Formal systems of constructive mathematics

1956 ◽  
Vol 21 (1) ◽  
pp. 63-75
Author(s):  
M. H. Löb

In the present paper we investigate in some detail two systems closely related to those contained in [4]. The system K1 of [4] is simplified by eliminating conjunction and disjunction, giving rise to a system Σ whose primitives are two individuals, concatenation, identity, existential and limited universal quantification. Any relation expressible in Σ may be expressed in the form (Eα1)(β)α1(Eα2)…(Eαn)(γ = δ). A further reduction of primitives leads to a system Σμ, which differs from Σ by containing the operator μ (i.e. “the first string, such that” with respect to a certain ordering) in place of the quantifiers. Reasons are given for restricting Myhill's definition of constructive ideas to those expressible in Σμ.Acquaintance with [4], [5], [6] is assumed. We deviate from [4] by dropping concatenation-signs and parentheses in chain matrices from the official notation, and, furthermore, by using Greek and German letters as metalinguistic variables standing for chain matrices and chains, respectively, while retaining small Roman letters (excluding a, b) for use as variables of the systems.

2018 ◽  
Vol 48 (2) ◽  
pp. 76-86 ◽  
Author(s):  
Jennie Shepheard ◽  
Elsa Lapiz ◽  
Carla Read ◽  
Terri J Jackson

Background: The Council of Australian Governments has focused the attention of health service managers and state health departments on a list of hospital-acquired complications (HACs) proposed as the basis of funding adjustments for poor quality of hospital inpatient care. These were devised for the Australian Commission on Safety and Quality in Health Care as a subset of their earlier classification of hospital-acquired complications (CHADx) and designed to be used by health services to monitor safety performance for their admitted patients. Objective: To improve uptake of both classification systems by clarifying their purposes and by reconciling the ICD-10-AM code sets used in HACs and the Victorian revisions to the CHADx system (CHADx+). Method: Frequency analysis of individual clinical codes with condition onset flag (COF 1) included in both classification systems using the Victorian Admitted Episodes Dataset for 2014/2015 ( n = 2,623,275 separations). Narrative description of the resulting differences in definition of “adverse events” embodied in the two systems. Results: As expected, a high proportion of ICD-10-AM codes used in the HACs also appear in CHADx+, and given the wider scope of CHADx+, it uses a higher proportion of all COF 1 diagnoses than HACs (82% vs. 10%). This leads to differing estimates of rates of adverse events: 2.12% of cases for HACs and 11.13% for CHADx+. Most CHADx classes (70%) are not covered by the HAC system; discrepancies result from the exclusion from HACs of several major CHADx+ groups and from a narrower definition of detailed HAC classes compared with CHADx+. Case exclusion criteria in HACs (primarily mental health admissions) resulted in a very small proportion of discrepancies (0.13%) between systems. Discussion: Issues of purpose and focus of these two Australian systems, HACs for clinical governance and CHADx+ for local quality improvement, explain many of the differences between them, and their approach to preventability, and risk stratification. Conclusion: A clearer delineation between these two systems using routinely coded hospital data will assist funders, clinicians, quality improvement professionals and health information managers to understand discrepancies in case identification between them and support their different information needs.


1958 ◽  
Vol 23 (2) ◽  
pp. 188-206 ◽  
Author(s):  
A. Grzegorczyk ◽  
A. Mostowski ◽  
C. Ryll-Nardzewski

We consider two formal systems for the theory of (natural) numbers, both of which are applied second-order functional calculi with equality and the description operator. The two systems have the same primitive symbols, rules of formation, and axioms, differing only in the rules of inference.The primitive logical symbols of the systems are the improper symbols (,), the prepositional connectives ∨, &, ⊃, ≡, ~, the quantifiers ( ), (E), the equality symbol =, the description operator ι,-infinitely many distinct individual (or number) variables, and for each positive integer k infinitely many distinct k-place function variables. Our systems have in addition the following four primitive nonlogical (or arithmetical) constants:0, 1, +, ×.The classes of “number formulas” (nfs) and “propositional formulas” (pfs) are defined inductively as the least classes of formal expressions (i.e. of concatenations of primitive symbols) satisfying the following conditions:(1) 0, 1, and the number variables are nfs.


