Results for 'Ramified type theory'

946 found
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  1.  71
    The Fact Semantics for Ramified Type Theory and the Axiom of Reducibility.Edwin D. Mares - 2007 - Notre Dame Journal of Formal Logic 48 (2):237-251.
    This paper uses an atomistic ontology of universals, individuals, and facts to provide a semantics for ramified type theory. It is shown that with some natural constraints on the sort of universals and facts admitted into a model, the axiom of reducibility is made valid.
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  2.  52
    Re-examining Russell's Paralysis: Ramified Type-Theory and Wittgenstein's Objection to Russell's Theory of Judgment.Graham Stevens - 2003 - Russell: The Journal of Bertrand Russell Studies 23 (1).
    It is well known that Russell abandoned his multiple-relation theory of judgment, which provided the philosophical foundations for _PM_'s ramified type-theory, in response to criticisms by Wittgenstein. Their exact nature has remained obscure. An influential interpretation, put forth by Sommerville and Griffin, is that Wittgenstein showed that the theory must appeal to the very hierarchy it is intended to generate and thus collapses into circularity. I argue that this rests on a mistaken interpretation of (...)-theory and suggest an alternative one to explain Russell's reaction. (shrink)
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  3. A weak ramified type theory.W. A. van der Moore - 1968 - In P. Braffort & F. van Scheepen, Automation in language translation and theorem proving. Brussels,: Commission of the European Communities, Directorate-General for Dissemination of Information.
     
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  4.  37
    A constructive examination of a Russell-style ramified type theory.Erik Palmgren - 2018 - Bulletin of Symbolic Logic 24 (1):90-106.
    In this article we examine the natural interpretation of a ramified type hierarchy into Martin-Löf type theory with an infinite sequence of universes. It is shown that under this predicative interpretation some useful special cases of Russell’s reducibility axiom are valid, namely functional reducibility. This is sufficient to make the type hierarchy usable for development of constructive mathematical analysis in the style of Bishop. We present a ramified type theory suitable for this (...)
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  5.  29
    (1 other version)Mathematical induction in ramified type theory.James R. Royse - 1969 - Mathematical Logic Quarterly 15 (1‐3):7-10.
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  6.  52
    A correspondence between Martin-löf type theory, the ramified theory of types and pure type systems.Fairouz Kamareddine & Twan Laan - 2001 - Journal of Logic, Language and Information 10 (3):375-402.
    In Russell''s Ramified Theory of Types RTT, two hierarchical concepts dominate:orders and types. The use of orders has as a consequencethat the logic part of RTT is predicative.The concept of order however, is almost deadsince Ramsey eliminated it from RTT. This is whywe find Church''s simple theory of types (which uses the type concept without the order one) at the bottom of the Barendregt Cube rather than RTT. Despite the disappearance of orders which have a strong (...)
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  7.  37
    Cumulative versus Noncumulative Ramified Types.Anthony F. Peressini - 1997 - Notre Dame Journal of Formal Logic 38 (3):385-397.
    In this paper I examine the nature of Russell's ramified type theory resolution of paradoxes. In particular, I consider the effect of construing the types in Church's cumulative sense, that is, the range of a variable of a given type includes the range of every variable of directly lower type. Contrary to what seems to be generally assumed, I show that the decision to make the levels cumulative and allow this to be reflected in the (...)
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  8. Ordinal Type Theory.Jan Plate - forthcoming - Inquiry: An Interdisciplinary Journal of Philosophy.
    Higher-order logic, with its type-theoretic apparatus known as the simple theory of types (STT), has increasingly come to be employed in theorizing about properties, relations, and states of affairs—or ‘intensional entities’ for short. This paper argues against this employment of STT and offers an alternative: ordinal type theory (OTT). Very roughly, STT and OTT can be regarded as complementary simplifications of the ‘ramified theory of types’ outlined in the Introduction to Principia Mathematica (on a (...)
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  9. Why Ramify?Harold T. Hodes - 2015 - Notre Dame Journal of Formal Logic 56 (2):379-415.
    This paper considers two reasons that might support Russell’s choice of a ramified-type theory over a simple-type theory. The first reason is the existence of purported paradoxes that can be formulated in any simple-type language, including an argument that Russell considered in 1903. These arguments depend on certain converse-compositional principles. When we take account of Russell’s doctrine that a propositional function is not a constituent of its values, these principles turn out to be too (...)
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  10. Russell´s Early Type Theory and the Paradox of Propositions.André Fuhrmann - 2001 - Principia: An International Journal of Epistemology 5 (1-2):19–42.
    The paradox of propositions, presented in Appendix B of Russell's The Principles of Mathematics (1903), is usually taken as Russell's principal motive, at the time, for moving from a simple to a ramified theory of types. I argue that this view is mistaken. A closer study of Russell's correspondence with Frege reveals that Russell carne to adopt a very different resolution of the paradox, calling into question not the simplicity of his early type theory but the (...)
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  11. Ramified structure.Gabriel Uzquiano - 2022 - Philosophical Studies 180 (5-6):1651-1674.
    The Russell–Myhill theorem threatens a familiar structured conception of propositions according to which two sentences express the same proposition only if they share the same syntactic structure and their corresponding syntactic constituents share the same semantic value. Given the role of the principle of universal instantiation in the derivation of the theorem in simple type theory, one may hope to rehabilitate the core of the structured view of propositions in ramified type theory, where the principle (...)
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  12.  74
    A modern elaboration of the ramified theory of types.Twan Laan & Rob Nederpelt - 1996 - Studia Logica 57 (2-3):243 - 278.
    The paper first formalizes the ramified type theory as (informally) described in the Principia Mathematica [32]. This formalization is close to the ideas of the Principia, but also meets contemporary requirements on formality and accuracy, and therefore is a new supply to the known literature on the Principia (like [25], [19], [6] and [7]).As an alternative, notions from the ramified type theory are expressed in a lambda calculus style. This situates the type system (...)
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  13. Fitchův paradox poznatelnosti a rozvětvená teorie typů [Fitch's Paradox of Knowability and Ramified Theory of Types].Jiri Raclavsky - 2013 - Organon F: Medzinárodný Časopis Pre Analytickú Filozofiu 20:144-165.
    It is already known that Fitch’s knowability paradox can be solved by typing knowledge within ramified theory of types. One of the aims of this paper is to provide a greater defence of the approach against recently raised criticism. My second goal is to make a sufficient support for an assumption which is needed for this particular application of typing knowledge but which is not inherent to ramified theory of types as such.
     
