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Modal logic with names
Journal of Philosophical Logic 22 (6):607  636 (1993)
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This is an extended version of the lectures given during the 12thConference on Applications of Logic in Philosophy and in the Foundationsof Mathematics in Szklarska Poręba. It contains a surveyof modal hybrid logic, one of the branches of contemporary modal logic. Inthe ﬁrst part a variety of hybrid languages and logics is presented with adiscussion of expressivity matters. The second part is devoted to thoroughexposition of proof methods for hybrid logics. The main point is to showthat application of hybrid logics (...) 

We study hybrid logics in topological semantics. We prove that hybrid logics of separation axioms are complete with respect to certain classes of finite topological models. This characterisation allows us to obtain several further results. We prove that aforementioned logics are decidable and PSPACEcomplete, the logics of T 1 and T 2 coincide, the logic of T 1 is complete with respect to two concrete structures: the Cantor space and the rational numbers. 

For branchingtime temporal logic based on an Ockhamist semantics, we explore a temporal language extended with two additional syntactic tools. For reference to the set of all possible futures at a moment of time we use syntactically designated restricted variables called fannames. For reference to all possible futures alternative to the actual one we use a modification of a difference modality, localized to the set of all possible futures at the actual moment of time.We construct an axiomatic system for this (...) 

This paper contributes to the principled construction of tableaubased decision procedures for hybrid logic with global, difference, and converse modalities. We also consider reflexive and transitive relations. For conversefree formulas we present a terminating control that does not rely on the usual chainbased blocking scheme. Our tableau systems are based on a new model existence theorem. 

In this paper we argue that hybrid logic is the deductive setting most natural for Kripke semantics. We do so by investigating hybrid axiomatics for a variety of systems, ranging from the basic hybrid language to the strong Priorean language . We show that hybrid logic offers a genuinely firstorder perspective on Kripke semantics: it is possible to define base logics which extend automatically to a wide variety of frame classes and to prove completeness using the Henkin method. In the (...) 

We consider algebras on binary relations with two main operators: relational composition and dynamic negation. Relational composition has its standard interpretation, while dynamic negation is an operator familiar to students of Dynamic Predicate Logic (DPL) (Groenendijk and Stokhof, 1991): given a relation R its dynamic negation R is a test that contains precisely those pairs (s,s) for which s is not in the domain of R. These two operators comprise precisely the propositional part of DPL.This paper contains a finite equational (...) 

We show that basic hybridization (adding nominals and @ operators) makes it possible to give straightforward Henkinstyle completeness proofs even when the modal logic being hybridized is higherorder. The key ideas are to add nominals as expressions of type t, and to extend to arbitrary types the way we interpret [email protected]_i$ in propositional and firstorder hybrid logic. This means: interpret [email protected]_i\alpha _a$ , where $\alpha _a$ is an expression of any type $a$ , as an expression of type $a$ that (...) 

Topic of the paper is Qlogic  a logic of agency in its temporal and modal context. Qlogic may be considered as a basal logic of agency since the most important stitoperators discussed in the literature can be defined or axiomatized easily within its semantical and syntactical framework. Its basic agent dependent operator, the Qoperator (also known as Δ or cstitoperator), which has been discussed independently by E v. Kutschera and B. E Chellas, is investigated here in respect of its (...) 



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): 5778, 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 (...) 

Hybrid languages have both modal and firstorder characteristics: a Kripke semantics, and explicit variable binding apparatus. This paper motivates the development of hybrid languages, sketches their history, and examines the expressive power of three hybrid binders. We show that all three binders give rise to languages strictly weaker than the corresponding firstorder language, that full firstorder expressivity can be gained by adding the universal modality, and that all three binders can force the existence of infinite models and have undecidable satisfiability (...) 

Many of the formalisms used in Attribute Value grammar are notational variants of languages of propositional modal logic, and testing whether two Attribute Value Structures unify amounts to testing for modal satisfiability. In this paper we put this observation to work. We study the complexity of the satisfiability problem for nine modal languages which mirror different aspects of AVS description formalisms, including the ability to express reentrancy, the ability to express generalisations, and the ability to express recursive constraints. Two main (...) 



This paper deals with two main topics: One is a semantical investigation for a bimodal language with a modal operator \blacksquare associated with the intersection of the accessibility relation R and the inequality ≠. The other is a generalization of some of the former results to general extended languages with modal operators. First, for our language L\sb{\square\blacksquare}, we prove that Segerberg's theorem (equivalence between finite frame property and finite model property) fails and establish both van Benthemstyle and GoldblattThomasonstyle characterizations. We (...) 

