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)

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)

This paper outlines a semantic account of attitude reports and deontic modals based on cognitive and illocutionary products, mental states, and modal products, as opposed to the notion of an abstract proposition or a cognitive act.

The paper studies many-dimensional modal logics corresponding to products of Kripke frames. It proves results on axiomatisability, the finite model property and decidability for product logics, by applying a rather elaborated modal logic technique: p-morphisms, the finite depth method, normal forms, filtrations. Applications to first order predicate logics are considered too. The introduction and the conclusion contain a discussion of many related results and open problems in the area.

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.

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)

In the present study, cross-modal (i.e., across sensory modalities such as smell and sound) iconicity (i.e., resemblance) and indexicality (i.e., contiguity) in lexical sensory and emotional signs in Finnish Sign Language will be considered from an articulatory perspective (i.e., the production of signs). Such cross-modal iconicity has not been extensively studied previously, so here, with the help of cognitive semiotics, I aim to carefully describe the cross-modal patterns observed across 118 signs, including 60 sensory signs and 58 (...) emotional signs. The analysis is framed within the theoretical model of Semiotic Hierarchy, which entails a non-reductionist view of meaning. In addition, a pheno-methodological triangulation will be applied: phenomenology (first-person method), literature of phenomenological and semiotic descriptions (second-person perspective) and experimental findings (third-person perspective). The results of this analysis show that (a) 71 of the 118 sensory and emotional signs are cross-modally indexical, (b) only 10 of the 71 signs can be regarded as cross-modally iconic, (c) cross-modal iconicity is highly diagrammatic, (d) iconicity and indexicality are highly integrated, and (e) articulatory feedback matters in the formation of semiotic patterns. This study contributes to our understanding of cross-modal iconicity in signed languages, as well as studies in semiotic systems more generally. (shrink)

In the first part of this paper we introduced products of modal logics and proved basic results on their axiomatisability and the f.m.p. In this continuation paper we prove a stronger result - the product f.m.p. holds for products of modal logics in which some of the modalities are reflexive or serial. This theorem is applied in classical first-order logic, we identify a new Square Fragment of the classical logic, where the basic predicates are binary and all (...) quantifiers are relativised, and for which we show the f.m.p. in the classical sense. Also we prove that SF not included in Guarded Fragment and that it can be embedded into the equational theory of relational algebras. (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.

Most everyday tasks involve multiple modalities, which raises the question of how the processing of these modalities is coordinated by the cognitive system. In this paper, we focus on the coordination of visual attention and linguistic processing during speaking. Previous research has shown that objects in a visual scene are fixated before they are mentioned, leading us to hypothesize that the scan pattern of a participant can be used to predict what he or she will say. We test this hypothesis (...) using a data set of cued scene descriptions of photo-realistic scenes. We demonstrate that similar scan patterns are correlated with similar sentences, within and between visual scenes; and that this correlation holds for three phases of the language production process (target identification, sentence planning, and speaking). We also present a simple algorithm that uses scan patterns to accurately predict associated sentences by utilizing similarity-based retrieval. (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)

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)

Natural language involves both speaking and listening. Recent models claim that production and comprehension share aspects of processing and are linked within individuals (Dell & Chang, 2014; MacDonald, 2013; Pickering & Garrod, 2004; 2013a). Evidence for this claim has come from studies of cross-modality structural priming, mainly examining processing in the direction of comprehension to production. The current study replicated these comprehension to production findings and developed a novel cross-modal structural priming paradigm from production to comprehension using a temporally-sensitive (...) online measure of comprehension, Event-Related Potentials. For Comprehension-to-Production priming, participants first listened to active or passive sentences and then described target pictures using either structure. In Production-to-Comprehension priming, participants first described a picture using either structure and then listened to target passive sentences while EEG was recorded. Comprehension-to-Production priming showed the expected passive sentence priming for syntactic choice, but not response time or average syllable duration. In Production-to-Comprehension priming, primed, versus unprimed, passive sentences elicited a reduced N400. These effects support the notion that production and comprehension share aspects of processing and are linked within the individual. Moreover, this paradigm can be used for the exploration priming at different linguistic levels as well as the influence of extra-linguistic factors on natural language use. (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)

Unifiability of terms (and formulas) and structural completeness in the variety of relation algebras RA and in the products of modal logic S5 is investigated. Nonunifiable terms (formulas) which are satisfiable in varieties (in logics) are exhibited. Consequently, RA and products of S5 as well as representable diagonal-free n-dimensional cylindric algebras, RDfn, are almost structurally complete but not structurally complete. In case of S5n a basis for admissible rules and the form of all passive rules are provided.

