The notion of a function from ℕ to ℕ defined by recursion on ordinal notations is fundamental in proof theory. Here this notion is generalized to functions on the universe of sets, using notations for well orderings longer than the class of ordinals. The generalization is used to bound the rate of growth of any function on the universe of sets that is Σ1-definable in Kripke–Platek admissible set theory with an axiom of infinity. Formalizing the argument provides an ordinal (...) class='Hi'>analysis. (shrink)

Multicultural environments require learning multiple number notations wherein some are encountered more frequently than others. This leads to differences in exposure and consequently differences in usage between notations. We find that differential notational usage imposes a significant neurocognitive load on number processing. Despite simultaneous acquisition, forty-two adult binumerate populations, familiar with two positional writing systems namely Hindu Nagari digits and Hindu Arabic digits, reported significantly lower preference and usage for Nagari as compared to Arabic. Twenty-four participants showed significantly increased (...) reaction times and reduced accuracy while performing magnitude comparison tasks in Nagari with respect to Arabic. Functional magnetic resonance imaging revealed that processing Nagari elicited significantly greater activity in number processing and attention networks. A direct subtraction of networks for Nagari and Arabic notations revealed a neural circuit comprising of bilateral intra-parietal sulcus, inferior and mid frontal gyri, fusiform gyrus and the anterior cingulate cortex (FDR p<0.005). Additionally, whole brain correlation analysis showed that activity in the left inferior parietal region was modulated by task performance in Nagari. We attribute the increased activation in Ng to increased task difficulty due to infrequent exposure and usage. Our results reiterate the role of the left intra-parietal sulcus in modulating performance in numeric tasks and highlight that of the attention network for monitoring symbolic notation mode in binumerates. (shrink)

This article was first published as “Notation atomique et hypothèses atomistiques”, Revue des questions scientifiques, 31 (1892), 391– 457. It is the second of a series of articles Duhem was to publish in the Catholic journal Revue des questions scientifiques, in which he presents his understanding of what can justifiably be said about the structure of chemical substances as captured by chemical formulas. The argument unfolds following a broadly historical development of events throughout the course of the century which was (...) coming to a close as he wrote. He later reflected in his classic The Aim and Structure of Physical Theories – based in large part on articles which had appeared in the Revue – that “To give the history of a physical principle is at the same time to make a logical analysis of it” (p. 269). Logical analysis clearly dominates in the present article. The historical context was elaborated considerably in a later work, Le mixte et la combinaison chimique: Essai sur l’évolution d’une idée, which did not lead him to retract any aspect of his earlier position but provided a broader setting in which it could be elaborated. In particular, the Aristotelian influence, which is only hinted at here in some of the formulations (see especially the beginning of section VII) without mentioning Aristotle by name, is explicit in the later work, making Duhem’s own ontological conception a little clearer. A discussion of stereoisomerism, conspicuously absent in the present article, is also integrated into the later book. The same holds for Avogadro’s hypothesis. (shrink)

In this paper, I attempt to address some of the themes of Ludwig Wittgenstein’s Tractatus logico-philosophicus with the aim of their deconstructionist interpretation. My analysis is based on David Gunkel’s book Deconstruction (MIT Press 2021). Based on some of its findings, I show how the Tractatus allows deconstruction and its practice to be thought. I show that the graphic structure of signs is crucial for the young Wittgenstein’s analysis and that it justifies the metaphysical findings in favor of (...) which he argues. ------------------------- Received:11/09/2022. Reviewed: 17/11/2022. Accepted: 1/12/2022. (shrink)

The primary purpose of the system of linear notation is to make the logic of the syllogism more convenient to use by eliminating many of the operations required by its traditional forms. Except for its employment of the distinction between symmetric and nonsymmetric relations and the distinction between transitive and nontransitive relations, linear notation introduces no new principles into syllogistic logic. It is new only as a system of notation. As a system of notation it radically simplifies the application of (...) the principles of syllogistic logic to the analysis of arguments.The use of linear notation enables us to state subject-predicate propositions in such a way that their formal relations with each other are revealed directly without the necessity of performing any of the separate operations required by the traditional methods of syllogistic logic. Very complex arguments, involving any number of related premises, can be stated in such form that all possible conclusions that are obtainable by any combination of syllogistic operations may be read directly from the premises. (shrink)

It is argued that standard music notation pictures Aristotle’s time (time, as Aristotle conceived of it) in a number of important respects, which concern its micro-structure. It is then argued that this allows us to see some features of Aristotle’s time more clearly. Most importantly, Aristotelian instants can be pictured by bar-lines. This allows us to see as how radically devoid of any content Aristotelian instants should be interpreted. Thus, attention to music notation may show why Aristotle was not a (...) presentist. (shrink)

