The paper provides an alternative interpretation of ‘pair points’, discussed in Beall et al., "On the ternary relation and conditionality", J. of Philosophical Logic 41(3), 595-612. Pair points are seen as points viewed from two different ‘perspectives’ and the latter are explicated in terms of two independent valuations. The interpretation is developed into a semantics using pairs of Kripke models (‘pair models’). It is demonstrated that, if certain conditions are fulfilled, pair models are validity-preserving copies of positive substructural models. This (...) yields a general soundness and completeness result for a variety of (positive) substructural logics with respect to multimodal Kripke frames with binary accessibility relations. In addition, an epistemic interpretation of pair models is provided. (shrink)

In the early 1960s, to prove undecidability of monadic fragments of sublogics of the predicate modal logic $$\textbf{QS5}$$ QS 5 that include the classical predicate logic $$\textbf{QCl}$$ QCl, Saul Kripke showed how a classical atomic formula with a binary predicate letter can be simulated by a monadic modal formula. We consider adaptations of Kripke’s simulation, which we call the Kripke trick, to various modal and superintuitionistic predicate logics not considered by Kripke. We also discuss settings where the Kripke trick does (...) not work and where, as a result, decidability of monadic modal predicate logics can be obtained. (shrink)

The paper considers the set ML 1 of first-order polymodal formulas the modal operators in which can be applied to subformulas of at most one free variable. Using a mosaic technique, we prove a general satisfiability criterion for formulas in ML 1 , which reduces the modal satisfiability to the classical one. The criterion is then used to single out a number of new, in a sense optimal, decidable fragments of various modal predicate logics.

In this paper1 we study admissible consecutions in multi-modal logics with the universal modality. We consider extensions of multi-modal logic S4n augmented with the universal modality. Admissible consecutions form the largest class of rules, under which a logic is closed. We propose an approach based on the context effective finite model property. Theorem 7, the main result of the paper, gives sufficient conditions for decidability of admissible consecutions in our logics. This theorem also provides an explicit algorithm for recognizing such (...) consecutions. Some applications to particular logics with the universal modality are given. (shrink)

We introduce the horizontal and vertical topologies on the product of topological spaces, and study their relationship with the standard product topology. We show that the modal logic of products of topological spaces with horizontal and vertical topologies is the fusion S4 ⊕ S4. We axiomatize the modal logic of products of spaces with horizontal, vertical, and standard product topologies.We prove that both of these logics are complete for the product of rational numbers ℚ × ℚ with the appropriate topologies.

We prove completeness and decidability results for a family of combinations of propositional dynamic logic and unimodal doxastic logics in which the modalities may interact. The kind of interactions we consider include three forms of commuting axioms, namely, axioms similar to the axiom of perfect recall and the axiom of no learning from temporal logic, and a Church–Rosser axiom. We investigate the influence of the substitution rule on the properties of these logics and propose a new semantics for the test (...) operator to avoid unwanted side effects caused by the interaction of the classic test operator with the extra interaction axioms. (shrink)

This paper considers a new class of agent dynamic logics which provide a formal means of specifying and reasoning about the agents activities and informational, motivational and practical aspects of the behaviour of the agents. We present a Hilbert-style deductive system for a basic agent dynamic logic and consider a number of extensions of this logic with axiom schemata formalising interactions between knowledge and commitment (expressing an agent s awareness of her commitments), and interactions between knowledge and actions (expressing no (...) learning and persistence of knowledge after actions). The deductive systems are proved sound and complete with respect to a Kripke-style semantics. Each of the considered logics is shown to have the small model property and therefore decidable. (shrink)

We consider the problem of axiomatizing various natural "successor" logics for 2-dimensional integral spacetime. We provide axiomatizations in monomodal and multimodal languages, and prove completeness theorems. We also establish that the irreflexive successor logic in the "standard" modal language (i.e. the language containing □ and ◊) is not finitely axiomatizable.

