Informatik 8, FAU Erlangen-Nürnberg [email protected] Subverting the Classical Picture: (Co)algebra, Constructivity and Modalities∗ Tadeusz Litak Contents 1. Subverting the Classical Picture 2 2. Beyond Frege and Russell: Modalities Instead of Predicates 6 3. Beyond Peirce, Tarski and Kripke: Coalgebras and CPL 11 3.1. Introduction to the Coalgebraic Framework ............. 11 3.2. Coalgebraic Predicate Logic ...................... 14 4. Beyond Cantor: Effectivity, Computability and Determinacy 16 4.1. Finite vs. Classical Model Theory: The Case of CPL ........ 17 4.2. Effectivity, Choice and Determinacy in Formal Economics ..... 19 4.3. Quasiconstructivity and Modal Logic ................. 21 5. Beyond Scott, Montague and Moss: Algebras and Possibilities 23 5.1. Dual Algebras, Neighbourhood Frames and CA-baes ........ 23 5.2. A Hierachy of (In)completeness Notions for Normal Modal Logics . 24 5.3. Possibility Semantics and Complete Additivity ........... 26 5.4. GQM: One Modal Logic to Rule Them All? ............. 28 6. Beyond Boole and Lewis: Intuitionistic Modal Logic 30 6.1. Curry-Howard Correspondence, Constructive Logic and 2 ..... 31 6.2. Guarded Recursion and Its Neighbours ................ 33 6.3. KM, mHC, LC and the Topos of Trees ................ 34 6.4. Between Boole and Heyting: Negative Translations ......... 35 7. Unifying Lewis and Brouwer 37 7.1. Constructive Strict Implication .................... 37 7.2. Relevance, Resources and (G)BI .................... 40 ∗Introductory overview chapter for my cumulative habilitation thesis as required by FAU Erlangen-Nürnberg regulations (submitted in 2018, finally approved in May 2019). Available online at https://www8.cs.fau. de/ext/litak/gitpipe/tadeusz_habilitation.pdf. 1 8. Beyond Brouwer, Heyting and Kolmogorov: Nondistributivity 43 8.1. Relational Lattices ........................... 43 8.2. Functorially Guarded Traces ...................... 46 9. References 47 9.1. Own Papers ............................... 47 9.2. Other References ............................ 49 A. Examples of Coalgebraic Structures 58 A.1. Some Set-endofunctors ......................... 58 A.2. Predicate Liftings ............................ 58 A.3. Rank-1 Axiomatizations ........................ 59 B. Sketch of a Proof System for Guarded Recursion 60 C. Kripke Semantics for 2 and J 63 D. Quine, Tarski and the Classical Picture 66 D.1. Afterword: Is It Possible at All to Justify The Boolean Laws? ... 69 1. Subverting the Classical Picture [L]ogic has permeated through computer science during the past thirty years much more than it has through mathematics during the past one hundred years. Indeed, at present concepts and methods of logic occupy a central place in computer science, insomuch that logic has been called “the calculus of computer science” [190] … [L]ogic provides computer science with both a unifying foundational framework and a powerful tool for modeling and reasoning about aspects of computation. Joseph Y. Halpern, Robert Harper, Neil Immerman, Phokion G. Kolaitis, Moshe Y. Vardi and Victor Vianu [124, pp. 215, 233] In academia, in industry, and in the commercial world, there is a widespread belief that computing science as such has been all but completed and that, consequently, computing has “matured” from a theoretical topic for the scientists to a practical issue for the engineers, the managers and the entrepreneurs, i.e. mostly people—and there are many of those!—who can accept the application of science for the obvious benefits, but feel rather uncomfortable with its creation because they don’t understand what the doing of research, with its intangible goals and its uncertain rewards, entails. This widespread belief, however, is only correct if we identify the goals of computing science with what has been accomplished and forget those goals that we have failed to reach, even if they are too important to be ignored. I would therefore like to posit that computing’s central challenge, viz. “How not to make a mess of it”, has not been met. Edsger W. Dijkstra [78] Logic as traditionally presented in typical introductory courses—in mathematics, philosophy or computer science—has four distinctive features: BOOL It relies on classical, boolean laws: law of excluded middle, the law of double negation and Peirce’s syllogism. In particular, any formula can be transformed to a suitable negation normal form, either a conjunctive one 2 (CNF) or a disjunctive one (DNF). It also entails a high degree of dependence between basic connectives: !, _, ^, :, >, ?, 8 and 9. SETAC Furthermore, in modelling and formalising any phenomenon, one can use not only these boolean assumptions, but also standard set-theoretic principles such as the Axiom of Choice or at least the Boolean Prime