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 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)

General frames are often used in classical modal logic. Since they are duals of modal algebras, completeness follows automatically as with algebras but the intuitiveness of Kripke frames is also retained. This paper develops basics of general frames for relevant modal logics by showing that they share many important properties with general frames for classical modal logic.

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)

This article is about Avicenna’s account of syllogisms comprising opposite premises. We examine the applications and the truth conditions of these syllogisms. Finally, we discuss the relation between these syllogisms and the principle of non-contradiction.

In this article, we tell a story about incompleteness in modal logic. The story weaves together an article of van Benthem, “Syntactic aspects of modal incompleteness theorems,” and a longstanding open question: whether every normal modal logic can be characterized by a class of completely additive modal algebras, or as we call them, ${\cal V}$-baos. Using a first-order reformulation of the property of complete additivity, we prove that the modal logic that starred in van Benthem’s article resolves the open question (...) in the negative. In addition, for the case of bimodal logic, we show that there is a naturally occurring logic that is incomplete with respect to ${\cal V}$-baos, namely the provability logic $GLB$. We also show that even logics that are unsound with respect to such algebras do not have to be more complex than the classical propositional calculus. On the other hand, we observe that it is undecidable whether a syntactically defined logic is ${\cal V}$-complete. After these results, we generalize the Blok Dichotomy to degrees of ${\cal V}$-incompleteness. In the end, we return to van Benthem’s theme of syntactic aspects of modal incompleteness. (shrink)

Part I of this paper is developed in the tradition of Stone-type dualities, where we present a new topological representation for general lattices (influenced by and abstracting over both Goldblatt's [17] and Urquhart's [46]), identifying them as the lattices of stable compact-opens of their dual Stone spaces (stability refering to a closure operator on subsets). The representation is functorial and is extended to a full duality.In part II, we consider lattice-ordered algebras (lattices with additional operators), extending the Jónsson and Tarski (...) representation results [30] for Boolean algebras with Operators. Our work can be seen as developing, and indeed completing, Dunn's project of gaggle theory [13, 14]. We consider general lattices (rather than Boolean algebras), with a broad class of operators, which we dubb normal, and which includes the Jónsson-Tarski additive operators. Representation of l-algebras is extended to full duality. (shrink)

David Lewis proved in 1974 that all logics without iterative axioms are weakly complete. In this paper we extend Lewis's ideas and provide a proof that such logics are canonical and so strongly complete. This paper also discusses the differences between relational and neighborhood frame semantics and poses a number of open questions about the latter.

In this paper, we are going to analyze the phenomenon of modal incompleteness from an algebraic point of view. The usual method of showing that a given logic L is incomplete is to show that for some L and some cannot be separated from by a suitably wide class of complete algebras — usually Kripke algebras. We are going to show that classical examples of incomplete logics, e.g., Fine logic, are not complete with respect to any class of complete BAOs. (...) Even above Grz it is possible to find a continuum of such logics, which immediately implies the existence of a continuum of neighbourhood-incomplete Grz logics. Similar results can be proved for Löb logics. In addition, completely incomplete logics above Grz may be found uniformly as a result of failures of some admissible rule of a special kind. (shrink)

We introduce a bimodal epistemic logic intended to capture knowledge as truth in all epistemically alternative states and belief as a generalised ‘majority’ quantifier, interpreted as truth in most of the epistemically alternative states. This doxastic interpretation is of interest in knowledge-representation applications and it also holds an independent philosophical and technical appeal. The logic comprises an epistemic modal operator, a doxastic modal operator of consistent and complete belief and ‘bridge’ axioms which relate knowledge to belief. To capture the notion (...) of a ‘majority’ we use the ‘large sets’ introduced independently by K. Schlechta and V. Jauregui, augmented with a requirement of completeness, which furnishes a ‘weak ultrafilter’ concept. We provide semantics in the form of possible-worlds frames, properly blending relational semantics with a version of general Scott–Montague frames and we obtain soundness and completeness results. We examine the validity of certain epistemic principles discussed in the literature, in particular some of the ‘bridge’ axioms discussed by W. Lenzen and R. Stalnaker, as well as the ‘paradox of the perfect believer’, which is not a theorem of. (shrink)

This paper looks at the concept of neighborhood canonicity introduced by BRIAN CHELLAS [2]. We follow the lead of the author's paper [9] where it was shown that every non-iterative logic is neighborhood canonical and here we will show that all logics whose axioms have a simple syntactic form-no intensional operator is in boolean combination with a propositional letter-and which have the finite model property are neighborhood canonical. One consequence of this is that KMcK, the McKinsey logic, is neighborhood canonical, (...) an interesting counterpoint to the results of ROBERT GOLDBLATT and XIAOPING WANG who showed, respectively, that KMcK is not relational canonical [5] and that KMcK is not relationally strongly complete [11]. (shrink)

We investigate the role of coalgebraic predicate logic, a logic for neighborhood frames first proposed by Chang, in the study of monotonic modal logics. We prove analogues of the Goldblatt–Thomason theorem and Fine’s canonicity theorem for classes of monotonic neighborhood frames closed under elementary equivalence in coalgebraic predicate logic. The elementary equivalence here can be relativized to the classes of monotonic, quasi-filter, augmented quasi-filter, filter, or augmented filter neighborhood frames, respectively. The original, Kripke-semantic versions of the theorems follow as a (...) special case concerning the classes of augmented filter neighborhood frames. (shrink)

We give an extension of the Jónsson‐Tarski representation theorem for both normal and non‐normal modal algebras so that it preserves countably many infinite meets and joins. In order to extend the Jónsson‐Tarski representation to non‐normal modal algebras we consider neighborhood frames instead of Kripke frames just as Došen's duality theorem for modal algebras, and to deal with infinite meets and joins, we make use of Q‐filters, which were introduced by Rasiowa and Sikorski, instead of prime filters. By means of the (...) extended representation theorem, we show that every predicate modal logic, whether it is normal or non‐normal, has a model defined on a neighborhood frame with constant domains, and we give a completeness theorem for some predicate modal logics with respect to classes of neighborhood frames with constant domains. Similarly, we show a model existence theorem and a completeness theorem for infinitary modal logics which allow conjunctions of countably many formulas. (shrink)