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The article delves into the concept of refutation in Gentzen calculi, specifically exploring the differences between intuitionistic and classical logic. It introduces R-formulas and derivability in calculi, emphasizing the need for a semantic foundation to compare various logical systems. The text discusses rules for proving and refuting derivability relations, leading to a deductively closed calculus. It also covers the replacement theorem for Gentzen calculi in the context of K∞, defining logical operators and proving the completeness of the system. The article further examines direct propositional logic in Gentzen calculi, excluding indirect inferences and proving the equivalence of formulas and completeness of operator systems through induction and logical rules. The distinctions between direct logic, intuitionistic logic, and classical propositional logic are discussed within the Gentzen semantics framework, with references to works by Ackermann, Gentzen, Schütte, and von Kutschera. [Extracted from the article]

The Exact Exposition of the Orthodox Faith is perhaps the most logically structured and inspired work not only in the oeuvre of the seventh-to-eighth-century theologian John Damascene, but most likely throughout the entire Greek Patristic literature. As such, the Exact Exposition definitely presents some quite intriguing features, such as the prolific use of logical distinctions, syllogisms, or full-fledged arguments, to name a few. Regarding the latter, John Damascene's use of certain arguments in order to prove the existence of God not only hold a unique place in Byzantine theology but have also exercised a tremendous influence on Eastern Orthodox apologetics. However, what I would call his rationalization agenda comes not only with merits but with faults as well. It is to both these that the present study draws attention by evaluating them logico-philosophically and interpreting them religio-hermeneutically. What is of special interest is the fact that John Damascene's logical faults are the most interesting parts of his theologizing. [ABSTRACT FROM AUTHOR]

In Islamic law, preponderance (tarjīḥ)—a practical method for mujtahid s to resolve legal contradictions (taʿāruḍ) between proofs—has been known in the uṣūl tradition from at least the 10th-century jurist al-Jaṣṣāṣ (d. 370/981). Yet, it is in several 17th and 18th-century Arabic and Ottoman manuscripts of Princeton's Garrett Collection that we encounter summary-like lists labelled "tarjīḥāt al-bayyināt" ("TB s"), which succinctly compile the complex rules of preponderance. Organized into three-columned lists, on loose leaves, as annotations in the margin or separate textual units, the TB s follow a grammatical and visual layout that made them predictable and recognizable for manuscript readers. This paper examines the TB s as a codicological phenomenon, arguing that they served as a shorthand for legal practitioners familiar with evidentiary law and that their presence suggests a broader transformative process of readers'/legal practitioners' relationship with codices of positive law at this critical moment in the history of Islamic law. [ABSTRACT FROM AUTHOR]

This is a comment on a translation of Franz von Kutschera's paper ‘Ein verallgemeinerter Widerlegungsbegriff für Gentzenkalküle’, which was published in German in 1969. The paper is an important predecessor of what is nowadays called ‘proof-theoretic semantics’, which describes the view that the meaning of logical connectives is determined by the rules governing their use in a proof system. Von Kutschera adopts this view in this paper, and more specifically, a bilateralist view on this subject in that his aim is to give a general framework that provides generalized rule schemata for arbitrary connectives both for proving and refuting and to use this as a reference to prove completeness of certain systems of operators purely proof-theoretically. The main logical system in focus here has been shown to be equivalent to N4, Nelson's constructive logic with strong negation. In order to understand the translated paper better, a contextualizing comment is offered referring and relating it to a preceding paper by von Kutschera. [ABSTRACT FROM AUTHOR]

'We are convinced most of all whenever we take something to have been demonstrated' (1355a5-6). The meaning and significance of this claim is a key point of dispute between those who take Aristotle's project in the Rhetoric to be defending his distinctively argument-centred kind of rhetoric on the grounds that it is most persuasively effective, and those for whom he does so on the more normatively-charged grounds that this is the most valuable kind of rhetoric, and best delivers rhetoric's distinctive benefits to civic communities. On the interpretation defended, the claim links being convinced (πιστεύειν) and the things that get us convinced (πίστεις) to the kind of epistemic merits possessed above all by demonstrations. This saves Aristotle from an implausible generalisation about the persuasive supremacy of deductive arguments. Since πίστεις are clearly central to Aristotelian rhetoric, this interpretation also lends support to the more normative understanding of Aristotle's project overall. [ABSTRACT FROM AUTHOR]

