It provides a general approach to modeling systems, allowing us to apply important results from coalgebras, universal algebra and category theory in novel ways. The semantics is as usual where \left\langle a \right\rangle \phi holds at a. A coalgebraic view on positive modal logic request pdf. The theory of coalgebra for a comprehensive introduction see. We introduce a novel realvalued endogenous logic for expressing properties. 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.
The language of basic modal logic is an extension of classical propositional logic. 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. Possible worlds semantics was first presented as a formal semantics for modal logic in kripke 1959, 1963 and for intuitionistic logic in kripke 1963. Goldblattthomason theorem for coalgebraic graded modal logic.
Basic concepts in modal logic1 stanford university. 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. Chellas provides a systematic introduction to the principal ideas and results in contemporary treatments of modality, including theorems on completeness and decidability. A modal is an expression like necessarily or possibly that is used to qualify the truth of a judgement. 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. A coalgebraic semantics for a modal logic consists of a signature functor and. 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. Saturated semantics for coalgebraic logic programming. This lays the grounds of investigation on coalgebraic semantics of intuitionistic modal logics such as intk square, intk, fs and mipc see 28.
This is a recommendable book in modal logic, written from a broad perspective. Every effort has been made to simplify the presentation by using diagrams in place of more complex mathematical apparatus. Coalgebraic modal logic tries to develop the theory of modal logic parametric in the type of transition, i. 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. A remark on normal modal logics 20 5 intuitionistic propositional calculus 23 6 generalizing the basic framework 29 1. From a categorical point of view, one moves from ordinary categories to enriched categories. Coalgebraic semantics for positive modal logic sciencedirect. Expressiveness of positive coalgebraic logic modal logic. This type of semantics also provides an excellent motivation and. Specifically, we consider the relationship between classes of coalgebras for. Coalgebraic semantics, on the other hand, provides a uniform and encompassing view on the large variety of speci c logics used in particular domains.
The geometric relational kripke semantics of modal logics are instances of coalgebraic semantics. This paper presents a first step towards completenessviacanonicity results for coalgebraic modal logics. Every effort is made to simplify the presentation by using diagrams instead of more complex mathematical apparatus. Formulas are interpreted over graphlike structures. Coalgebras can be seen as a natural abstraction of kripke frames. In this chapter, we sketch some of the thematically related mathematical developments that followed. 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. It combines ideas from the theory of dynamical systems and from the theory of statebased computation. Bialgebraic methods and modal logic in structural operational.
Coalgebras and modal logic functionallogic development and. Request pdf a coalgebraic view on positive modal logic positive modal logic is the restriction of the modal local consequence relation defined by the class of all kripke models to the. In the same sense, coalgebraic logics are generalised. Goldblattthomason theorem for coalgebraic graded modal. 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. 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. Bezhanishvili, gabelaia 2010 more questions like this e. Modal logic for philosophers second edition t his book on modal logic is especially designed for philosophy students. However, he moved on to consider nonextensional logics in meaning and necessity. 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. Carnap 1947 was an important precursor to possible worlds semantics. Chapter 2 provides preliminaries for later chapters. Modal logic, coalgebraic semantics, knowledge representation, mobility and concurrency.
Our strategy is based on conjoining two wellknown approaches. In our overview paper we focus on two related approaches to modal logics in a coalgebraic setting and discuss their common, categorical abstraction. Neighborhood semantics for modal logic an introduction. Hintikka 1962, 1967 develops the possible worlds semantics and applies it to epistemic concepts. 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. We present a coalgebraic treatment of iterationfree dynamic modal logics. A semantics for the basic modal language was developed by saul kripke, stag kanger, jaakko hinitkka and others in the 1960s and 1970s. First, we have the coalgebraic approach to modal logic, where we build on the duality between stone spaces and boolean algebras. The third section is an introduction to modal logics for coalgebras.
The class of positive modal algebras is the one canonically associated with pml according to the theory of the algebrization of logics lecture notes in logic, springer. Some sahlqvist completeness results for coalgebraic logics. Institute for logic, language and computation, university of amsterdam, science park 107, nl1098xg amsterdam. In the same sense, coalgebraic logics are generalised modal logics. This book is a very good overview of the state of the art in modal logic. A semantic perspective 3 chapters in this handbook. Modal logic as a mathematical discipline has a long history. A view of its evolution 5 was a variable neither always true nor always false. In section 2, we give a brief introduction to the generic coalgebraic semantics of modal logic.
