On multiple conclusion deductions in classical logic

Kneale observed that Gentzen’s calculus of natural deductions NK for classical logic is not symmetric and has unnecessarily complicated hypothetical inference rules. Kneale proposed inference rules with multiple conclusions as a basis for a symmetric natural deduction calculus for classical logic. However, Kneale’s informally presented calculus is not complete. In this paper, we define a calculus of multiple conclusion natural deductions (MCD) for classical propositional logic based on Kneale’s multiple conclusion inference rules. For MCD we present elementary proof search that produces proofs in normal form. MCD proof search is motivated and explained as being a notational variant of Smullyan’s analytic tableaux method in its initial part and a notational variant of refutation proofs based on Robinson’s resolution in its final part. We consider MCD to have semantic motivation of both its inference rules and its proof search. This is unusual for the natural deduction calculi as they are syntactically motivated. Syntactic motivation is adequate for intuitionistic logic but not a natural fit for truth-functional classical propositional logic

Geometrija za analitike učenja

Learning analytics is focused on the educational challenge of optimizing opportunities for meaningful learning. Assessment deeply influences learning, but at the same time data about assessment are rarely considered and utilized by learning analytics. Current approaches to analysis and reasoning about peer-assessment lack rigor and appropriate measures of reliability assessment. Our paper addresses these issues with a geometrical model based on the taxicab geometry and the use of the scoring rubrics. We propose and justify measures for calculation of the final grade in peer-assessment and related inter-rater and intra-rater reliability measures. We present and discuss a geometrical model for two important peer-assessment scenarios.Analitike učenja usredotočene su na obrazovne izazove vezane uz postizanje svrsishodnog učenja. Vrednovanje postizanja ishoda učenja izrazito utječe na učenje. Međutim, podaci o procesu vrednovanja vrlo rijetko se koriste u postojećim analitikama učenja. Nadalje, postojeće implementacije i analize procesa istorazinskog (vršnjačkog) vrednovanja nisu zadovoljavajuće. Ovaj rad predstavlja izradu i upotrebu matematičkog modela za opis i računanje vezano uz istorazinsko vrednovanje. Razvijeni model zasniva se na Manhattan (taxicab) metrici te korištenju rubrika za vrednovanje ishoda učenja. U radu su opisane i opravdane metode računanja konačne ocjene vršnjačkog vrednovanja, mjere pouzdanosti takvog vrednovanja kao i ocjene za pojedine vrednovatelje. Razvijeni geometrijski model razmatran je u kontekstu dva važna scenarija istorazinskog vrednovanja

On multiple conclusion deductions in classical logic

Kneale observed that Gentzen’s calculus of natural deductions NK for classical logic is not symmetric and has unnecessarily complicated hypothetical inference rules. Kneale proposed inference rules with multiple conclusions as a basis for a symmetric natural deduction calculus for classical logic. However, Kneale’s informally presented calculus is not complete. In this paper, we define a calculus of multiple conclusion natural deductions (MCD) for classical propositional logic based on Kneale’s multiple conclusion inference rules. For MCD we present elementary proof search that produces proofs in normal form. MCD proof search is motivated and explained as being a notational variant of Smullyan’s analytic tableaux method in its initial part and a notational variant of refutation proofs based on Robinson’s resolution in its final part. We consider MCD to have semantic motivation of both its inference rules and its proof search. This is unusual for the natural deduction calculi as they are syntactically motivated. Syntactic motivation is adequate for intuitionistic logic but not a natural fit for truth-functional classical propositional logic

Learning Analytics for Peer-assessment: (Dis)advantages, Reliability and Implementation

Learning analytics deals with the data that occurs from students\u27 interaction with ICT: collecting data, analyzing and reporting that can influence learning and teaching. Analysis of validity and reliability of assessment lags behind other applications of learning analytics. We present here mathematical modeling of learning analytics for assessment, especially for peer-assessment. In addition, we analyze and categorize students\u27 recognition of advantages and disadvantages of peer-assessment. Finally implementations of reliability check of peer-assessment in Moodle Workshop module are explained

On Translation Curves and Geodesics in <inline-formula><math display="inline"><semantics><msubsup><mrow><mi>Sol</mi></mrow><mn>1</mn><mn>4</mn></msubsup></semantics></math></inline-formula>

A translation curve in a homogeneous space is a curve such that for a given unit vector at the origin, translation of this vector is tangent to the curve in its every point. Translation curves coincide with geodesics in most Thurston spaces, but not in twisted product Thurston spaces. Moreover, translation curves often seem more intuitive and simpler than geodesics. In this paper, we determine translation curves in Sol14 space. Their curvature properties are discussed and translation spheres are presented. Finally, characterization of geodesics in Sol14 space is given

Evaluating Compromise in Social Choice Functions

We investigate the notion of compromise in the strict preferential voting setting. We introduce divergence as an inverse measure of compromise in a collection of strict preferential votes. Classical functions of social choice theory are analyzed with respect to divergence. New social welfare functions and new social choice functions with the objective of compromise are defined directly from optimization of divergence and later analyzed with respect to the common desiderata of social choice theory. For a very natural function, a simple divergence minimizer, we prove it satisfies the properties of anonymity, neutrality, consistence, and continuity. Consequently, according to Young’s theorem of characterization it follows that this function is a scoring point function. Its scoring point vector is also given. Finally, we discuss the parameter p in the divergence measure which was introduced to address vagueness and fuzziness of compromise and to control for a variety of intended levels of compromise