Minimal and Subminimal Logic of Negation MSc Thesis (Afstudeerscriptie) written by Almudena Colacito (born November 29th, 1992 in Fabriano, Italy) under the supervision of Dr. Marta Bílková, Prof. Dr. Dick de Jongh, and submitted to the Board of Examiners in partial fulfillment of the requirements for the degree of MSc in Logic at the Universiteit van Amsterdam. Date of the public defense: Members of the Thesis Committee: August 26th, 2016 Dr. Floris Roelofsen (Chair) Dr. Benno van den Berg Dr. Marta Bílková Drs. Julia Ilin Prof. Dr. Dick de Jongh Prof. Dr. Yde Venema Abstract Starting from the original formulation of minimal propositional logic proposed by Johansson, this thesis aims to investigate some of its relevant subsystems. The main focus is on negation, defined as a primitive unary operator in the language. Each of the subsystems considered is defined by means of some ‘axioms of nega- tion’: different axioms enrich the negation operator with different properties. The basic logic is the one in which the negation operator has no properties at all, ex- cept the property of being functional. A Kripke semantics is developed for these subsystems, and the clause for negation is completely determined by a function between upward closed sets. Soundness and completeness with respect to this se- mantics are proved, both for Hilbert-style proof systems and for defined sequent calculus systems. The latter are cut-free complete proof systems and are used to prove some standard results for the logics considered (e.g., disjunction property, Craig’s interpolation theorem). An algebraic semantics for the considered sys- tems is presented, starting from the notion of Heyting algebras without a bottom element. An algebraic completeness result is proved. By defining a notion of de- scriptive frame and developing a duality theory, the algebraic completeness result is transferred into a frame-based completeness result which has a more generalized form than the one with respect to Kripke semantics. Contents 1 Introduction 1 1.1 Minimal Propositional Logic . .1 1.1.1 Equivalence of the two formulations of MPC .........7 1.2 Weak Negation . .9 1.2.1 Absorption of Negation: a Further Analysis . 14 1.3 Historical Notes . 15 2 Subminimal Systems 17 2.1 A Basic Logic of a Unary Operator . 17 2.1.1 Kripke-style Semantics . 17 2.1.2 Soundness and Completeness Theorems . 22 2.2 Intermediate Systems between N and MPC .............. 25 2.2.1 Negative ex Falso Logic . 25 2.2.2 Contraposition Logic . 29 3 Finite Models and the Disjunction Property 35 3.1 Finite Model Property . 35 3.1.1 Decidability via Finite Models . 39 3.2 Disjunction Property . 39 3.2.1 Slash Relation . 42 3.3 Filtration Method . 44 4 Algebraic Semantics 47 4.1 Generalized Heyting Algebras . 47 4.1.1 Compatible Functions . 48 4.2 Algebraic Completeness . 49 4.2.1 The Lindenbaum-Tarski Construction . 51 4.3 Descriptive Frames . 53 4.3.1 From Frames to Algebras . 56 4.3.2 From Algebras to Frames . 57 4.4 Duality . 58 4.5 Completeness . 66 5 Sequent Calculi 71 5.1 The G1-systems . 71 5.2 Absorbing the Structural Rules . 74 5.3 The Cut Rule . 86 5.4 Equivalence of G-systems and Hilbert systems . 86 6 Cut Elimination and Applications 91 6.1 Closure under Cut . 91 6.2 Some Applications . 103 6.2.1 Decidability via Sequent Calculi . 106 6.2.2 Craig’s Interpolation Theorem . 107 6.3 Translating MPC into CoPC ...................... 115 7 Conclusions 123 Acknowledgements I would like to start by expressing my gratitude to Dick de Jongh. For the last two years, he has been a guiding light for my career as a MoL student as well as for my life at the ILLC. He has been an incredibly caring academic mentor, and he has believed in me and in the work we were carrying out together. He has always given me an enormous amount of advice, by making himself available for discussions and clarifications. He has motivated me with ever-changing questions and he made me capable of such a work I can be proud of. Although we met for the first time less than a year ago, Marta Bílková has surely been a fundamental presence for the production of this thesis. In the moment I met her, I found myself in the presence of an amazing and inspiring logician. Her guidance, her competence and her love for the subject have been motivating me throughout these months. I would also like to thank the members of my Master’s thesis committee, Benno van den Berg, Julia Ilin, Yde Venema and Floris Roelofsen, for having made the defense a useful moment of debate and discussion with their stimulating questions and comments. I am grateful to Nick