Multi-modal Logic Research Articles

A Note on Constructive Interpolation for the Multi-Modal Logic Km

The Craig Interpolation Theorem is a well-known property in the mathematical logic curricula, with many domain applications, such as in the modularization of formal specifications and ontologies. This property states the following: given an implication, say formula ϕ implies another formula ψ, then there is a formula β, called the interpolant, in the common language of ϕ and ψ, such that ϕ also implies β, as well as β implies ψ. Although it is already known that the propositional multi-modal logic Km enjoys Craig interpolation, we are not aware of method providing an explicit construction of interpolants. We describe in this paper a constructive proof of the Craig interpolation property on the multi-modal logic Km. Interpolants can be explicitly computed from the proof. Furthermore, we also describe an upper bound for the computation of interpolants. The proof is based on the application of Maehara technique on a tree-hypersequent calculus. As a corollary of interpolation, we also show Beth definability and Robinson joint consistency.

An infinitary axiomatization of dynamic topological logic

Abstract Dynamic topological logic ($\textsf{DTL}$) is a multi-modal logic that was introduced for reasoning about dynamic topological systems, i.e. structures of the form $\langle{\mathfrak{X}, f}\rangle $, where $\mathfrak{X}$ is a topological space and $f$ is a continuous function on it. The problem of finding a complete and natural axiomatization for this logic in the original tri-modal language has been open for more than one decade. In this paper, we give a natural axiomatization of $\textsf{DTL}$ and prove its strong completeness with respect to the class of all dynamic topological systems. Our proof system is infinitary in the sense that it contains an infinitary derivation rule with countably many premises and one conclusion. It should be remarked that $\textsf{DTL}$ is semantically non-compact, so no finitary proof system for this logic could be strongly complete. Moreover, we provide an infinitary axiomatic system for the logic ${\textsf{DTL}}_{\mathcal{A}}$, i.e. the $\textsf{DTL}$ of Alexandrov spaces, and show that it is strongly complete with respect to the class of all dynamical systems based on Alexandrov spaces.

Atom-canonicity in varieties of cylindric algebras with applications to omitting types in multi-modal logic

Fix and let denote the class of cylindric algebras of dimension n. Roughly, is the algebraic counterpart of the proof theory of first-order logic restricted to the first n variables which we denote by . The variety of representable s reflects algebraically the semantics of . A variety of Boolean algebras with operators is atom-canonical, if whenever is atomic, then its Dedekind–MacNeille completion is also in . We show using a so-called blow up and blur construction that for any , any variety of the form containing (and including) (when ) is not atom-canonical. We show that a restricted form of that the celebrated Henkin–Orey omitting types theorem, which we refer to below as Vaught's Theorem (), fails dramatically for even if we allow only so-called m-square models for . We deduce that many multi-modal logics, like , the m clique guarded fragments and packed fragments of are not Sahlqvist. In contrast, we prove a positive for theories with respect to usual semantics, by imposing extra conditions (such as quantifier elimination) on possibly uncountable theories considered and/or the non-principal types omitted (that are also allowed to be possibly uncountable) such as completeness.ast is a maximality condition delineating the edge of an independent statement to a provable one. is shown that the maximality condition cannot be removed even to prove the weaker for uncountable theories; and this construction is used to reprove a celebrated result of Hirsch and Hodkinson, namely, that the class of completely representable s is not el.

MOORE’S PARADOX AND THE LOGIC OF BELIEF

Moore’s Paradox is a test case for any formal theory of belief. In Knowledge and Belief, Hintikka developed a multimodal logic for statements that express sentences containing the epistemic notions of knowledge and belief. His account purports to offer an explanation of the paradox. In this paper I argue that Hintikka’s interpretation of one of the doxastic operators is philosophically problematic and leads to an unnecessarily strong logical system. I offer a weaker alternative that captures in a more accurate way our logical intuitions about the notion of belief without sacrificing the possibility of providing an explanation for problematic cases such as Moore’s Paradox.