In recent years combinations of tense and modality have moved intothe focus of logical research. From a philosophical point of view, logical systems combining tense and modality are of interest because these logics have a wide field of application in original philosophical issues, for example in the theory of causation, of action, etc. But until now only methods yielding completeness results for propositional languages have been developed. In view of philosophical applications, analogous results with respect to languages of predicate logic (...) 

In this paper we argue that Prior and Reichenbach are best viewed as allies, not antagonists. We do so by combining the central insights of Prior and Reichenbach in the framework of hybrid tense logic. This overcomes a wellknown defect of Reichenbach’s tense schema, namely that it gives multiple representations to sentences in the future perfect and the futureinthepast. It also makes it easy to define an iterative schema for tense that allows for multiple points of reference, a possibility noted (...) 



Our concern is the axiomatisation problem for modal and algebraic logics that correspond to various fragments of twovariable firstorder logic with counting quantifiers. In particular, we consider modal products with Diff, the propositional unimodal logic of the difference operator. We show that the twodimensional product logic $Diff \times Diff$ is nonfinitely axiomatisable, but can be axiomatised by infinitely many Sahlqvist axioms. We also show that its ‘square’ version (the modal counterpart of the substitution and equality free fragment of twovariable firstorder (...) 

We survey main developments, results, and open problems on interval temporal logics and duration calculi. We present various formal systems studied in the literature and discuss their distinctive features, emphasizing on expressiveness, axiomatic systems, and (un)decidability results. 



We introduce and study hierarchies of extensions of the propositional modal and temporal languages with pairs of new syntactic devices: point of referencereference pointer which enable semantic references to be made within a formula. We propose three different but equivalent semantics for the extended languages, discuss and compare their expressiveness. The languages with reference pointers are shown to have great expressive power (especially when their frugal syntax is taken into account), perspicuous semantics, and simple deductive systems. For instance, Kamp's and (...) 

This papers gives a survey of recent results about simulations of one class of modal logics by another class and of the transfer of properties of modal logics under extensions of the underlying modal language. We discuss: the transfer from normal polymodal logics to their fusions, the transfer from normal modal logics to their extensions by adding the universal modality, and the transfer from normal monomodal logics to minimal tense extensions. Likewise, we discuss simulations of normal polymodal logics by normal (...) 



Viewing the language of modal logic as a language for describing directed graphs, a natural type of directed graph to study modally is one where the nodes are sets and the edge relation is the subset or superset relation. A wellknown example from the literature on intuitionistic logic is the class of Medvedev frames $\langle W,R\rangle$ where $W$ is the set of nonempty subsets of some nonempty finite set $S$, and $xRy$ iff $x\supseteq y$, or more liberally, where $\langle W,R\rangle$ (...) 

We introduce and study a variety of modal logics of parallelism, orthogonality, and affine geometries, for which we establish several completeness, decidability and complexity results and state a number of related open, and apparently difficult problems. We also demonstrate that lack of the finite model property of modal logics for sufficiently rich affine or projective geometries (incl. the real affine and projective planes) is a rather common phenomenon. 

A complete axiomatic system CTL$_{rp}$ is introduced for a temporal logic for finitely branching $\omega^+$trees in a temporal language extended with so called reference pointers. Syntactic and semantic interpretations are constructed for the branching time computation tree logic CTL* into CTL$_{rp}$. In particular, that yields a complete axiomatization for the translations of all valid CTL*formulae. Thus, the temporal logic with reference pointers is brought forward as a simpler (with no path quantifiers), but in a way more expressive medium for reasoning (...) 

On the 4th of December 1967, Hans Kamp sent his UCLA seminar notes on the logic of ‘now’ to Arthur N. Prior. Kamp’s twodimensional analysis stimulated Prior to an intense burst of creativity in which he sought to integrate Kamp’s work into tense logic using a onedimensional approach. Prior’s search led him through the work of Castañeda, and back to his own work on hybrid logic: the first made temporal reference philosophically respectable, the second made it technically feasible in a (...) 

We show that basic hybridization makes it possible to give straightforward Henkinstyle completeness proofs even when the modal logic being hybridized is higherorder. 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}[email protected]_i$\end{document} in propositional and firstorder hybrid logic. This means: interpret \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{69pt} \begin{document}[email protected]_i\alpha _a$\end{document}, where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} (...) 

The paper introduces a firstorder theory in the language of predicate tense logic which contains a single simple axiom. It is shewn that this theory enables times to be referred to and sentences involving ‘now’ and ‘then’ to be formalised. The paper then compares this way of increasing the expressive capacity of predicate tense logic with other mechanisms, and indicates how to generalise the results to other modal and tense systems. 