Mathematics is often treated as different from other disciplines, since arguments in the field rely on deductive proof rather than empirical evidence as in the natural sciences. A mathematical paper can therefore, at least in principle, be replicated simply by reading it. While this distinction is sometimes taken as the basis to claim that the results in mathematics are therefore certain, mathematicians themselves know that the published literature contains many mistakes. Reading a proof is not easy, and checking whether an argument constitutes a proof is surprisingly difficult. This article uses peer review of submissions to mathematics journals as a site where referees are explicitly concerned with checking whether a paper is correct and therefore could be published. Drawing on 95 qualitative interviews with mathematics journal editors, as well as a collection of more than 100 referee reports and other correspondence from peer review processes, this article establishes that while mathematicians acknowledge that peer review does not guarantee correctness, they still value it. For mathematicians, peer review 'adds a bit of certainty', especially in contrast to papers only submitted to preprint servers such as arXiv. Furthermore, during peer review there can be disagreements not just regarding the importance of a result, but also whether a particular argument constitutes a proof or not (in particular, whether there are substantial gaps in the proof). Finally, the mathematical community is seen as important when it comes to accepting arguments as proofs and assigning certainty to results. Publishing an argument in a peer-reviewed journal is often only the first step in having a result accepted. Results get accepted if they stand the test of time and are used by other mathematicians. [ABSTRACT FROM AUTHOR]

Geometric argumentation has an important role in solving mathematical problems in geometric material, so students must have this ability. Each student has different thoughts, including when stating arguments. Each student's arguments will vary and be at different levels. This study aims to determine the levels of lower secondary school students' geometric argumentation. This research was conducted in a lower secondary school involving 20 ninth-grade students. Students participating in this study were asked to work on geometry problems related to proof. Through the proofs carried out, the argumentation structure owned by students is visible. The structures of argumentation given by the students were then analysed using Toulmin's model of argumentation. The components of the Toulmin model used consist of claim, data, warrant, and backing. The results of the analysis of the proof prepared by the students stated that some of the students have been able to reach a high level of geometric argumentation and can compile a series of proofs. But not a few of them also have difficulty compiling the proof, have difficulty providing the components of the Toulmin model, and make some mistakes. Errors made by students include symbol writing errors, calculation errors, and others.

We survey a collection of proofs towards equality, inequality, or independence of the relation of P to NP. Since the problem has attracted much attention from experts, amateurs, and in-betweens, this work is intended as a pointer into directions to enable a “self-assessment” of ideas laid out by people interested in the problem. To this end, we identify the most popular approaches to proving equality, inequality, or independence. Since the latter category appears to be without any attempts to follow the necessary proof strategies, we devote a Section to an intuitive outline of how independence proofs would work. In the other cases of proving equality or inequality, known barriers like (affine) relativization, algebrization, and others are to be avoided. The most important and powerful technique available in this regard is a formalization of arguments in automated proof assistants. This allows an objective self-check of a proof before presenting it to the scientific community.

The Exact Exposition of the Orthodox Faith is perhaps the most logically structured and inspired work not only in the oeuvre of the seventh-to-eighth-century theologian John Damascene, but most likely throughout the entire Greek Patristic literature. As such, the Exact Exposition definitely presents some quite intriguing features, such as the prolific use of logical distinctions, syllogisms, or full-fledged arguments, to name a few. Regarding the latter, John Damascene’s use of certain arguments in order to prove the existence of God not only hold a unique place in Byzantine theology but have also exercised a tremendous influence on Eastern Orthodox apologetics. However, what I would call his rationalization agenda comes not only with merits but with faults as well. It is to both these that the present study draws attention by evaluating them logico-philosophically and interpreting them religio-hermeneutically. What is of special interest is the fact that John Damascene’s logical faults are the most interesting parts of his theologizing.

It is prima facie uncontroversial that the justification of an assertion amounts to a collection of other (inferentially related) assertions. In this paper, we point at a class of assertions, i.e. mathematical assertions, that appear to systematically flout this principle. To justify a mathematical assertion (e.g. a theorem) is to provide a proof—and proofs are sequences of directives. The claim is backed up by linguistic data on the use of imperatives in proofs, and by a pragmatic analysis of theorems and their proofs. Proofs, we argue, are sequences of instructions whose performance inevitably gets one to truth. It follows that a felicitous theorem, i.e. a theorem that has been correctly proven, is a persuasive theorem. When it comes to mathematical assertions, there is no sharp distinction between illocutionary and perlocutionary success. [ABSTRACT FROM AUTHOR]

Helping learners to develop a solid grasp of geometric concepts poses a challenge for teachers. Therefore, it is important that teachers have a sound understanding of the geometry they teach. The aim of this qualitative study was to explore pre-service teachers' (PST's) understanding of the concept of similarity in Euclidean geometry and to use van Hiele's theory to explain misconceptions evidenced by the PSTs. Data in this study were collected from 34 first-year PSTs studying for a Bachelor of Education degree in high school mathematics. The authors analysed the written responses to a 13-item worksheet and also conducted interviews with seven of the participants. The analysis of the data was guided by van Hiele's theory which was used to identify misconceptions amongst PST's who had not yet developed the appropriate reasoning skills linked to particular van Hiele levels of geometric thought. It was found that these students used reasoning that is characteristic of the elementary levels to make judgments. Many PST's faced challenges with similarity notation and the process of proving the similarity between two figures. This study recommends that PST's should be given more opportunities to connect visual and analytic representations of similarity. [ABSTRACT FROM AUTHOR]

