The theory of coalgebra for a comprehensive introduction see. This chapter introduces the theory of consequence relations and matrix semantics. Stochastic coalgebraic logic ernsterich doberkat springer. It combines ideas from the theory of dynamical systems and from the theory of statebased computation. Aczels book 2 \nonwell founded set theory where he gives a description of the final system for the signature pa. Formulas are interpreted over graphlike structures. As described in stone coalgebras we can derive an endofunctor on the category of boolean algebras, ba ba, from a modal operator, algebras for which are modal algebras think lindenbaum algebra of a propositional logic having the necessary operator \box. Specifically, we look at stone duality for the vietoris hyperspace and the vietoris powerlocale, and at recent work combining coalgebraic modal logic and the vietoris functor. We study hybrid logic in the broad context of coalgebraic semantics, where. Coalgebraic semantics, on the other hand, provides a uniform and encompassing view on the large variety of speci c logics used in particular domains. A semantics for the basic modal language was developed by saul kripke, stag kanger, jaakko hinitkka and others in the 1960s and 1970s.
Goldblattthomason theorem for coalgebraic graded modal logic. Structural operational semantics and modal logic, revisited. Modal logic is a textbook on modal logic, intended for readers already acquainted with the elements of formal logic. Expressiveness of positive coalgebraic logic modal logic. Chapter 1 presents the basics of algebra and general propositional logic inasmuch as they are essential for understanding modal logic. Goldblattthomason theorem for coalgebraic graded modal. A coalgebraic semantics for a modal logic consists of a signature functor and. Carnap 1947 was an important precursor to possible worlds semantics. First, we have the coalgebraic approach to modal logic, where we build on the duality between stone spaces and boolean algebras. We argue that coalgebras unify the semantics of a large range of different modal logics such as probabilistic, graded, relational, conditional and discuss unifying approaches to reasoning at this level of generality. In this paper, we give an overview of the basic tools, techniques and results that connect coalgebras and modal logic. What more can a modal logic say about the topology of r in csemantics.
A modal is an expression like necessarily or possibly that is used to qualify the truth of a judgement. This is very comparative to the case of classical modal logic, to which kripke semantics provides a quite clear, intuitive way of viewing the logic in consideration. Modal logic for philosophers designed for use by philosophy students, this book provides an accessible yet technically sound treatment of modal logic and its philosophical applications. Modal logic for philosophers second edition t his book on modal logic is especially designed for philosophy students. Several kinds of semantics for modal logic have been proposed, the most popular of which is kripke semantics. Hintikka 1962, 1967 develops the possible worlds semantics and applies it to epistemic concepts. In the same sense, coalgebraic logics are generalised.
Applications of modal logics are abundant in computer science, and a large number of structurally di erent modal logics have been successfully employed in a diverse spectrum of application contexts. Coalgebraic semantics, on the other hand, provides a uniform and encompassing view on the large variety of specific logics used in particular domains. Therefore, modal logic, through its kripke semantics, can be considered as part of secondorder logic. A view of its evolution 5 was a variable neither always true nor always false. Chapter 2 provides preliminaries for later chapters. Coalgebraic semantics, on the other hand, provides a uniform and encompassing view on the large variety of speci. Goldblattthomason theorem for coalgebraic graded modal logic minghui ma department of philosophy, tsinghua university, beijing graded modal logic gml was originally presented by kit fine 1972 to make the modal analogue to counting quanti. Part of the lecture notes in computer science book series lncs, volume 11425. Saturated semantics for coalgebraic logic programming. Neighborhood semantics for modal logic an introduction.
