We provide a direct method for proving Craig interpolation for a range of modal and intuitionistic logics, including those containing a "converse" modality. We demonstrate this method for classical tense logic, its extensions with path axioms, and for bi-intuitionistic logic. These logics do not have straightforward formalisations in the traditional Gentzen-style sequent calculus, but have all been shown to have cut-free nested sequent calculi. The proof of the interpolation theorem uses these calculi and is purely syntactic, without resorting to embeddings, (...) semantic arguments, or interpreted connectives external to the underlying logical language. A novel feature of our proof includes an orthogonality condition for defining duality between interpolants. (shrink)

Free-variable semantic tableaux are a well-established technique for first-order theorem proving where free variables act as a meta-linguistic device for tracking the eigenvariables used during proof search. We present the theoretical foundations to extend this technique to propositional modal logics, including non-trivial rigorous proofs of soundness and completeness, and also present various techniques that improve the efficiency of the basic naive method for such tableaux.

We define cut-free display calculi for knowledge logics wherean indiscernibility relation is associated to each set of agents, andwhere agents decide the membership of objects using thisindiscernibility relation. To do so, we first translate the knowledgelogics into polymodal logics axiomatised by primitive axioms and thenuse Kracht's results on properly displayable logics to define thedisplay calculi. Apart from these technical results, we argue thatDisplay Logic is a natural framework to define cut-free calculi for manyother logics with relative accessibility relations.

In 1983, Valentini presented a syntactic proof of cut elimination for a sequent calculus GLSV for the provability logic GL where we have added the subscript V for “Valentini”. The sequents in GLSV were built from sets, as opposed to multisets, thus avoiding an explicit contraction rule. From a syntactic point of view, it is more satisfying and formal to explicitly identify the applications of the contraction rule that are ‘hidden’ in these set based proofs of cut elimination. There is (...) often an underly ing assumption that the move to a proof of cut elimination for sequents built from multisets is easy. Recently, however, it has been claimed that Valentini’s arguments to eliminate cut do not terminate when applied to a multiset formulation of GLSV with an explicit rule of contraction. The claim has led to much confusion and various authors have sought new proofs of cut elimination for GL in a multiset setting. Here we refute this claim by placing Valentini’s arguments in a formal setting and proving cut elimination for sequents built from multisets. The formal setting is particularly important for sequents built from multisets, in order to accurately account for the interplay between the weakening and contraction rules. Furthermore, Valentini’s original proof relies on a novel induction parameter called “width” which is computed ‘globally’. It is diffi cult to verify the correctness of his induction argument based on “width”. In our formulation however, verification of the induction argument is straight forward. Finally, the multiset setting also introduces a new complication in the the case of contractions above cut when the cut formula is boxed. We deal with this using a new transformation based on Valentini’s original arguments. Finally, we show that the algorithm purporting to show the non termi nation of Valentini’s arguments is not a faithful representation of the original arguments, but is instead a transformation already known to be insufficient. (shrink)

We present a sound and complete tableau calculus for the class of regular grammar logics. Our tableau rules use a special feature called automaton-labelled formulae, which are similar to formulae of automaton propositional dynamic logic. Our calculus is cut-free and has the analytic superformula property so it gives a decision procedure. We show that the known EXPTIME upper bound for regular grammar logics can be obtained using our tableau calculus. We also give an effective Craig interpolation lemma for regular grammar (...) logics using our calculus. (shrink)

We show that the polynomial translation of the classical propositional normal modal logic S4 into the intuitionistic propositional logic Int from Fernández is incorrect. We give a modified translation and prove its correctness, and provide implementations of both translations to allow others to test our results.