Provisional draft, pre-production copy of my book “The Modal Future” (forthcoming with Cambridge University Press).

Perceptual modalities have been traditionally considered the product of dedicated biological systems producing information for higher cognitive processing. Psychological and neuropsychological evidence is offered which undermines this point of view and an alternative account of modality from the enactive approach to understanding cognition is suggested. Under this view, a perceptual modality is a stable form of perception which is structured not just by the biological sensitivities of the agent, but by their goals and the set of skills or expertise (...) which they are deploying at a given time. Such a view suggests that there is no such thing as an experience that is purely visual, auditory, or otherwise modal and that our attempts to understand consciousness and the mind must be conducted within a framework that provides an account of embodied, goal-directed adaptive coping with the world. (shrink)

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 (...) class='Hi'>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)

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.

Modal logics, originally conceived in philosophy, have recently found many applications in computer science, artificial intelligence, the foundations of mathematics, linguistics and other disciplines. Celebrated for their good computational behaviour, modal logics are used as effective formalisms for talking about time, space, knowledge, beliefs, actions, obligations, provability, etc. However, the nice computational properties can drastically change if we combine some of these formalisms into a many-dimensional system, say, to reason about knowledge bases developing in time or moving objects. (...) To study the computational behaviour of many-dimensional modal logics is the main aim of this book. On the one hand, it is concerned with providing a solid mathematical foundation for this discipline, while on the other hand, it shows that many seemingly different applied many-dimensional systems (e.g., multi-agent systems, description logics with epistemic, temporal and dynamic operators, spatio-temporal logics, etc.) fit in perfectly with this theoretical framework, and so their computational behaviour can be analyzed using the developed machinery. We start with concrete examples of applied one- and many-dimensional modal logics such as temporal, epistemic, dynamic, description, spatial logics, and various combinations of these. Then we develop a mathematical theory for handling a spectrum of 'abstract' combinations of modal logics - fusions and products of modal logics, fragments of first-order modal and temporal logics - focusing on three major problems: decidability, axiomatizability, and computational complexity. Besides the standard methods of modal logic, the technical toolkit includes the method of quasimodels, mosaics, tilings, reductions to monadic second-order logic, algebraic logic techniques. Finally, we apply the developed machinery and obtained results to three case studies from the field of knowledge representation and reasoning: temporal epistemic logics for reasoning about multi-agent systems, modalized description logics for dynamic ontologies, and spatio-temporal logics. The genre of the book can be defined as a research monograph. It brings the reader to the front line of current research in the field by showing both recent achievements and directions of future investigations (in particular, multiple open problems). On the other hand, well-known results from modal and first-order logic are formulated without proofs and supplied with references to accessible sources. The intended audience of this book is logicians as well as those researchers who use logic in computer science and artificial intelligence. More specific application areas are, e.g., knowledge representation and reasoning, in particular, terminological, temporal and spatial reasoning, or reasoning about agents. And we also believe that researchers from certain other disciplines, say, temporal and spatial databases or geographical information systems, will benefit from this book as well. Key Features: Integrated approach to modern modal and temporal logics and their applications in artificial intelligence and computer science Written by internationally leading researchers in the field of pure and applied logic Combines mathematical theory of modal logic and applications in artificial intelligence and computer science Numerous open problems for further research Well illustrated with pictures and tables. (shrink)

In his great poem, Parmenides uses an argument by elimination to select the correct "way of inquiry" from a pool of two, the ways of is and of is not , joined later by a third, "mixed" way of is and is not . Parmenides' first two ways are soon given modal upgrades - is becomes cannot not be , and is not becomes necessarily is not (B2, 3-6) - and these are no longer contradictories of one another. And (...) is the common view right, that Parmenides rejects the "mixed" way because it is a contradiction? I argue that the modal upgrades are the product of an illicit modal shift. This same shift, built into two Exclusion Arguments, gives Parmenides a novel argument to show that the "mixed" way fails. Given the independent failure of the way of is not , Parmenides' argument by elimination is complete. (shrink)