Usually, proofs of mathematical statements involve both algebraic rearrangements and logical reasoning. But there are mathematical statements whose truth is obvious at first glance when there is a diagram illustrating that proof. Although the proofs based on the drawing are not necessarily full and complete, but the drawing helps to notice facts that are then easily supported by algebra and logic. The paper presents proofs of mathematical propositions where, upon careful study of the drawing, the main idea of the proof can be seen from the drawing, and the proof itself becomes beautiful and clear.

This article presents several activities suitable for a transition to proofs course. In addition, this article surveys literature in support of active learning in the transition to proofs course and discusses how these activities have been successfully implemented in one such course. [ABSTRACT FROM AUTHOR]

Copyright of Naša Škola (Sarajevo. Online) is the property of Udruzenje Nastavnika Federacije BiH and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

Copyright of Lietuvos Matematikos Rinkinys is the property of Vilnius University and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

Copyright of Naša Škola (Sarajevo. Online) is the property of Udruzenje Nastavnika Federacije BiH and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

One of the promising techniques to address the dependability of a system is to apply, at early design stages, domain-specific languages (DSLs) with execution semantics. Indeed, an executable DSL would not only represent the expected system's structure, but it is intended to itself behave as the system should run. In order to make executable DSLs a powerful asset in the development of safety-critical systems, not only a rigorous development process is required but the domain expert should also have confidence in the execution semantics provided by the DSL developer. To this aim, we recently developed the Meeduse tool and showed how to bridge the gap between MDE and a proof-based formal approach. In this work, we apply our approach to the Petri-net DSL and we present MeeNET, a proved Petri-net designer and animator powered by Meeduse. MeeNET is built on top of PNML (Petri-Net Markup Language), the international standard ISO/IEC 15909 for Petri-nets, and provides underlying formal static and dynamic semantics that are verified by automated reasoning tools. This paper first presents simplified MDE implementations of Petri-nets applying Java, QVT, Kermeta and fUML that we experimented in order to debug a safety-critical system and summarises the lessons learned from this study. Then, it provides formal alternatives, based on the B method and process algebra, which are well-established techniques allowing interactive animation on the one hand and reasoning about the behaviour correctness, on the other hand. [ABSTRACT FROM AUTHOR]

Helping learners to develop a solid grasp of geometric concepts poses a challenge for teachers. Therefore, it is important that teachers have a sound understanding of the geometry they teach. The aim of this qualitative study was to explore pre-service teachers’ (PST’s) understanding of the concept of similarity in Euclidean geometry and to use van Hiele’s theory to explain misconceptions evidenced by the PSTs. Data in this study were collected from 34 first-year PSTs studying for a Bachelor of Education degree in high school mathematics. The authors analysed the written responses to a 13-item worksheet and also conducted interviews with seven of the participants. The analysis of the data was guided by van Hiele’s theory which was used to identify misconceptions amongst PST’s who had not yet developed the appropriate reasoning skills linked to particular van Hiele levels of geometric thought. It was found that these students used reasoning that is characteristic of the elementary levels to make judgments. Many PST’s faced challenges with similarity notation and the process of proving the similarity between two figures. This study recommends that PST’s should be given more opportunities to connect visual and analytic representations of similarity.

We propose a systematic approach to the construction of invariant union of polytopes (IUP) in expanding piecewise affine mappings whose linear components are isotropic scalings. The approach relies on using empirical information embedded in trajectories in order to infer, and then to solve, a so-called conditioning problem for some generating collection of polytopes. A conditioning problem consists of a series of requirements on the polytopes’ localisation and on the dynamical transitions between these elements. The core element of the approach is a reformulation of the problem as a set of piecewise linear inequalities for some matrices which encapsulate geometric constraints. In that way, the original topological puzzle is converted into a standard problem in computational geometry. This transformation involves an optimisation procedure that ensures that both problems are equivalent. As a proof of concept, the approach is applied to the study of the loss of ergodicity in basic examples of globally coupled maps. The study explains, completes and substantially extends previous achievements about asymmetric IUP in these systems. Comparison with the numerics reveals sharp existence conditions depending on the map parameters, and accurate fits of the empirical ergodic components. In addition, this application also reveals unanticipated features about conditioning problem solutions, especially as the dependence on the set of admissible face directions is concerned. [ABSTRACT FROM AUTHOR]

By framing Aristotle’s dialectic in the broader context of scientific inquiry and demonstration, this paper is aimed at showing of what use the “reputable opinions” can be for grasping the principles of sciences, as declared in Topics I.2. It argues that such a use cannot imply ‒ at any stage of inquiry ‒ a replacement of the logic and intrinsic goals of demonstration by those proper to dialectic. However, it also defends a substantive (but still modest) contribution of dialectic ‒ beyond its well-attested methodological role in discarding contradictory opinions and its (possible though not germane to the context of Topics I.2) application to proving the principle of non-contradiction by means of refutation. This contribution consists in providing the preliminary accounts of facts in order to have scientific inquiry started, as required in Posterior Analytics II.8. To better appreciate how the proposed location of dialectic in a pre-demonstrative stage of inquiry is operational, the paper finally examines Physics IV.1-5.