We investigate transfer of interpolation in such combinations of modal logic which lead to interaction of the modalities. Combining logics by taking products often blocks transfer of interpolation. The same holds for combinations by taking unions, a generalization of Humberstone's inaccessibility logic. Viewing first-order logic as a product of modal logics, we derive a strong counterexample for failure of interpolation in the finite variable fragments of first-order logic. We provide a simple condition stated only in terms of frames and bisimulations (...) which implies failure of interpolation. Its use is exemplified in a wide range of cases. (shrink)

We investigate an extension of the formalism of interpreted systems by Halpern and colleagues to model the correct behaviour of agents. The semantical model allows for the representation and reasoning about states of correct and incorrect functioning behaviour of the agents, and of the system as a whole. We axiomatise this semantic class by mapping it into a suitable class of Kripke models. The resulting logic, KD45ni-j, is a stronger version of KD, the system often referred to as Standard Deontic (...) Logic. We extend this formal framework to include the standard epistemic notions defined on interpreted systems, and introduce a new doubly-indexed operator representing the knowledge that an agent would have if it operates under the assumption that a group of agents is functioning correctly. We discuss these issues both theoretically and in terms of applications, and present further directions of work. (shrink)

Diversity of agents occurs naturally in epistemic logic, and dynamic logics of information update and belief revision. In this paper we provide a systematic discussion of different sources of diversity, such as introspection ability, powers of observation, memory capacity, and revision policies, and we show how these can be encoded in dynamic epistemic logics allowing for individual variation among agents. Next, we explore the interaction of diverse agents by looking at some concrete scenarios of communication and learning, and we propose (...) a logical methodology to deal with these as well. We conclude with some further questions on the logic of diversity and interaction. (shrink)

We discuss a simple logic to describe one of our favourite games from childhood, hide and seek, and show how a simple addition of an equality constant to describe the winning condition of the seeker makes our logic undecidable. There are certain decidable fragments of first-order logic which behave in a similar fashion with respect to such a language extension, and we add a new modal variant to that class. We discuss the relative expressive power of the proposed logic in (...) comparison to the standard modal counterparts. We prove that the model checking problem for the resulting logic is \(\textsf{P}\) -complete. In addition, by exploring the connection with related product logics, we gain more insight towards having a better understanding of the subtleties of the proposed framework. (shrink)

It is shown that the many-dimensional modal logic K n , determined by products of n-many Kripke frames, is not finitely axiomatisable in the n-modal language, for any $n > 2$ . On the other hand, K n is determined by a class of frames satisfying a single first-order sentence.

We show the first examples of recursively enumerable (even decidable) two-dimensional products of finitely axiomatisable modal logics that are not finitely axiomatisable. In particular, we show that any axiomatisation of some bimodal logics that are determined by classes of product frames with linearly ordered first components must be infinite in two senses: It should contain infinitely many propositional variables, and formulas of arbitrarily large modal nesting-depth.

There are two known general results on the finite model property of commutators [L0,L1]. If L is finitely axiomatizable by modal formulas having universal Horn first-order correspondents, then both [L,K] and [L,S5] are determined by classes of frames that admit filtration, and so they have the fmp. On the negative side, if both L0 and L1 are determined by transitive frames and have frames of arbitrarily large depth, then [L0,L1] does not have the fmp. In this paper we show that (...) commutators with a “weakly connected” component often lack the fmp. Our results imply that the above positive result does not generalize to universally axiomatizable component logics, and even commutators without “transitive” components such as [K3,K] can lack the fmp. We also generalize the above negative result to cases where one of the component logics has frames of depth one only, such as [S4.3,S5] and the decidable product logic S4.3×S5. We also show cases when already half of commutativity is enough to force infinite frames. (shrink)

