Notations are cognitive systems involving distinctive psychological functions, behaviors, and material forms. Seen through this lens, two main types – semasiography and visible language – are fundamentally differentiated by their material prehistories, emphasis on iconography, and the centrality of language's combinatorial faculty. These fundamental differences suggest that key qualities (iconicity, expressiveness, concision) are difficult to conjoin in a single system.

In this article we investigate whether the following conjecture is true or not: does the addition-free theta functions form a canonical notation system for the linear versions of Friedman’s well-partial-orders with the so-called gap-condition over a finite set of n labels. Rather surprisingly, we can show this is the case for two labels, but not for more than two labels. To this end, we determine the order type of the notation systems for addition-free theta functions in terms of ordinals (...) less than ε0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varepsilon _0$$\end{document}. We further show that the maximal order type of the Friedman ordering can be obtained by a certain ordinal notation system which is based on specific binary theta functions. (shrink)

G. Jäger gave in Arch. Math. Logik Grundlagenforsch. 24 , 49-62, a recursive notation system on a basis of a hierarchy Iαß of α-inaccessible regular ordinals using collapsing functions following W. Buchholz in Ann. Pure Appl. Logic 32 , 195-207. Jäger's system stops, when ordinals α with Iα0 = α enter. This border is now overcome by introducing additional a hierarchy Jαß of weakly inaccessible Mahlo numbers, which is defined similarly to the Jäger hierarchy. An ordinal μ is called Mahlo, (...) if every normal-function f : μ → μ has regular fixpoints. Collapsing is defined for both Mahlo and simply regular ordinals such that for every Mahlo ordinal μ out of the J-hierarchy Ψμα is a regular σ such that Iσ0 = σ. For these regular σ again collapsing functions Ψσ are defined. To get a proper systematical order into the collapsing procedure, a pair of ordinals is associated to σ and α, and the definition of Ψσα is given by recursion on a suitable well-ordering of these pairs. Thus a fairly large system of ordinal notations can be established. It seems rather straightforward, how to extend this setting further. (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)

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)

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)

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)

The standard method of generating countable ordinals from uncountable ordinals can be replaced by a use of fixed point extractors available in the term calculus of Howard’s system. This gives a notion of the intrinsic complexity of an ordinal analogous to the intrinsic complexity of a function described in Gödel’s T.

In his Languages of Art, Nelson Goodman proposes a theory of artistic notation that includes foundational requirements for any system of symbols we might use to specify and communicate the features of an artwork, in architecture or any other art form. Goodmans' theory usefully explains how notation can reveal linguistic-like phenomena of various art forms. But not all art forms can enjoy benefits of a full-blown notational system, in Goodman's view, and he suggests that architecture's symbol systems fall (...) short in this regard. It is a shortcoming of architecture, he believes, that its notation cannot communicate the sum of a given work's essential features. Against this view it may be argued that artworks are generally and inimically historical in character, such that an inability to capture this dimension may result in a failure to pick out the identity of a work. As a consequence, the suggestion of that inability looks like a shortcoming of the general notational theory rather than of the art form or its notation per se. I defend Goodman's background formalism against the criticism that an ahistoricist notation cannot possibly be adequate. But I reject his view that no actual architectural notation can satisfy the formal criteria of his theory. In particular, I propose that the foundational theory for Computer Aided Design (CAD) described by the architect William Mitchell satisfies Goodman's criteria and so yields a notation that enables communication of an architectural work's essential features. If, as I suggest, such a notation is feasible, then we have an additional result that Goodman foresees: a means of abstracting future architectural works from their historical contexts. This in turn yields a consequence at which Goodman only hints-that such architectural works can be intellectually grasped and physically constructed along purely formalist lines. One practical result here is that we introduce economical solutions to architectural design. On a theoretical level, we also expose the inessential character of historical properties to creating or understanding architecture, at least with respect to future possible works. (shrink)

Buchholz presented a method to build notation systems for infinite sequent-style derivations, analogous to well-known systems of notation for ordinals. The essential feature is that from a notation one can read off by a primitive recursive function its n th predecessor and, e.g. the last rule applied. Here we extend the method to the more general setting of infinite terms, in order to make it applicable in other proof-theoretic contexts as well as in recursion theory. As examples, we (...) use the method to 1. give a new proof of a well-known trade-off theorem , which says that detours through higher types can be eliminated by the use of transfinite recursion along higher ordinals, and 2. construct a continuous normalization operator with an explicit modulus of continuity. (shrink)

Spekkens has introduced a toy theory (Spekkens in Phys. Rev. A 75(3):032110, 2007) in order to argue for an epistemic view of quantum states. I describe a notation for the theory (excluding certain joint measurements) which makes its similarities and differences with the quantum mechanics of stabilizer states clear. Given an application of the qubit stabilizer formalism, it is often entirely straightforward to construct an analogous application of the notation to the toy theory. This assists calculations within the toy theory, (...) for example of the number of possible states and transformations, and enables superpositions to be defined for composite systems. (shrink)

This paper argues that many so-called digital technologies can be construed as notational technologies, explored through the example of Monegraph, an art and digital asset management platform built on top of the blockchain system originally developed for the cryptocurrency bitcoin. As the paper characterizes it, a notational technology is the performance of syntactic notation within a field of reference, a technologized version of what Nelson Goodman called a “notational system.” Notational technologies produce abstracted entities through positive (...) and reliable, or constitutive, tests of socially acceptable meaning. Accordingly, this account deviates from typical narratives of blockchains, instead demonstrating that blockchain technologies are effective at managing digital assets because they produce abstracted identities through the performance of notation. Since notational technologies rely on configurations of socially acceptable meaning, this paper also provides a philosophical account of how blockchain technologies are socially embedded. (shrink)

A linear notation for Charles S. Peirce's alpha and beta diagrammatic systems of existential graphs is presented. These two systems are equivalent to propositional and first-order logic. Some differences between the linear and graphical notation are analyzed, revealing some of the strengths and weaknesses of Peirce's system.

