University of Connecticut OpenCommons@UConn Doctoral Dissertations University of Connecticut Graduate School 11-28-2018 New Directions in Justification Logic Joseph Lurie University of Connecticut - Storrs, [email protected] Follow this and additional works at: https://opencommons.uconn.edu/dissertations Recommended Citation Lurie, Joseph, "New Directions in Justification Logic" (2018). Doctoral Dissertations. 2018. https://opencommons.uconn.edu/dissertations/2018 New Directions in Justification Logic Joseph Andrew Lurie, Ph.D. University of Connecticut, 2018 ABSTRACT Justification logics are constructive analogues of modal logics. As such, they pro- vide perspicuous models of those modalities that have inherently constructive char- acter, such as intuitionistic mathematical provability or the knowledge operator of evidentialist epistemology. In this dissertation, I examine a variety of positions in epistemology, along with their associated ontological commitments, and develop var- ious classical and non-classical justification logics that are suitable for use as models of these positions. New Directions in Justification Logic Joseph Andrew Lurie B.S., University of Pittsburgh, 2008 A Dissertation Submitted in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy at the University of Connecticut 2018 Copyright by Joseph Andrew Lurie 2018 APPROVAL PAGE Doctor of Philosophy Dissertation New Directions in Justification Logic Presented by Joseph Andrew Lurie, B.S. Major Advisor Jc Beall Associate Advisor Marcus Rossberg Associate Advisor D. Reed Solomon University of Connecticut 2018 ii ACKNOWLEDGMENTS No project as involved as a doctoral dissertation can be completed without the assistance of many parties. First, I must thank my advisors, Jc Beall, Marcus Ross- berg, and Reed Solomon, who have provided helpful and encouraging feedback at every stage of this project. Additional thanks are due to all of the faculty and of my fellow graduate students at UConn, for everything that I have learned from them in various seminars and colloquia, and for helpful questions and suggestions given on those occasions when I have presented drafts of portions of this work. Thanks are also due to certain individuals outside of UConn: to Bob Milnikel, whose logic group talk in 2009 began my involvement with justification logics; to Che-Ping Su, for sharing with me his research into paraconsistent justification logics, and for some helpful criticism of my own work in that area; and to Greg Restall, who provided some helpful suggestions toward the writing of a section that unfortunately had to be omitted from the final draft of this dissertation. Additional thanks are due to the UConn Logic Group, for bringing these and many other excellent logicians to Storrs, and so greatly enriching the work of the entire community. A version of Chapter 6 was previously published in the e-journal Philosophies [70]. Thanks are due to the editors of that journal, for agreeing to publish my work, and to the anonymous reviewers that the journal solicited, whose critiques impelled me to make enormous improvements in the quality of the material. Note that permission to reprint content in this dissertation is not required, as the Philosophies publication iii iv is open-access under the terms of the CC-BY license, version 4.0 [29]. Any errors that may remain in this dissertation, as well as the fatal flaws that lead to the removal of the section to which Greg contributed, are entirely the fault of the author, and not due to any of the contributors acknowledged above. Finally, the greatest thanks of all are due to my parents, Richard and Mary Lurie, without whose financial and motivational support this project could not possibly have been completed. Contents Ch. 1. History and Formal Presentation of Artemov's JT4 and Other Normal Justification Logics 1 1.1 Logic of Intuitionistic Provability. 2 1.2 Justification Logic . 9 1.2.1 Axiomatic Systems . 9 1.2.2 On the Use of Justification Logic in Intuitionism . 16 1.3 Sequent Calculus for JT4 . 19 1.4 Realization of S4 in JT4 . 20 1.5 Model Theory for Normal Justification Logics . 22 1.6 Motivation as Epistemic Logic . 30 Ch. 2. Non-Normal Justification Logic 42 2.1 The Justification System JS0.5 . 44 2.1.1 Axiomatic Development . 44 2.1.2 The Realization Theorem . 46 2.1.3 Model Theory for JS0.5 . 54 2.1.4 Epistemic Application of JS0.5 . 60 2.2 Justification Analogues for Other Non-Normal Modal Logics . 60 Ch. 3. Paraconsistent Justification Logic 64 3.1 Motivation . 64 3.2 The Paraconsistent System LP . 70 3.3 Simple Axiomatizable Paraconsistent Systems . 73 3.4 Sequent Calculi and Realization for the Axiomatizable Paraconsistent Justification Logics . 80 3.5 Other Axiomatizable Paraconsistent Systems . 92 v Ch. 4. Non-Applicative Justification Logic 94 4.1 Introduction . 94 4.2 Che-Ping Su's Non-Applicative System PJF . 96 4.3 LP-Based Non-Applicative Systems . 99 4.3.1 The System pLPJT4 . 101 4.3.2 The System tLPJT4 . 109 4.3.3 Philosophical Applications of the LP-based Systems . 