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Hypersequent Calculi for Modal Logics

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Hypersequent Calculi for Modal Logics University of Calgary PRISM: University of Calgary's Digital Repository Graduate Studies The Vault: Electronic Theses and Dissertations 2018-04-23 Hypersequent Calculi for Modal Logics Burns, Samara Elizabeth Burns, S. A. (2018). Hypersequent Calculi for Modal Logics (Unpublished master's thesis). University of Calgary, Calgary, AB. doi:10.11575/PRISM/31825 http://hdl.handle.net/1880/106539 master thesis University of Calgary graduate students retain copyright ownership and moral rights for their thesis. You may use this material in any way that is permitted by the Copyright Act or through licensing that has been assigned to the document. For uses that are not allowable under copyright legislation or licensing, you are required to seek permission. Downloaded from PRISM: https://prism.ucalgary.ca UNIVERSITY OF CALGARY Hypersequent Calculi for Modal Logics by Samara Elizabeth Burns A THESIS SUBMITTED TO THE FACULTY OF GRADUATE STUDIES IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF ARTS GRADUATE PROGRAM IN PHILOSOPHY CALGARY, ALBERTA April, 2018 © Samara Elizabeth Burns 2018 Abstract This thesis surveys and examines hypersequent approaches to the proof theory of modal logics. Traditional sequent calculi for modal logics often fail to have many of the desirable properties that we expect of a sequent calculus. Cut cannot be eliminated from the system for S5, the axioms of each logic are not straightforwardly related to the sequent rules, and vari- ation between modal sequent calculi occurs in the presence and absence of logical rules, rather than structural rules, which violates Došen’s princi- ple. The hypersequent framework is beneficial as we can provide Cut-free complete treatments of many modal logics. However, hypersequent ap- proaches often lack generality, or do not conform to Došen’s principle. A recent development in the proof theory of modal logics, called relational hypersequents, appears to overcome many of these issues. Relational hy- persequents provide a unified proof theory for many modal logics, where the logical rules are held constant between modal systems. This thesis pro- vides some preliminary results for relational hypersequents by providing a Cut-free completeness proof for the modal logic K. ii Acknowledgements I would like to thank everyone in the Department of Philosophy at the Uni- versity of Calgary for their support and kindness throughout my time in the department. In particular, I would like to thank my thesis supervisor Richard Zach. Your endless encouragement, pep talks and wealth of knowl- edge have been invaluable to me. This thesis would not have been possible without you. I would also like to thank my partner, Mike, who has been the greatest support for me throughout the writing process. Thank you for believing in me, and for everything that you do. Finally, thank you to my parents, who have always encouraged me to do my best. iii Table of Contents Abstract ............................................. ii Acknowledgements .................................... iii Table of Contents ...................................... iv List of Tables .......................................... vi 1 Introduction ....................................... 1 2 Gentzen Systems for Modal Logics ....................... 7 2.1 Syntax and Semantics ............................. 7 2.2 The Sequent Calculus ............................. 8 2.3 Natural Deduction ............................... 16 2.4 Modal Sequent Calculi ............................ 19 2.5 Modal Natural Deduction .......................... 24 2.6 Conclusion .................................... 27 3 Hypersequent Systems ............................... 28 3.1 Introduction ................................... 28 3.2 Hypersequent Calculi ............................. 29 3.3 Hypersequents and Modal Logic ..................... 30 3.4 The Development of Hypersequent Calculi . 32 3.5 Hypersequent Rules from Frame Properties . 39 3.6 Related Sequent Systems .......................... 44 3.7 Conclusion .................................... 53 4 Relational Hypersequents ............................. 55 4.1 Introduction ................................... 55 iv Table of Contents 4.2 Relational Hypersequents .......................... 56 4.3 Soundness ..................................... 61 4.4 Completeness for RK ............................. 70 5 Conclusion ........................................ 85 Bibliography ......................................... 88 v List of Tables 2.1 Sequent Rules for the System LK ....................... 11 2.2 Natural Deduction Rules for the System NK . 17 2.3 Modal Sequent Calculus Rules ......................... 20 2.4 Modal Natural Deduction Rules ........................ 