Neighborhood Semantics Research Articles

Peritopological Spaces and Bisimulations

Generalizing ordinary topological and pretopological spaces, we introduce the notion of peritopology where neighborhoods of a point need not contain that point, and some points might even have an empty neighborhood. We briefly describe various intrinsic aspects of this notion. Applied to modal logic, it gives rise to peritopological models, a generalization of topological models, a spacial case of neighborhood semantics. (In a last section, the relation between the latter and the former is discussed, cursorily). A new cladding for bisimulation is presented. The concept of Alexandroff peritopology is used in order to determine the logic of all peritopological spaces, and we prove that the minimal logic K is strongly complete with respect to the class of all peritopological spaces. We also show that the classes of T0, T1 and T2-peritopological spaces are not modal definable, and that D is the logic of all proper peritopological spaces. AMS subject classification : 03B45, 54A05

Neighborhood Semantics for Basic and Intuitionistic Logic

In this paper we present a neighborhood semantics for Intuitionistic Propositional Logic (IPL). We show that for each Kripke model of the logic there is a pointwise equivalent neighborhood model and vice versa. In this way, we establish soundness and completeness of IPL with respect to the neighborhood semantics. The relation between neighborhood and topological semantics are also investigated. Moreover, the notions of bisimulation and n-bisimulation between neighborhood models of IPL are defined naturally and some of their basic properties are proved. We also consider Basic Propositional Logic (BPL), a logic weaker than IPL introduced by Albert Visser, and introduce and study its neighborhood models in the same manner.

Representing qualitative capacities as families of possibility measures

This paper studies the structure of qualitative capacities, that is, monotonic set-functions, when they range on a finite totally ordered scale equipped with an order-reversing map. These set-functions correspond to general representations of uncertainty, as well as importance levels of groups of criteria in multiple-criteria decision-making. We show that any capacity or fuzzy measure ranging on a qualitative scale can be viewed both as the lower bound of a set of possibility measures and the upper bound of a set of necessity measures (a situation somewhat similar to the one of quantitative capacities with respect to imprecise probability theory). We show that any capacity is characterized by a non-empty class of possibility measures having the structure of an upper semi-lattice. The lower bounds of this class are enough to reconstruct the capacity, and the number of them is characteristic of its complexity. An algorithm is provided to compute the minimal set of possibility measures dominating a given capacity. This algorithm relies on the representation of the capacity by means of its qualitative Möbius transform, and the use of selection functions of the corresponding focal sets. We provide the connection between Sugeno integrals and lower possibility measures. We introduce a sequence of axioms generalizing the maxitivity property of possibility measures, and related to the number of possibility measures needed for this reconstruction. In the Boolean case, capacities are closely related to non-regular modal logics and their neighborhood semantics can be described in terms of qualitative Möbius transforms.

A note on Gettier cases in epistemic logic

The paper explains how Gettier’s conclusion can be reached on general theoretical grounds within the framework of epistemic logic, without reliance on thought experiments. It extends the argument to permissive conceptions of justification that invalidate principles of multi-premise closure and require neighbourhood semantics rather than semantics of a more standard type. The paper concludes by recommending a robust methodology that aims at convergence in results between thought experimentation and more formal methods. It also warns against conjunctive definitions as sharing several of the drawbacks of disjunctive definitions.

Dualities for modal N4-lattices

We introduce a new Priestley-style topological duality for N4-lattices, which are the algebraic counterpart of paraconsistent Nelson logic. Our duality differs from the existing one, due to S. Odintsov, in that we only rely on Esakia duality for Heyting algebras and not on the duality for De Morgan algebras of Cornish and Fowler. A major advantage of our approach is that we obtain a simple description for our topological structures, which allows us to extend the duality to other algebraic structures such as N4-lattices with monotonic modal operators, and also to provide a neighborhood semantics for the non-normal modal logic corresponding to these algebras.

Evidence and plausibility in neighborhood structures

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.

