Logical Extension Research Articles

On structural proof theory of the modal logic K+ extended with infinitary derivations

Abstract We consider an extension of the modal logic of transitive closure $\textsf{K}^{+}$ with certain infinitary derivations and present a sequent calculus for this extension, which allows non-well-founded proofs. We establish continuous cut-elimination for the given calculus using fixed-point theorems for contractive mappings. The infinitary derivations mentioned above are well founded and countably branching, while the non-well-founded proofs of the sequent calculus can only be finitely branching. The ordinary derivations in $\textsf{K}^{+}$, as we show additionally, correspond to the non-well-founded proofs of the calculus that are regular and cut-free. Therefore, in this article, we explore the relationship between deductive systems for $\textsf{K}^{+}$ with well-founded infinitely branching derivations and (regular) non-well-founded finitely branching proofs.

Axiomatization of Boolean Connexive Logics with syncategorematic negation and modalities

Abstract In the article we investigate three classes of extended Boolean Connexive Logics. Two of them are extensions of Modal and non-Modal Boolean Connexive Logics with a property of closure under an arbitrary number of negations. The remaining one is an extension of Modal Boolean Connexive Logic with a property of closure under the function of demodalization. In our work we provide a formal presentation of mentioned properties and axiom schemata that allow us to incorporate them into Hilbert-style calculi. The presented axiomatic systems are provided with proofs of soundness, completeness and decidability. The properties of closure under negation and demodalization are motivated by the syncategorematic view on the connectives of negation and modalities, which is discussed in the paper.

Anti-Diodorean Quantum Logic of Observables

The conception of “Anti-Diodorean” logic is based on the reversal of the definitions of Diodorus Cronus, who proposed to define modal concepts in terms of temporal ones. In case of quantum logic this is done by using the semantic interpretation of anti-Diodorean definitions, i.e. semantically defining spatio-temporal concepts by means of a quantum accessibility relation. Since Quantum Logic of Observables could be treated as the many-valued extension of the usual Quantum Logic, we can consider the Anti-Diodorean Logic of Observables in the same way.

Spatiotemporal evolution of post-seismic deformation caused by large earthquakes around the Pacific and its impact on plate movement

SUMMARY The fully relaxed deformation caused by an earthquake refers to the total deformation of the earth caused by the earthquake, including the coseismic deformation and the total effect of post-seismic stress relaxation in the mantle on crustal deformation. An in-depth investigation into post-seismic and fully relaxed deformation resulting from large earthquakes is conducive to comprehending the feedback effect of earthquakes on plate motion. In this paper, we first calculated theoretically the coseimic, post-seicmic and fully relaxed deformations caused by the Tohoku-Oki Mw9.0 earthquake using the spherical dislocation theory. The post-seismic displacement caused by the great earthquake leads to the continuous convergence of plates on both sides of the seismogenic fault, which will certainly facilitate the downward insertion of the subduction plate. As time goes by the influence range of seismic deformations becomes larger and larger. The Tohoku-Oki Mw9.0 earthquake can produce >5 cm of cumulative post-seismic horizontal displacement at a far place like the East Pacific Rise over 100 millenniums. Then, we built slip models of 375 earthquakes with Mw7.0 and above in the circum-Pacific seismic belt, calculated the cumulative post-seismic deformations of them, and found that significant post-seismic horizontal displacements covered the entire Pacific Ocean. The post-seismic deformation field caused by the 2011 Tohoku-Oki Mw9.0 earthquake, the 2010 Chile Mw8.8 earthquake, the 1964 Prince William Sound Mw9.1 earthquake and the 1960 Chile Mw(9.3–9.4) earthquake determines the overall distribution pattern of the deformation field in the Pacific region. Those large earthquakes around the Pacific make the total Pacific Plate present a tension strain in the northwest–southeast direction. The one at the East Pacific Rise reaches 300–400 nstr, with orientation of principle strain approximately perpendicular to the Rise. This tensile strain is bound to encourage transverse expansion of the mid-ocean ridge. By a logical extension, the post-seismic stress relaxation in the mantle caused by past earthquakes should be an important driving force for the current plate movement, in addition to the classic driving force like negative buoyancy and plate material phase transition. This study proves theoretically that there is a two-way relationship between great earthquakes and plate movement, and the viscous structure of mantle plays a key role in the relationship.