A number of accounts of expert programming behavior have been advanced. These models of the programming activity have served to highlight the range of factors that are thought to underpin programming strategy. However, such accounts have tended to emphasize either the effects of the organization of the programmer's knowledge representation or the role played by features of the notation of the task language on the emergence, development, and support of particular forms of strategy. Such work has neglected to (a) provide (...) an account of the way in which these factors might interact to determine programming strategy, and (b) shed light upon the nature of the development of particular strategies as programming skill increases.This article presents the results of an empirical analysis of the strategies employed by programmers of varying skill levels using different programming languages. The development of particular forms of strategy is hypothesized to be related both to the development of systematic asymmetries in programmers' generic plan‐based representations of programming knowledge and to the way in which features of the notation of particular programming languages might differentially support particular strategies. However, the results of the study reported here suggest that these effects must be interpreted within a broad developmental context. Hence, as programming skill develops, it appears that the notation of the task language tends to take precedence as a determinant of strategy, but has less relevance at beginning stages of skill development and within the context of expert performance. This finding would not be predicted by current models of the programming activity because such models suggest that any notational effects will tend to be consistent regardless of the skill level attained by the programmer. This article endeavors to extend current models of programming by emphasizing the need to consider in detail not only the development of programming strategy, but also the way in which knowledge representation and features of the task langauge interact to give rise to particular forms of programming behavior. (shrink)

Peano was one of the driving forces behind the development of the current mathematical formalism. In this paper, we study his particular approach to notational design and present some original features of his notations. To explain the motivations underlying Peano's approach, we first present his view of logic as a method of analysis and his desire for a rigorous and concise symbolism to represent mathematical ideas. On the basis of both his practice and his explicit reflections on notations, (...) we discuss the principles that guided Peano's introduction of new symbols, the choice of characters, and the layout of formulas. Finally, we take a closer look, from a systematic and historical perspective, at one of Peano's most striking innovations, his use of dots for the grouping of subformulas. (shrink)

The nature of time is perceived by intellectuals variedly. An attempt is made in this paper to reconcile such varied views in the light of the Upanishads and related Indian spiritual and philosophical texts. The complex analysis of modern mathematics is used to represent the nature and presentation physical and psychological times so differentiated. Also the relation between time and energy is probed using uncertainty relations, forms of energy and phases of matter.

This paper introduces a formal concept of ideology and ideological system. The formalization takes ideologies and ideological systems to be situated in agent societies. An ideological system is defined as a system of operations able to create, maintain, and extinguish the ideologies adopted by the social groups of agent societies. The concepts of group ideology, ideological contradiction, ideological dominance, and dominant ideology of an agent society, are defined. An ideology-based concept of social group is introduced. Relations between the proposed formal (...) concept of ideology and the classical concepts of ideology elaborated in social sciences are examined. A computational notation is presented, to support the realization of ideological systems in computationally implemented agent societies. The adequacy of the approach for the formal modeling and analysis of ideological issues is illustrated through three case studies. (shrink)

This study explores the relationship between the scoring structure and the win or loss of a badminton match, while providing quantitative analytic data using binary entropy to determine the uncertainty of said win or loss. Scoring structure data were collected from the official match records of the top 16 events of the World Badminton Championships from 2006 to 2020 as collection objects and were analyzed by means of notational analysis. Our entropy analysis showed that the main factor (...) affecting the certainty of win or loss in men’s singles, men’s doubles and mixed doubles comes from the number of leading points, and in women’s singles and women’s doubles from whether the current point is closer to the match point. Our binary-entropy analysis based on scoring structure showed that, to maintain high uncertainty so that players stay competitive, the scoring points of two sides should differ in less than 5; in addition, the decisive factors for victory strongly depend on gender, also justifying research results of previous studies. (shrink)

The aim of the present study was to determine statistical differences in a set of badminton competition matches in five different modalities with regard to competition level (Group Phase versus Eliminatory Phase). Data from 453 sets (125 in men’s singles; 108 sets in women’s singles; 77 sets in men’s doubles; 73 in women’s doubles and 70 in mixed doubles) from the RIO 2016 Olympics Games were recorded and classified in two groups of variables to analyse variables related to match (5) (...) and set (15). A descriptive analysis and univariate test (Mann Whitney U) for non-parametric data were conducted. The results show in men’s and women’s singles all the variables related to match were higher in the Elimination Phase than in the Group Phase (p<0.01). In Sets 1 and 3, the longest set duration, rally and average rally were found in the Elimination Phase than Group Stage (p<0.05). In women’s singles, these differences were also recorded in Set 2. For doubles, the results are more stable among groups. Men’s doubles had a longer duration of the match and set (set 1 and set 2) (p<0.01), and also scored highest for average rally strokes (set 1 and set 3) (p<0.05) and shuttles used in the Elimination Phase versus the Group Phase along the match (p<0.01). In women’s doubles, more shuttles were used in a match in the Elimination than in the Group Phase. Moreover, the same results are established for Set 2, including for average rally. Mixed doubles saw no match going to three sets. However, the greatest differences showed a longer rally and average rally being registered in the Elimination than in the Group Phase. In conclusion, the timing factors of the badminton singles and doubles games were different in the Elimination and Group Phases. This information may help players and coaches prepare and administer different types of workouts or, more specifically, competition schedules adapted to the characteristics of modern badminton. (shrink)