A Novel Multimodal Radiomics Model for Preoperative Prediction of Lymphovascular Invasion in Rectal Cancer

Objective: To explore a new predictive model of lymphatic vascular infiltration (LVI) in rectal cancer based on magnetic resonance (MR) and computed tomography (CT).Methods: A retrospective study was conducted on 94 patients with histologically confirmed rectal cancer, they were randomly divided into training cohort (n = 65) and validation cohort (n = 29). All patients underwent MR and CT examination within 2 weeks before treatment. On each slice of the tumor, we delineated the volume of interest on T2-weighted imaging, diffusion weighted imaging, and enhanced CT images, respectively. A total of 1,188 radiological features were extracted from each patient. Then, we used the student t-test or Mann–Whitney U-test, Spearman's rank correlation and least absolute shrinkage and selection operator (LASSO) algorithm to select the strongest features to establish a single and multimodal logic model for predicting LVI. Receiver operating characteristic (ROC) curves and calibration curves were plotted to determine how well they explored LVI prediction performance in the training and validation cohorts.Results: An optimal multi-mode radiology nomogram for LVI estimation was established, which had significant predictive power in training (AUC, 0.884; 95% CI, 0.803–0.964) and validation (AUC, 0.876; 95% CI, 0.721–1.000). Calibration curve and decision curve analysis showed that the multimodal radiomics model provides greater clinical benefits.Conclusion: Multimodal (MR/CT) radiomics models can serve as an effective visual prognostic tool for predicting LVI in rectal cancer. It demonstrated great potential of preoperative prediction to improve treatment decisions.

Expressivity results for deontic logics of collective agency

We use a deontic logic of collective agency to study reducibility questions about collective agency and collective obligations. The logic that is at the basis of our study is a multi-modal logic in the tradition of stit (‘sees to it that’) logics of agency. Our full formal language has constants for collective and individual deontic admissibility, modalities for collective and individual agency, and modalities for collective and individual obligations. We classify its twenty-seven sublanguages in terms of their expressive power. This classification enables us to investigate reducibility relations between collective deontic admissibility, collective agency, and collective obligations, on the one hand, and individual deontic admissibility, individual agency, and individual obligations, on the other.

Modal logics with hard diamond-free fragments

Abstract We investigate the complexity of modal satisfiability for a family of multi-modal logics with interdependencies among the modalities. In particular, we examine four characteristic multi-modal logics with dependencies and demonstrate that, even if we restrict the formulae to be diamond-free and to have only one propositional variable, these logics still have a high complexity. This result identifies and isolates two sources of complexity: the presence of axiom $D$ for some of the modalities and certain modal interdependencies. We then further investigate and characterize the complexity of the diamond-free, 1-variable fragments of multi-modal logics in a general setting.

A flexible logic-based approach to closeness using order of magnitude qualitative reasoning

Abstract In this paper, we focus on a logical approach to the important notion of closeness, which has not received much attention in the literature. Our notion of closeness is based on the so-called proximity intervals, which will be used to decide the elements that are close to each other. Some of the intuitions of this definition are explained on the basis of examples. We prove the decidability of the recently introduced multimodal logic for closeness and, then, we show some capabilities of the logic with respect to expressivity in order to denote particular positions of the proximity intervals.

Modal Resolution

Resolution-based provers for multimodal normal logics require pruning of the search space for a proof to ameliorate the inherent intractability of the satisfiability problem for such logics. We present a clausal modal-layered hyper-resolution calculus for the basic multimodal logic, which divides the clause set according to the modal level at which clauses occur to reduce the number of possible inferences. We show that the calculus is complete for the logics being considered. We also show that the calculus can be combined with other strategies. In particular, we discuss the completeness of combining modal layering with negative and ordered resolution and provide experimental results comparing the different refinements.