We present a syntactic proof of cut-elimination for weak Grzegorczyk logic Go. The logic has a syntactically similar axiomatisation to Gödel–Löb logic GL (provability logic) and Grzegorczyk’s logic Grz. Semantically, GL can be viewed as the irreflexive counterpart of Go, and Grz can be viewed as the reflexive counterpart of Go. Although proofs of syntactic cut-elimination for GL and Grz have appeared in the literature, this is the first proof of syntactic cut-elimination for Go. The proof is technically interesting, requiring (...) a deeper analysis of the derivation structures than the proofs for GL and Grz. New transformations generalising the transformations for GL and Grz are developed here. (shrink)

We propose a new sequent calculus for bi intuitionistic logic which sits somewhere between display calculi and traditional sequent calculi by using nested sequents. Our calculus enjoys a simple (purely syntactic) cut elimination proof as do display calculi. But it has an easily derivable variant calculus which is amenable to automated proof search as are (some) traditional sequent calculi. We first present the initial calculus and its cut elimination proof. We then present the derived calculus, and then present a proof (...) search strategy which allows it to be used for automated proof search. We prove that this search strategy is terminating and complete by showing how it can be used to mimic derivations obtained from an existing calculus GBiInt for bi intuitionistic logic. As far as we know, our new calculus is the first sequent calculus for bi intuitionistic logic which uses no semantic additions like labels, which has a purely syntactic cut elimination proof, and which can be used naturally for backwards proof search. Keywords: Bi-intuitionistic logic, display calculi, proof search. (shrink)

We present sound, (weakly) complete and cut-free tableau systems for the propositional normal modal logicsS4.3, S4.3.1 andS4.14. When the modality is given a temporal interpretation, these logics respectively model time as a linear dense sequence of points; as a linear discrete sequence of points; and as a branching tree where each branch is a linear discrete sequence of points.Although cut-free, the last two systems do not possess the subformula property. But for any given finite set of formulaeX the superformulae involved (...) are always bounded by a finite set of formulaeX* L depending only onX and the logicL. Thus each system gives a nondeterministic decision procedure for the logic in question. The completeness proofs yield deterministic decision procedures for each logic because each proof is constructive. (shrink)

We present a cut-free tableau calculus with histories and variables for the EXPTIME-complete multi-modal logic of common knowledge. Our calculus constructs the tableau using only one pass, so proof-search for testing theoremhood of ϕ does not exhibit the worst-case EXPTIME-behaviour for all ϕ as in two-pass methods. Our calculus also does not contain a “finitized ω-rule” so that it detects cyclic branches as soon as they arise rather than by worst-case exponential branching with respect to the size of ϕ. Moreover, (...) by retaining the rooted-tree form from traditional tableaux, our calculus becomes amenable to the vast array of optimisation techniques which have proved essential for “practical” automated reasoning in very expressive description logics. Our calculus forms the basis for developing a uniform framework for the family of all fix-point logics of common knowledge. However, there is still no “free lunch” as, in the worst case, our method exhibits 2EXPTIME-behaviour. A prototype implementation can be found at twb.rsise.anu.edu.au which allows users to test formulae via a simple graphical interface. (shrink)

We use a deep embedding of the display calculus for relation algebras RA in the logical framework Isabelle/HOL to formalise a machine-checked proof of cut-admissibility for RA. Unlike other “implementations”, we explicitly formalise the structural induction in Isabelle/HOL and believe this to be the first full formalisation of cutadmissibility in the presence of explicit structural rules.

We study a propositional bimodal logic consisting of two S4 modalities and [a], together with the interaction axiom scheme a ϕ → a ϕ. In the intended semantics, the plain..

The main technical errors are in the literature survey. On pages 44, 93-94, 131 and 133 I claim that Fitting's and/or Rautenberg's systems are incomplete because they omit contraction. The claim is wrong because contraction is implicit in their set notation. Their systems are complete because they allow contraction on any formula whereas the systems in this technical report explicitly build contraction into certain rules, allowing contraction only on certain types of formulae. Please accept my apologies for any confusion this (...) causes you. (shrink)

The Tableaux Work Bench (TWB) is a meta tableau system designed for logicians with limited programming or automatic reasoning knowledge to experiment with new tableau calculi and new decision procedures. It has a simple interface, a history mechanism for controlling loops or pruning the search space, and modal simplification.

In this paper I address the vexed question of the relationship between secular states and religious education.