The simplest bimodal combination of unimodal logics \ and \ is their fusion, \, axiomatized by the theorems of \ for \ and of \ for \, and the rules of modus ponens, necessitation for \ and for \, and substitution. Shehtman introduced the frame product \, as the logic of the products of certain Kripke frames: these logics are two-dimensional as well as bimodal. Van Benthem, Bezhanishvili, ten Cate and Sarenac transposed Shehtman’s idea to the topological semantics and introduced (...) the topological product \, as the logic of the products of certain topological spaces. For almost all well-studies logics, we have \, for example, \. Van Benthem et al. show, by contrast, that \. It is straightforward to define the product of a topological space and a frame: the result is a topologized frame, i.e., a set together with a topology and a binary relation. In this paper, we introduce topological-frame products \ of modal logics, providing a complete axiomatization of \, whenever \ is a Kripke complete Horn axiomatizable extension of the modal logic D: these extensions include \ and \, but not \ or \. We leave open the problem of axiomatizing \, \, and other related logics. When \, our result confirms a conjecture of van Benthem et al. concerning the logic of products of Alexandrov spaces with arbitrary topological spaces. (shrink)

Shehtman introduced bimodal logics of the products of Kripke frames, thereby introducing frame products of unimodal logics. Van Benthem, Bezhanishvili, ten Cate and Sarenac generalize this idea to the bimodal logics of the products of topological spaces, thereby introducing topological products of unimodal logics. In particular, they show that the topological product of S4 and S4 is S4 \ S4, i.e., the fusion of S4 and S4: this logic is strictly weaker than the frame product S4 × S4. Indeed, van (...) Benthem et al. show that S4 \ S4 is the bimodal logic of the particular product space , leaving open the question of whether S4 \ S4 is also complete for the product space . We answer this question in the negative. (shrink)

The simplest combination of unimodal logics \ into a bimodal logic is their fusion, \, axiomatized by the theorems of \. Shehtman introduced combinations that are not only bimodal, but two-dimensional: he defined 2-d Cartesian products of 1-d Kripke frames, using these Cartesian products to define the frame product \. Van Benthem, Bezhanishvili, ten Cate and Sarenac generalized Shehtman’s idea and introduced the topological product \, using Cartesian products of topological spaces rather than of Kripke frames. Frame products have been (...) extensively studied, but much less is known about topological products. The goal of the current paper is to give necessary and sufficient conditions for the topological product to match the frame product, for Kripke complete extensions of \. (shrink)

In this paper we consider the normal modal logics of elementary classes defined by first-order formulas of the form \. We prove that many properties of these logics, such as finite axiomatisability, elementarity, axiomatisability by a set of canonical formulas or by a single generalised Sahlqvist formula, together with modal definability of the initial formula, either simultaneously hold or simultaneously do not hold.

This paper is predicated on the idea that some modal operators are better understood as quantificational expressions over worlds that determine not only first-order facts but modal facts also. In what follows, we will present a framework in which these two types of facts are brought closer together. Structural features will be located in the worlds themselves. This result will be achieved by decomposing worlds into parts, where some of these parts will have “modal import” in the sense that they (...) will determine structure on other worldly parts. The main upshot of this will be a clearer grasp of the interaction between modal notions. (shrink)

In this paper, we introduce a new fragment of the first-order temporal language, called the monodic fragment, in which all formulas beginning with a temporal operator have at most one free variable. We show that the satisfiability problem for monodic formulas in various linear time structures can be reduced to the satisfiability problem for a certain fragment of classical first-order logic. This reduction is then used to single out a number of decidable fragments of first-order temporal logics and of two-sorted (...) first-order logics in which one sort is intended for temporal reasoning. Besides standard first-order time structures, we consider also those that have only finite first-order domains, and extend the results mentioned above to temporal logics of finite domains. We prove decidability in three different ways: using decidability of monadic second-order logic over the intended flows of time, by an explicit analysis of structures with natural numbers time, and by a composition method that builds a model from pieces in finitely many steps. (shrink)