#MeToo in Sweden: Museum Collections, Digital Archiving and Hashtag Visuality

ABSTRACT In October 2017, the Nordic Museum in Stockholm launched its #metoo collection. The aim was to capture the viral #MeToo campaign that in Sweden has been likened to a (feminist) revolution. Based on archival research, interviews and media analysis, this article explores public submissions to the #metoo collection and analyses the museum’s rationale for collecting what is considered to be difficult cultural heritage. Noting the absence of images in the collection, the article argues that the iconic hashtag #MeToo constitutes an alternative form of digital visuality, here termed hashtag visuality. Hashtag visuality, the article suggests, is an emerging form of visual representation that captures the multimodal logic of social media, blurring distinctions between texts and images. In Sweden, #MeToo hashtag visuality reveals the contradictory prevalence of structural sexism and sexual violence in a country with a national self-image of gender equality and a self-proclaimed feminist government, while affirming feminist agency.

From Oughts to Goals: A Logic for Enkrasia

This paper focuses on (an interpretation of) the Enkratic principle of rationality, according to which rationality requires that if an agent sincerely and with conviction believes she ought to X, then X-ing is a goal in her plan. We analyze the logical structure of Enkrasia and its implications for deontic logic. To do so, we elaborate on the distinction between basic and derived oughts, and provide a multi-modal neighborhood logic with three characteristic operators: a non-normal operator for basic oughts, a non-normal operator for goals in plans, and a normal operator for derived oughts. We prove two completeness theorems for the resulting logic, and provide a dynamic extension of the logic by means of product updates. We illustrate how this setting informs deontic logic by considering issues related to the filtering of inconsistent oughts, the restricted validity of deontic closure, and the stability of oughts and goals under dynamics.

Modularisation of Sequent Calculi for Normal and Non-normal Modalities

In this work, we explore the connections between (linear) nested sequent calculi and ordinary sequent calculi for normal and non-normal modal logics. By proposing local versions to ordinary sequent rules, we obtain linear nested sequent calculi for a number of logics, including, to our knowledge, the first nested sequent calculi for a large class of simply dependent multimodal logics and for many standard non-normal modal logics. The resulting systems are modular and have separate left and right introduction rules for the modalities, which makes them amenable to specification as bipole clauses. While this granulation of the sequent rules introduces more choices for proof search, we show how linear nested sequent calculi can be restricted to blocked derivations, which directly correspond to ordinary sequent derivations.

Geometrical representation theorems for cylindric-type algebras

In this paper, we give new proofs of the celebrated Andréka-Resek-Thompson representability results of certain axiomatized cylindric-like algebras. Such representability results provide completeness theorems for variants of first order logic, that can also be viewed as multi-modal logics. The proofs herein are combinatorial and we also use some techniques from game theory.

A Resolution-Based Theorem Prover for {textsf {K}}_{n}^{}: Architecture, Refinements, Strategies and Experiments

In this paper we describe the implementation of , a resolution-based prover for the basic multimodal logic {textsf {K}}_{n}^{}. The prover implements a resolution-based calculus for both local and global reasoning. The user can choose different normal forms, refinements of the basic resolution calculus, and strategies. We describe these options in detail and discuss their implications. We provide experiments comparing some of these options and comparing the prover with other provers for this logic.

Frames for fusions of modal logics

Let us consider multimodal logics and . We assume that is characterised by a class of connected frames, and there exists an -frame with a so-called -starting point. Similarly, the logic is characterised by a class of connected frames, and there exists an -frame with a -starting point. Using isomorphic copies of the frames and , we construct a connected frame which characterises the fusion . The frame thus obtained has some useful properties. Among others, is countable if both and are countable, and there is a special world of the frame such that any formula is valid in the frame if and only if it is valid at the point . We also describe a similar construction where we assume the existence of two classes consisting of rooted frames only. Using those classes and frames with so-called -roots, we construct a rooted frame adequate for the fusion of modal logics characterised by the classes under consideration. Both constructions can be used interchangeably. The selection of the construction depends on the classes characterising the logics under consideration.

Digital Play as Purposeful Productive Literacies in African American Boys

AbstractDigital literacies abound in playing a foundational role in the rhythm and pattern of our lives, yet debates continue about how to harness them to teach and learn literacy. In an effort to humanize digital literacies, this department column offers a vast array of topics, from participatory work that pushes educators and researchers to communicate in local and global spaces to ways of redefining core reading and literacy skills through a screen‐based, multimodal logic. This column also provides a venue for research and practical applications that depict technology use as a part of the fabric of being human. The column helps educators reconceptualize the ways that children learn with technology, media, and new communication systems; honors educator success stories and burning questions and issues; and reimagines literacy futures.