2003 ◽  
Vol 23 (1) ◽  
pp. 221-229 ◽  
Author(s):  
Guillermo Durán ◽  
Agustín Gravano ◽  
Marina Groshaus ◽  
Fábio Protti ◽  
Jayme L. Szwarcfiter

We say that G is an e-circle graph if there is a bijection between its vertices and straight lines on the cartesian plane such that two vertices are adjacent in G if and only if the corresponding lines intersect inside the circle of radius one. This definition suggests a method for deciding whether a given graph G is an e-circle graph, by constructing a convenient system S of equations and inequations which represents the structure of G, in such a way that G is an e-circle graph if and only if S has a solution. In fact, e-circle graphs are exactly the circle graphs (intersection graphs of chords in a circle), and thus this method provides an analytic way for recognizing circle graphs. A graph G is a Helly circle graph if G is a circle graph and there exists a model of G by chords such that every three pairwise intersecting chords intersect at the same point. A conjecture by Durán (2000) states that G is a Helly circle graph if and only if G is a circle graph and contains no induced diamonds (a diamond is a graph formed by four vertices and five edges). Many unsuccessful efforts - mainly based on combinatorial and geometrical approaches - have been done in order to validate this conjecture. In this work, we utilize the ideas behind the definition of e-circle graphs and restate this conjecture in terms of an equivalence between two systems of equations and inequations, providing a new, analytic tool to deal with it.


Incarceration ◽  
2020 ◽  
Vol 1 (1) ◽  
pp. 263266632093643
Author(s):  
Alice Ievins

Late-modern penal power has been described as ‘tight’. Through the increasing use of indeterminate sentences and psychological assessment, and the growing insistence that prisoners engage in self-government, the prison monitors and seeks to change those it holds. This tight and disciplinarian power is often described as contributing to the increasing fragmentation and atomisation of the prisoner community. However, this article, which is based on research conducted in a English medium-security prison for men convicted of sex offences, argues that tightness can operate through the prisoner community, in a process which it terms ‘lateral regulation’. It shows that prisoners spend a lot of time observing, categorising and policing their peers, in ways which replicate and often uphold the more formal systems of power. However, the relationship between these two systems of power is complex, and prisoners’ collective self-regulation can conflict with and challenge the demands of the penal institution, in a way which reveals some of the weaknesses in the institution’s disciplinary gaze, and indicates the normative motivations underlying this regulation.


2016 ◽  
Vol 42 (3) ◽  
pp. 527-535 ◽  
Author(s):  
Johan Bos

The syntax of abstract meaning representations (AMRs) can be defined recursively, and a systematic translation to first-order logic (FOL) can be specified, including a proper treatment of negation. AMRs without recurrent variables are in the decidable two-variable fragment of FOL. The current definition of AMRs has limited expressive power for universal quantification (up to one universal quantifier per sentence). A simple extension of the AMR syntax and translation to FOL provides the means to represent projection and scope phenomena.


1973 ◽  
Vol 8 (2) ◽  
pp. 221-232 ◽  
Author(s):  
John Staples

In constructive mathematics the Dedekind cut definition of real number is not equivalent to the definition of real number by Cauchy sequences, and the Dedekind real numbers do not satisfy Heyting's axioms for constructive fields. A more general notion of constructive field is proposed which includes the Dedekind real numbers; some linear algebra is given which applies to such fields.