This led to his publication of the introduction of semantics 1942, a work restricted to exclusively extensional logic, as was the subsequent volume, formalization of semantics 1943. This book will be useful for students, researchers, and professionals in all of these and related disciplines. First, in the setting of coalgebraic modal logic, we introduce the new. Structural operational semantics and modal logic, revisited bartek klin1 university of cambridge, warsaw university abstract a previously introduced combination of the bialgebraic approach to structural operational semantics with coalgebraic modal logic is reexamined and improved in some aspects. In section 2, we give a brief introduction to the generic coalgebraic semantics of modal logic. The predicate lifting approach 75 can be seen as a direct generalisation of basic modal logic. From the point of view of coalgebra, posets arise if one is interested in simulations as opposed to bisimulations. On a categorical framework for coalgebraic modal logic liangting chen institute of information science academia sinica taipei, taiwan achim jungy school of computer science university of birmingham birmingham, united kingdom 28th june 2014 abstract a category of onestep semantics is introduced to unify di erent ap. Applications of modal logics are abundant in computer science, and a large number of structurally di. A coalgebraic view on positive modal logic sciencedirect. This book offers a stateoftheart introduction to the basic techniques and results of neighborhood semantics for modal logic. However, he moved on to consider nonextensional logics in meaning and necessity. Coalgebras and modal logic functionallogic development and.
Advanced topics topological semantics for modal logic, some model theory. The third section is an introduction to modal logics for coalgebras. The chellas text in uenced me the most, though the order of presentation is inspired more by goldblatt. The main goal of the corse is to understand the basic techniques, results and applications of neighborhood semantics for modal logic and to understand the exact relationship with the standard relational semantics. Download citation coalgebraic semantics of modal logics. It provides an accessible yet technically sound treatment of modal logic and its philosophical applications. Coalgebraic modal logic tries to develop the theory of modal logic parametric in the type of transition, i. In this chapter, we sketch some of the thematically related mathematical developments that followed. On a categorical framework for coalgebraic modal logic. Coalgebras can be seen as a natural abstraction of kripke frames. This lays the grounds of investigation on coalgebraic semantics of intuitionistic modal logics such as intk square, intk, fs and mipc see 28. Bialgebraic methods and modal logic in structural operational. Kripke semantics also known as relational semantics or frame semantics, and often confused with possible world semantics is a formal semantics for nonclassical logic systems created in the late 1950s and early 1960s by saul kripke and andre joyal. A coalgebraic view on positive modal logic request pdf.
In the same sense, coalgebraic logics are generalised modal logics. All of the s1s5 modal logics of lewis and langford, among others, are constructed. Coalgebraic modal logic i was unable to continue this yesterday, so let me give a bit more precision to my ideas. From the point of view of modal logic, coalgebraic logic over posets is the natural coalgebraic generalisation of positive modal logic. Modal logic as a mathematical discipline has a long history. It was first conceived for modal logics, and later adapted to intuitionistic logic and other nonclassical systems. Coalgebraic logic is an important research topic in the areas of concurrency theory, semantics, transition systems and modal logics. Institute for logic, language and computation, university of amsterdam, science park 107, nl1098xg amsterdam. Author links open overlay panel alessandra palmigiano 1. Every effort is made to simplify the presentation by using diagrams instead of more complex mathematical apparatus. The proofs of some statements that appear in this paper are. To put it another way, this chapter is devoted to what is known as the relational or kripke semantics for modal logic. Our strategy is based on conjoining two wellknown approaches. Strong completeness for iterationfree coalgebraic dynamic logics.
While we do not pretend to work speci cally on one of the ukcrc grand challenges, it is. Bezhanishvili, gabelaia 2010 more questions like this e. Since the late 1970s, it has become clear that modal logics are a fundamental conceptual and methodological tool in nearly all areas of science. The geometric relational kripke semantics of modal logics are instances of coalgebraic semantics. The area of coalgebra has emerged within theoretical computer science with a unifying claim. Applications of modal logics are abundant in computer science, and a large number of structurally different modal logics have been successfully employed in a diverse spectrum of application contexts. We introduce a novel realvalued endogenous logic for expressing properties. Modal logic is, strictly speaking, the study of the deductive behavior of the expressions it is necessary that and it is possible that. The language of basic modal logic is an extension of classical propositional logic. Coalgebraic logic, automata theory, fixed point logics coalgebraic logic for structural operational semantics applied coalgebraic logic moss gave a presentation on new developments on the logic of recursion, which is one of the oldest topics in coalgebraic logic going back to the book vicious circles by barwise and moss 1996. I was trying to describe a multiagent system and as david pointed out you can do that coalgebraically if the number of agents is fixed a a is the set of agent labels.