Name der Zeitschrift: Semiotica Jahrgang: 2015 Heft: 204 Seiten: 419-428.

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)

The pragmatic notion of assertion has an important inferential role in logic. There are also many notational forms to express assertions in logical systems. This paper reviews, compares and analyses languages with signs for assertions, including explicit signs such as Frege’s and Dalla Pozza’s logical systems and implicit signs with no specific sign for assertion, such as Peirce’s algebraic and graphical logics and the recent modification of the latter termed Assertive Graphs. We identify and discuss the main (...) ‘points’ of these notations on the logical representation of assertions, and evaluate their systems from the perspective of the philosophy of logical notations. Pragmatic assertions turn out to be useful in providing intended interpretations of a variety of logical systems. (shrink)

Past and present societies world-wide have employed well over 100 distinct notational systems for representing natural numbers, some of which continue to play a crucial role in intellectual and cultural development today. The diversity of these notations has prompted the need for classificatory schemes, or typologies, to provide a systematic starting point for their discussion and appraisal. The present paper provides a general framework for assessing the efficacy of these typologies relative to certain desiderata, and it uses this (...) framework to discuss the two influential typologies of Zhang & Norman and Chrisomalis. Following this, a new typology is presented that takes as its starting point the principles by which numerical notations represent multipliers (the principles of cumulation and cipherization), and bases (those of integration, parsing, and positionality). Many different examples show that this new typology provides a more refined classification of numerical notations than the ones put forward previously. In addition, the framework provided here can be used to assess typologies not only of numerical notations, but also of many other domains. (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)

Summary In the early years of the nineteenth century, the English chemist John Dalton (1766–1844) developed his atomic theory, a set of theoretical commitments describing the nature of atoms and the rules guiding their interactions and combinations. In this paper, I examine a set of conceptual and illustrative tools used by Dalton in developing his theory as well as in presenting it to the public in printed form as well as in his many public lectures. These tools—the concept of ‘atmosphere’, (...) the pile of shot analogy, and Dalton's system of chemical notation—served not just to guide Dalton's own thinking and to make his theories clear to his various audiences, but also to bind these theories together into a coherent system, presented in its definitive form in the three volumes of A New System of Chemical Philosophy (1808, 1810, and 1827). Despite these links, Dalton's contemporaries tended to pick and choose which of his theories to accept; his system of notation failed to be adopted in part because it embodied the whole of his system indivisibly. (shrink)

The design process in Kashmiri carpet weaving is distributed over a number of actors and artifacts and is mediated by a weaving notation called talim. The script encodes entire design in practice-specific symbols. This encoded script is decoded and interpreted via design-specific conventions by weavers to weave the design embedded in it. The cognitive properties of this notational system are described in the paper employing cognitive dimensions (CDs) framework of Green (People and computers, Cambridge University Press, Cambridge, 1989) and (...) Blackwell et al. (Cognitive technology: instruments of mind—CT 2001, LNAI 2117, Springer, Berlin, 2001). After introduction to the practice, the design process is described in ‘The design process’ section which includes coding and decoding of talim. In ‘Cognitive dimensions of talim’ section, after briefly discussing CDs framework, the specific cognitive dimensions possessed by talim are described in detail. (shrink)

The present paper expounds the logic notation proposed by Augustus De Morgan in 1850 from within the original context of De Morgan’s account of syllogistic logic and his approach to quantification. The notational system of 1850 is shown to be a flexible tool to state inferences, to prove their validity and to derive formulæ of the respective system by ‘blind’ application of transformation rules. These pertain to the swapping of operator signs, which are of inverse ‘character’ in a two-fold (...) sense: The signs both represent inverse operations and exemplify oppositeness due to their very shapes. Thus, while De Morgan’s initial choice of symbol shapes is due to certain diagrammatic features, his notational system of 1850 leads to an account of logical operations as effected by manipulation of symbol tokens. It is concluded that as an instantiation of what De Morgan terms a ‘calculus of opposite relations’, his notation of 1850 may also be conceived of as a ‘graphicism’ affording for the working of a ‘symbolic machine’ on ‘de-semantification’. (shrink)

The purpose of this study is to compile a selection of the various formalisms found in conversation theory to introduce readers to Pask's discursive algebra. In this way, the text demonstrates how concept sharing and concept formation by means of the interaction of two participants may be formalized. The approach taken in this study is to examine the formal notation system used by Pask and demonstrate how such formalisms may be used to represent concept sharing and concept formation through conversation. (...) The compilation of the discursive algebra using the framework provided by conversation theory could potentially be used as an auxiliary framework to study conversational interactions in multi-agent systems theory. (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)

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)

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)

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 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)