2018 ◽  
Vol 12 (1) ◽  
pp. 113-126
Author(s):  
Thomas Petraschka

Abstract This essay asks whether the attribution of unreliability to the narrator of a literary text is always dependent upon interpretation. The bulk of narratological research answers with »yes«. Yet the content of the term »interpretation-dependent« is understood in radically different ways. As a minimal consensus, it is commonly accepted that the attribution of unreliability cannot be described as »interpretation-neutral«, in the way that, for instance, the statement »The narrator in text T is a homodiegetic narrator« is interpretation-neutral. Following a few preliminary explanatory remarks on terminology, I propose two arguments for why this majority opinion is false. I argue that the statement »Text T is unreliably narrated« is not always interpretation-dependent. Within the framework of the first argument, I attempt to show that the criterion of »interpretation neutrality« depends upon some meta-theoretical assumptions. If one assumes that basic linguistic characteristics are valid independent of their interpretation and argues that a sentence such as »Call me Ishmael« establishes a homodiegetic narrator because the word »me« signals that he belongs to the narrated story, then one implicitly excludes as inadequate certain idiosyncratic theories of meaning that would ascribe a different meaning to »me«. That is not problematic in and of itself. But it shows that there are conditions of adequacy for theories of meaning that are fundamentally negotiable. And the set of statements which can be attributed the attribute of being »interpretation-neutral« can vary depending upon how these conditions of adequacy are defined. In a corresponding adaptation of the conditions of adequacy for theories of meaning and interpretation, it is therefore inherently possible that even statements about the reliability of a narrator could be granted the status of being interpretation-neutral. The second argument focuses on the praxis of interpretation. I seek to reconstruct how exactly the qualification of a narrator as homodiegetic (an attribute that is usually considered as interpretation-neutral) and as unreliable (an attribute that is usually not considered as interpretation-neutral) can come about in a process of interpretation. There appear to be cases in which criteria commonly cited to qualify a statement as an interpretation-neutral description of a text are also applicable for the attribution of narrative unreliability. Such cases are literary texts like Agatha Christie’s The Murder of Roger Ackroyd or Ambrose Bierce’s An Occurrence at Owl Creek Bridge, in which the unreliability of the narrator is apparent. The knowledge that the narrators in these texts at least temporarily withhold facts relevant to the plot, tell lies, make mistakes, hallucinate, etc. can just as much be attained on the basis of an unreflective understanding of the linguistic meanings of words as can the knowledge that the narrators are part of the stories they tell. If one wishes to not relinquish the interpretation-neutral status of statements about the ontology of a narrator (the qualification, that is, of a narrator as homo- or heterodiegetic) to a relativism that includes linguistic interpretations, then one is forced in principle to also retain the status of interpretation neutrality for statements about the reliability of a narrator. Both arguments lead me to conclude that the universal quantification that all determinations of the reliability of a narrator are dependent upon interpretation is false. I propose that we limit ourselves to more modest existential quantifications and that we do not attribute the attributes »interpretation-dependent« and »interpretation-neutral« to entire literary categories or types of statements in general, but rather to individual statements. Moreover, I give a short and tentative definition of the criterion »interpretation neutrality« that follows from these considerations.