In the propositional modal treatment of two-variable first-order logic equality is modelled by a ‘diagonal’ constant, interpreted in square products of universal frames as the identity relation. Here we study the decision problem of products of two arbitrary modal logics equipped with such a diagonal. As the presence or absence of equality in two-variable first-order logic does not influence the complexity of its satisfiability problem, one might expect that adding a diagonal to product logics in general is similarly harmless. We (...) show that this is far from being the case, and there can be quite a big jump in complexity, even from decidable to the highly undecidable. Our undecidable logics can also be viewed as new fragments of first-order logic where adding equality changes a decidable fragment to undecidable. We prove our results by a novel application of counter machine problems. While our formalism apparently cannot force reliable counter machine computations directly, the presence of a unique diagonal in the models makes it possible to encode both lossy and insertion-error computations, for the same sequence of instructions. We show that, given such a pair of faulty computations, it is then possible to reconstruct a reliable run from them. (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)

Larry Moss and Rohit Parikh used subset semantics to characterize a family of logics for reasoning about knowledge. An important feature of their framework is that subsets always decrease based on the assumption that knowledge always increases. We drop this assumption and modify the semantics to account for logics of knowledge that handle arbitrary changes, that is, changes that do not necessarily result in knowledge increase, such as the update of our knowledge due to an action. We present a system (...) which is complete for subset spaces and prove its decidability. (shrink)

In this paper we improve the results of [2] by proving the product f.m.p. for the product of minimal n-modal and minimal n-temporal logic. For this case we modify the finite depth method introduced in [1]. The main result is applied to identify new fragments of classical first-order logic and of the equational theory of relation algebras, that are decidable and have the finite model property.

We solve a major open problem concerning algorithmic properties of products of ‘transitive’ modal logics by showing that products and commutators of such standard logics asK4,S4,S4.1,K4.3,GL, orGrzare undecidable and do not have the finite model property. More generally, we prove that no Kripke complete extension of the commutator [K4, K4] with product frames of arbitrary finite or infinite depth (with respect to both accessibility relations) can be decidable. In particular, ifl1andl2are classes of transitive frames such that their depth cannot be (...) bounded by any fixedn< ω, then the logic of the class {5ℑ1× ℑ2∣ ℑ1∈l1, ℑ2, ∈l2} is undecidable. (On the contrary, the product of, say,K4and the logic of all transitive Kripke frames of depth ≤n, for some fixedn< ω, is decidable.) The complexity of these undecidable logics ranges from r.e. to co-r.e. and Π11-complete. As a consequence, we give the first known examples of Kripke incomplete commutators of Kripke complete logics. (shrink)

We show that—unlike products of ‘transitive’ modal logics which are usually undecidable—their ‘expanding domain’ relativisations can be decidable, though not in primitive recursive time. In particular, we prove the decidability and the finite expanding product model property of bimodal logics interpreted in two-dimensional structures where one component—call it the ‘flow of time’—is • a finite linear order or a finite transitive tree and the other is composed of structures like • transitive trees/partial orders/quasi-orders/linear orders or only finite such structures expanding (...) over time. The decidability proof is based on Kruskal’s tree theorem, and the proof of non-primitive recursiveness is by reduction of the reachability problem for lossy channel systems. The result is used to show that the dynamic topological logic interpreted in topological spaces with continuous functions is decidable if the number of function iterations is assumed to be finite. (shrink)

This paper is concerned with a propositional modal logic with operators for necessity, actuality and apriority. The logic is characterized by a class of relational structures defined according to ideas of epistemic two-dimensional semantics, and can therefore be seen as formalizing the relations between necessity, actuality and apriority according to epistemic two-dimensional semantics. We can ask whether this logic is correct, in the sense that its theorems are all and only the informally valid formulas. This paper gives outlines of two (...) arguments that jointly show that this is the case. The first is intended to show that the logic is informally sound, in the sense that all of its theorems are informally valid. The second is intended to show that it is informally complete, in the sense that all informal validities are among its theorems. In order to give these arguments, a number of independently interesting results concerning the logic are proven. In particular, the soundness and completeness of two proof systems with respect to the semantics is proven (Theorems 2.11 and 2.15), as well as a normal form theorem (Theorem 3.2), an elimination theorem for the actuality operator (Corollary 3.6), and the decidability of the logic (Corollary 3.7). It turns out that the logic invalidates a plausible principle concerning the interaction of apriority and necessity; consequently, a variant semantics is briefly explored on which this principle is valid. The paper concludes by assessing the implications of these results for epistemic two-dimensional semantics. (shrink)