Fuzzy Linear Temporal Logic with Quality Constraints

As an extension of quantitative temporal logic, uncertain temporal logic essentially describes the temporal behavior of uncertain and incomplete systems, thus better solving search and decision-making problems in such systems. Fuzzy linear temporal logic (FLTL) is a focal point in uncertain temporal logic research. However, there are evident shortcomings in the current research outcomes. First, in previous FLTL studies, the practice of obtaining path reachability and formula satisfaction values independently and subsequently selecting the smaller of the two as the satisfaction value metric led to information loss. Furthermore, this simplistic information fusion approach fails to reflect the varying importance of these two types of information to the requirements. Second, computing path reachability and temporal logic formula satisfaction values separately may result in a mismatch between the two pieces of information with respect to the same path segment. Thus, the primary challenge lies in accurately integrating the satisfaction values of temporal logic formulas with the path reachability of the segments that yields these satisfaction values, utilizing various reasonable information synthesis methods, to ensure synchronization between path reachability and formula satisfaction values without incurring information loss. Additionally, it is crucial to reflect the different preference requirements for these two types of information. Moreover, the temporal logic formula characterizes system properties, with its sub-formulas delineating distinct sub-properties. Consequently, considering the varying importance preferences of sub-formulas is also significant. To address these deficiencies, we introduced quality constraint operators into FLTL, resulting in quality-constrained fuzzy linear temporal logic (QFLTL). This incorporation enables the synchronization and comprehensive fusion of path-reachability information and formula satisfaction values within the final semantic metric, thereby resolving the issues related to information synchronization and loss. Furthermore, it can accommodate the differing preference requirements between the two types of information and sub-properties during the information synthesis process. We defined the syntax and semantics of QFLTL and examined its expressive power and properties. Notably, we investigated the decidability of logical decision problems in QFLTL, encompassing validity, satisfiability, and model-checking issues. We proposed corresponding solution algorithms and analyzed their complexities.

Games for hybrid logic from semantic games to analytic calculi

Abstract Game semantics and winning strategies offer a potential conceptual bridge between semantics and proof systems of logics. We illustrate this link for hybrid logic—an extension of modal logic that allows for explicit reference to worlds within the language. We lift a Hintikka-style semantic game to a disjunctive game. The disjunctive game adequately models entailment and validity over classes of frames characterizable in the hybrid language. The search for winning strategies in this game can be reformulated as a sequent-style proof system.

A hybrid-fuzzy-decision-making framework for digital technologies selection

In the era of Industry X.0, manufacturers are increasingly adopting digital technologies such as robotics, big data, mobile communications, and information technology to enhance productivity and profitability. However, navigating this digital revolution presents significant challenges, particularly in selecting the most suitable technologies amidst uncertainties and the lack of precise quantitative data. Fuzzy logic, initially introduced to manage vagueness and imprecision, has proven to be an essential tool in various domains for handling these uncertainties. However, as systems and environments become more complex, traditional fuzzy logic exhibits limitations in effectively representing and managing deeper levels of uncertainty. This realization has driven the development of several extensions to fuzzy logic, designed to address challenges associated with uncertainty representation. Yet, a central question remains: Which fuzzy logic extension is the most relevant for better dealing with uncertainty? This study introduces a hybrid-fuzzy decision-making framework for technology selection, which allows for more than one representation of fuzzy information. Using a multi-expert, multi-criteria decision-making (MCDM) strategy, the study seeks to support organizations in selecting the most suitable digital technology for their operations, particularly when precise quantitative data is unavailable, and decisions must rely on expert opinions. The suggested framework is designed to handle decision-makers' hesitancy, subjectivity, and ambiguity in the decision-making process. To showcase the practical application of the proposed methodology, a case study was conducted to select appropriate technologies in the Moroccan automotive sector. The results demonstrate that, regardless of the fuzzy logic extension used, digital automation and automatic identification consistently rank among the top technologies, underscoring their importance according to most experts. Our proposed approach is particularly effective in resolving conflicts between different fuzzy logic extensions, thereby facilitating the ranking of the remaining technologies. Notably, additive manufacturing ranked third, followed by the other technologies.