The scope of social policy today is extensive. With the changing global scenario, there is a rediscovery of “social” in it. Indubitably, there is a gender perspective on social policy globally. The world Economic Forum states that there are only six countries in the world (Belgium, Denmark, France, Latvia, Luxembourg and Sweden) where women have equal work rights to men. It is noted that the situation in different countries varies when it comes to the working benefits of different genders whether (...) for native or expatriate workers in those places. Though there are rooms to enjoy attractive income and favourable working conditions such as job security, generous retirement plan, and other welfare benefits, there are still various factors that might lead to gender-based differences or gender discrimination in the unique labour market context of the GCC countries. The scenario is distinct in many Scandinavian nations like Sweden. Those countries are often considered the role model for gender equal work allowances. When it comes to India there is still disparity and difference in many areas despite the social security system the country offers in its policies. This paper aims at a descriptive and qualitative study on the causes, consequences and conclusion of the gender disparity in employee allowances in these nations. The study would imply a simple random method of interrogation to examine gender differences empirically within the labour market of the mentioned nations. The economic benefits of a gender-equal nation in the framing of social policy will be emphasized and focused on. (shrink)

The concept of marriage has undergone a transition presently when compared with the past. Norms, customs and traditions have also changed. Attitudes, choices and preferences of individuals contribute to these changes accompanied by education and modernization. Equality of women, social changes, and liberalized economy can be a few determinants contributing to the choices and preferences, yet fertility issues remain a nagging problem after marriage. The present trend highlights late marriages, and stress at home and works front for both genders, contributing (...) to instability in marriages. These being certain issues, on one hand, increase in divorce cases, maintenance and custody of children, single parenting is another set of challenges, on the other hand. Hence, marriages are also riskier and those who embark on this journey are the successful lot. This paper primarily aims to identify the perception of youth about marriage as terminologies such as “Live in relationships”, “Friends with benefits”, etc., are in vogue. Secondly, the objective of this paper is also to study if “mid-life crisis” in both the genders accompanied by emotional, physical and psychological issues can be the causal determinants for various issues in marriage. Thirdly, the paper aims to study the causal effects between marriage issues, family laws and the country’s economy as there is a direct nexus between the three forces using appropriate research methodology tools and techniques. (shrink)

Over the years, the urban heat vulnerability has evolved as a pressing global concern for researchers and policymakers alike. Numerous studies have aimed at mitigating the adverse effects of urban heat vulnerability on public health and safety. However, the critical task of selecting the most fitting indicator for urban heat islands in public spaces is not emphasized in the existing studies, considering the diverse indices available. Beyond identification, studies that delve into the prioritization of these indices and the determination of (...) optimal variables along with their respective weights are scanty. The present study attempts to address these gaps in urban heat vulnerability research through the lens of natural solutions. The study employs a fuzzy analytical hierarchy approach to ascertain the weights of selected criteria, providing a nuanced understanding of the multifaceted factors influencing their applicability. Real-time impact estimation using the fuzzy technique for order of preference by similarity to ideal solutions enhances the practicality of the findings. The results reveal a set of pivotal characteristics crucial for constructing a robust heat vulnerability index, facilitating the prioritization of selected criteria. The implementation of this hybrid methodology culminates in a streamlined, precise, and practical tool for evaluating heat vulnerability in hot urban public spaces. These findings contribute valuable insights to the academic discourse and offer a tangible resource for policymakers and practitioners. The proactive approach provides a roadmap for addressing impending challenges related to urban heat vulnerability, marking a positive step toward creating resilient and sustainable urban environments. (shrink)

Two notions of reconciliation exist. The weak or thin conception is akin to ‘resignation’. It is sought by groups that have waged war against one another but have come to the realization that neither can win. Reconciliation in this sense results from an enforced lowering of expectations. In the stronger sense, reconciliation means a virtual cancellation of enmity or estrangement via a morally grounded forgiveness, achievable only when conflicting groups acknowledge collective responsibility for past injustice, and shed their deep prejudices (...) by a profound and painful transformation in their identities. It is because this process is not possible without a somewhat brutal confrontation with oneself and a painful recognition of one’s own moral degradation that reconciliation is difficult to achieve. (shrink)

Background As noise brings great error in the analysis of metallographic images, an adaptive weighted mean filtering method proposed to overcome the shortcomings of the standard mean filtering method. Methods The method used to detect the pulse noise points in the image, and then the modified mean method used to filter out the detected noise points. Patents on metallographic image processing have discussed for the development of the proposed methodology. Results It is shown that filter window can be filtered in (...) comparison with the conventional 3×3, 5×5 and 7×7 filt window to reduce noise detection and reduce the complexity of the weight calculation. Conclusion It can be concluded that this method can better protect the details of the image, has better filtering effect than the standard mean filtering, and its processing speed is faster than the median filtering of the large window, which has profound significance for the edge detection and processing of the metallographic image. (shrink)