Bezhanishvili, for having introduced me to the world(s) of modal logic, and for being an enthusiastic and passionate professor. I also wish to express my gratitude to the ToLo V community. I want to thank each and every researcher who was in Tbilisi, for sharing with me a huge amount of advice which helped me to improve the thesis as well as my general attitude as a researcher. I want to acknowledge Stefano Baratella’s contribution too – it’s thanks to his course and under his supervision that I have discovered logic, fallen in love with the subject and eventually made it the scope of my life. A huge amount of my gratitude goes to Valeria Pierdominici, who has been the mirror I needed to remind myself what I was looking for. A fundamental role was covered by all those people who have lived these two years from far away by means of thousands of texts and Skype calls. They have been my safe place, always ready to celebrate with me and dry my tears when needing to. In particular, I want to thank Davide, because if it is difficult to support someone day by day, it is even more difficult doing so when you live 1000 km apart. I am grateful to the people who made all of this possible: my family. First, for teaching me to always fight for my dreams; second, for being so supportive throughout these years. My gratitude goes to my mother, for being my role model: I have never had to look too far to find a woman who could inspire me. An equally special thanks goes to my father, for always reminding me that rationality and logic become fundamental only when combined with passion and love. Last but not least, I am grateful to my sister Alice: the way she has always believed in me has represented my major source of strength and courage. If I think of a place where to study logic, I cannot imagine a better place than the ILLC. I am extremely grateful to every person who is, in a way or another, part of this huge family. It is mainly for such lively and enthusiastic environment that my MoL experience has been so happy and motivating. For similar reasons, I want to thank my Master of Logic colleagues, all the people with whom I shared all-night crazy assignments as well as long and deep discussions about improbable proof steps. Thanks to Albert, Andrés, Anna B., Anna F., Frederik and Hugo. I am grateful to Arianna, because – I swear – I lived. To Thomas, for all the proof checking and for our several discussions, as well as for our countless laughs in Tbilisi. A special thanks goes to Marco, who has represented my daily MoL life, while being a friend, a flatmate, a chef, a colleague, and much more than this. Last but surely not least, I owe a profound thanks to Lorenzo, for always reminding me how good I am in playing foosball. Overview This thesis is concerned with subminimal logics, with particular focus on the negation operator. With subminimal logics we want to denote subsystems of minimal propositional logic obtained by weakening the negation operator. We give here a quick overview of the thesis, presenting its structure as well as its contents. Chapter 1. The first chapter introduces our conceptual starting point: minimal propositional logic. After an introduction of the syntax, and a brief presentation of an associated Kripke semantics, the relation between the two versions of minimal logic is made formal. In particular, we give a proof of the fact that the two considered definitions of minimal logic are indeed equivalent. Later, the main axioms of negation are introduced, with some motivations. Their formal relation with the minimal system is studied. The chapter is concluded with some historical notes concerning minimal logic, as well as the study of minimal logic with focus on the negation operator. Chapter 2. We start introducing the core of the thesis. We present a Kripke semantics for each of the three main subminimal systems, and we introduce some of the relevant notions (p-morphism, generated subframe, disjoint union). Com- pleteness proofs are carried out via canonical models. The analysis goes here ‘bottom-up’: we start from the basic system of an arbitrary unary negation op- erator with no special properties, and we go on by adding axioms (and hence, properties) for the negation operator. In defining a Kripke semantics for those systems, the semantics of negation is given by a function on upward closed sub- sets of a partially ordered set. This reflects the fact that the negation is basically seen as a unary functional operator. The reader may notice that the axioms of negation happen to be equivalent to properties of the considered function.