Students´ language in computer-assisted tutoring of mathematical proofs

Truth and proof are central to mathematics. Proving (or disproving) seemingly simple statements often turns out to be one of the hardest mathematical tasks. Yet, doing proofs is rarely taught in the classroom. Studies on cognitive difficulties in learning to do proofs have shown that pupils and students not only often do not understand or cannot apply basic formal reasoning techniques and do not know how to use formal mathematical language, but, at a far more fundamental level, they also do not understand what it means to prove a statement or even do not see the purpose of proof at all. Since insight into the importance of proof and doing proofs as such cannot be learnt other than by practice, learning support through individualised tutoring is in demand. This volume presents a part of an interdisciplinary project, set at the intersection of pedagogical science, artificial intelligence, and (computational) linguistics, which investigated issues involved in provisioning computer-based tutoring of mathematical proofs through dialogue in natural language. The ultimate goal in this context, addressing the above-mentioned need for learning support, is to build intelligent automated tutoring systems for mathematical proofs. The research presented here has been focused on the language that students use while interacting with such a system: its linguistic propeties and computational modelling. Contribution is made at three levels: first, an analysis of language phenomena found in students´ input to a (simulated) proof tutoring system is conducted and the variety of students´ verbalisations is quantitatively assessed, second, a general computational processing strategy for informal mathematical language and methods of modelling prominent language phenomena are proposed, and third, the prospects for natural language as an input modality for proof tutoring systems is evaluated based on collected corpora

Pseudo-contractions as Gentle Repairs

Updating a knowledge base to remove an unwanted consequence is a challenging task. Some of the original sentences must be either deleted or weakened in such a way that the sentence to be removed is no longer entailed by the resulting set. On the other hand, it is desirable that the existing knowledge be preserved as much as possible, minimising the loss of information. Several approaches to this problem can be found in the literature. In particular, when the knowledge is represented by an ontology, two different families of frameworks have been developed in the literature in the past decades with numerous ideas in common but with little interaction between the communities: applications of AGM-like Belief Change and justification-based Ontology Repair. In this paper, we investigate the relationship between pseudo-contraction operations and gentle repairs. Both aim to avoid the complete deletion of sentences when replacing them with weaker versions is enough to prevent the entailment of the unwanted formula. We show the correspondence between concepts on both sides and investigate under which conditions they are equivalent. Furthermore, we propose a unified notation for the two approaches, which might contribute to the integration of the two areas

Topics in Programming Languages, a Philosophical Analysis through the case of Prolog

[EN]Programming languages seldom find proper anchorage in philosophy of logic, language and science. is more, philosophy of language seems to be restricted to natural languages and linguistics, and even philosophy of logic is rarely framed into programming languages topics. The logic programming paradigm and Prolog are, thus, the most adequate paradigm and programming language to work on this subject, combining natural language processing and linguistics, logic programming and constriction methodology on both algorithms and procedures, on an overall philosophizing declarative status. Not only this, but the dimension of the Fifth Generation Computer system related to strong Al wherein Prolog took a major role. and its historical frame in the very crucial dialectic between procedural and declarative paradigms, structuralist and empiricist biases, serves, in exemplar form, to treat straight ahead philosophy of logic, language and science in the contemporaneous age as well. In recounting Prolog's philosophical, mechanical and algorithmic harbingers, the opportunity is open to various routes. We herein shall exemplify some: - the mechanical-computational background explored by Pascal, Leibniz, Boole, Jacquard, Babbage, Konrad Zuse, until reaching to the ACE (Alan Turing) and EDVAC (von Neumann), offering the backbone in computer architecture, and the work of Turing, Church, Gödel, Kleene, von Neumann, Shannon, and others on computability, in parallel lines, throughly studied in detail, permit us to interpret ahead the evolving realm of programming languages. The proper line from lambda-calculus, to the Algol-family, the declarative and procedural split with the C language and Prolog, and the ensuing branching and programming languages explosion and further delimitation, are thereupon inspected as to relate them with the proper syntax, semantics and philosophical élan of logic programming and Prolog

Conceptual analysis knowledge management and conceptual graph theory

There exists an impressive quantity of literature dealing with knowledge Representation that covers highly technical contributions as well as more philosophical ones or again those that have a more or less explicit "cognitive" orientation. So, it is not very astonishing to notice that the definition of what knowledge representation is, is quite vague. It is not our intention to give a historical survey of that notion nor to proceed to a critical enumeration of the several topics that are covered by it. Our objective is, rather, to develop a conceptual framework that should permit us to handle the major descriptive problems in the conception of knowledge based systems. In order to be able to put forth in a systematic way our conception of knowledge representation (KR), we will discuss in the first section some central problems of knowledge description. In the second section, we will introduce the conceptual graph theory developed mainly by Sowa (1984) and try to give a more formal account of KR