Propositional quantifiers are added to a propositional modal language with two modal operators. The resulting language is interpreted over so-called products of Kripke frames whose accessibility relations are equivalence relations, letting propositional quantifiers range over the powerset of the set of worlds of the frame. It is first shown that full second-order logic can be recursively embedded in the resulting logic, which entails that the two logics are recursively isomorphic. The embedding is then extended to all sublogics containing the logic (...) of so-called fusions of frames with equivalence relations. This generalizes a result due to Antonelli and Thomason, who construct such an embedding for the logic of such fusions. (shrink)

We solve a major open problem concerning algorithmic properties of products of ‘transitive’ modal logics by showing that products and commutators of such standard logics as K4, S4, S4.1, K4.3, GL, or Grz are undecidable and do not have the finite model property. More generally, we prove that no Kripke complete extension of the commutator [K4,K4] with product frames of arbitrary finite or infinite depth (with respect to both accessibility relations) can be decidable. In particular, if.

This work introduces a two-dimensional modal logic to represent agents' Concurrent Common Knowledge in distributed systems. Unlike Common Knowledge, Concurrent Common Knowledge is a kind of agreement reachable in asynchronous environments. The formalization of such type of knowledge is based on a model for asynchronous systems and on the definition of Concurrent Knowledge introduced before in paper [5]. As a proper semantics, we review our concept of closed sub-product of modal logics which is based on the product of modal logics. (...) The key idea is to reason about Concurrent Knowledge under a two-dimensional approach, regarding asynchronous runs and consistent global states as dimensions. We present an axiomatic system for the logic and issue the corresponding soundness and completeness proofs. (shrink)

We present an extension of the mosaic method aimed at capturing many-dimensional modal logics. As a proof-of-concept, we define the method for logics arising from the combination of linear tense operators with an “orthogonal” S5-like modality. We show that the existence of a model for a given set of formulas is equivalent to the existence of a suitable set of partial models, called mosaics, and apply the technique not only in obtaining a proof of decidability and a proof of completeness (...) for the corresponding Hilbert-style axiomatization, but also in the development of a mosaic-based tableau system. We further consider extensions for dealing with the case when interactions between the two dimensions exist, thus covering a wide class of bundled Ockhamist branching-time logics, and present for them some partial results, such as a non-analytic version of the tableau system. (shrink)

We extend stit logic by adding a spatial dimension. This enables us to distinguish between powers and opportunities of agents. Powers are agent-specific and do not depend on an agent’s location. Opportunities do depend on locations, and are the same for every agent. The central idea is to define the real possibility to see to the truth of a condition in space and time as the combination of the power and the opportunity to do so. The focus on agent-relative powers (...) and space-relative opportunities firmly roots effectivity of an autonomous choice making agent in a space–time picture. Our space–time view will be classically Newtonian, since we will assume relativistic phenomena do not play a role in agentive effectivity. We show how our semantics naturally distinguishes between different kinds of histories; histories that reflect real possibilities and histories that reflect counterfactual possibilities. Furthermore, we discuss how the spatial picture sheds light on conceptual problems plaguing the central stit property of ‘independence of agency’. At several points in the article we will emphasise and defend the differences with Belnap’s theory of agency in relativistic branching space–times. (shrink)

