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 terms of validity in Kripke frames, a modal formula expresses a universal monadic second-order condition. Those modal formulae which are equivalent to first-order 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 (...) results, and outline further research perspectives. (shrink)

This book provides a detailed exposition of one of the most practical and popular methods of proving theorems in logic, called Natural Deduction. It is presented both historically and systematically. Also some combinations with other known proof methods are explored. The initial part of the book deals with Classical Logic, whereas the rest is concerned with systems for several forms of Modal Logics, one of the most important branches of modern logic, which has wide applicability.

This volume is dedicated to Leo Esakia's contributions to the theory of modal and intuitionistic systems. Consisting of 10 chapters, written by leading experts, this volume discusses Esakia’s original contributions and consequent developments that have helped to shape duality theory for modal and intuitionistic logics and to utilize it to obtain some major results in the area. Beginning with a chapter which explores Esakia duality for S4-algebras, the volume goes on to explore Esakia duality for Heyting algebras and its generalizations (...) to weak Heyting algebras and implicative semilattices. The book also dives into the Blok-Esakia theorem and provides an outline of the intuitionistic modal logic KM which is closely related to the Gödel-Löb provability logic GL. One chapter scrutinizes Esakia’s work interpreting modal diamond as the derivative of a topological space within the setting of point-free topology. The final chapter in the volume is dedicated to the derivational semantics of modal logic and other related issues. (shrink)

Our preferences determine how we act and think, but exactly what the mechanics are and how they work is a central cause of concern in many disciplines. This book uses techniques from modern logics of information flow and action to develop a unified new theory of what preference is and how it changes. The theory emphasizes reasons for preference, as well as its entanglement with our beliefs. Moreover, the book provides dynamic logical systems which describe the explicit triggers driving preference (...) change, including new information, suggestions, and commands. In sum, the book creates new bridges between many fields, from philosophy and computer science to economics, linguistics, and psychology. For the experienced scholar access to a large body of recent literature is provided and the novice gets a thorough introduction to the action and techniques of dynamic logic. (shrink)

In the present paper, we continue the research in Zhao (2021, Logic J. IGPL) to develop the Sahlqvist completeness theory for hybrid logic with satisfaction operators and downarrow binders |$\mathcal {L}( @, {\downarrow })$|⁠. We define the class of restricted Sahlqvist formulas for |$\mathcal {L}( @, {\downarrow })$| following the ideas in Conradie and Robinson (2017, J. Logic Comput., 27, 867–900), but we follow a different proof strategy which is purely proof-theoretic, namely showing that for every restricted Sahlqvist formula |$\varphi (...) $| and its hybrid pure correspondence |$\pi $|⁠, |$\textbf {K}_{\mathcal {H}( @, {\downarrow })}+\varphi $| proves |$\pi $|⁠; therefore, |$\textbf {K}_{\mathcal {H}( @, {\downarrow })}+\varphi $| is complete with respect to the class of frames defined by |$\pi $|⁠, using a modified version |$\textsf {ALBA}^{{\downarrow }}_{\textsf {Modified}}$| of the algorithm |$\textsf {ALBA}^{{\downarrow }}$| defined in Zhao (2021, Logic J. IGPL). (shrink)

In the present paper, we develop the algorithmic correspondence theory for hybrid logic with binder |$\mathcal {H}(@, \downarrow )$|⁠. We define the class of Sahlqvist inequalities for |$\mathcal {H}(@, \downarrow )$|⁠, and each inequality of which is shown to have a first-order frame correspondent effectively computable by an algorithm |$\textsf {ALBA}^{\downarrow }$|⁠.

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 (...) are desirable, and in this paper I present two such results. The main developments in this area can be split into two directions, differing in the question whether the ordering of time is world-independent or not. Semantically, this difference appears in the discussion whether T x W-frames or Kamp-frames (resp. Ockham-frames) provide a suitable semantics for combinations of tense and modality. Here, two calculi are presented, the first adequate with respect to Kamp-semantics, the second to T x Wsemantics. (Both calculi contain an appropriate version of Gabbay's irreflexivity rule.) Furthermore, the proposed constructions of canonical frames simplify some of those which have hitherto been discussed. (shrink)

Topic of the paper is Q-logic - a logic of agency in its temporal and modal context. Q-logic 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 Q-operator (also known as Δ- or cstit-operator), which has been discussed independently by E v. Kutschera and B. E Chellas, is investigated here in respect of its (...) relation to other temporal and modal operators. The main result of the paper, then, is a completeness result for a calculus of Q-logic with respect to a semantics defined on the tree-approach to agency as introduced and developed by, among others, E v. Kutschera and N. D. Belnap. (shrink)

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 well-established first-order topological language.

