This book offers a state-of-the-art introduction to the basic techniques and results of neighborhood semantics for modal logic. 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 logic. In addition, the book discusses a broad range of topics, including standard modal logic results ; bisimulations for (...) class='Hi'>neighborhood models and other model-theoretic constructions; comparisons with other semantics for modal logic ; neighborhood semantics for first-order modal logic, applications in game theory ; applications in epistemic logic ; and non-normal modal logics with dynamic modalities. The book can be used as the primary text for seminars on philosophical logic focused on non-normal modal logics; as a supplemental text for courses on modal logic, logic in AI, or philosophical logic ; or as the primary source for researchers interested in learning about the uses of neighborhood semantics in philosophical logic and game theory. (shrink)

In this paper, we give an alternative semantics to the non-normal logic of knowing how proposed by Fervari et al., based on a class of Kripke neighborhood models with both the epistemic relations and neighborhood structures. This alternative semantics is inspired by the same quantifier alternation pattern of ∃∀\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\exists \forall $$\end{document} in the semantics of the know-how modality and the neighborhood semantics for the standard (...) modality. We show that this new semantics is equivalent to the original Kripke semantics in terms of the validities. A key result is a representation theorem showing that the more abstract Kripke neighborhood models can be represented by the concrete Kripke models with action transitions modulo the valid formulas. We prove the completeness of the logic for the neighborhood semantics. The neighborhood semantics can be adapted to other variants of logics of knowing how. It provides us a powerful technical tool to study these logics while preserving the basic semantic intuition. (shrink)

This article outlines a semantic approach to the logics of unknown truths, and the logic of false beliefs, using neighborhood structures, giving results on soundness, completeness, and expressivity. Relational semantics for the logics of unknown truths are also addressed, specically the conditions under which sound axiomatizations of these logics might be obtained from their normal counterparts, and the relationship between refexive insensitive logics and logics containing the provability operator as the primary modal operator.

Towards NonBeing (Priest, 2005) gives a noneist account of the semantics of intentional operators and predicates. The semantics for intentional operators are modelled on those for the , is given and assessed.

The paper presents a semantics for quantified modal logic which has a weaker axiomatization than the usual Kripke semantics. In particular, the Barcan Formula and its converse are not valid with the proposed semantics. Subclasses of models which validate BF and other interesting formulas are presented. A completeness theorem is proved, and the relation between this result and completeness with respect to Kripke models is investigated.

As we’ve seen in the last chapter, there is good linguistic reason to categorize negations (and negative operators in general) by which De Morgan laws they support. The weakest negative operators (merely downward monotonic) support only two De Morgan laws;1 medium-strength negative operators support a third;2 and strong negative operators support all four. As we’ve also seen, techniques familiar from modal logic are of great use in giving unifying theories of negative operators. In particular, Dunn’s (1990) distributoid theory allows us (...) to generate relational semantics for many negations. However, the requirements of distributoid theory are a bit too strict for use in modeling the weakest negations. For a relational semantics to work, an operator must either distribute or antidistribute over either conjunction or disjunction; but the merely downward monotonic operators do not. Thus, a unifying semantics cannot be had in distributoid theory. In the (more familiar) study of positive modalities, there is a parallel result. Normal necessities distribute over conjunction, and normal possibilities over disjunction. When these distributions break down, a relational semantics is no longer appropriate. Here, there is a somewhat familiar solution: neighborhood semantics. In this chapter, I’ll adapt neighborhood semantics to the less familiar case of negative modalities, showing how it can be used to give a single semantic framework appropriate to all the pertinent sorts of negative operators. (shrink)

In a mathematical perspective, neighborhood models for modal logic are generalized quantifiers, parametrized to points in the domain of objects/worlds. We explore this analogy further, connecting generalized quantifier theory and modal neighborhood logic. In particular, we find interesting analogies between conservativity for linguistic quantifiers and the locality of modal logic, and between the role of invariances in both fields. Moreover, we present some new completeness results for modal neighborhood logics of linguistically motivated classes of generalized quantifiers, and (...) raise new types of open problem. With the bridges established here, many further analogies might be explored between the two fields to mutual benefit. (shrink)