117 Ch. 5. Paracomplete Justification Logic 120 5.1 Introduction . 120 5.2 Paracomplete Base Systems K3 andL 3, With Their Modal Extensions 122 5.3 Justification Extensions ofL 3 . 127 5.4 Justification Extensions of K3 . 133 5.5 Justification Extensions of FDE . 143 Ch. 6. Probabilistic Justification Logic 145 6.1 Introduction . 145 6.2 Probability Theory and Fuzzy Logic . 147 6.3 Previous Justification Logic Approaches to the Vagueness of Epistemic Justification . 148 6.3.1 Milnikel's Logic of Uncertain Justifications . 148 6.3.2 Kokkinis' Probabilistic Justification Logic . 151 6.3.3 Ghari's H´ajek-Pavelka-Style Justification Logics . 152 6.4 Probabilistic Justification Logic . 154 Appendix A: Axiomatic Systems for Classical and Paraconsistent Logics 167 A.1 Introduction . 167 A.2 Classical Logic . 168 A.3 Paraconsistent Logics . 175 Appendix B: Sequent Calculus for Classical and Intuitionist Logics 181 B.1 Introduction . 181 B.2 Formal Characterization of the Sequent Calculus . 182 B.3 Cut Elimination . 188 Bibliography 196 vi Chapter 1 History and Formal Presentation of Artemov's JT4 and Other Normal Justification Logics Convention 1.0.1 (Originality of Results). In this chapter and in the appendices, all of the formal results that are presented were previously published by some other author. In all other chapters, any result which is formally presented as a numbered theorem is original. Results that were previously published by some other author will either be stated in the text without an assigned theorem number, or else will be incorporated into a (possibly numbered) formal definition. 1 2 1.1 Logic of Intuitionistic Provability. Intuitionism is a philosophy of mathematics founded by L. E. J. Brouwer in the 1920s.1 Its metaphysical basis is the claim that mathematical objects have no independent existence, but rather are constructed (in a neo-Kantian sense) by mathematicians through proofs. In particular, mathematical existence claims are true only if there is a (possibly indeterministic)2 algorithm which generates an object with the desired property. Because mathematical entities have no existence independent of mathemat- ical proof, no proposition is true or false of them unless it has been so proven. This metaphysical position invalidates many of the classical rules of inference in mathe- matical contexts, and has led to a new branch of mathematical inquiry that operates without such inferences. The pure logic required for work in intuitionistic mathematics, known as intuition- istic logic, was formalized by Brouwer's student, Arend Heyting. Proof theoretically, intuitionistic logic is just classical logic with one of the axioms or rules governing negation removed.3 As there are countless proof systems equivalent to classical logic, the missing axiom or rule will vary. The most common choices are double negation elimination, the excluded middle, and positive conclusion reductio ad absurdum| all of which are equivalent to each other, given suitable axioms and rules governing disjunction and conditionals.4 1The following interpretation of Brouwer's ideas is essentially my own; it is supported by evidence in sources such as the first chapter of [25]. 2The possibility of indeterministic constructions, or free choice sequences, is required to recon- cile the facts that Cantor's diagonal argument proceeds in an intuitionistically acceptable manner and that there is only a countable infinity of deterministic algorithms for constructing numerical quantities. 3This is a slight oversimplification; many of the common axiomatizations of classical logic are unsuitable for this procedure, as in intuitionistic logic, separate axioms or rules are needed for conjunction, disjunction, and the conditional; they are not interdefinable in the usual manner. 4Obviously, given a base system which is equivalent to intutitionistic logic, all possibilities for the 3 However, in mathematics, there are many other concepts to be formalized in addi- tion to those which are contained in a logical system proper. One important example is the concept of provability. Usually this is defined metalinguistically, but in math- ematics it is sometimes needed as an object-language notion. There are two major strategies for accomplishing this, both developed by Kurt G¨odel. One strategy is to map the sentences of the language onto objects in the language's first-order do- main using a metalinguistic function known as a “G¨odelcoding," and then to find the first-order predicate which is true of exactly those objects with which the G¨odel coding represents provable sentences. This technique leads to some powerful results (notably, G¨odel'sincompleteness theorems), but also imposes significant limitations. Obviously, it can only be used in a language containing at least first-order predi- cation. Moreover, aside from degenerate cases, the requirement that sentences be mapped onto objects forces the first-order domain to be infinite, and in order to be sure of the existence of a provability predicate, the language must allow complete first-order comprehension.