What is common to all languages is notation, so Universal Grammar can be understood as a system of notational types. Given that infants acquire language, it can be assumed to arise from some a priori mental structure. Viewing language as having the two layers of calculus and protocol, we can set aside the communicative habits of speakers. Accordingly, an analysis of notation results in the three types of Identifier, Modifier and Connective. Modifiers are further interpreted as Quantifiers and (...) Qualifiers. The resulting four notational types constitute the categories of Universal Grammar. Its ontology is argued to consist in the underlying cognitive schema of Essence, Quantity, Quality and Relation. The four categories of Universal Grammar are structured as polysemous fields and are each constituted as a radial network centred on some root concept which, however, need not be lexicalized. The branches spread out along troponymic vectors and together map out all possible lexemes. The notational typology of Universal Grammar is applied in a linguistic analysis of the ‘parts of speech’ using the English language. The analysis constitutes a ‘proof of concept’ in (1) showing how the schema of Universal Grammar is capable of classifying the so-called ‘parts of speech’, (2) presenting a coherent analysis of the verb, and (3) showing how the underlying cognitive schema allows for a sub-classification of the auxiliaries. (shrink)

Gentzen’s singular sequential system of first-order logic was an alternative notation for his system of natural deductions. His multiple sequential system was his symmetric generalization that was more appropriate to classical logic. Beth’s tableaus system was a system that was derived directly from the semantic analysis of connectives and quantifiers. It was soon realized that the Beth’s system and the Gentzen’s multiple system were only notational variants of each other. Kneale’s system of multiple natural deductions was a generalization (...) of Gentzen’s system of natural deductions. We prove that Kneale’s natural deductions are also a notational variant of Beth’s tableaus. (shrink)

As one of the tools used to express feelings and thoughts, music has been utilised by people in many fields throughout history. Music is seen as a means of expressing religious feelings, being used as an educational tool, as a way for military bands to invoke heroic feelings in soldiers, and as way of expressing emotions in joyful and melancholic days. Music was especially born and shaped by rituals of religious origin. With the spread of the religion of Islam, music (...) became an issue that paved the way for many religious discussions. It should be clearly noted that while many verses in the Qur'an are cited as a source for these discussions, it is very difficult to deduce from these verses whether music is halal (permissible) or haram (prohibited). Additionally, many permissible matters in Islam become prohibited when used maliciously or out of their intended purpose, however, if everything that has a possibility of being used maliciously or outside of its intended purpose is considered haram, this will lead to a path of denying many blessings that Allah has bestowed upon people in this world. Restricting issues out of the possibility of their wrongful misuse and depriving others of this blessings cannot be just. Nevertheless, although the discussion of music as permissible or prohibited has largely existed within the religion of Islam, those involved with Tasawwuf (Islamic Mysticism), also known as Sufis, have generally perceived music as nutrient for the soul. Consequently, Sufis have incorporated music into their worship in order to invoke religious feelings over the centuries. Religious music is divided into two; the first is Mosque Recitations, and the second is Tekke music (also known as Tasawwuf Music). Mosque Recitation is that which is performed without instrumentals using only vocals and an improvised approach. On the other hand, Tekke music is performed with an instrument and a composed written piece. Within the Mosque Recitation is the Qur’an recitation, which when read in accordance with its rules has a natural phonetic style, meaning that when the Qur’an is recited according to the Maqam (Melodic modes of recitation) requirements, it evokes spiritual feelings to the listeners. Accordingly, there is a connection between the recitation of the Qur'an and the scientific understanding of music. When a connection between the Qur’an and disciplined recitation is established, this causes an artistic sensation to the listener, which experts in the field have name “Fem-i Muhsin”.Hafız (a title used for a person who has memorised the Qur’an in its entirety) Bekir Sıdkı Sezgin is one of the important musicians to have left his mark on Turkish music, where a discipline was formed after him. He has a very high status due to the value he gave to Turkish music as he became one of the leading names in Turkish music, especially considering the large yet unknown number of students that he personally trained. Hafız Bekir states in interviews that he especially owed his success in the science of music to the Qur'an education he received at a young age, and he emphasises that those who are on the way to becoming good musicians must receive a Qur'an education; additionally, because classical Turkish music lyrics are Arabic and Persian, he argues that the way to correctly pronounce these lyrics is to also know Ottoman Turkish to a high degree. Hafız Bekir was also able to recite the Qur'an to a degree of perfection and was proud of being a Hafız, where he believed that the Qur'an harmonised between his memory and excellent musical knowledge. Hafız Bekir recited the Qur’an according to the suitable Istanbul Maqam of reading, and by using the Maqam befitting the Qur’an recitation, Hafız Bekir was able to take the listeners on a spiritual journey.A survey of the works produced about Hafız Bekir shows that although there have been many theses, articles, and interviews published about him, there is no specific study on his Qur’an recitation according to the Istanbul Maqam style. In this study, Hafız Bekir’s dual identity as both a musician and Qur’an Hafız is underlined as a way to contribute to the field by revealing this side of Hafız Bekir, and also examining the Qur'an education that he recommends to his students. The recitation of verses 189-194 of the chapter Âli İmrân, which Hafız Bekir recited in the Istanbul Maqam style, is notated with the Mus notation system, and analysed according to the Maqam reading style. The notation is formed based on a sensory evaluation with assistance from experts in the field. This study, as a musical analysis, was produced to reveal the importance of this subject to interested parties. (shrink)