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 PSPACE-complete, 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.

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 Benthem-style and Goldblatt-Thomason-style characterizations. We (...) extract the notion of \blacksquare-realizer (a generalization of bulldozing) as an essence from the proofs of our results. Second, we generalize the notion of \blacksquare-realizer and prove quite general versions of these semantical characterization results. The known and previously unknown characterization results for almost all of the languages extended with modal operators already proposed will be immediate corollaries. (shrink)

In this paper I present a dynamic-epistemic hybrid logic for reasoning about information and intention changes in situations of strategic interaction. I provide a complete axiomatization for this logic, and then use it to study intentions-based transformations of decision problems.

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.

In [14], we studied the computational behaviour of various first-order 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) two-variable fragment of first-order logic with binary pred-icates interpreting the (...) metric. The frame conditions needed correspond rather directly with a Boolean modal logic that is, again, of the same expressivity as the two-variable fragment. We use this representation to derive an axiomatisation of the modal hybrid variant of the two-variable fragment, discuss the compactness property in distance logics, and derive some results on (the failure of) interpolation in distance logics of various expressive power. (shrink)

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 (...) monomodal logics, of nominals and the difference operator by normal operators, of monotonic monomodal logics by normal bimodal logics, of polyadic normal modal logics by polymodal normal modal logics, and of intuitionistic modal logics by normal bimodal logics. (shrink)

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

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.

This is an extended version of the lectures given during the 12-thConference 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 (...) may remarkably improve the situation inmodal proof theory. (shrink)

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 well-known 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 $\langle W,R\rangle$ (...) is isomorphic as a directed graph to $\langle \wp(S)\setminus\{\emptyset\},\supseteq\rangle$. Prucnal [32] proved that the modal logic of Medvedev frames is not finitely axiomatizable. Here we continue the study of Medvedev frames with extended modal languages. Our results concern definability. We show that the class of Medvedev frames is definable by a formula in the language of tense logic, i.e., with a converse modality for quantifying over supersets in Medvedev frames, extended with any one of the following standard devices: nominals (for naming nodes), a difference modality (for quantifying over those $y$ such that $x\not= y$), or a complement modality (for quantifying over those $y$ such that $x\not\supseteq y$). It follows that either the logic of Medvedev frames in one of these tense languages is finitely axiomatizable---which would answer the open question of whether Medvedev's [31] "logic of finite problems'' is decidable---or else the minimal logics in these languages extended with our defining formulas are the beginnings of infinite sequences of frame-incomplete logics. (shrink)

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)

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 (...) axiomatization for these dynamic relation algebras. The completenessresult uses techniques from modal logic. We also lookat the variety generated by the class of dynamic relation algebras and note that there exist nonrepresentable algebras in this variety, ones which cannot be construedas spaces of relations. These results are also proved for an extension to a signature containing atomic tests and union. (shrink)

Our concern is the axiomatisation problem for modal and algebraic logics that correspond to various fragments of two-variable first-order logic with counting quantifiers. In particular, we consider modal products with Diff, the propositional unimodal logic of the difference operator. We show that the two-dimensional product logic $Diff \times Diff$ is non-finitely 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 two-variable first-order (...) logic with counting to two) is non-finitely axiomatisable over $Diff \times Diff$ , but can be axiomatised by adding infinitely many Sahlqvist axioms. These are the first examples of products of finitely axiomatisable modal logics that are not finitely axiomatisable, but axiomatisable by explicit infinite sets of canonical axioms. (shrink)

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 (...) about branching time. (shrink)

We introduce and study hierarchies of extensions of the propositional modal and temporal languages with pairs of new syntactic devices: point of reference-reference 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 (...) Stavi's temporal operators, as well as nominals (names, clock variables), are definable in them. Universal validity in these languages is proved undecidable. The basic modal and temporal logics with reference pointers are uniformly axiomatized and a strong completeness theorem is proved for them and extended to some classes of their extensions. (shrink)

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 (...) method of axiomatization hinges upon the fact that a "difference" operator is definable in hyperboolean algebras, and makes use of additional non-Hilbert-style rules. Finally, we discuss a number of open questions and directions for further research. (shrink)