In the literature, different frameworks of probabilistic epistemic logic have been proposed. Most of these frameworks define knowledge or belief by relational structure. In this paper, we explore the relationship between probability and belief, based on the Lockean thesis, and adopt neighborhood semantics that defines belief directly using probability. We provide a sound and weakly complete axiomatization for our framework. We also try to explain the lottery paradox by modelling it within our framework. Moreover, the paper presents findings (...) concerning the relationship between knowledge, belief and probability within various frameworks. (shrink)

We propose a new perspective on logics of computation by combining instantial neighborhood logic \ with bisimulation safe operations adapted from \. \ is a recent modal logic, based on an extended neighborhood semantics which permits quantification over individual neighborhoods plus their contents. This system has a natural interpretation as a logic of computation in open systems. Motivated by this interpretation, we show that a number of familiar program constructors can be adapted to instantial neighborhood (...) class='Hi'>semantics to preserve invariance for instantial neighborhood bisimulations, the appropriate bisimulation concept for \. We also prove that our extended logic \ is a conservative extension of dual-free game logic, and its semantics generalizes the monotone neighborhood semantics of game logic. Finally, we provide a sound and complete system of axioms for \, and establish its finite model property and decidability. (shrink)

In this article, we study logics of unknown truths and false beliefs under neighborhood semantics. We compare the relative expressivity of the two logics. It turns out that they are incomparable over various classes of neighborhood models, and the combination of the two logics are equally expressive as standard modal logic over any class of neighborhood models. We propose morphisms for each logic, which can help us explore the frame definability problem, show a general soundness and (...) completeness result, and generalize some results in the literature. We axiomatize the two logics over various classes of neighborhood frames. Most importantly, by adopting the intersection semantics and the subset semantics in the literature, we extend the results to the case of public announcements, which gives us the desired reduction axioms and has good applications related to Moore sentences, successful formulas and self-refuting formulas. Also, we can say something about the comparative merits of the intersection semantics and the subset semantics. (shrink)

We propose a new perspective on logics of computation by combining instantial neighborhood logic INL\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathsf {INL}$$\end{document} with bisimulation safe operations adapted from PDL\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathsf {PDL}$$\end{document}. INL\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathsf {INL}$$\end{document} is a recent modal logic, based on an extended neighborhood semantics which permits quantification over individual neighborhoods plus their contents. This system has a natural (...) interpretation as a logic of computation in open systems. Motivated by this interpretation, we show that a number of familiar program constructors can be adapted to instantial neighborhood semantics to preserve invariance for instantial neighborhood bisimulations, the appropriate bisimulation concept for INL\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathsf {INL}$$\end{document}. We also prove that our extended logic IPDL\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathsf {IPDL}$$\end{document} is a conservative extension of dual-free game logic, and its semantics generalizes the monotone neighborhood semantics of game logic. Finally, we provide a sound and complete system of axioms for IPDL\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathsf {IPDL}$$\end{document}, and establish its finite model property and decidability. (shrink)

Intellectual humility is an interesting but underexplored disposition. The claim “I am (intellectually) humble” seems paradoxical in that someone who has the disposition in question would not typically volunteer it. There is an explanatory gap between the meaning of the sentence and the meaning the speaker expresses by uttering it. We therefore suggest analyzing intellectual humility semantically, using a psycholexical approach that focuses on both synonyms and antonyms of ‘intellectual humility’. We present a thesaurus-based method to map the semantic space (...) of intellectual humility as a heuristic to support philosophical and psychological analysis of this disposition. We find three semantic clusters that compose intellectual humility: the sensible self, the discreet self, and the inquisitive self; likewise, we find three clusters that compose its contraries: the overrated self, the underrated other, and the underrated self. (shrink)

This paper presents a neighborhood semantics for logics of entailment. It begins with a minimal system Min that expresses the most fundamental assumptions about the entailment relation, and continues by examining various extensions that reflect further assumptions that might be made about entailment. This leads first to the logic B that is the basic relevant logic, and then to more powerful systems. All of these logics are proved to be sound and strongly complete. With B the neighborhood (...) semantics meets the Routley-Meyer relational semantics for relevant logic; these connections are examined. The minimal and basic entailment logics are shown to have the finite model property, and hence to be decidable. (shrink)

