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The Proscriptive Principle and Logics of Analytic Implication City University of New York (CUNY) CUNY Academic Works All Dissertations, Theses, and Capstone Projects Dissertations, Theses, and Capstone Projects 2-2017 The Proscriptive Principle and Logics of Analytic Implication Thomas M. Ferguson The Graduate Center, City University of New York How does access to this work benefit ou?y Let us know! More information about this work at: https://academicworks.cuny.edu/gc_etds/1882 Discover additional works at: https://academicworks.cuny.edu This work is made publicly available by the City University of New York (CUNY). Contact: [email protected] The Proscriptive Principle and Logics of Analytic Implication by Thomas Macaulay Ferguson A dissertation submitted to the Graduate Faculty in Philosophy in partial fulfillment of the requirements for the degree of Doctor of Philosophy, The City University of New York 2017 ii © 2017 Thomas Macaulay Ferguson All Rights Reserved iii This manuscript has been read and accepted by the Graduate Faculty in Philosophy in satisfaction of the dissertation requirement for the degree of Doctor of Philosophy. Professor Sergei Artemov Date Chair of Examining Committee Professor Iakovos Vasiliou Date Executive Officer Professor Graham Priest Professor Heinrich Wansing Professor Kit Fine Supervisory Committee The City University of New York iv Abstract The Proscriptive Principle and Logics of Analytic Implication by Thomas Macaulay Ferguson Adviser: Professor Graham Priest The analogy between inference and mereological containment goes at least back to Aristotle, whose discussion in the Prior Analytics motivates the validity of the syllogism by way of talk of parts and wholes. On this picture, the application of syllogistic is merely the anal- ysis of concepts, a term that presupposes—through the root ἀνά + λύω —a mereological background. In the 1930s, such considerations led William T. Parry to attempt to codify this notion of logical containment in his system of analytic implication AI. Parry’s system AI was later expanded to the system PAI. The hallmark of Parry’s systems—and of what may be thought of as containment logics or Parry systems in general—is a strong relevance property called the ‘Proscriptive Principle’ (PP) described by Parry as the thesis that: No formula with analytic implication as main relation holds universally if it has a free variable occurring in the consequent but not the antecedent. This type of proscription is on its face justified, as the presence of a novel parameter in the consequent corresponds to the introduction of new subject matter. The plausibility of the thesis that the content of a statement is related to its subject matter thus appears also to support the validity of the formal principle. Primarily due to the perception that Parry’s formal systems were intended to accurately model Kant’s notion of an analytic judgment, Parry’s deductive systems—and the suitability v of the Proscriptive Principle in general—were met with severe criticism. While Anderson and Belnap argued that Parry’s criterion failed to account for a number of prima facie analytic judgments, others—such as Sylvan and Brady—argued that the utility of the criterion was impeded by its reliance on a ‘syntactical’ device. But these arguments are restricted to Parry’s work qua exegesis of Kant and fail to take into account the breadth of applications in which the Proscriptive Principle emerges. It is the goal of the present work to explore themes related to deductive systems satisfying one form of the Proscriptive Principle or other, with a special emphasis placed on the rehabilitation of their study to some degree. The structure of the dissertation is as follows: ∗ In Chapter 2 we identify and develop the relationship between Parry-type deductive sys- tems and the field of ‘logics of nonsense.’ Of particular importance is Dmitri Bochvar’s ‘internal’ nonsense logic Σ0, and we observe that two ⊢-Parry subsystems of Σ0—Harry Deutsch’s Sfde and Frederick Johnson’s RC—can be considered to be the products of partic- ular ‘strategies’ of eliminating problematic inferences from Bochvar’s system. ∗ The material of Chapter 3 considers Kit Fine’s program of state space semantics in the context of Parry logics. Fine—who had already provided the first intuitive semantics for Parry’s PAI—has offered a formal model of truthmaking (and falsemaking) that provides one of the first natural semantics for Richard B. Angell’s logic of analytic containment AC, itself a ⊢-Parry system. After discussing the relationship between state space semantics and nonsense, we observe that Fabrice Correia’s weaker framework—introduced as a semantics for a containment logic weaker than AC—tacitly endorses an implausible feature of allowing hypernonsensical statements. By modelling Correia’s containment logic within the stronger setting of Fine’s semantics, we are able to retain Correia’s intuitions about factual equivalence without such a commitment. As a further application, we observe that Fine’s setting can resolve some ambiguities in Greg