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 (...) are elementary canonical and thus extend Sahlqvist’s theorem over them. In particular, we give a simple example of an inductive formula which is not frame-equivalent to any Sahlqvist formula. Then, after a deeper analysis of the inductive formulae as set-theoretic operators in descriptive and Kripke frames, we establish a somewhat stronger model-theoretic characterization of these formulae in terms of a suitable equivalence to syntactically simpler formulae in the extension of the language with reversive modalities. Lastly, we study and characterize the elementary canonical formulae in reversive languages with nominals, where the relevant notion of persistence is with respect to discrete frames. (shrink)

ABSTRACT A complete axiomatic system CTLrp is introduced for a temporal logic for finitely branching ω+ -trees in a language extended with so called reference pointers. Syntactic and semantic interpretations are constructed for the branching time computation tree logic CTL* into CTLrp. 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 about branching time. (shrink)

A certain type of inference rules in modal logics, generalizing Gabbay's Irreflexivity rule, is introduced and some general completeness results about modal logics axiomatized with such rules are proved.

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.

Many powerful logics exist today for reasoning about multi-agent systems, but in most of these it is hard to reason about an infinite or indeterminate number of agents. Also the naming schemes used in the logics often lack expressiveness to name agents in an intuitive way.To obtain a more expressive language for multi-agent reasoning and a better naming scheme for agents, we introduce a family of logics called term-modal logics. A main feature of our logics is the use of modal (...) operators indexed by the terms of the logics. Thus, one can quantify over variables occurring in modal operators. In term-modal logics agents can be represented by terms, and knowledge of agents is expressed with formulas within the scope of modal operators. (shrink)

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.

This work is divided in two papers (Part I and Part II). In Part I, we introduced the class of Rare-logics for which the set of terms indexing the modal operators are hierarchized in two levels: the set of Boolean terms and the set of terms built upon the set of Boolean terms. By investigating different algebraic properties satisfied by the models of the Rare-logics, reductions for decidability were established by faithfully translating the Rare-logics into more standard modal logics (some (...) of them contain the universal modal operator).In Part II, we push forward the results from Part I. For Rare-logics with nominals (present at the level of formulae and at the level of modal expressions), we show that the constructions from Part I can be extended although it is technically more involved. We also characterize a class of standard modal logics for which the universal modal operator can be eliminated as far as satifiability is concerned. Although the previous results have a semantic flavour, we are also able to define proof systems for Rare-logics from existing proof systems for the corresponding standard modal logics. Last, but not least, decidability results for Rare-logics are established uniformly, in particular for information logics derived from rough set theory. (shrink)

The paper introduces a first-order 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.

For branching-time 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 fan-names. 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 (...) extended branching-time logic and prove its soundness and completeness with respect to bundle tree semantics. Finally, we show how our axiomatic system can be extended with a variety of important additional operators, such as Since and Until, a global difference operator, operators for undivided and divided histories, reference pointers, etc. (shrink)

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 so-called 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 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 well-known defect of Reichenbach’s tense schema, namely that it gives multiple representations to sentences in the future perfect and the future-in-the-past. It also makes it easy to define an iterative schema for tense that allows for multiple points of reference, a possibility noted (...) by Prior and demanded by Comrie, and we sketch how this schema can be generalized to a shift-and-restrict pattern in which special propositional symbols act as restrictors on the range of tense operators. (shrink)

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. (shrink)

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 (a decidable system with the same complexity as orthodox propositional modal logic) to the strong Priorean language (which offers full first-order expressivity).We show that hybrid logic offers a genuinely first-order 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 weaker languages, this requires the use of non-orthodox rules. We discuss these rules in detail and prove non-eliminability and eliminability results. We also show how another type of rule, which reflects the structure of the strong Priorean language, can be employed to give an even wider coverage of frame classes. We show that this deductive apparatus gets progressively simpler as we work our way up the expressivity hierarchy, and conclude the paper by showing that the approach transfers to first-order hybrid logic. (shrink)

The title reflects my conviction that, viewed semantically,modal logic is fundamentally dialogical; this conviction is based on the key role played by the notion of bisimulation in modal model theory. But this dialogical conception of modal logic does not seem to apply to modal proof theory, which is notoriously messy. Nonetheless, by making use of ideas which trace back to Arthur Prior (notably the use of nominals, special proposition symbols which ‘name’ worlds) I will show how to lift the dialogical (...) conception to modal proof theory. I argue that this shift to hybrid logic has consequences for both modal and dialogical logic, and I discuss these in detail. (shrink)

Hybrid languages have both modal and first-order 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 first-order language, that full first-order 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 (...) problems. (shrink)