Probabilistic stit logic and its decomposition

We define an extension of stit logic that encompasses subjective probabilities representing beliefs about simultaneous choice exertion of other agents. This semantics enables us to express that an agent sees to it that a condition obtains under a minimal chance of success. We first define the fragment of XSTIT where choice exertion is not collective. Then we add lower bounds for the probability of effects to the stit syntax, and define the semantics of the newly formed stit operator in terms of subjective probabilities concerning choice exertion of other agents. We show how the resulting probabilistic stit logic faithfully generalizes the non-probabilistic XSTIT fragment. In a second step we analyze the defined probabilistic stit logic by decomposing it into an XSTIT fragment and a purely epistemic fragment. The resulting epistemic logic for grades of believes is a weak modal logic with a neighborhood semantics combining probabilistic and modal logic theory.

<b>A Neighbourhood Semantics for the Logic TK</b><br>DOI:10.5007/1808-1711.2011v15n2p287

The logic TK was introduced as a propositional logic extending the classical propositional calculus with a new unary operator which interprets some conceptions of Tarski’s consequence operator. TK-algebras were introduced as models to TK . Thus, by using algebraic tools, the adequacy (soundness and completeness) of TK relatively to the TK-algebras was proved. This work presents a neighbourhood semantics for TK , which turns out to be deductively equivalent to the non-normal modal logic EMT4 . DOI:10.5007/1808-1711.2011v15n2p287

Neighborhood-Sheaf Semantics for First-Order Modal Logic

This paper extends neighborhood semantics for propositional modal logic to the first-order case, by unifying topological-sheaf semantics (in [Awodey, S. and K. Kishida, Topology and modality: the topological interpretation of first-order modal logic, Review of Symbolic Logic 1 (2008), pp. 146–66]) for first-order S4 and Kripke-sheaf semantics (see [Goldblatt, R., “Topoi: The Categorial Analysis of Logic”, North-Holland, Amsterdam and New York, 1979] and [Gabbay, D. M., V. Shehtman and D. Skvortsov, “Quantification in Nonclassical Logic”, Volume 1, Elsevier, Oxford, 2009], just for instance) for quantified K. It will be shown how to take a sheaf-like structure over a neighborhood frame, and the resulting semantics properly generalizes the two preceding sheaf semantics; it has a weaker modal logic (in which the rule N fails) sound and complete, while accommodating classical, full first-order logic with equality and function symbols.

Dynamic Logics of Evidence-Based Beliefs

This paper adds evidence structure to standard models of belief, in the form of families of sets of worlds. We show how these more fine-grained models support natural actions of “evidence management”, ranging from update with external new information to internal rearrangement. We show how this perspective leads to new richer languages for existing neighborhood semantics for modal logic. Our main results are relative completeness theorems for the resulting dynamic logic of evidence.

NEIGHBORHOOD SEMANTICS FOR INTENTIONAL OPERATORS

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 □ in normal modal logics. In this paper an alternative semantics, modelled on neighborhood semantics for □, is given and assessed.

Anselm’s Logic of Agency

The origins of treating agency as a modal concept go back at least to the 11th century when Anselm, Archbishop of Canterbury, provided a modal explication of the Latin facere ‘to do’, which can be formalized within the context of modern modal logic and neighborhood semantics. The agentive logic induced by his conception satisfies the traditional square of opposition, but also has some unique properties which reflect the fact that Anselm’s modal view of agency is grounded strongly in non-logical philosophical and theological considerations. We show that the logic modeling Anselm’s theory of agency provides an interesting alternative to standard logics of agency based on stit-theory.

On a Modal Epistemic Axiom Emerging from McDermott-Doyle Logics

An important question in modal nonmonotonic logics concerns the limits of propositional definability for logics of the McDermott-Doyle family. Inspired by this technical question we define a variant of autoepistemic logic which provably corresponds to the logic of the McDermott-Doyle family that is based on themodal axiom p5: ◊p⊃ (¬sp⊃s¬sp). This axiomis a naturalweakening of classical negative introspection restricting its scope to possible facts. It closely resembles the axiom w5: p⊃ (¬sp⊃s¬sp) which restricts the effect of negative introspection to true facts. We examine p5 in the context of classical possible-worlds Kripke models, providing results for correspondence, completeness and the finite model property. We also identify the corresponding condition for p5 in the context of neighbourhood semantics. Although rather natural epistemically, this axiom has not been investigated in classical modal epistemic reasoning, probably because its addition to S4 gives the well-known strong modal system S5.