The intuitive notion of evidence has both semantic and syntactic features. In this paper, we develop an evidence logic for epistemic agents faced with possibly contradictory evidence from different sources. The logic is based on a neighborhood semantics, where a neighborhood N indicates that the agent has reason to believe that the true state of the world lies in N. Further notions of relative plausibility between worlds and beliefs based on the latter ordering are then defined in (...) terms of this evidence structure, yielding our intended models for evidence-based beliefs. In addition, we also consider a second more general flavor, where belief and plausibility are modeled using additional primitive relations, and we prove a representation theorem showing that each such general model is a p-morphic image of an intended one. This semantics invites a number of natural special cases, depending on how uniform we make the evidence sets, and how coherent their total structure. We give a structural study of the resulting ‘uniform’ and ‘flat’ models. Our main result are sound and complete axiomatizations for the logics of all four major model classes with respect to the modal language of evidence, belief and safe belief. We conclude with an outlook toward logics for the dynamics of changing evidence, and the resulting language extensions and connections with logics of plausibility change. (shrink)

This article proposes the axiomatizations of contingency logics of various natural classes of neighborhood frames. In particular, by defining a suitable canonical neighborhood function, we give sound and complete axiomatizations of monotone contingency logic and regular contingency logic, thereby answering two open questions raised by Bakhtiari, van Ditmarsch, and Hansen. The canonical function is inspired by a function proposed by Kuhn in 1995. We show that Kuhn’s function is actually equal to a related function originally given by Humberstone.

As part of an approach to the liar paradox and the other paradoxes affecting truth, I have proposed replacing our concept of truth with two concepts: ascending truth and descending truth.1 I am not going to discuss why I think this is the best approach or how it solves the paradoxes; instead, I concentrate on the theory of ascending and descending truth. I formulate an axiomatic theory of ascending truth and descending truth (ADT) and provide a possible-worlds semantics for (...) it (which I dub xeno semantics). Xeno semantics is a generalization of the familiar neighborhood semantics, which itself is a generalization of the standard relational semantics. Once the details of ADT have been presented, it is easy to show that neither relational semantics nor neighborhood semantics will work for it; thus, the move to a more general framework is required. The main result is a fixed point theorem that guarantees the existence of an acceptable first-order constant-domain xeno model. From this result it follows that ADT is sound with respect to the class of such models. The upshot is that ADT is consistent relative to the background set theory. (shrink)

I first provide an accessible overview of the DWE (Doing Well Enough) logical and semantic framework for representing going beyond the call and its family of kindred concepts in a tightly intergraded way. Next, a module, for representing some basic agent-evaluative notions is developed (“AA” for “Aretaic Assessment”), and then it is integrated with the more act-evaluative notions of DWE, thereby allowing for a representation of suberogation and supererogation (as distinct from going beyond the call) and many other combined deontic (...) and agent-evaluative notions. I then probe more deeply into how the semantic structures for the DWE framework might themselves be generated and then offer some brief reflections on the classic supererogation puzzle, as well as that of supererogatory holes and the all or nothing problem, framing options via the preceding reflections. (shrink)

We define the notion of a model of higher-order modal logic in an arbitrary elementary topos E. In contrast to the well-known interpretation of higher-order logic, the type of propositions is not interpreted by the subobject classifier ΩE, but rather by a suitable complete Heyting algebra H. The canonical map relating H and ΩE both serves to interpret equality and provides a modal operator on H in the form of a comonad. Examples of such structures arise from surjective geometric morphisms (...) f : F → E, where H = f∗ΩF. The logic differs from non-modal higher-order logic in that the principles of functional and propositional extensionality are not longer valid but may be replaced by modalized versions. The usual Kripke, neighborhood, and sheaf semantics for propositional and first-order modal logic are subsumed by this notion. (shrink)

In "A Problem in Possible-World Semantics," David Kaplan presented a consistent and intelligible modal principle that cannot be validated by any possible world frame (in the terminology of modal logic, any neighborhood frame). However, Kaplan's problem is tempered by the fact that his principle is stated in a language with propositional quantification, so possible world semantics for the basic modal language without propositional quantifiers is not directly affected, and the fact that on careful inspection his principle does (...) not target the world part of possible world semantics—the atomicity of the algebra of propositions—but rather the idea of propositional quantification over a complete Boolean algebra of propositions. By contrast, in this paper we present a simple and intelligible modal principle, without propositional quantifiers, that cannot be validated by any possible world frame precisely because of their assumption of atomicity (i.e., the principle also cannot be validated by any atomic Boolean algebra expansion). It follows from a theorem of David Lewis that our logic is as simple as possible in terms of modal nesting depth (two). We prove the consistency of the logic using a generalization of possible world semantics known as possibility semantics. We also prove the completeness of the logic (and two other relevant logics) with respect to possibility semantics. Finally, we observe that the logic we identify naturally arises in the study of Peano Arithmetic. (shrink)