We prove that everyn-modal logic betweenKnandS5nis undecidable, whenever n ≥ 3. We also show that each of these logics is non-finitely axiomatizable, lacks the product finite model property, and there is no algorithm deciding whether a finite frame validates the logic. These results answer several questions of Gabbay and Shehtman. The proofs combine the modal logic technique of Yankov–Fine frame formulas with algebraic logic results of Halmos, Johnson and Monk, and give a reduction of the representation problem (...) of finite relation algebras. (shrink)

We prove that everyn-modal logic betweenKnandS5nis undecidable, whenever n ≥ 3. We also show that each of these logics is non-finitely axiomatizable, lacks the product finite model property, and there is no algorithm deciding whether a finite frame validates the logic. These results answer several questions of Gabbay and Shehtman. The proofs combine the modal logic technique of Yankov–Fine frame formulas with algebraic logic results of Halmos, Johnson and Monk, and give a reduction of the representation problem (...) of finite relation algebras. (shrink)

We prove that everyn-modal logic betweenKnandS5nis undecidable, whenever n ≥ 3. We also show that each of these logics is non-finitely axiomatizable, lacks the product finite model property, and there is no algorithm deciding whether a finite frame validates the logic. These results answer several questions of Gabbay and Shehtman. The proofs combine the modal logic technique of Yankov–Fine frame formulas with algebraic logic results of Halmos, Johnson and Monk, and give a reduction of the (undecidable) representation (...) problem of finite relation algebras. (shrink)

The purpose of this paper is to explore how nurses are enrolled into and take part in programmes of efficiency and effectiveness. Using the philosophical theorizing about desire as a force or power, I focus specifically on what is understood as relations between desire and productivity in current Westernized health‐care systems. Use is made of the idea from Spinoza that human emotions consist only of pleasure, pain, and desire as these act as a motive force. This is then linked with (...) more contemporary work on the politics and discourses of desire. A report on the implementation of a productivity programme in the United Kingdom, The Productive Ward: Releasing time to care™, is explored for the ways its developers set about motivating nurses to endorse and enact the programme. In exploring the mechanics of desire in these processes, a view of desire as productive is promoted. Looking at desire as assembling actions, and an assemblage, moves the analysis to an interrogation of actions and practices used to enable and bring nurses to the process. Moreover, in working through the various modalities and operations of desire, the potential and limits of such projects are abstracted. Such potentials and limits are necessarily set by the intensification of power and desire in the capitalist economy, saturating areas of nursing, and health‐care provision. (shrink)

We prove that every n-modal logic between K n and S5 n is undecidable, whenever n ≥ 3. We also show that each of these logics is non- finitely axiomatizable, lacks the product finite model property, and there is no algorithm deciding whether a finite frame validates the logic. These results answer several questions of Gabbay and Shehtman. The proofs combine the modal logic technique of Yankov-Fine frame formulas with algebraic logic results of Halmos, Johnson and Monk, (...) and give a reduction of the (undecidable) representation problem of finite relation algebras. (shrink)

Kant famously argued that, from experience, we can only learn how something actually is, but not that it must be so. In this paper, I defend an improved version of Kant's argument for the existence of a priori knowledge, the Modal Argument , against recent objections by Casullo and Kitcher. For the sake of the argument, I concede Casullo's claim that we may know certain counterfactuals in an empirical way and thereby gain epistemic access to some nearby, nomologically possible (...) worlds. But I maintain that our beliefs about metaphysical necessities still cannot be justified empirically. Furthermore, I reject Casullo's deflationary thesis about the significance of such justification. Kitcher's most troublesome objection is that we can gain any modal justification whatsoever through testimony , i.e. in an experiential way. This can be countered by distinguishing between productive sources of justification, like perception, and merely reproductive sources, like testimony. Thus, some productive a priori source will always be needed somewhere. (shrink)