Rewriting the Infinite Chase for Guarded TGDs

Guarded tuple-generating dependencies (GTGDs) are a natural extension of description logics and referential constraints. It has long been known that queries over GTGDs can be answered by a variant of the chase —a quintessential technique for reasoning with dependencies. However, there has been little work on concrete algorithms and even less on implementation. To address this gap, we revisit Datalog rewriting approaches to query answering, where a set of GTGDs is transformed to a Datalog program that entails the same base facts on each base instance. We show that a rewriting consists of “shortcut” rules that circumvent certain chase steps, we present several algorithms that compute a rewriting by deriving such “shortcuts” efficiently, and we discuss important implementation issues. Finally, we show empirically that our techniques can process complex GTGDs derived from synthetic and real benchmarks and are thus suitable for practical use.

Logics with Concrete Domains: First-Order Properties, Abstract Expressive Power, and (Un)Decidability

Concrete domains have been introduced in description logic (DL) to enable reference to concrete objects (such as numbers) and predefined predicates on these objects (such as numerical comparisons) when defining concepts. The primary research goal in this context was to find restrictions on the concrete domain such that its integration into certain DLs preserves decidability or tractability. In this article, we complement these investigations by studying the abstract expressive power of both first-order and description logics extended with concrete domains, i.e., we analyze which classes of first-order interpretations can be expressed using these logics, compared to what their counterparts without concrete domains can express. We demonstrate that, under natural conditions on the concrete domain D (which also play a role for decidability), extensions of first-order logic (FOL) or the well-known DL ALC with D share important formal characteristics with FOL, such as the compactness and the Löwenheim-Skolem properties. Nevertheless, these conditions do not ensure that the abstract expressive power of the extensions we consider is contained in that of FOL, though in some cases it is. To be more precise, we show, on the one hand, that unary concrete domains leave the abstract expressive power within FOL if we are allowed to introduce auxiliary predicates. As a by-product, we obtain (semi-)decidability results for some fragments of FOL extended with the concrete domains considered in this article. On the other hand, we show that the ability to express equality between elements of D , another condition employed in the context of showing decidability of ALC ( D ), is sufficient to push the abstract expressive power of most first-order fragments with concrete domains beyond that of FOL. While for such concrete domains D decidability is retained for ALC ( D ), we show that the availability of equality in D causes undecidability of the two-variable fragment of FOL( D ), although the two-variable fragment of FOL is decidable.

Seeking asylum during a pandemic: A postcolonial media discourse analysis

The beginning of the COVID-19 pandemic in 2020 saw an increase in people seeking asylum in the UK via small boats across the English Channel. The small boat arrivals reported in the news were analyzed using critical discourse analysis to investigate how different outlets framed people seeking sanctuary both visually and linguistically, as well as how postcolonial theory assists an understanding of such discourses. This research conceptualizes common findings in the literature of forced migrants as dehumanized and “othered,” arguing that humanitarian securitization is not an oxymoron but an extension of colonial logic in post-Brexit Britain. The article ends with recommendations for critical reporting of forced migration and for representing people seeking sanctuary in more humanizing ways.

Interpolation in Linear Logic and Related Systems

We prove that there are continuum-many axiomatic extensions of the full Lambek calculus with exchange that have the deductive interpolation property. Further, we extend this result to both classical and intuitionistic linear logic as well as their multiplicative-additive fragments. None of the logics we exhibit have the Craig interpolation property, but we show that the exhibited extensions of classical and intuitionistic linear logic all enjoy a guarded form of Craig interpolation. We also give continuum-many axiomatic extensions of classical linear logic without the deductive interpolation property.

Two semantic interpretations of probabilities in description logics of typicality