Computational modeling and eye-tracking were used to investigate how phonological and semantic information interact to influence the time course of spoken word recognition. We extended our recent models to account for new evidence that competition among phonological neighbors influences activation of semantically related concepts during spoken word recognition . The model made a novel prediction: Semantic input modulates the effect of phonological neighbors on target word processing, producing an approximately inverted-U-shaped pattern with a high phonological density advantage at an intermediate (...) level of semantic input—in contrast to the typical disadvantage for high phonological density words in spoken word recognition. This prediction was confirmed with a new analysis of the Apfelbaum et al. data and in a visual world paradigm experiment with preview duration serving as a manipulation of strength of semantic input. These results are consistent with our previous claim that strongly active neighbors produce net inhibitory effects and weakly active neighbors produce net facilitative effects. (shrink)

The main purpose of this paper is to define and study a particular variety of Montague-Scott neighborhood semantics for modal propositional logic. We call this variety the first-order neighborhood semantics because it consists of the neighborhood frames whose neighborhood operations are, in a certain sense, first-order definable. The paper consists of two parts. In Part I we begin by presenting a family of modal systems. We recall the Montague-Scott semantics and apply it to (...) some of our systems that have hitherto be uncharacterized. Then, we define the notion of a first-order indefinite semantics, along with the more specific notion of a first-order uniform semantics, the latter containing as special cases the possible world semantics of Kripke. In Part II we prove consistency and completeness for a broad range of the systems considered, with respect to the first-order indefinite semantics, and for a selected list of systems, with respect to the first-order uniform semantics. The completeness proofs are algebraic in character and make essential use of the finite model property. A by-product of our investigations is a result relating provability in S-systems and provability in T-systems, which generalizes a known theorem relating provability in the systems S 2° and C 2. (shrink)

Listed buildings, even if they are not top landmarks, are increasingly attracting visitors. People express interest in hidden gems in their neighborhood or along their travel itinerary, and in the history of the building they live in. All required data has been meticulously collected by the offices for historical monuments but is not flexibly accessible. In Bremen, the database of buildings (with location, map of the estate, construction history, architect, photos) is searchable and browsable online3, but that only helps (...) users who know how to use a rigidly structured database search form. Our beginning BauDenkMalNetz effort (“listed building web”) aims at a wider purpose: modeling the semantic structure of these data, starting in Bremen but open for other data, and exposing them via a semantic web interface with enhanced querying and presentation capabilities. Requirements beyond interactive browsing comprise auto-generation of customized printed guides (e. g. “Bauhaus villas in my neighborhood”), on-demand presentation on mobile devices (e. g. “medieval churches along my travel itinerary”), and serving linked data for usage by other services. (shrink)

The impressive volume before us started out as an attempt to write “the history of epistemology since Kant, the way Carnap would have written it had he been Hegel.” Coffa began his project in 1981 while a fellow at the Center for Philosophy of Science in Pittsburgh and had finished a “good penultimate draft” when he suddenly died, after a brief illness, on 30 Dec., 1984. The title alludes to Edmund Wilson’s classic study of revolutionary ideology, To the Finland Station. (...) This is no accident and would be presumptuous in a lesser work. The two studies cover roughly the same period, from the late eighteenth century to the first part of the present one, and they are both sparkling narratives that boldly project a large picture and then fill it in with canny detail. Coffa begins with Kant, the villain of the piece, and ends with a group of thinkers “in the neighborhood of Vienna” who came to terms with the restrictive framework they had inherited from Kant, finally overcoming it in favor of a new philosophy in which meaning, rather than mind, played the central role. (shrink)