On the relationship between fuzzy description logics and many-valued modal logics

In this paper we study the relationships between a family of ALC-like fuzzy description logics (FDLs) defined over left-continuous t-norms and some many-valued multi-modal logics (MMLs). We analyze these relationships in both directions, that is, how to merge FDLs into MMLs and vice-versa. The analysis starts from the relationships between the languages to reach systematically the deeper level of reasoning tasks. At this level we are able to truly investigate both formalisms from each other point of view. Finally, the results concerning translations between reasoning tasks are applied in order to get decidability and complexity bounds.

A multimodal logic for closeness

We introduce a multimodal logic for order of magnitude reasoning which considers a new logic-based alternative to the notion of closeness, we provide an axiom system and prove its soundness and completeness.

Opening the Case of theiPad: What Matters, and Where Next?

AbstractDigital literacies abound in playing a foundational role in the rhythm and pattern of our lives, yet debates continue about how to harness them to teach and learn literacy. In an effort to humanize digital literacies, this department column offers a vast array of topics, from participatory work that pushes educators and researchers to communicate in local and global spaces to ways of redefining core reading and literacy skills through a screen‐based, multimodal logic. This column also provides a venue for research and practical applications that depict technology use as a part of the fabric of being human. The column helps educators reconceptualize the ways that children learn with technology, media, and new communication systems; honors educator success stories and burning questions and issues; and reimagines literacy futures.

Propositional Epistemic Logics with Quantification Over Agents of Knowledge

The paper presents a family of propositional epistemic logics such that languages of these logics are extended by quantification over modal (epistemic) operators or over agents of knowledge and extended by predicate symbols that take modal (epistemic) operators (or agents) as arguments. Denote this family by \({\mathcal {P}\mathcal {E}\mathcal {L}}_{({ QK})}\). There exist epistemic logics whose languages have the above mentioned properties (see, for example Corsi and Orlandelli in Stud Log 101:1159–1183, 2013; Fitting et al. in Stud Log 69:133–169, 2001; Grove in Artif Intell 74(2):311–350, 1995; Lomuscio and Colombetti in Proceedings of ATAL 1996. Lecture Notes in Computer Science (LNCS), vol 1193, pp 71–85, 1996). But these logics are obtained from first-order modal logics, while a logic of \({\mathcal {P}\mathcal {E}\mathcal {L}}_{({ QK})}\) can be regarded as a propositional multi-modal logic whose language includes quantifiers over modal (epistemic) operators and predicate symbols that take modal (epistemic) operators as arguments. Among the logics of \({\mathcal {P}\mathcal {E}\mathcal {L}}_{({ QK})}\) there are logics with a syntactical distinction between two readings of epistemic sentences: de dicto and de re (between ‘knowing that’ and ‘knowing of’). We show the decidability of logics of \({\mathcal {P}\mathcal {E}\mathcal {L}}_{({ QK})}\) with the help of the loosely guarded fragment (LGF) of first-order logic. Namely, we generalize LGF to a higher-order decidable loosely guarded fragment. The latter fragment allows us to construct various decidable propositional epistemic logics with quantification over modal (epistemic) operators. The family of this logics coincides with \({\mathcal {P}\mathcal {E}\mathcal {L}}_{({ QK})}\). There are decidable propositional logics such that these logics implicitly contain quantification over agents of knowledge, but languages of these logics are usual propositional epistemic languages without quantifiers and predicate symbols (see Grove and Halpern in J Log Comput 3(4):345–378, 1993). Some logics of \({\mathcal {P}\mathcal {E}\mathcal {L}}_{({ QK})}\) can be regarded as counterparts of logics defined in Grove and Halpern (J Log Comput 3(4):345–378, 1993). We prove that the satisfiability problem for these logics of \({\mathcal {P}\mathcal {E}\mathcal {L}}_{({ QK})}\) is Pspace-complete using their counterparts in Grove and Halpern (J Log Comput 3(4):345–378, 1993).