Within many approaches to the interpretation of quantum mechanics, especially modal interpretations, one singles out a particular decomposition of the state vector in order to fix the properties that are well-defined for the system. We present a novel proposal for this preferred decomposition. Given a distinguished factorization of the Hilbert space, it is the decomposition that minimizes the Ingarden–Urbanik entropy from among all product decompositions with respect to the distinguished factorization. We incorporate this choice of preferred decomposition into (...) a framework for modal interpretations and investigate in detail the extent to which it provides a solution to the measurement problem and the extent to which it ensures that measurements whose outcomes are predictable with probability 1 reveal pre-existing properties of the system under investigation. (shrink)

In this paper, I claim that two ways of defining validity for modal languages (“real-world” and “general” validity), corresponding to distinction between a correct and an incorrect way of defining modal valid- ity, correspond instead to two substantive ways of conceiving modal truth. At the same time, I claim that the major logical manifestation of the real- world/general validity distinction in modal propositional languages with the actuality operator should not be taken seriously, but simply as a (...) by-product of the way in which the semantics of such an operator is usually 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. (shrink)

This paper explores consequences of the claim that phenomenal experiences are physical events of great descriptive complexity. This claim is attractive both because it can explain our most perplexing intuitions about the quality of consciousness and also because it is suggestive of very productive research opportunities. I illustrate the former by showing that two of the most compelling anti-physicalist arguments about phenomenal experience – the modal argument of Kripke and the conceivability argument of Chalmers – are not sound if (...) this claim is true. I illustrate the latter by showing that significant empirical predictions are a consequence of this claim. (shrink)

In this paper I question the primacy of argumentation relying solely on logic by showing how the body and mind are deeply connected and as a result how communication and argumentation are a product of this mind/body connection. In particular, I explore the physicality of argumentation through the research and writings on gestures and the embodied mind. Michael Gilbert’s theory of multi-modal argumentation provides the general approach for this elaboration.

In 1988, Michael Nyman composed the score for Peter Greenaway’s film Drowning by Numbers (or did something that we would ordinarily think of as composing that score). We can think of Nyman’s compositional activity as a “generative performance” and of the sound structure that Nyman indicated (or of some other abstract object that is appropriately related to that sound structure) as the product generated by that performance (ix).1 According to one view, Nyman’s score for Drowning by the Numbers—the musical (...) work—is the product generated by Nyman’s compositional activity (namely, an abstract object) and, more generally, artworks are identified with the products generated by compositional or other creative activities. Let’s call this view The Product Theory. By contrast, according to another view, Nyman’s score for Drowning by Numbers is the generative performance itself (namely, Nyman’s compositional activity) and, more generally, artworks are identified with generative performances themselves. Following David Davies in Art as Performance, let’s call this view The Performance Theory (80). In that book, Davies argues for The Performance Theory and against The Product Theory. (shrink)

A major source of latter-day skepticism about necessity is the work of David Hume. Hume is widely taken to have endorsed the Humean claim: there are no necessary connections between distinct existences. The Humean claim is defended on the grounds that necessary connections between wholly distinct things would be mysterious and inexplicable. Philosophers deploy this claim in the service of a wide variety of philosophical projects. But Saul Kripke has argued that it is false. According to Kripke, there are necessary (...) connections between distinct existences; in particular, there are necessary connections between material objects and their material origins. In this paper I argue that the primary motivation for the Humean claim, Hume's datum, also motivates the key premise in an argument for the necessity of origins. The very considerations that the Humean takes to show that necessary connections between wholly distinct things would be mysterious and inexplicable indicate that there must be some such necessary connections. Thus, in the absence of alternative support, there is no reason to believe the Humean claim. (shrink)

Some philosophers of science have suggested that contemporary science should be the source of inspiration to the new analytic metaphysics (A. Chakra vartty, C. Callender, S. French, J. Ladyman, T. Maudlin, etc.). This paper explores the prospect of a string metaphysics: a research program in analytic metaphysics based on string theory. Different forms of fundamentalism and pluralism are discussed in this context. The paper focus on string metaphysics with S-dualities (a relation between models of string theory at different coupling regimes) (...) and argues that fundamentality and compositionality have to be reconceptualized. String metaphysics with dualities is better couched in terms of metaphysical pluralism. Grounding, as well as a sketch of a string modality, are briefly discussed. The paper concludes with a suggestion for future work: the metaphysician may find a productive ground for research in discussing other dualities (especially the T-dualities, or the AdS/CFT duality), the emergence of spacetime, the concept of time in string theory, the multiverse etc. (shrink)