One of the most expected properties of a logical system is that it can be algebraizable, in the sense that an algebraic counterpart of the deductive machinery could be found. Since the inception of da Costa's paraconsistent calculi, an algebraic equivalent for such systems have been searched. It is known that these systems are non self-extensional (i.e., they do not satisfy the replacement property). More than this, they are not algebraizable in the sense of Blok-Pigozzi. The same negative results hold (...) for several systems of the hierarchy of paraconsistent logics known as Logics of Formal Inconsistency (LFIs). Because of this, these logics are uniquely characterized by semantics of non-deterministic kind. This paper offers a solution for two open problems in the domain of paraconsistency, in particular connected to algebraization of LFIs, by obtaining several LFIs weaker than C1, each of one is algebraizable in the standard Lindenbaum-Tarski's sense by a suitable variety of Boolean algebras extended with operators. This means that such LFIs satisfy the replacement property. The weakest LFI satisfying replacement presented here is called RmbC, which is obtained from the basic LFI called mbC. Some axiomatic extensions of RmbC are also studied, and in addition a neighborhood semantics is defined for such systems. It is shown that RmbC can be defined within the minimal bimodal non-normal logic E+E defined by the fusion of the non-normal modal logic E with itself. Finally, the framework is extended to first-order languages. RQmbC, the quantified extension of RmbC, is shown to be sound and complete w.r.t. BALFI semantics. (shrink)

The paper focuses on extending to the first order case the semantical program for modalities first introduced by Dana Scott and Richard Montague. We focus on the study of neighborhood frames with constant domains and we offer in the first part of the paper a series of new completeness results for salient classical systems of first order modal logic. Among other results we show that it is possible to prove strong completeness results for normal systems without the Barcan Formula (...) (like FOL + K)in terms of neighborhood frames with constant domains. The first order models we present permit the study of many epistemic modalities recently proposed in computer science as well as the development of adequate models for monadic operators of high probability. Models of this type are either difficult of impossible to build in terms of relational Kripkean semantics [40].We conclude by introducing general first order neighborhood frames with constant domains and we offer a general completeness result for the entire family of classical first order modal systems in terms of them, circumventing some well-known problems of propositional and first order neighborhood semantics (mainly the fact that many classical modal logics are incomplete with respect to an unmodified version of either neighborhood or relational frames). We argue that the semantical program that thus arises offers the first complete semantic unification of the family of classical first order modal logics. (shrink)

We present a generalization of Segerberg's onion semantics for belief revision, in which the linearity of the spheres need not occur. The resulting logic is called broccoli logic. We provide a minimal relational logic, with a bi-modal neighborhood semantics. We then show that broccoli logic is a well-known conditional logic, the Burgess-Veltman minimal conditional logic.

Firstly, the authors analyzed the properties of primary-onditionals and secondary-conditionals, establishthe minimum system $C2L_{m}$ of primary-conditionals and secondary-conditionals, and then prove some of the formal theorems of the system which have important intuitive meanings. Secondly, the authors constructed the neighborhood semantics, prove the soundness of $C2L_{m}$ , introduce a general concept of canonical model by the neighborhood semantics, and then prove the completeness of $C2L_{m}$ by the canonical model. Finally, according to the technical results of the (...) minimum system $C2L_{m}$ , the authors discuss some of the important problems concerning primary-conditionals and secondary- onditionals. (shrink)

We take Carnap’s problem to be to what extent standard consequence relations in various formal languages fix the meaning of their logical vocabulary, alone or together with additional constraints on the form of the semantics. This paper studies Carnap’s problem for basic modal logic. Setting the stage, we show that neighborhood semantics is the most general form of compositional possible worlds semantics, and proceed to ask which standard modal logics (if any) constrain the box operator to (...) be interpreted as in relational Kripke semantics. Except when restricted to finite domains, no modal logic characterizes all the Kripkean interpretation of P. Moreover, we show that, in contrast with the case of first-order logic, the obvious requirement of permutation invariance is not adequate in the modal case. After pointing out some known facts about modal logics that nevertheless force the Kripkean interpretation, we focus on another feature often taken to embody the gist of modal logic: locality. We show that invariance under point-generated subframes (properly defined) does single out the Kripkean interpretations, but only among topological interpretations, not in general. Finally, we define a notion of bisimulation invariance — another aspect of locality — that, together with a reasonable closure condition, gives the desired general result. Along the way, we propose a new perspective on normal neighborhood frames as filter frames, consisting of a set of worlds equipped with an accessibility relation, and a free filter at every world. (shrink)

We construct a Hilbert style system RPL for the notion of plausibility measure introduced by Halpern J, and we prove the soundness and completeness with respect to a neighborhood style semantics. Using the language of RPL, we demonstrate that it can define well-studied notions of necessity, conditionals and propositional identity.