We consider five ordinal notation systems of ε0 which are all well-known and of interest in proof-theoretic analysis of Peano arithmetic: Cantor’s system, systems based on binary trees and on countable tree-ordinals, and the systems due to Schütte and Simpson, and to Beklemishev. The main point of this paper is to demonstrate that the systems except the system based on binary trees are equivalent as structured systems, in spite of the fact that they have their origins in different views (...) and trials in proof theory. This is true while Weiermann’s results based on Friedman-style miniaturization indicate that the system based on binary trees is of different character than the others. (shrink)

Which function is computed by a Turing machine will depend on how the symbols it manipulates are interpreted. Further, by invoking bizarre systems of notation it is easy to define Turing machines that compute textbook examples of uncomputable functions, such as the solution to the decision problem for first-order logic. Thus, the distinction between computable and uncomputable functions depends on the system of notation used. This raises the question: which systems of notation are the relevant ones for determining whether a (...) function is computable? These are the acceptable notations. I argue for a use criterion of acceptability: the acceptable notations for a domain of objects D\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {D}}$$\end{document} are the notations that we can use for the usual D\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {D}}$$\end{document}-activities. Acceptable notations for natural numbers are ones that we can count with. Acceptable notations for logical formulas are the ones that we can use in inference and logical analysis. And so on. This criterion makes computability a relative notion—whether a function is computable depends on which notations are acceptable, which is relative to our interests and abilities. I argue that this is not a problem, however, since there is independent reason for taking the extension of computable function to be relative to contingent facts. Similarly, the fact that the use criterion makes a mathematical distinction depend on practical concerns is also not a problem because it dovetails with similar phenomena in other areas of computability theory, namely the roles of notation in computation over real numbers and of Extended Church’s Thesis in complexity theory. (shrink)

This paper provides an analysis of the notational difference between Beta Existential Graphs, the graphical notation for quantificational logic invented by Charles S. Peirce at the end of the 19th century, and the ordinary notation of first-order logic. Peirce thought his graphs to be “more diagrammatic” than equivalently expressive languages for quantificational logic. The reason of this, he claimed, is that less room is afforded in Existential Graphs than in equivalently expressive languages for different ways of representing the (...) same fact. The reason of this, in turn, is that Existential Graphs are a non-linear, occurrence-referential notation. As a non-linear notation, each graph corresponds to a class of logically equivalent but syntactically distinct sentences of the ordinary notation of first-order logic that are obtained by permuting those elements that in the graphs lie in the same area. As an occurrence-referential notation, each Beta graph corresponds to a class of logically equivalent but syntactically distinct sentences of the ordinary notation of first-order logic in which the identity of reference of two or more variables is asserted. In brief, Peirce’s graphs are more diagrammatic than the linear, type-referential notation of first-order logic because the function that translates the latter to the graphs does not define isomorphism between the two notations. (shrink)

We introduce the appropriate iterated version of the system of ordinal notations from [G1] whose order type is the familiar Howard ordinal. As in [G1], our ordinal notations are partly inspired by the ideas from [P] where certain crucial properties of the traditional Munich' ordinal notations are isolated and used in the cut-elimination proofs. As compared to the corresponding “impredicative” Munich' ordinal notations (see e.g. [B1, B2, J, Sch1, Sch2, BSch]), our ordinal notations arearbitrary terms in the appropriate simple term (...) algebra based on the notion of collapsing functions (which we would rather identify as projective functions). In Sect. 1 below we define the systems of ordinal notationsPRJ( ), for any primitive recursive limit wellordering . In Sect. 2 we prove the crucial well-foundness property by using the appropriate well-quasi-ordering property of the corresponding binary labeled trees [G3]. In Sect. 3 we interprete inPRJ( ) the familiar Veblen-Bachmann hierarchy of ordinal functions, and in Sect. 4 we show that the corresponding Buchholz's system of ordinal notationsOT( ) is a proper subsystem ofPRJ( ), although it has the same order type according to [G3] together with the interpretation from Sect. 2 in the terms of labeled trees. In Sect. 5 we use Friedman's approach in order to obtain an appropriate purely combinatorial statement which is not provable in the theory of iterated inductive definitions ID< λ, for arbitrarily large limit ordinalλ. Formal theories, axioms, etc. used below are familiar in the proof theory of subsystems of analysis (see [BFPS, T, BSch]). (shrink)

Methods of representing sentences containing mass terms (e.g. "gold") and amount terms (e.g. "three gallons") within the predicate calculus are given, and the semantics of the resulting sentences is discussed. the appendix sketches a way to systematically translate english sentences into the logical notation, exploiting some results of transformational grammar.