Rank-1 Modal Logics are Coalgebraic

Coalgebras provide a unifying semantic framework for a wide variety of modal logics. It has previously been shown that the class of coalgebras for an endofunctor can always be axiomatized in rank 1. Here we establish the converse, i.e. every rank-1 modal logic has a sound and strongly complete coalgebraic semantics. This is achieved by constructing for a given modal logic a canonical coalgebraic semantics, consisting of a signature functor and interpretations of modal operators, which turns out to be final among all such structures. The canonical semantics may be seen as a coalgebraic reconstruction of neighbourhood semantics, broadly construed. A finitary restriction of the canonical semantics yields a canonical weakly complete semantics which moreover enjoys the Hennessy–Milner property. As a consequence, the machinery of coalgebraic modal logic, in particular generic decision procedures and upper complexity bounds, becomes applicable to arbitrary rank-1 modal logics, without regard to their semantic status; we thus obtain purely syntactic versions of such results. As an extended example, we apply our framework to recently defined deontic logics. In particular, our methods lead to the new result that these logics are strongly complete.

QUANTIFIED LOGIC OF AWARENESS AND IMPOSSIBLE POSSIBLE WORLDS

Among the many possible approaches to dealing with logical omniscience, I consider here awareness and impossible worlds structures. The former approach, pioneered by Fagin and Halpern, distinguishes between implicit and explicit knowledge, and avoids logical omniscience with respect to explicit knowledge. The latter, developed by Rantala and by Hintikka, allows for the existence of logically impossible worlds to which the agents are taken to have “epistemological” access; since such worlds need not behave consistently, the agents’ knowledge is fallible relative to logical omniscience. The two approaches are known to be equally expressive in propositional systems interpreted over Kripke semantics. In this paper I show that the two approaches are equally expressive in propositional systems interpreted over Montague-Scott (neighborhood) semantics. Furthermore, I provide predicate systems of both awareness and impossible worlds structures interpreted on neighborhood semantics and prove the two systems to be equally expressive.

Logics from Galois connections

In this paper, Information Logic of Galois Connections (ILGC) suited for approximate reasoning about knowledge is introduced. In addition to the three classical propositional logic axioms and the inference rule of modus ponens, ILGC contains only two auxiliary rules of inference mimicking the performance of Galois connections of lattice theory, and this makes ILGC comfortable to use due to the flip-flop property of the modal connectives. Kripke-style semantics based on information relations is defined for ILGC. It is also shown that ILGC is equivalent to the minimal tense logic K t , and decidability and completeness of ILGC follow from this observation. Additionally, relationship of ILGC to the so-called classical modal logics is studied. Namely, a certain composition of Galois connection mappings forms a lattice-theoretical interior operator, and this motivates us to axiomatize a logic of these compositions. It turns out that this logic satisfies the axioms of the non-normal logic EMT4. Hence, EMT4 can be viewed to be embedded in ILGC. EMT4 is complete with respect to the neighbourhood semantics. Here, we introduce an alternative semantics for EMT4. This is done by defining the so-called interior models, and completeness of EMT4 is proved with respect to the interior semantics.

Restricted Classical Modal Logics

We consider a family of noncongruential modal logics obtained by restricting the smallest classical modal logic E and some of its extensions. We show that these logics are also properly contained in Lemmon's S0.5; semantics for them are adapted from Cresswell's semantics for S0.5 and from neighborhood semantics for classical modal logics. Some extensions of these logics by means of usual modal logical axioms are also considered, and determination results proved. As a further example, we also show how to obtain restricted versions of some of Chellas and Segerberg's prenormal modal logics.

From onions to broccoli: generalizing Lewis' counterfactual logic

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.

Logic of primary-conditionals and secondary-conditionals

Firstly, the authors analyzed the properties of primary-onditionals and secondary-conditionals, establish the minimum system C2Lm 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 C2Lm, introduce a general concept of canonical model by the neighborhood semantics, and then prove the completeness of C2Lm by the canonical model. Finally, according to the technical results of the minimum system C2Lm, the authors discuss some of the important problems concerning primary-conditionals and secondary-onditionals.

First-Order Classical Modal Logic

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.