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Neighborhood semantics for modal logic
Pacuit E., Springer International Publishing, New York, NY, 2017. 154 pp. Type: Book (978-3-319671-48-2)
Date Reviewed: Feb 14 2019

Reading and writing a review of this wonderful book has been a pleasure. Knowing the basics of propositional modal logic may explain why I enjoyed reading it. The author has gathered and surveyed many papers in writing this book. This is a must-read for those who want to do research on neighborhood semantics--after having acquired a basic knowledge of modal logic.

Modal logic uses labeled transition systems to study the semantics and possible computations of a program. Modal logic is also used for verification of reactive systems. The term “neighborhood” is inherited from point-set topology. In neighborhood semantics, each state is associated with a set of sets of states that are considered necessary. Neighborhood models are generalizations of standard Kripke models, and are used to reason about non-normal modal logics that don’t include some of the axioms or rules of inference of normal modal logic. For example, in deontic logic, the monotonicity rule “From φ → ψ, infer ∎ φ → ∎ ψ” is not valid. In logics of high probability, the axiom “(∎ φ ∧ ∎ ψ) → ∎(φ ∧ ψ)” is not true. To deal with such weak systems of modal logic, we need neighborhood semantics.

Next is a brief summary of the book’s contents.

Section 1.1 introduces subset space logic, which is used to describe neighborhood semantics. Note that neighborhood semantics is different from relational semantics. Then, in subsection 1.2.1, the author gives a neighborhood semantics for a variety of modal operators apart from basic modal operators. He then describes the syntax and semantics of a simple logic with two modalities to reason about subset spaces. In section 1.3, he describes why and how non-normal logics are created and provides examples of some different kinds of non-normal logics. In section 1.4, he explains why neighborhood structures are studied in the literature and gives examples of how neighborhood structures are related to other mathematics structures.

Section 2.1 describes the structural equivalence, that is, bisimulation, of two monotonic neighborhood models. It also describes the bisimulation of any two arbitrary neighborhood models using bounded morphisms, and explains that modal satisfaction is invariant under disjoint union in neighborhood models. As neighborhood semantics may not be the best semantics for weak modal logics, the author explores alternative semantics such as relational models, generalized relational models, multi-relational models, and impossible worlds to study non-normal modal logics. Section 2.3 covers the axiomatic proof theory of different non-normal modal logics and raises two questions: Which axioms are valid on all neighborhood models? Are all axioms and rules independent? The author then adopts the standard approach of the canonical model method to prove its completeness with respect to neighborhood semantics. Section 2.4 shows that the satisfiability problem for non-normal modal logics is decidable and discusses its complexity. In subsection 2.4.3, Pacuit introduces the sequent calculus for non-normal modal logics. In section 2.5, he looks at correspondence theory with respect to neighborhood frames. Section 2.6 describes a connection between neighborhood models and relational models, and shows that every non-normal modal logic can be simulated by a normal modal logic; a translation of non-normal modal logic into first-order logic is included.

In the last chapter (3), the author studies different extensions of the basic modal language interpreted on neighborhood structures. In section 3.1, he describes the extension of basic modal logic with universal modality and nominals. In subsection 3.1.1, he proves the completeness of an axiomatic system for non-normal modal logic with universal modality. In section 3.2, he describes the syntax and semantics of first-order neighborhood structures and discusses the interaction between modal operators and quantifiers using the Barcan and converse Barcan axiom schemes. He then discusses the axiomatization and completeness of first-order neighborhood frames with constant domains. Section 3.3 introduces multiagent neighborhood models and shows how to define various notions of group beliefs in multiagent neighborhood models. Section 3.4 introduces the syntax and neighborhood semantics of game logic and discusses the outcome of a game using fixed points. The last section (3.5) introduces dynamics in neighborhood structures. Subsection 3.5.1 introduces public announcement in neighborhood models, and subsection 3.5.2 describes the dynamic logics of “evidence management” in neighborhood models.

I was happy to write a review of what I believe will become a standard book in the field.

Reviewer:  Manoj K. Raut Review #: CR146435 (1905-0158)
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