2010 ◽  
Vol 52 ◽  
pp. 38-54
Author(s):  
Rimvydas Laužikas

XX amžiuje prasidėjusi informacinės ir komunikacinės paradigmų plėtra ir besiformuojanti tinklaveikos visuomenė neišvengiamai turi įtakos įvairių veiklos sričių, taip pat paveldo sampratai, paveldosaugos, paveldotvarkos ir paveldo informacijos bei komunikacijos procesams. Šio straipsnio objektas – informacinės ir komunikacinės paradigmų raiška paveldo erdvėje. Straipsnio tikslas – suformuluoti paveldo informacijos ir komunikacijos sampratą, parodant sąsajas su komunikacijos ir informacijos mokslų teorijomis bei pademonstruojant informacinės paradigmos taikymo paveldo tyrimuose galimybes. Straipsnyje pateikiami teoriniai svarstymai iliustruojami konkretaus paveldo objekto – Dubingių piliavietės archeologinių tyrimų pavyzdžiais.Straipsnyje pristatomo tyrimo metu nustatyta, kad paveldo informacijos ir komunikacijos požiūriu pagrįstuose tyrimuose galima tarpdiscipliniškai pritaikyti svarbiausias informacijos ir komunikacijos mokslų teorijas ir metodus. Jų taikymas paveldo erdvėje gali būti kreipiamas dviem lygiavertėmis tyrimų kryptimis: tyrimuose akcentuojant išskirtinius paveldo objektus ir artefaktus bei tyrimuose orientuojantis į masinės medžiagos pažinimą. Abi tyrimų kryptys leidžia kiekybiškai įvertinti paveldo objektus, nustatant paveldo kompleksų ir jų dalių santykį, lyginant paveldo objektus ir kompleksus, siekiant identifikuoti kompleksą sukūrusių visuomenių ypatumus.Pagrindiniai žodžiai: Paveldas, paveldo informacija ir komunikacija, informacijos teorija, komunikacijos teorija, komunikacijos ir informacijos mokslai, Dubingiai, archeologija, entropija.Uniqueness and Entropy in the Information and Communication on Cultural HeritageRimvydas Laužikas SummaryThe technologies that have been developing in the 20th–21st century gradually permeate all fields of human life and activities. These technologies open new opportunities for communication, management, information processing as well as for the development of new, based on the information and communication paradigms research methods, models of practical activities, and interdisciplinary research. In this context, it is promising to use the basic theories of communication and information science (McLuhan’s, Shannon’s, Lotman’s, Prigogine’s) and the definition of the communication and information on cultural heritage.The purpose of the article is to formulate the concept of information and communication regarding cultural heritage, to show links between heritage and information and communication science theories and the examples of using the information paradigm for heritage studies. The theoretical considerations are illustrated by examples from a particular heritage site – the Dubingiai (Moletai reg., Lithuania) castle site archaeological excavations.The concept of information and communication on cultural heritage is based on the definition of cultural heritage. If “cultural heritage – the entire corpus of material signs – either artistic or symbolic – handed on by the past to each culture…” (HEREIN…., 2009), then a single heritage object should be regarded as a sign (a minimum, indivisible, atomic information transmission unit) and the heritage complex as a text (medium, message). The signs and texts point to mental ideas and are part of a system applied by members of specific (past) culture for intercommunication. If we accept this definition, then heritage research means the decoding, i.e. “reading”the primary (archaeological complexes from past cultures) texts and the encoding, i.e. creating a secondary (scientifically for scientists) and tertiary (for general public) text in contemporary culture. Research of cultural heritage is the foundation for the further communicative use of the cultural heritage in our times (as museum exhibits, cultural monuments, etc.).This position allows customizing the quantitative parameters of information theory to the heritage research and extending the “toolbox” of heritage quantitative analysis. In this approach, a heritage complex as a text is an evolved, open (for external influence) and dissipative system. Such systems have all features of evolving systems determined by Ilya Prigogine (Prigogine, 2006): they evolve; their evolution is based on an objective arrow of time which ensures irreversibility of processes; the variability of system components, going on for a long time, causes changes in the whole system; the evolution of a system is a process that can be forecasted only in part; sometimes an evolving system experiences disturbances that change it essentially (system mutations); both systems evolve at a different speed; two systems the evolution of which started in different points of space and time, more and more recede from each other; two systems are not inter-integrating.In this way, we find two approaches to “reading the past”: a research of exclusive (unique) artifacts and research of mass artifacts. Both approaches say that unique (or mass) artifacts are markers of social structure, of trading connections, of urbanization, of cultural influences, etc. However, to use this method, we need statistically significant samples, the methodology of triangulation by several benchmark examples, and a relative scale of references. From the first point of view, perspective methods of “reading” the cultural heritage information are the content analysis and summaries of a complex information. In archaeology, it is the coefficient of graves with findings, the average number of unique finds in a grave, the average number of semantically valuable finds, the complex (text) density index, the complex readability index and mapping the network of finds on the basis of binary oppositions theory. From the second point of view, promising methods are information entropy modeling and calculation. The main problem of calculating entropy for heritage complexes is that the original text (the primary state of complex) is only hypothetical; it cannotbe reconstructed and it is problematical to record the entropy change. In this context, a productive way is the visualization and relative calculation of the general object-level entropy in the three-dimensional space-time model (X and Y as the geographical coordinates of archaeological object and Z as the chronological parameter). Also, the entropy model can be calibrated by the heterogeneity of a cultural space (differences in the cultural dependence of creators (authors) and “readers” (researchers) of the archaeological complex as a text). When the creator and the “reader” belong to the same culture (the same macrosystem, the same cluster), entropy variation during the same period of time is less than when the creator and the “reader” come from related or different cultures. Another promising direction of research is calculation of entropy for specific complexes of heritage. By this method, we can calculate the entropy of an individual object (e.g. finding), of the whole heritage complex or a subcomplex (e.g. grave) and a complex (e.g. necropolis) relationship. The entropy is calculated using the entropy formula (Голдман, 1957). The resulting entropy is directly (inversely) proportional to the possibilities of the investigator’s ability to understand (“read”) the complex as a text.


1975 ◽  
Vol 40 (3) ◽  
pp. 347-382 ◽  
Author(s):  
John Myhill

This paper is the third in a series collectively entitled Formal systems of intuitionistic analysis. The first two are [4] and [5] in the bibliography; in them I attempted to codify Brouwer's mathematical practice. In the present paper, which is independent of [4] and [5], I shall do the same for Bishop's book [1]. There is a widespread current impression, due partly to Bishop himself (see [2]) and partly to Goodman and the author (see [3]) that the theory of Gödel functionals, with quantifiers and choice, is the appropriate formalism for [1]. That this is not so is seen as soon as one really tries to formalize the mathematics of [1] in detail. Even so simple a matter as the definition of the partial function 1/x on the nonzero reals is quite a headache, unless one is prepared either to distinguish nonzero reals from reals (a nonzero real being a pair consisting of a real x and an integer n with ∣x∣ > 1/n) or, to take the Dialectica interpretation seriously, by adjoining to the Gödel system an axiom saying that every formula is equivalent to its Dialectica interpretation. (See [1, p. 19], [2, pp. 57–60] respectively for these two methods.) In more advanced mathematics the complexities become intolerable.


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