Patrick Blackburn

In this article we define a logical system called Hybrid Partial Type Theory ( $\mathcal {HPTT}$ ). The system is obtained by combining William Farmer’s partial type theory with a strong form of hybrid logic. William Farmer’s system is a version of Church’s theory of types which allows terms to be non-denoting; hybrid logic is a version of modal logic in which it is possible to name worlds and evaluate expressions with respect to particular worlds. We motivate this combination of ideas in the introduction, and devote the rest of the article to defining, axiomatising, and proving a completeness result for $\mathcal {HPTT}$.

In a study involving 62 Danish children with autism spectrum disorder, we obtained results showing that the mastery of linguistic recursion is a significant predictor of success in second-order false belief tasks. The same study also showed that the mastery of linguistic recursion was not significantly correlated with success in a task involving three heavily used Danish discourse particles. This calls for further explanation, as the reasoning involved in both types of tasks seems similar. In this paper, we discuss second-order false belief reasoning, the reasoning underlying the use of the three Danish discourse particles, say what we know about them experimentally, and discuss what they do have in common.

With the rise of the internet and the proliferation of technology to gather and organize data, our era has been defined as "the information age." With the prominence of information as a research concept, there has arisen an increasing appreciation of the intertwined nature of fields such as logic, linguistics, and computer science that answer the questions about information and the ways it can be processed. The many research traditions do not agree about the exact nature of information. By bringing together ideas from diverse perspectives, this book presents the emerging consensus about what a conclusive theory of information should be. The book provides an introduction to the topic, work on the underlying ideas, and technical research that pins down the richer notions of information from a mathematical point of view. The book contains contributions to a general theory of information, while also tackling specific problems from artificial intelligence, formal semantics, cognitive psychology, and the philosophy of mind. There is focus on the dynamics of information flow, and also a consideration of static approaches to information content; both quantitative and qualitative approaches are represented

This paper is about the good side of modal logic, the bad side of modal logic, and how hybrid logic takes the good and fixes the bad.In essence, modal logic is a simple formalism for working with relational structures . But modal logic has no mechanism for referring to or reasoning about the individual nodes in such structures, and this lessens its effectiveness as a representation formalism. In their simplest form, hybrid logics are upgraded modal logics in which reference to individual nodes is possible.But hybrid logic is a rather unusual modal upgrade. It pushes one simple idea as far as it will go: represent all information as formulas. This turns out to be the key needed to draw together a surprisingly diverse range of work . Moreover, it displays a number of knowledge representation issues in a new light, notably the importance of sorting

In their simplest form, hybrid languages are propositional modal languages which can refer to states. They were introduced by Arthur Prior, the inventor of tense logic, and played an important role in his work: because they make reference to specific times possible, they remove the most serious obstacle to developing modal approaches to temporal representation and reasoning. However very little is known about the computational complexity of hybrid temporal logics.In this paper we analyze the complexity of the satisfiability problem of a number of hybrid temporal logics: the basic hybrid language over transitive trees; nominal Until logic; and referential interval logic. We discuss the effects of including nominals, the @ operator, the somewhere modality E, and the difference operator D. Adding nominals to tense logic leads for several frame-classes to an increase in complexity of the satisfiability problem from PSPACE to EXPTIME. On transitive trees, however, the satisfiability problem for this language can be decided in PSPACE. Along the way we make a detour through hybrid propositional dynamic logic: we establish upper bounds for a number of temporal logics by generalizing results due to Passy and Tinchev [29] and De Giacomo [16]. We conclude with some remarks on the relevance of our results to description logic, and draw attention to the utility of the spypoint technique for proving upper and lower bounds

This LNCS book is part of the FOLLI book series and constitutes the proceedings of the 7th International Workshop on Logic, Rationality, and Interaction, LORI 2019, held in Chongqing, China, in October 2019. The 31 papers presented in this book were carefully reviewed and selected from 56 submissions. They focus on the following topics: agency; argumentation and agreement; belief revision and belief merging; belief representation; cooperation; decision making and planning; natural language; philosophy and philosophical logic; and strategic reasoning.

The Handbook of Modal Logic contains 20 articles, which collectively introduce contemporary modal logic, survey current research, and indicate the way in which the field is developing. The articles survey the field from a wide variety of perspectives: the underling theory is explored in depth, modern computational approaches are treated, and six major applications areas of modal logic (in Mathematics, Computer Science, Artificial Intelligence, Linguistics, Game Theory, and Philosophy) are surveyed. The book contains both well-written expository articles, suitable for beginners approaching the subject for the first time, and advanced articles, which will help those already familiar with the field to deepen their expertise. Please visit: http://people.uleth.ca/~woods/RedSeriesPromo_WP/PubSLPR.html - Compact modal logic reference - Computational approaches fully discussed - Contemporary applications of modal logic covered in depth

We show that basic hybridization makes it possible to give straightforward Henkin-style completeness proofs even when the modal logic being hybridized is higher-order. The key ideas are to add nominals as expressions of type t, and to extend to arbitrary types the way we interpret \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$@_i$\end{document} in propositional and first-order hybrid logic. This means: interpret \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$@_i\alpha _a$\end{document}, where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\alpha _a$\end{document} is an expression of any type \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$a$\end{document}, as an expression of type \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$a$\end{document} that rigidly returns the value that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\alpha_a$\end{document} receives at the i-world. The axiomatization and completeness proofs are generalizations of those found in propositional and first-order hybrid logic, and we automatically obtain a wide range of completeness results for stronger logics and languages. Our approach is deliberately low-tech. We don’t, for example, make use of Montague’s intensional type s, or Fitting-style intensional models; we build, as simply as we can, hybrid logicover Henkin’s logic.

Hybrid languages are expansions of propositional modal languages which can refer to worlds. The use of strong hybrid languages dates back to at least [Pri67], but recent work has focussed on a more constrained system called $\mathscr{H}$. We show in detail that $\mathscr{H}$ is modally natural. We begin by studying its expressivity, and provide model theoretic characterizations and a syntactic characterization. The key result to emerge is that $\mathscr{H}$ corresponds to the fragment of first-order logic which is invariant for generated submodels. We then show that $\mathscr{H}$ enjoys interpolation, provide counterexamples for its finite variable fragments, and show that weak interpolation holds for the sublanguage $\mathscr{H}$. Finally, we provide complexity results for $\mathscr{H}$ and other fragments and variants, and sharpen known undecidability results for $\mathscr{H}$.

In this note we show that the classical modal technology of Sahlqvist formulas gives quick proofs of the completeness theorems in [8] (D. Gregory, Completeness and decidability results for some propositional modal logics containing “actually” operators, Journal of Philosophical Logic 30(1): 57–78, 2001) and vastly generalizes them. Moreover, as a corollary, interpolation theorems for the logics considered in [8] are obtained. We then compare Gregory's modal language enriched with an “actually” operator with the work of Arthur Prior now known under the name of hybrid logic. This analysis relates the “actually” axioms to standard hybrid axioms, yields the decidability results in [8], and provides a number of complexity results. Finally, we use a bisimulation argument to show that the hybrid language is strictly more expressive than Gregory's language.