This paper presents a method for the study of the influence of stability of a power transformer on the power system based on the vibration principle. Traditionally, the EMD and EEMD algorithms are employed to test the box vibration signal data of the power transformer under three working conditions. The proposed method utilizes a partial EMD screening along with MPEEMD method for the online monitoring of power transformer. A complete online monitoring system is designed by using the STM32 processor and (...) LabVIEW system. The proposed system is compared with EMD and EEMD algorithms in terms of the number of IMFs obtained by decomposition, maximum correlation coefficient, and mean square error. The inherent mode correlation, when compared with the mean square error of the reconstructed signal, shows that the reconstruction error of MPEEMD algorithm is 4.762×10−15 which is better than the traditional EMD algorithm. It is observed from the results that the proposed method outperforms both EMD and EEMD algorithms. (shrink)

This book puts together the most important contemporary writings in the debate on secularism. It deals with conceptual, normative and explanatory issues in secularism and addresses urgent questions, including the relevance of secularism to non-Western societies and the question of minority rights.

India is one of the most culturally, philosophically and religiously diverse countries in the world. The roots, not only of these diversities but also of morally appropriate responses to them, i.e. to pluralism, go very deep. This presentation substantiates this claim by looking at the relevant edicts of Emperor Asoka who reigned in India in the 3rd century BCE. Asoka not only advises people with deeply divergent worldviews to live together face to face but also suggests what the basis for (...) this coexistence could be. He claims that resources exist in all traditions to exercise self-restraints. These self-restraints are of two kinds: self-related and other-related. Everyone should exercise both these self-restraints, particularly in speech. This ‘control of tongue’ is crucial for morally legitimate and principled coexistence. In the article, I try to explicate the meaning of these edicts and flesh out this argument by providing a vivid, quasi-phenomenological account of what public life in Asokan times was like. (shrink)

The literature on methodological individualism is characterized by a widely held view that if the doctrine were stated with sufficient care it would be seen to be trivially true. Professor Bhargava questions this view. He begins by carefully disentangling the various formulations of the doctrine, identifies its most plausible version, and finally locates the principal assumption underlying it, namely that beliefs are attitudes individuated entirely in terms of what lies within the individual mind. Bhargava argues that once this individualist assumption (...) is challenged it is possible to rehabilitate a non-individualist methodology which permits a contextual study of beliefs and actions, and even a study of social context relatively independent of the beliefs and actions of individuals. (shrink)

This Paper is divided into three sections. In the first section I provide a brief historical overview of Hindu-Muslim relations in India and of the condition of Indian Muslims today. I conclude by claiming that Indian Muslims are a marginalized minority who have been persistently underrepresented in political institutions, particularly in the Indian Parliament. This section is important for those who are less informed about these issues—and I assume that most readers fall in this category. In the second section, I (...) examine the case for political representation for Muslims. This was a much debated issue in pre-independent India. It was debated with subtlety and in considerable detail in the Constituent Assembly debates on the Indian constitution. However, with the partition of the country and the formation of the separate state of Pakistan, all debate on the political representation of Muslims ceased. I examine the merits and demerits of the case for the political representation of Indian Muslims. I also attempt a brief explanation of why this issue has virtually disappeared from the public arena in India. I conclude in the section that although political representation of Muslims qua Muslims is desirable, it is still unfeasible in the prevailing situation in India. In other words, I would support the recommendation to the Indian State that political rights not to be granted to any religious community. If political theory was to remain a handmaiden of state policy, then the matter ends right here. However, since I believe that political theory must think for the long run and design just institutions and policies for the future, and since, there is, I claim, no principled objection to the political representation of Muslims, in the third and final section I briefly outline which of the several electoral mechanisms are best suited to ensure fair political representation for Muslims in the future. In my view, the principle of fair political representation for Indian Muslims is best fulfilled by a complex mechanism consisting of preferential voting in multi-member constituencies with intra-party quotas in proportion to the overall population of Muslims in the country. (shrink)