This paper introduces a formal concept of ideology and ideological system. The formalization takes ideologies and ideological systems to be situated in agent societies. An ideological system is defined as a system of operations able to create, maintain, and extinguish the ideologies adopted by the social groups of agent societies. The concepts of group ideology, ideological contradiction, ideological dominance, and dominant ideology of an agent society, are defined. An ideology-based concept of social group is introduced. Relations between the proposed formal (...) concept of ideology and the classical concepts of ideology elaborated in social sciences are examined. A computational notation is presented, to support the realization of ideological systems in computationally implemented agent societies. The adequacy of the approach for the formal modeling and analysis of ideological issues is illustrated through three case studies. (shrink)

This paper introduces a formal concept of ideology and ideological system. The formalization takes ideologies and ideological systems to be situated in agent societies. An ideological system is defined as a system of operations able to create, maintain, and extinguish the ideologies adopted by the social groups of agent societies. The concepts of group ideology, ideological contradiction, ideological dominance, and dominant ideology of an agent society, are defined. An ideology-based concept of social group is introduced. Relations between the proposed formal (...) concept of ideology and the classical concepts of ideology elaborated in social sciences are examined. A computational notation is presented, to support the realization of ideological systems in computationally implemented agent societies. The adequacy of the approach for the formal modeling and analysis of ideological issues is illustrated through three case studies. (shrink)

What is common to all languages is notation, so Universal Grammar can be understood as a system of notational types. Given that infants acquire language, it can be assumed to arise from some a priori mental structure. Viewing language as having the two layers of calculus and protocol, we can set aside the communicative habits of speakers. Accordingly, an analysis of notation results in the three types of Identifier, Modifier and Connective. Modifiers are further interpreted as Quantifiers and (...) Qualifiers. The resulting four notational types constitute the categories of Universal Grammar. Its ontology is argued to consist in the underlying cognitive schema of Essence, Quantity, Quality and Relation. The four categories of Universal Grammar are structured as polysemous fields and are each constituted as a radial network centred on some root concept which, however, need not be lexicalized. The branches spread out along troponymic vectors and together map out all possible lexemes. The notational typology of Universal Grammar is applied in a linguistic analysis of the ‘parts of speech’ using the English language. The analysis constitutes a ‘proof of concept’ in (1) showing how the schema of Universal Grammar is capable of classifying the so-called ‘parts of speech’, (2) presenting a coherent analysis of the verb, and (3) showing how the underlying cognitive schema allows for a sub-classification of the auxiliaries. (shrink)

This work has been selected by scholars as being culturally important, and is part of the knowledge base of civilization as we know it. This work was reproduced from the original artifact, and remains as true to the original work as possible. Therefore, you will see the original copyright references, library stamps (as most of these works have been housed in our most important libraries around the world), and other notations in the work. This work is in the public domain (...) in the United States of America, and possibly other nations. Within the United States, you may freely copy and distribute this work, as no entity (individual or corporate) has a copyright on the body of the work. As a reproduction of a historical artifact, this work may contain missing or blurred pages, poor pictures, errant marks, etc. Scholars believe, and we concur, that this work is important enough to be preserved, reproduced, and made generally available to the public. We appreciate your support of the preservation process, and thank you for being an important part of keeping this knowledge alive and relevant. (shrink)

To solve the probability problem of the Many Worlds Interpretation of Quantum Mechanics, D. Wallace has presented a formal proof of the Born rule via decision theory, as proposed by D. Deutsch. The idea is to get subjective probabilities from rational decisions related to quantum measurements, showing the non-probabilistic parts of the quantum formalism, plus some rational constraints, ensure the squared modulus of quantum amplitudes play the role of such probabilities. We provide a new presentation of Wallace’s proof, reorganized to (...) simplify some arguments, and analyze it from a formal perspective. Similarities with classical decision theory are made explicit, to clarify its structure and main ideas. A simpler notation is used, and details are filled in, making it easier to follow and verify. Some problems have been identified, and we suggest possible corrections. (shrink)

We propose a reconstruction of the constellation of problems and philosophical positions on the nature and number of the primitives of logic in four authors of the nineteenth century logical scene: Peano, Padoa, Frege and Peirce. We argue that the proposed reconstruction forces us to recognize that it is in at least four different senses that a notation can be said to be simpler than another, and we trace the origins of these four senses in the writings of these authors. (...) We conclude that Frege, and even more so Peirce, developed new notations not to make drawing logical conclusions easier but in order to answer the needs of logical analysis. (shrink)