Restall’s own truthmaker semantics. vi ∗ Chapter 4 we consider interpretations of disjunction that accord with the characteristic failure of Addition in which the evaluation of a disjunction A ∨˙ B requires not only the truth of one disjunct, but also that both disjuncts satisfy some further property. In the setting of computation, such an analysis requires the existence of some procedure tasked with ensuring the satisfaction of this property by both disjuncts. This observation leads to a computational analysis of the relationship between Parry logics and logics of nonsense in which the semantic category of ‘nonsense’ is associated with catastrophic faults in computer programs. In this spirit, we examine semantics for several ⊢-Parry logics in terms of the successful execution of certain types of programs and the consequences of extending this analysis to dynamic logic and constructive logic. ∗ Chapter 5 considers these faults in the particular case in which Nuel Belnap’s ‘artificial reasoner’ is unable to retrieve the value assigned to a variable. This leads not only to a natural ⋆ interpretation of Graham Priest’s semantics for the ⊢-Parry system Sfde but also a novel, many-valued semantics for Angell’s AC, completeness of which is proven by establishing a correspondence with Correia’s semantics for AC. These many-valued semantics have the additional benefit of allowing us to apply the material in Chapter 2 to the case of AC to define intensional extensions of AC in the spirit of Parry’s PAI. ∗ One particular instance of the type of disjunction central to Chapter 4 is Melvin Fitting’s cut-down disjunction. Chapter 6 examines cut-down operations in more detail and provides bilattice and trilattice semantics for the ⊢-Parry systems Sfde and AC in the style of Ofer Arieli and Arnon Avron’s logical bilattices. The elegant connection between these systems and logical multilattices supports the fundamentality and naturalness of these logics and, additionally, allows us to extend epistemic interpretation of bilattices in the tradition of artificial intelligence to these systems. ∗ Finally, the correspondence between the present many-valued semantics for AC and those vii of Correia is revisited in Chapter 7. The technique that plays an essential role in Chapter 4 is used to characterize a wide class of first-degree calculi intermediate between AC and classical logic in Correia’s setting. This correspondence allows the correction of an incorrect charac- terization of classical logic made by Correia and leads to the question of how to characterize hybrid systems extending Angell’s AC∗. Finally, we consider whether this correspondence aids in providing an interpretation to Correia’s first semantics for AC. Personal Acknowledgments The dissertation that follows would not have existed were it not for the support of a legion of friends, family, and colleagues. Above all, I am grateful for the patience and persistence of my wife Virginia and our chil- dren Mirabel and Rowan. They bore the trials that accompany the writing of a dissertation— the inevitable sleepless nights, long absences, and too-frequently empty bank accounts— alongside me year after year. Their encouragement has remained resolute even during the most difficult times and I would have long ago surrendered to my anxieties had it not been for their optimism and loyalty. I also acknowledge the essential contributions made to me by many other family members. Much of the work in this dissertation—and outside it—benefited from the constructive and detailed criticism of my father, Gillum Ferguson, who diligently read drafts of my work. As I wrote the dissertation, my grandfather Richard Ferguson and grandmother Marjorie Munn were sources of inspiration and it is my hope that I have made them proud. The support of my mother, Marsha Smith, and her husband Bill has also been critical, as has been the support of my stepmother Susan Ferguson and my parents-in-law Richard and Carol Kropp. Whether lending us their roofs between moves or taking care of the children while I attended conferences, any successes I may have achieved were built on the foundation of their collective support. Being able to study logic in New York—home to so many outstanding logicians—has viii ix been a remarkable privilege and I owe a significant debt to every member of the New York logic community. In particular, I cannot overstate the importance of the opportunity to learn from and interact with the members of my committee—Professors Graham Priest, Heinrich Wansing, and Kit Fine—whose encouragement, advice, and confidence has been indispensable during the past few years. Furthermore, I remain grateful to Professors Sergei Artemov and Melvin Fitting for agreeing to serve as examiners during my defense and for the thoughtful remarks they shared with me. Beyond the input of the foregoing individuals, the contents of the dissertation have been sharpened and refined by the many pages of comments and criticism I have received from anonymous referees; that they are anonymous does not lessen my gratitude for their help.
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