Abstract In this work we focus on extensions of Description Logics (DLs) of typicality by means of probabilities. We introduce a novel extension of the logic of typicality $\mathcal{ALC}+\mathbf{T}_{\mathbf{R}}$, able to represent and reason about typical properties and defeasible inheritance in DLs. The novel logic ($\mathcal{ALC}\mathbf{T}^{\textsf{P}}$: Typical $\mathcal{ALC}$ with Probabilities as Proportions) allows inclusions of the form $\mathbf{T}(C) \sqsubseteq ^{p} D$, with probability $p$ representing a proportion, meaning that “all the typical $C$s are $D$s, and the probability that a $C$ element is not a $D$ element is $1-p$”. We also compare and confront this novel logic with a similar one already presented in the literature ($\mathbf{T}^{\textsf{CL}}$, introduced in Lieto and Pozzato (2020, J. Exp. Theor. Artif. Intell., 32, 769–804)), inspired by the DISPONTE semantics and that allows inclusions of the form $p \:: \ \mathbf{T}(C) \sqsubseteq D$ with probability $p$, where $p$ represents a degree of belief, whose meaning is that “we believe with a degree $p$ that typical $C$s’ are also $D$s.”. We then show that the proposed $\mathcal{ALC}\mathbf{T}^{\textsf{P}}$ extension (like the previous $\mathbf{T}^{\textsf{CL}}$) can be applied in order to tackle a specific and challenging problem in the field of common-sense reasoning, namely the combination of prototypical concepts, that have been shown to be problematic to model for other symbolic approaches like fuzzy logic. We show that, for the proposed extension, the complexity of reasoning remains ExpTime-complete as for the underlying standard monotonic DL $\mathcal{ALC}$.

Absolutely Fuzzy Lipschitz p-summing Maps between Fuzzy Pointed Metric Spaces

Motivated by Lipschitz ideals in the conventional (crisp) theory, we are constructing a new Lipschitz ideal in fuzzy theory. We introduce the notion of fuzzy Lipschitz ideals and give some elementary illustrations. The class of absolutely fuzzy Lipschitz -summing maps between arbitrary fuzzy pointed metric spaces is a significant category of fuzzy Lipschitz ideals. It is a logical extension of the concept of absolutely (crisp) Lipschitz p-summing maps between arbitrary pointed metric spaces, as established by Farmer Jeffrey and William Johnson. We establish that the fuzzy Lipschitz norm of the previously specified concept is a fuzzy real number. We demonstrate that a complete fuzzy normed fuzzy operator ideal is the resulting class of fuzzy Lipschitz operators between arbitrary fuzzy pointed metric spaces and complete fuzzy normed spaces. Next, we define a basic characterisation of a Lipschitz p-summing map that is completely fuzzy. By demonstrating a fuzzy variant of the nonlinear Pietsch Domination Theorem, this is accomplished. Lastly, we bring forth a few unsolved problems that we find intriguing.

Himalayan Buddhism as Human Geological Agency: Rethinking the Novelty of "the Anthropocene"

This article uses a Himalayan Buddhist lens to critically interrogate a fundamental premise of "the Anthropocene"---that the epoch commemorates a "newfound" capacity of humans to mobilise Earth forces. Rather, Himalayan Buddhism has long held that humans wield geological agency, mobilised through relationships with territorial landscape deities, which inflict severe weather in retaliation for human moral infractions. Offering an alternative model of anthropogenic climate change, Buddhist and Indigenous lifeworlds challenge Western convictions that "the Anthropocene" is a novel planetary epoch. Since the term has gained a vibrant discursive life beyond geology, its cultural assumptions---rather than biophysical thresholds---are primarily evaluated, revealing an extension of Eurocentric colonial logic into this new planetary chapter. Alternatively, I suggest the Himalayan Buddhist term "kawa nyampa" (degenerate era) better encapsulates our transition towards environmental breakdown. There was no need to "invent" the Anthropocene as a new epoch of thought---it had long already existed.

Enjoying what comes naturally: The Church of England and Sexuality in the 1930s

This article begins by outlining the changing approach in Anglican attitudes to contraception at the Lambeth Conference of 1930, where birth control was permitted for married couples and sex separated from the possibility of procreation. The logical extension of this teaching, as was noted by Bishop Charles Gore, was that other forms of sexual pleasure, including homosexuality, which was increasingly seen as a ‘natural’ condition, might eventually be sanctioned by the church. Later in the 1930s, a series of letters by Robert Reid to Cosmo Gordon Lang, archbishop of Canterbury, shows the beginnings of a campaign for a change of policy. At the outbreak of the Second World War, the Anglican writer Kenneth Ingram published Sex-Morality Tomorrow, which pressed for full homosexual equality and provoked calls to William Temple to suppress the book. 1940 proved an inopportune moment for reform of church teaching on homosexuality, which continues to elicit widespread controversy.