Motiivide semiootika: Kenneth Burke ja Deely-Tartu semiootika

Kõrvutades Kenneth Burke'i (1897–1993) ja tänapäeva semiootika võtmekontseptsioonie, uurib käesolev töö motiivi kriitilist rolli tähenduse arenemisel ja avastamisel inimkogemuses. Analüüsime Burke’i töid motiivi rollist inimese tähendusloomes, mida ta kirjeldas kui aktiivset, tahtelist protsessi. Tuginedes eriti Burke’i nn Motivorumi seeria teostele (A Grammar of Motives, A Rhetoric of Motives ja Symbolic of Motives), täiendame Burke'i kontseptuaalset skeemi semiootilise raamiga, tuletades selle John Deely üldsemiootika ja Tartu koolkonna teooria, mida esindab käesolevas eelkõige Kalevi Kull, ristumiskohast. Tulemuseks on analüütiline raamistik, mis jälgib motiivi vajalikkust alates bioloogilisele eesmärgile orienteeritusest kuni inimmõtte formaalsete piirideni. Kasutades Burke'i terministlike ekraanide mõistet, mida käsitletakse siin kui hierarhilise modelleerimise vormi, näitame, et tegelikkuse kohta järelduste tegemiseks peame selle kirjeldamisel kasutama valitud mõisteid, mis samas tähendab nendega kokkusobimatute võimaluste tagasilükkamist. Mõned tähendused tehakse võimalikuks, teised aga sõelutakse semiootiliselt välja. Nii võib tähenduse ja kogemuse osas saada väga erinevaid tulemusi– eriti kontekstides, mida juhivad nn jumalaterminid ehk terministlike ekraanide maksimaalsed vormid. Meie valikud, mis konkretiseerivad motiive, tekitavad kogemuse semiootilise filtri, misläbi kujuneme aja jooksul ise oma valikute ja motiivide märgiks. See on vastuolus modernismi paradigmaga, mis käsitleb motiveeritud järeldusi põhjuslike protsesside tagajärgedena. See vastuolu tuleneb modernismi semiootilisest vundamendist; me kasutame oma Burke'i-semiootilist raamistikku selle aluse, tema motiivide ja tulenevate tähenduste analüüsimiseks. Me väidame, et modernismi sisemised vastuolud põhjustavad realismi ja idealismi kollapsi, mille tulemuseks on kaks tavaliselt seisukohta: (1) naiivne realism, mida praktikas dikteerivad võimuinstitutsioonid, ja (2) naiivne relativism, mille dikteerib üksikagent. Alternatiivina pakume välja teleoloogilise meetodi, mis vaatleb kokkusobimatute tähendusvõimaluste erinevaid trajektoore, jälgib nende (entelehhiaalseid) lõpp-punkte ja näeb neid lõpp-punkte agendi valikute alusena – eriti suure olulisusega tähenduskontekstides. Küsimused on seega järgmised: „Milleni see valik viib? Millise kogemusliku maailma see avab või sulgeb? Milline motiiv selle aluseks on? Mille märgiks ma tahan olla?”By laying in parallel key concepts from Kenneth Burke (1897–1993) and contemporary semiotics, this work explores the critical role that motive plays in the development and discovery of meaning within human experience. We highlight Burke’s thought regarding the role of motive in human meaning-making, which he highlighted as an active, volitional process. Drawing especially from the so-called Motivorum works (A Grammar of Motives, A Rhetoric of Motives, and the nebulous Symbolic of Motives), we enrich Burke’s conceptual schema within an explicitly semiotic frame, derived from an intersection between John Deely’s general semiotics and the Tartu School approach, represented especially by Kalevi Kull. The result is an analytical framework that traces out the necessity of motive from biological goal-orientation to the formal boundaries of human thought. Employing Burke’s notion of terministic screens, treated here as a form of hierarchical modeling, we show that in order to make conclusions about reality, we must use some terms to describe it, which also entails rejecting other incompatible options. Some meanings are made possible, while others are semiotically screened out, which can lead to quite different results for meaning and experience – especially in contexts governed by God-terms, the maximal form of terministic screens. Our choices, which concretize motives, result in a semiotic filter for experience, and over time we become signs of our choices and motives. This runs counter to the Modern paradigm, which treats its motivated conclusions as deterministic entailments. That contradiction arises from Modernism’s semiotic foundation, and we employ our Burkean-semiotic framework to analyze this foundation, its underlying motives, and its meaning results. We argue that Modernism’s internal contradictions result in a collapsing of realism and idealism, with two commonly resulting stances: (1) naive realism, in practice dictated by institutions of power, and (2) naive relativism, dictated by the individual agent. Contrastively, we suggest a teleological method that looks at the various trajectories of incompatible meaning possibilities, traces out their (entelechial) end points, and uses those end points as the basis for which the agent will choose – especially in high-stakes meaning contexts. The questions asked are thus: “What does this choice lead to? What kind of experiential world will it open or close? What kind of motive underlies it? What do I want to be a sign of?”https://www.ester.ee/record=b548547