We show that monadic intuitionistic quantifiers admit the following temporal interpretation: “always in the future” (for$\forall $) and “sometime in the past” (for$\exists $). It is well known that Prior’s intuitionistic modal logic${\sf MIPC}$axiomatizes the monadic fragment of the intuitionistic predicate logic, and that${\sf MIPC}$is translated fully and faithfully into the monadic fragment${\sf MS4}$of the predicate${\sf S4}$via the Gödel translation. To realize the temporal interpretation mentioned above, we introduce a new tense extension${\sf TS4}$of${\sf S4}$and provide a full and faithful translation (...) of${\sf MIPC}$into${\sf TS4}$. We compare this new translation of${\sf MIPC}$with the Gödel translation by showing that both${\sf TS4}$and${\sf MS4}$can be translated fully and faithfully into a tense extension of${\sf MS4}$, which we denote by${\sf MS4.t}$. This is done by utilizing the relational semantics for these logics. As a result, we arrive at the diagram of full and faithful translations shown in Figure 1 which is commutative up to logical equivalence. We prove the finite model property (fmp) for${\sf MS4.t}$using algebraic semantics, and show that the fmp for the other logics involved can be derived as a consequence of the fullness and faithfulness of the translations considered. (shrink)

We prove that each countable rooted K4 -frame is a d-morphic image of a subspace of the space $\mathbb{Q}$ of rational numbers. From this we derive that each modal logic over K4 axiomatizable by variable-free formulas is the d-logic of a subspace of $\mathbb{Q}$ . It follows that subspaces of $\mathbb{Q}$ give rise to continuum many d-logics over K4 , continuum many of which are neither finitely axiomatizable nor decidable. In addition, we exhibit several families of modal logics finitely axiomatizable (...) by variable-free formulas over K4 that d-define interesting classes of topological spaces. Each of these logics has the finite model property and is decidable. Finally, we introduce quasi-scattered and semi-scattered spaces as generalizations of scattered spaces, develop their basic properties, axiomatize their corresponding modal logics, and show that they also arise as the d-logics of some subspaces of $\mathbb{Q}$. (shrink)

We introduce the horizontal and vertical topologies on the product of topological spaces, and study their relationship with the standard product topology. We show that the modal logic of products of topological spaces with horizontal and vertical topologies is the fusion ${\bf S4}\oplus {\bf S4}$ . We axiomatize the modal logic of products of spaces with horizontal, vertical, and standard product topologies. We prove that both of these logics are complete for the product of rational numbers ${\Bbb Q}\times {\Bbb Q}$ (...) with the appropriate topologies. (shrink)

The product of two |$\textbf {Alt}$| logics possesses the polynomial product finite model property and its membership problem is |$\textbf {coNP}$|-complete. Using a reduction from an undecidable domino-tiling problem, we prove that its admissibility problem is undecidable.

This paper sets out a new way of combining Kripke-complete modal logics: lexicographic product. It discusses some basic properties of the lexicographic product construction and proves axiomatization/completeness results.

Shehtman introduced bimodal logics of the products of Kripke frames, thereby introducing frame products of unimodal logics. Van Benthem, Bezhanishvili, ten Cate and Sarenac generalize this idea to the bimodal logics of the products of topological spaces, thereby introducing topological products of unimodal logics. In particular, they show that the topological product of S4 and S4 is S4 ⊗ S4, i.e., the fusion of S4 and S4: this logic is strictly weaker than the frame product S4 × S4. In this (...) paper, we axiomatize the topological product of S4 and S5, which is strictly between S4 ⊗ S5 and S4 × S5. We also apply our techniques to (1) proving a conjecture of van Benthem et al concerning the logic of products of Alexandrov spaces with arbitrary topological spaces; and (2) solving a problem in quantified modal logic: in particular, it is known that standard quantified S4 without identity, QS4, is complete in Kripke semantics with expanding domains; we show that QS4 is complete not only in topological semantics with constant domains (which was already shown by Rasiowa and Sikorski), but wrt the topological space Q with a constant countable domain. (shrink)

This paper, we propose a modal logic satisfying minimal requirements for reasoning about diagrams via collection of sets and relations between them, following Harel's proposal. We first give an axiomatics of such a theory and then provide its Kripke semantics. Then we extend previous works of ours in order to obtain a decision procedure based on tableaux for this logic. Beside soundness and completeness of our tableaux, we manage to define a strategy of rule application ensuring termination by extending the (...) usual loop test of modal logic S4 to whole sub-structures of the model being computed. (shrink)