A matrix, or manyvalued semantics, for sentential modal logic is formalized, and an important. The usual algebraic semantics of modal logic is in terms of boolean algebras with operators and is described in the entry algebraic models for modal logic. This type of semantics also provides an excellent motivation and. It provides a general approach to modeling systems, allowing us to apply important results from coalgebras, universal algebra and category theory in novel ways. Overview 1 a brief introduction to category 2 coalgebra 3 logical languages and semantics coalgebraic logics via predicate liftings cover modality 4 summary wang yunsong sms coalgebraic modal logic may 28th, 2019235. The agenda introduction basic modal logic normal systems of modal logic metatheorems of normal systems variants of modal logic conclusion. Bialgebraic methods and modal logic in structural operational semantics bartek klin1 university of edinburgh, warsaw university abstract bialgebraic semantics, invented a decade ago by turi and plotkin, is an approach to formal reasoning about wellbehaved structural operational semantics sos. From a categorical point of view, one moves from ordinary categories to enriched categories. In this paper, we give an overview of the basic tools. Chellas provides a systematic introduction to the principal ideas and results in contemporary treatments of modality, including theorems on completeness and decidability.
In addition to presenting the relevant technical background, it highlights both the pitfalls and potential uses of neighborhood models an interesting class of mathematical structures that were originally introduced to provide a semantics for weak systems of modal. Coalgebraic semantics for positive modal logic sciencedirect. Can we import resultsideas from model theory for modal logic with respect to kripke semantics topological semantics. This is a recommendable book in modal logic, written from a broad perspective.
Notes on modal logic notes for philosophy 151 eric pacuit january 28, 2009. This is the best known and with the exception of algebraic semantics the best explored style of modal semantics. Coalgebraic semantics for positive modal logic article pdf available in electronic notes in theoretical computer science 821. Basic concepts in modal logic1 stanford university. Part of the lecture notes in computer science book series lncs, volume 8705. Overview personal story three gracious ladies completeness in csemantics quasiorders as topologies finite connected spaces are interior images of the real line connected logics completeness in dsemantics. Some sahlqvist completeness results for coalgebraic logics. Possible worlds semantics was first presented as a formal semantics for modal logic in kripke 1959, 1963 and for intuitionistic logic in kripke 1963. The semantics is as usual where \left\langle a \right\rangle \phi holds at a. This book has been cited by the following publications. Pdf coalgebraic modal logic in cocasl researchgate. Goldblattthomason theorem for coalgebraic gml 3 proposition 4. This paper presents a first step towards completenessviacanonicity results for coalgebraic modal logics. Yv 2017 do not distribute abstract these notes give an introduction to the theory of universal coalgebra and coalgebraic modal logic.
Specifically, we consider the relationship between classes of coalgebras for. Stochastic coalgebraic logic is a detailed study devoted to the modal logic of general probability spaces. A semantic perspective 3 chapters in this handbook. Positive modal logic is the restriction of the modal local consequence relation defined by the class of all kripke models to the propositional negationfree modal language. This is sometimes called possible worlds semantics, although the formal semantics doesnt require us to think of the entities in its domain as possible worlds. Hybrid logic extends modal logic with support for reasoning about individual states, designated by socalled nominals. In our overview paper we focus on two related approaches to modal logics in a coalgebraic setting and discuss their common, categorical abstraction.
We present a coalgebraic treatment of iterationfree dynamic modal logics. This book is a very good overview of the state of the art in modal logic. A remark on normal modal logics 20 5 intuitionistic propositional calculus 23 6 generalizing the basic framework 29 1. Every effort has been made to simplify the presentation by using diagrams in place of more complex mathematical apparatus. From it we deduce the basic completeness results in modal logic. Topological semantics of modal logic david gabelaia.1237 150 594 874 79 256 461 335 1021 1193 1485 578 1081 339 1148 717 916 591 761 252 1062 602 854 812 999 69 286 336 138 1178 1312 579 147 1069 887 218 235