m honored that I worked under the supervision of this committee of professors, who I would like to express my profound acknowledgement towards. Without them this dissertation would not be written. During my PhD studies, I received a great amount of financial support in the forms of TA, RA, and teaching as the main instructor. I sincerely appreciate the significance of the efforts the administrative staff of the department put into bestowing this support. I would like to mention Professor Bruce Hunter, the former chair, Professor Jack Zupko, the present chair of the department, and especially Professor Amy Schmitter, the graduate coordinator. Their hard work and positive attitude provided a worry-free life for me to spend my time on my study and research. Among the faculty members of the department who were not on my committee, Professor Philip Corkum showed the most interest in my work, which led to some effective discussions. From outside the department, Professor John Corcoran, of the University at Buffalo, provided me with many comments and suggestions. I have never met him in person, but we have had flourishing discussions through email since 2007. I am certainly indebted to both of them. And last but not least, I really appreciate the constant and unconditional support from my family: my wife, Zahra, and my son, Soroush. They have always put my work prior to their own interests and given up so many vacations, picnics, parties, and even family times to help me carry out my study and research. I would like to express thanks towards my parents and my brothers from Iran, who have encouraged me to follow my aspiration and have always inspired me through their nice words. I have to thank them all. vi Table of contents Chapter 1. Introduction ................................................................................................................ 1 1.1. The thesis statement .................................................................................................. 1 1.2. An outline of the dissertation .................................................................................... 2 1.3. The scope of the study .............................................................................................. 4 Chapter 2. Characterization of natural deduction ...................................................................... 12 2.1. Background: axiomatic systems ............................................................................. 12 2.2. The history of natural deduction ............................................................................. 22 2.3. Some remarks .......................................................................................................... 32 2.4. Logicians’ reluctance to embrace natural deduction .............................................. 43 2.5. Various characterizations of natural deduction ....................................................... 45 Chapter 3. Epistemic rigor vs. formal rigor ............................................................................... 48 3.1. Euclid and the axiomatic method ............................................................................ 48 3.2. Frege and the ideal of a gapless deduction ............................................................. 51 3.3. Formal rigor ............................................................................................................ 53 3.4. The cogency of a deduction .................................................................................... 56 3.5. Epistemic rigor ........................................................................................................ 63 Chapter 4. Representation of natural deduction derivations ...................................................... 70 4.1. Structured derivations vs. linear derivations ........................................................... 70 4.2. Two-dimensionality of natural deduction ............................................................... 76 4.3. Different forms of representing natural deduction derivations ............................... 81 4.4. The epistemological significance of graphical representations .............................. 91 vii Chapter 5. Real mathematical reasoning ................................................................................. 100 5.1. Introduction ........................................................................................................... 100 5.2. Examples of mathematical proofs ......................................................................... 101 5.3. A proposal to categorize the components of deductive reasoning ........................ 122 Chapter 6. Natural deduction and the epistemological approach to logic ............................... 125 6.1. Three classes of logical systems ..........................................................................
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