Ideal Theorem. RELST It is supposed to deal with various types of relational structures, either in the form of transition systems in operational semantics of programming languages or in the form of the relational model/algebra in database theory. Furthermore, these relations are supposed to be modelled in a standard set- theoretic way. SYFOL Finally, its syntactic paradigm is supposed to be provided by standard (mostly first-order) predicate logic, which reflects and enforces its relational orientation mentioned in the preceding point. The syntax of predicate logic underlies not only languages used in logic programming, but also the concept of dependent types. Such an understanding of logic goes back to authors brought up in the nineteenth century and reacting to philosophical and mathematical challenges of their day. Its respective components were provided, in varying contexts and for different reasons, by the great pioneers of the field: Gottlob Frege, Bertrand Russell, Charles S. Peirce, George Boole and Georg Cantor. In a later generation of logicians, Willard Van Orman Quine can be seen as a militant proponent of an entire set of views combining these ingredients.1 In what follows, I am going to call it the Classical Picture. It is not nearly adequate for the evolving needs of contemporary computer science; we are going to recall the reasons for subverting SYFOL from § 2 on, for subverting RELST from § 3 on, for subverting SETAC from § 4 on and for subverting BOOL from § 6 on. This overview summarizes my recent (as of mid- 2018) contributions to an ever-growing body of work illustrating the fruitfulness of lifting each of the above assumptions. Remark 1.1. As a community, we still seem far away from having an adequate synthesis of this emerging new picture, something akin to a new Organon of logic in computer science. This is related to another question: how should the logical educa- tion of CS students evolve to close the growing gap between teaching and research? Apart from presenting my own work, this overview provides some insight into the choice of subjects which in my opinion such a synthesis should incorporate. These include elements of category theory, type theory, proof theory, finite model theory, (co)algebraic methods, the study of effectiveness, computability and complexity, and last but not the least, automated reasoning and interactive theorem proving. I focus on my papers accepted or published between 2014 and the first half of 2018, with occasional references to earlier ones. In more detail: I Even in the case of relational structures with classical metatheory, modalities have been well-understood to provide an attractive alternative to predicate 1To avoid getting into finer historical points too early in this introduction, I provide a somewhat more detailed discussion of Quine’s (and Tarski’s) views in Appendix D. 3 first- and higher-order syntax. This perspective lies at the heart ofwhat one may call the Amsterdam approach to modal logic [17, 38] and is broadly accepted in contemporary CS [292, 124, 192]. Some discussion of my work on related subjects prior to the time interval covered by this overview [O12, O7, O6, O2, O3, O1] can be found at the end of § 2. I Furthermore, the research of the past two decades has shown that modal formalisms arise naturally in the context of coalgebraic logic (§ 3), which provides a categorical umbrella for various state-based but non-relational sys- tems. Incorporating modalities directly into predicate syntax—not in the sense of creating a variant of quantified modal logic!—is at the center ofour generalization of FOL to the coalgebraic setting: coalgebraic predicate logic (CPL)[I7, S3, S2, I11] (§§ 3.2 and 4.1). I In recent years, another generalization of Kripke semantics is gaining popu- larity, namely so-called possibility semantics. The research on the subject is led mostly by the team of Wesley H. Holliday at UC Berkeley. While Kripke coalgebras are dual to Boolean Algebras with operators (baos) which may fail to be completely additive, possibility frames are dual to baos which may fail to be atomic (§ 5, especially § 5.3). My 2005 PhD Thesis [O8] and a series of related publications [O10, O9, O4] investigate a whole hierarchy of “sub- Kripkean” notions of completeness (§ 5.2). The insights provided by research on possibility semantics have helped Holliday and myself [I1] to solve an open question posed in my earlier work: the existence of logics incomplete with respect to completely additive baos (§ 5.3). I Moreover, our investigation of modal definability/axiomatizability of com- plete additivity [I1, §§ 8–9] has lead us to the invention of the global quan- tificational modality [8p]. Unlike the propositional quantifier 8p by itself, the global