From the point of view of modal logic, coalgebraic logic over posets is the natural coalgebraic generalisation of positive modal logic. 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. 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. Since the late 1970s, it has become clear that modal logics are a fundamental conceptual and methodological tool in nearly all areas of science. Pdf coalgebraic modal logic in cocasl researchgate.
Yv 2017 do not distribute abstract these notes give an introduction to the theory of universal coalgebra and coalgebraic modal logic. Strong completeness for iterationfree coalgebraic dynamic logics. Goldblattthomason theorem for coalgebraic gml 3 proposition 4. It provides an accessible yet technically sound treatment of modal logic and its philosophical applications. This is the best known and with the exception of algebraic semantics the best explored style of modal semantics. The proofs of some statements that appear in this paper are. What more can a modal logic say about the topology of r in csemantics. We study hybrid logic in the broad context of coalgebraic semantics, where. Therefore, modal logic, through its kripke semantics, can be considered as part of secondorder logic. Hybrid logic extends modal logic with support for reasoning about individual states, designated by socalled nominals. This book offers a stateoftheart introduction to the basic techniques and results of neighborhood semantics for modal logic. Modal logic is a textbook on modal logic, intended for readers already acquainted with the elements of formal logic.
Author links open overlay panel alessandra palmigiano 1. This book has been cited by the following publications. Stochastic coalgebraic logic is a detailed study devoted to the modal logic of general probability spaces. While we do not pretend to work speci cally on one of the ukcrc grand challenges, it is.
From it we deduce the basic completeness results in modal logic. From the point of view of coalgebra, posets arise if one is interested in simulations as opposed to bisimulations. Notes on modal logic notes for philosophy 151 eric pacuit january 28, 2009. A matrix, or manyvalued semantics, for sentential modal logic is formalized, and an important.
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. A coalgebraic view on positive modal logic sciencedirect. To put it another way, this chapter is devoted to what is known as the relational or kripke semantics for modal logic. It was first conceived for modal logics, and later adapted to intuitionistic logic and other nonclassical systems. The area of coalgebra has emerged within theoretical computer science with a unifying claim. Download citation coalgebraic semantics of modal logics. Several kinds of semantics for modal logic have been proposed, the most popular of which is kripke semantics.
Coalgebraic logic is an important research topic in the areas of concurrency theory, semantics, transition systems and modal logics. Modal logic is, strictly speaking, the study of the deductive behavior of the expressions it is necessary that and it is possible that. All of the s1s5 modal logics of lewis and langford, among others, are constructed. 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. 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. Chapter 1 presents the basics of algebra and general propositional logic inasmuch as they are essential for understanding modal logic. On a categorical framework for coalgebraic modal logic. Stochastic coalgebraic logic ernsterich doberkat springer. Topological semantics of modal logic david gabelaia. 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. Coalgebraic modal logic i was unable to continue this yesterday, so let me give a bit more precision to my ideas.
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. Part of the lecture notes in computer science book series lncs, volume 11425. Coalgebraic semantics, on the other hand, provides a uniform and encompassing view on the large variety of specific logics used in particular domains. 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. In this paper, we give an overview of the basic tools, techniques and results that connect coalgebras and modal logic. Structural operational semantics and modal logic, revisited. 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. In this paper, we give an overview of the basic tools.
Coalgebraic semantics for positive modal logic article pdf available in electronic notes in theoretical computer science 821. Can we import resultsideas from model theory for modal logic with respect to kripke semantics topological semantics. Aczels book 2 \nonwell founded set theory where he gives a description of the final system for the signature pa. The agenda introduction basic modal logic normal systems of modal logic metatheorems of normal systems variants of modal logic conclusion.
The chellas text in uenced me the most, though the order of presentation is inspired more by goldblatt. This chapter introduces the theory of consequence relations and matrix semantics. Advanced topics topological semantics for modal logic, some model theory. Part of the lecture notes in computer science book series lncs, volume 8705. 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 speci.
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