A hybrid logic is obtained by adding to an ordinary modal logic further expressive power in the form of a second sort of propositional symbols called nominals and by adding socalled satisfaction operators. In this paper we consider hybridized versions of S5 (“the logic of everywhere”) and the modal logic of inequality (“the logic of elsewhere”). We give natural deduction systems for the logics and we prove functional completeness results. 

In [14], we studied the computational behaviour of various firstorder and modal languages interpreted in metric or weaker distance spaces. [13] gave an axiomatisation of an expressive and decidable metric logic. The main result of this paper is in showing that the technique of representing metric spaces by means of Kripke frames can be extended to cover the modal (hybrid) language that is expressively complete over metric spaces for the (undecidable) twovariable fragment of firstorder logic with binary predicates interpreting the (...) 

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 firstorder 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 terms of validity in Kripke frames, a modal formula expresses a universal monadic secondorder condition. Those modal formulae which are equivalent to firstorder conditions are called elementary. Modal formulae which have a certain persistence property which implies their validity in all canonical frames of modal logics axiomatized with them, and therefore their completeness, are called canonical. This is a survey of a recent and ongoing study of the class of elementary and canonical modal formulae. We summarize main ideas and (...) 

We study the general problem of axiomatizing structures in the framework of modal logic and present a uniform method for complete axiomatization of the modal logics determined by a large family of classes of structures of any signature. 





In this paper we study the expressive power and definability for modal languages interpreted on topological spaces. We provide topological analogues of the van Benthem characterization theorem and the Goldblatt–Thomason definability theorem in terms of the wellestablished firstorder topological language. 

In the paper we present a relatively simple proof of cut elimination theorem for variety of hybrid logics in the language with satisfaction operators and universal modality. The proof is based on the strategy introduced originally in the framework of hypersequent calculi but it works well also for standard sequent calculi. Sequent calculus examined in the paper works on so called satisfaction formulae and cover all logics adequate with respect to classes of frames defined by so called geometric conditions. 

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 firstorder logic which is invariant for generated (...) 

ABSTRACT This paper presents the axioinatization—without the rule of irreflexivity—of the modal logic of inequality as well as a method for proving its completeness. This method uses the technics of the frame of subordination. 

Hyperboolean algebras are Boolean algebras with operators, constructed as algebras of complexes (or, power structures) of Boolean algebras. They provide an algebraic semantics for a modal logic (called here a {\em hyperboolean modal logic}) with a Kripke semantics accordingly based on frames in which the worlds are elements of Boolean algebras and the relations correspond to the Boolean operations. We introduce the hyperboolean modal logic, give a complete axiomatization of it, and show that it lacks the finite model property. The (...) 

A complete axiomatic system CTL$_{rp}$ is introduced for a temporal logic for finitely branching $\omega^+$trees in a temporal language extended with so called reference pointers. Syntactic and semantic interpretations are constructed for the branching time computation tree logic CTL$^{*}$ into CTL$_{rp}$. In particular, that yields a complete axiomatization for the translations of all valid CTL$^{*}$formulae. Thus, the temporal logic with reference pointers is brought forward as a simpler (with no path quantifiers), but in a way more expressive medium for reasoning (...) 

We consider some modal languages with a modal operator $D$ whose semantics is based on the relation of inequality. Basic logical properties such as definability, expressive power and completeness are studied. Also, some connections with a number of other recent proposals to extend the standard modal language are pointed at. 

We study term modal logics, where modalities can be indexed by variables that can be quantified over. We suggest that these logics are appropriate for reasoning about systems of unboundedly many reasoners and define a notion of bisimulation which preserves propositional fragment of term modal logics. Also we show that the propositional fragment is already undecidable but that its monodic fragment is decidable, and expressive enough to include interesting assertions. 

Several extensions of the basic modal language are characterized in terms of interpolation. Our main results are of the following form: Language ℒ' is the least expressive extension of ℒ with interpolation. For instance, let ℳ be the extension of the basic modal language with a difference operator [7]. Firstorder logic is the least expressive extension of ℳ with interpolation. These characterizations are subsequently used to derive new results about hybrid logic, relation algebra and the guarded fragment. 



We generalize and extend the class of Sahlqvist formulae in arbitrary polyadic modal languages, to the class of so called inductive formulae. To introduce them we use a representation of modal polyadic languages in a combinatorial style and thus, in particular, develop what we believe to be a better syntactic approach to elementary canonical formulae altogether. By generalizing the method of minimal valuations à la Sahlqvist–van Benthem and the topological approach of Sambin and Vaccaro we prove that all inductive formulae (...) 