This study aims to evaluate the changes in shot characteristics of elite women table tennis players through the longitudinal analysis of women's singles finals and semifinals from 2004 to 2021 Olympic Games. A total of 13 games were selected, and the stroke position, stroke type, ball placement, and stroke efficacy of 5,877 shots were analyzed using the notational analysis method. A chi-square test was used to test whether the shot characteristics had changed between game years, and the (...) adjusted residual was calculated to judge these changes. In the four dimensions, namely, stroke position, stroke type, ball placement, and stroke efficacy, the shot characteristics of women table tennis players have changed with the year of the Olympic Games. The backhand stroke position has an upward trend from 55.0 to 58.1%, and the type of flip stroke has a downward trend from 7.6 to 2.1%. The ball placements of forehand and middle long have two fluctuations first rising from 11.2 to 12.7% and 20.3 to 23.7% and then falling from 12.7 to 5.3% and 23.7 to 16.0%. Moreover, backhand long is being increasingly used. In the year of equipment change, the proportion of neutral stroke increases from 40.0 to 61.5%. Once the players have become adapted to the equipment, the proportion of neutral stroke decreases from 61.5 to 48.5%. The development trend of women table tennis players is that the extreme stroke will be further increasingly used. Results suggest that elite women table tennis coaches and players should focus on the technical training in the forehand and backhand positions, strengthen the practice of topspin and block, and pay attention to the backhand long landing area. Women's table tennis coaches and players also need to pay close attention to the possible impact of changes in rules and equipment and carry out targeted training in advance to minimize the impact on the stroke efficacy of players. (shrink)

The article is an overview of the main philosophical ideas of Gottlob Frege. The author stresses that Frege has built important tools for the analysis of language: the logic of sentences and the logic of predicates. He stresses also the importance of Frege’s work on the foundations of mathematics. He argues that his greatest achievement is the fact that thanks to Frege language has become main object of investigations in contemporary philosophy, and that theory of meaning has replaced the (...) traditional theory of knowledge. Frege’s ideas, taken together, constitute a well-defined method, a particular style of approach to philosophical problems and of their resolution. The effectiveness and enduring pertinence of Frege’s method is demonstrated by means of an analysis of several typical category mistakes, frequent in philosophical argumentation. Key words FREGE, LANGUAGE. (shrink)

This work has been selected by scholars as being culturally important, and is part of the knowledge base of civilization as we know it. This work was reproduced from the original artifact, and remains as true to the original work as possible. Therefore, you will see the original copyright references, library stamps (as most of these works have been housed in our most important libraries around the world), and other notations in the work.This work is in the public domain in (...) the United States of America, and possibly other nations. Within the United States, you may freely copy and distribute this work, as no entity (individual or corporate) has a copyright on the body of the work.As a reproduction of a historical artifact, this work may contain missing or blurred pages, poor pictures, errant marks, etc. Scholars believe, and we concur, that this work is important enough to be preserved, reproduced, and made generally available to the public. We appreciate your support of the preservation process, and thank you for being an important part of keeping this knowledge alive and relevant. (shrink)

We explore the technical details and historical evolution of Charles Peirce's articulation of a truth table in 1893, against the background of his investigation into the truth-functional analysis of propositions involving implication. In 1997, John Shosky discovered, on the verso of a page of the typed transcript of Bertrand Russell's 1912 lecture on ?The Philosophy of Logical Atomism? truth table matrices. The matrix for negation is Russell's, alongside of which is the matrix for material implication in the hand of (...) Ludwig Wittgenstein. It is shown that an unpublished manuscript identified as composed by Peirce in 1893 includes a truth table matrix that is equivalent to the matrix for material implication discovered by John Shosky. An unpublished manuscript by Peirce identified as having been composed in 1883?1884 in connection with the composition of Peirce's ?On the Algebra of Logic: A Contribution to the Philosophy of Notation? that appeared in the American Journal of Mathematics in 1885 includes an example of an indirect truth table for the conditional. (shrink)

KPM is a subsystem of set theory designed to formalize a recursively Mahlo universe of sets. In this paper we show that a certain ordinal notation system is sufficient to measure the proof-theoretic strength ofKPM. This involves a detour through an infinitary calculus RS(M), for which we prove several cutelimination theorems. Full cut-elimination is available for derivations of $\Sigma (L_{\omega _1^c } )$ sentences, whereω 1 c denotes the least nonrecursive ordinal. This paper is self-contained, at least from a technical (...) point of view. (shrink)

This work has been selected by scholars as being culturally important, and is part of the knowledge base of civilization as we know it. This work was reproduced from the original artifact, and remains as true to the original work as possible. Therefore, you will see the original copyright references, library stamps (as most of these works have been housed in our most important libraries around the world), and other notations in the work. This work is in the public domain (...) in the United States of America, and possibly other nations. Within the United States, you may freely copy and distribute this work, as no entity (individual or corporate) has a copyright on the body of the work.As a reproduction of a historical artifact, this work may contain missing or blurred pages, poor pictures, errant marks, etc. Scholars believe, and we concur, that this work is important enough to be preserved, reproduced, and made generally available to the public. We appreciate your support of the preservation process, and thank you for being an important part of keeping this knowledge alive and relevant. (shrink)