Copyright of IUG Journal of Islamic Studies is the property of Islamic University of Gaza and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

It is well known that Arbitration agreements play a major and significant role in the arbitration process and helps to set the general framework for the arbitration dispute and to resolve the arguments without going to courts. In this article, cases of deviations of the general principle that prevents the court from interfering with the arbitral agreement have been studied and illustrated. This is presented in three sections; the first section examines the court's role in Calling and questioning witnesses, The second section studies the court's role in summoning the expert, and The third section Bring a document or a copy of it or view it, all of that by referring to both the Jordanian arbitration law the UNCITRAL Law, the International Chamber of Commerce in Paris (ICC) and some other comparative Arabic laws, as well as, the amount that these legislations take into account the benefit of the parties to the arbitral agreement, as they are the basis for resorting the arbitration. To conclude, this paper ends with a conclusion whereby results and recommendations will be highlighted. [ABSTRACT FROM AUTHOR]

In this note, I argue for and discuss my experiences with explicitly incorporating principles of critical and creative thinking in a transitions course which serves mathematics, mathematics education, and statistics majors. I describe several specific assignments and classroom tasks designed to enhance critical and creative thinking. I also discuss some evidence from survey instruments. [ABSTRACT FROM AUTHOR]

Copyright of Divulgaciones Matematicas is the property of Universidad del Zulia, Facultad Experimental de Ciencias and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

This paper suggests that Herbert Simon's concept of proof and predictions, in the solution of problems by human's, considered as Information Processing Agents subject to boundedly rational behaviour and satisficing objectives, is to be interpreted in terms of constructive mathematics. [ABSTRACT FROM AUTHOR]

Problematizing some of the mains aspects of the sociologies of the individual is the aim of this article with the intention of collaborating for a better understanding of the concepts and methods that have been mobilized for the sociological objectification of individual existence. Three central perspectives of this field are discussed: Bernard Lahire's theory of action, Jean-Claude Kaufmann's theory of habits, and Danilo Martuccelli's theory of individuation. It is considered in that choice the similar type of sociological practice that these authors carry out, which is sustained by a continuous relationship between research and original theorization. The article presents the contributions of the sociologies of the individual to contemporary sociological theory, underlining them as part of a heterogeneous theoretical-methodological field still in development. [ABSTRACT FROM AUTHOR]

The administrative proceeding in the matter of the declaration of the nullity of sacred ordination is currently regulated by Regulae servandae instruction of 16 October 2001 issued by the Congregation for Divine Worship and the Discipline of the Sacraments, which since 1 October 2011 has been applied in accordance with the changes introduced by Pope Benedict XV in the motu proprio Querit semper of 30 August 2011. This instruction replaced the previous Regulae servandae issued on 9 June 1931 by the Congregation for the Discipline of the Sacraments. The whole procedure consists of two stages: the case instruction, conducted by an ordinary authorized by the head of the Administrative Office of the Roman Rota (here the dean of the Roman Rota) and the legal decision of the case (in the competences of the Office). The studied regulations, adjusted to the Canon Law Code of 1983, guarantee protection of the rights of the priest who lodged a claim concerning declaration the nullity of sacred ordination and lead to forming moral certainty by the commissioners and the dean of the Roman Rota while dealing with the case. Efficient case instruction is possible due to the ordinary and instructor eligibility regarding designation of clerks and assistants as well as dispensing with the procedure when it is demanded by a rightful cause. Another factor is the ordinary's liability to watch over the observance of the law and application of the prescribed norms. [ABSTRACT FROM AUTHOR]

This study investigated the effects of students' constructions of geometrical concepts related to the circle on their construction and validation of proofs. The participants were 110 high school students. A questionnaire was administered; both qualitative and quantitative methods were used to analyze the results. Afterwards, in-depth interviews were conducted with some of the participants. The findings from the interviews enriched and strengthened the findings from the questionnaire. Together, the findings highlight the impact of two factors on the ability to construct or evaluate proofs: (1) the use of the self-attributes of a single presented drawing instead of the critical attributes of the concept; and (2) the use of prototypical or non-prototypical examples. In this study, the position of the drawing attached to the assignment affected students' construction of proofs. [ABSTRACT FROM AUTHOR]