Misstatements, misperceptions, and mistakes in controlling for covariates in observational research.

We discuss 12 misperceptions, misstatements, or mistakes concerning the use of covariates in observational or nonrandomized research. Additionally, we offer advice to help investigators, editors, reviewers, and readers make more informed decisions about conducting and interpreting research where the influence of covariates may be at issue. We primarily address misperceptions in the context of statistical management of the covariates through various forms of modeling, although we also emphasize design and model or variable selection. Other approaches to addressing the effects of covariates, including matching, have logical extensions from what we discuss here but are not dwelled upon heavily. The misperceptions, misstatements, or mistakes we discuss include accurate representation of covariates, effects of measurement error, overreliance on covariate categorization, underestimation of power loss when controlling for covariates, misinterpretation of significance in statistical models, and misconceptions about confounding variables, selecting on a collider, and p value interpretations in covariate-inclusive analyses. This condensed overview serves to correct common errors and improve research quality in general and in nutrition research specifically.

Exploring Non-Regular Extensions of Propositional Dynamic Logic with Description-Logics Features

We investigate the impact of non-regular path expressions on the decidability of satisfiability checking and querying in description logics extending ALC. Our primary objects of interest are ALCreg and ALCvpl, the extensions of with path expressions employing, respectively, regular and visibly-pushdown languages. The first one, ALCreg, is a notational variant of the well-known Propositional Dynamic Logic of Fischer and Ladner. The second one, ALCvpl, was introduced and investigated by Loding and Serre in 2007. The logic ALCvpl generalises many known decidable non-regular extensions of ALCreg. We provide a series of undecidability results. First, we show that decidability of the concept satisfiability problem for ALCvpl is lost upon adding the seemingly innocent Self operator. Second, we establish undecidability for the concept satisfiability problem for ALCvpl extended with nominals. Interestingly, our undecidability proof relies only on one single non-regular (visibly-pushdown) language, namely on r#s# := { r^n s^n | n in N } for fixed role names r and s. Finally, in contrast to the classical database setting, we establish undecidability of query entailment for queries involving non-regular atoms from r#s#, already in the case of ALC-TBoxes.

Metric Temporal Equilibrium Logic over Timed Traces

AbstractIn temporal extensions of answer set programming (ASP) based on linear time, the behavior of dynamic systems is captured by sequences of states. While this representation reflects their relative order, it abstracts away the specific times associated with each state. However, timing constraints are important in many applications like, for instance, when planning and scheduling go hand in hand. We address this by developing a metric extension of linear-time temporal equilibrium logic, in which temporal operators are constrained by intervals over natural numbers. The resulting Metric Equilibrium Logic (MEL) provides the foundation of an ASP-based approach for specifying qualitative and quantitative dynamic constraints. To this end, we define a translation of metric formulas into monadic first-order formulas and give a correspondence between their models in MEL and Monadic Quantified Equilibrium Logic, respectively. Interestingly, our translation provides a blue print for implementation in terms of ASP modulo difference constraints.

On undecidability of unary non-nested PFP-operators for one successor function theory

We investigate the decidability of first-order logic extensions. For example, it is established in A. S. Zolotov’s works that a logic with a unary transitive closure operator for the one successor theory is decidable. We show that in a similar case, a logic with a unary partial fixed point operator is undecidable. For this purpose, we reduce the halting problem for the counter machine to the problem of truth of the underlying formula. This reduction uses only one unary non-nested partial fixed operator that is applied to a universal or existential formula.

Quantified Modal Logics: One Approach to Rule (Almost) them All!

We present a general approach to quantified modal logics that can simulate most other approaches. The language is based on operators indexed by terms which allow to express de re modalities and to control the interaction of modalities with the first-order machinery and with non-rigid designators. The semantics is based on a primitive counterpart relation holding between n-tuples of objects inhabiting possible worlds. This allows an object to be represented by one, many, or no object in an accessible world. Moreover by taking as primitive a relation between n-tuples we avoid some shortcoming of standard individual counterparts. Finally, we use cut-free labelled sequent calculi to give a proof-theoretic characterisation of the quantified extensions of each first-order definable propositional modal logic. In this way we show how to complete many axiomatically incomplete quantified modal logics.