It is well-known that natural axiomatic theories are well-ordered by consistency strength. However, it is possible to construct descending chains of artificial theories with respect to consistency strength. We provide an explanation of this well-orderedness phenomenon by studying a coarsening of the consistency strength order, namely, the$\Pi ^1_1$reflection strength order. We prove that there are no descending sequences of$\Pi ^1_1$sound extensions of$\mathsf {ACA}_0$in this ordering. Accordingly, we can attach a rank in this order, which we call reflection rank, to any$\Pi (...) ^1_1$sound extension of$\mathsf {ACA}_0$. We prove that for any$\Pi ^1_1$sound theoryTextending$\mathsf {ACA}_0^+$, the reflection rank ofTequals the$\Pi ^1_1$proof-theoretic ordinal ofT. We also prove that the$\Pi ^1_1$proof-theoretic ordinal of$\alpha $iterated$\Pi ^1_1$reflection is$\varepsilon _\alpha $. Finally, we use our results to provide straightforward well-foundedness proofs of ordinal notation systems based on reflection principles. (shrink)

Nowadays, football goalkeepers play an important role in the team's organization, namely, considering the offensive and defensive processes. The purpose of our investigation focuses on the notational and T-pattern analysis of the offensive and defensive actions of elite young football GKs. The participating GKs presented 8 years of experience in the specific position, were internationally selected for the national team of Portugal, and competed in the national U-17 championship of Portugal. Thirty football matches were observed. The observational sample (...) consisted of defensive actions and offensive actions. Two observational instruments were used to codify the actions: the observation system of defensive technical-tactical actions of GKs and the observation system of offensive technical-tactical actions of GKs. Both instruments underwent a validation process, and inter- and intra-observer reliability was tested. The codification of the actions was performed with the LINCE program, and later the data were exported to Microsoft Excel and THEME 5.0. The notational data were analyzed in SPSS, and T-pattern detection analysis was performed in THEME 5.0. The predominant actions of young observed GKs were fundamentally goal defense and participation in the team's offensive process construction through actions performed with the foot and hand. The analysis of T-patterns allowed to identify T-pattern actions in the defensive actions of goal defense and exit of the goal, as well as related to the defensive set pieces. In the offensive process, the analysis of T-patterns reinforced the participation of the GK in the team's first phase of construction and in the execution of goal kicks and actions that start as a result of the actions of the opponent. The GK's defensive actions are mostly focused on the objective of goal defense and offensive actions with the hands and feet are important for their participation in positional attack construction. Our study contributes to a better knowledge of the GK's actions in the competition and is relevant to be considered by the specific position coaches in the training process organization. (shrink)

This work has been selected by scholars as being culturally important, and is part of the knowledge base of civilization as we know it. This work was reproduced from the original artifact, and remains as true to the original work as possible. Therefore, you will see the original copyright references, library stamps (as most of these works have been housed in our most important libraries around the world), and other notations in the work. This work is in the public domain (...) in the United States of America, and possibly other nations. Within the United States, you may freely copy and distribute this work, as no entity (individual or corporate) has a copyright on the body of the work.As a reproduction of a historical artifact, this work may contain missing or blurred pages, poor pictures, errant marks, etc. Scholars believe, and we concur, that this work is important enough to be preserved, reproduced, and made generally available to the public. We appreciate your support of the preservation process, and thank you for being an important part of keeping this knowledge alive and relevant. (shrink)

In the first part we show why ordinals and ordinal notations are naturally connected with proof theoretical research. We introduce the program of ordinal analysis. The second part gives examples of applications of ordinal analysis.

The present article proposes an Informational-Theoretic interpretation of logical analysis applied to natural language in Tractatus Logico-Philosophicus. Natural language is characterized by descriptive definitions in order to compress information according to empirical regularities. However, notations fitted to empirical patterns do not explicitly reflect the logical structure of language that enables it to represent those very patterns. I argue that logical analysis is the process of obtaining incompressible and uniformly distributed codes, best fitted to express the possible combinations of (...) facts instead of eventual empirical regularities. That implies mutual translatability between different representations of the same facts, which justifies the possibility to decompose natural language and to rebuild it in mathematized ideographical languages, such as physics. Logical analysis decomposition process, then, results in a reductionism that sets the basis for conceptions of logic and language not only in Tractatus, but also in Some Remarks on Logical Form, which, in turn, is shown to be unattainable in Philosophical Remarks. That posterior book introduces a fragmentation of linguistic systems and analytical methods that can no longer be interpreted under computation theory alone. (shrink)