25 3.1 Hypersequent Rules for the System HLK . 31 3.2 Mints’ Sequent Rules ................................ 35 3.3 Simple Frame Properties and their Normal Descriptions . 40 3.4 Tree Hypersequent Rules for the Base Calculus THSK . 46 3.5 Modal Tree Hypersequent Rules ........................ 46 3.6 Structural Tree Hypersequent Rules ..................... 46 3.7 Modal 2-Sequent Calculus Rules ....................... 49 3.8 Linear Nested Sequent Rules for the Base Calculus LNSK . 50 3.9 Modal Linear Nested Sequent Rules ..................... 51 3.10 Basic Rules for the Labelled System G3K . 51 3.11 Modal Labelled Sequent Rules ......................... 52 4.1 Relational Hypersequent Rules for the system RK . 57 4.2 External Structural Rules for Relational Hypersequents . 58 5.1 Modal Proof Systems and their Important Properties . 86 vi Chapter 1 Introduction Gerhard Gentzen introduced the sequent calculus and natural deduction systems in his “Untersuchungen über das logische Schließen I-II" [Investi- gations into Logical Deduction I-II] (1935a/1935b). Gentzen’s natural deduc- tion systems NJ and NK (for intuitionistic logic and classical logic, respec- tively) were the centerpiece of his investigation. He developed these calculi as a way of formally mimicking the structure of mathematicians’ “natural” reasoning. Gentzen’s natural deduction was a departure from the axiomatic deduction systems that were used at the time. Axiomatic proofs proceed top-down from a set of axioms with a limited set of rules (often just modus ponens), whereas natural deduction allows one to make and discharge as- sumptions within a formal derivation. In addition, there are two rules that govern each logical operator, and no axioms. Since mathematicians do not usually reason from axioms alone, natural deduction appears to be a better analogue to actual mathematical reasoning. At the same time as Gentzen’s calculi NJ and NK were developed, Stanisław Jaskowski´ presented similar ideas in his paper “On the Rules of Suppositions in Formal Logic” (1934), 1 1. Introduction which was inspired by lectures from Jan Lukasiewicz (Szabo, 1969, 4). A history of natural deduction can be found in Pelletier and Hazen (2012). The sequent calculus systems LJ and LK do not have the same natural structure as Gentzen’s natural deduction systems. Gentzen developed the sequent calculus as a meta-calculus for his natural deduction systems. At the time, Gentzen theorized that every natural deduction derivation could be converted into a determinate normal form. He thought that no formula need enter a derivation that is not a component of the end-formula. In other words, every natural deduction derivation has a presentation that is “not roundabout" (Gentzen, 1935a, 289). This technical result would come to be known as the normalization theorem. This theorem was not proved directly for classical logic until much later. Gentzen did prove normalization directly for intuitionistic logic, but did not publish the result (von Plato, 2008). Instead, Gentzen proved an analogous result in the sequent calculus. In the context of the sequent calculus, Gentzen was able to prove the Hauptsatz (or main theorem), also known as the cut elimination theorem. The cut rule is a unique structural rule of the sequent calculus, as it is the only rule where a formula can be “removed” from a derivation. Γ ∆,'',⇤ ⇥ Cut ) Γ ,⇤ ∆,⇥ ) ) Gentzen showed that any sequent calculus derivation can be algorith- mically converted into a derivation with the same end-formula but without using the Cut rule. Due to the correspondence between natural deduc- tion and the sequent calculus, the Cut elimination theorem for the sequent calculus implies that every proof in the corresponding natural deduction 2 1. Introduction system has a normal form. The Cut elimination theorem also has important philosophical and technical consequences. Any sequent calculus where Cut can be eliminated, and where the formulas in the premise of all other rules are subformulas of the formulas in the conclusion, is analytic: derivations can be constructed from the bottom-up simply by breaking formulas down into their subformulas. In some cases, the Cut elimination theorem can also be used to prove consistency and decidability. In computer science, the Cut elimination theorem theorem makes derivation systems amenable to proof search. Perhaps another important feature of Gentzen’s calculi, and one that is especially pertinent given the variety of logics we encounter in a contem- porary context, is its usefulness and applicability. Avron (1996) uses the sequent calculus as a paradigm example of a successful logical framework. It can handle a great many logics, and in fact, it is almost expected that a logic will have a well-behaved sequent calculus. It says something
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