A uniform analysis is offered for the source of the locations specified by all references in English to spatial properties including location and movement. This source is argued to be the location of events and states of affairs. These locations are specified by sets showing spaces momentarily occupied. Descriptions of motion are accounted for through a variety of ways of referencing these sets. Some classes of simple clauses are identified as requiring semantic analysis involving multiple events and states (...) of affairs. This importantly allows spatial references to be associated with different events and states of affairs. Identification of object location is accounted for by use of extra inference rules or meaning postulates. A number of other explanations are suggested for more limited and derivative phenomena. The analysis is developed within a Case-like notation presented in the first-order predicate calculus. It is supported by an extensive analysis of spatial reference phenomena and by identification of ancillary benefits. (shrink)

In this note we give a simplified ordinal analysis of first-order reflection. An ordinal notation system$OT$is introduced based on$\psi $-functions. Provable$\Sigma _{1}$-sentences on$L_{\omega _{1}^{CK}}$are bounded through cut-elimination on operator controlled derivations.

Turing’s analysis of computability has recently been challenged; it is claimed that it is circular to analyse the intuitive concept of numerical computability in terms of the Turing machine. This claim threatens the view, canonical in mathematics and cognitive science, that the concept of a systematic procedure or algorithm is to be explicated by reference to the capacities of Turing machines. We defend Turing’s analysis against the challenge of ‘deviant encodings’.Keywords: Systematic procedure; Turing machine; Church–Turing thesis; Deviant encoding; (...) Acceptable encoding; Turing’s analysis of computability; Turing’s Notational Thesis. (shrink)

Peirce considered the principal business of logic to be the analysis of reasoning. He argued that the diagrammatic system of Existential Graphs, which he had invented in 1896, carries the logical analysis of reasoning to the furthest point possible. The present paper investigates the analytic virtues of the Alpha part of the system, which corresponds to the sentential calculus. We examine Peirce’s proposal that the relation of illation is the primitive relation of logic and defend the view that (...) this idea constitutes the fundamental motive of philosophy of notation both in algebraic and graphical logic. We explain how in his algebras and graphs Peirce arrived at a unifying notation for logical constants that represent both truth-function and scope. Finally, we show that Shin’s argument for multiple readings of Alpha graphs is circular. (shrink)

The myriad ways in which spatiality, or socially produced space, impinges on media texts is the overarching concern of this study. Responding to Edward Soja’s call for an assertive foregrounding of a critical spatial perspective, this article is an ontological reassertion of space in relation to news media discourse and argues that the socially constructed spatiality of a journalistic text is just as revealingly significant as its historicality and sociality. Introduced here is Critical Spatial Discourse Analysis (CSDA), a methodological (...) framework that employs the spatiocultural theory of Bill Richardson to enable a mapping of the various aspects of spatiality informing media texts. Edward Said’s imaginative geographies is also drawn upon to advance a geographical notation and mapping of the territorial imaginations underlying news media texts. The fundamental assumption of CSDA is that space is not only actively constituted in media texts but also constitutes and that it can, and indeed must be read and interpreted as any other text. A sample column from Turkey’s Sabah newspaper will be used to illustrate the application of CSDA to press coverage of democracy to see what might be revealed of those real-and-imagined spaces that are interposed and obscured in the text’s background. (shrink)

This article proposes that Goffman's "Frame Analysis" can be interpreted as a step toward unpacking the idea of context. His analysis implies a recursive model involving frames within frames. The key problem is that neither Goffman nor anyone else has clearly defined what is meant by a frame. I propose that it can be represented by a word, phrase, or proposition. A subjective context can be represented as an assembly of these items, joined together by operators such as (...) and, since, if, not, and then. Furthermore, this model can be combined with the recursive levels of mutual awareness in earlier approaches to consensus. The combination would represent the inter subjective context: it can be used to find the minimum amount of background that would allow consensual interpretations of discourse. It could also construct a chain that links discourse to the institutional level, the micro-macro pathway from word and gesture to social structure. Goffman hinted that mathematical notation might be used to represent a frame assembly. By adding levels of awareness to such notation, it could represent social facts. Because the use of vernacular words rather than concepts is a problem in social science, Goffman's approach has a general as well as a particular significance. (shrink)

This work has been selected by scholars as being culturally important, and is part of the knowledge base of civilization as we know it. This work was reproduced from the original artifact, and remains as true to the original work as possible. Therefore, you will see the original copyright references, library stamps (as most of these works have been housed in our most important libraries around the world), and other notations in the work.This work is in the public domain in (...) the United States of America, and possibly other nations. Within the United States, you may freely copy and distribute this work, as no entity (individual or corporate) has a copyright on the body of the work.As a reproduction of a historical artifact, this work may contain missing or blurred pages, poor pictures, errant marks, etc. Scholars believe, and we concur, that this work is important enough to be preserved, reproduced, and made generally available to the public. We appreciate your support of the preservation process, and thank you for being an important part of keeping this knowledge alive and relevant. (shrink)