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  14. A Semantic Analysis of Russellian Simple Type Theory.Sten Lindström - 1986 - In Paul Needham & Jan Odelstad, Changing Positions: Essays Dedicated to Lars Lindahl. Uppsala: Department of Philosophy, Uppsala University.
    As emphasized by Alonzo Church and David Kaplan (Church 1974, Kaplan 1975), the philosophies of language of Frege and Russell incorporate quite different methods of semantic analysis with different basic concepts and different ontologies. Accordingly we distinguish between a Fregean and a Russellian tradition in intensional semantics. The purpose of this paper is to pursue the Russellian alternative and to provide a language of intensional logic with a model-theoretic semantics. We also discuss the so-called Russell-Myhill paradox that threatens simple Russellian (...)
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  15.  19
    Cut-Elimination Theorem Concerning a Formal System for Ramified Theory of Types Which Admits Quantifications on Types.Shôji Maehara - 1962 - Annals of the Japan Association for Philosophy of Science 2 (2):55-64.
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  16. Explikace a deukce: of jednoduché k rozvětvené teorii typů [Explication and Deduction: From Simple to Ramified Theory of Types].Jiri Raclavsky - 2012 - Organon F: Medzinárodný Časopis Pre Analytickú Filozofiu 20 (4):37-53.
  17.  44
    Russell's Zigzag Path to the Ramified Theory of Types.Alasdair Urquhart - 1988 - Russell: The Journal of Bertrand Russell Studies 8 (1):82-91.
  18.  23
    Cut-Elimination Theorem Concerning a Formal System for Ramified Theory of Types Which Admits Quantifications on Types.Sh^|^Ocirc Maehara & Ji - 1962 - Annals of the Japan Association for Philosophy of Science 2 (2):55-64.
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  19.  47
    Quantification Theory in *9 of Principia Mathematica.Gregory Landini - 2000 - History and Philosophy of Logic 21 (1):57-77.
    This paper examines the quantification theory of *9 of Principia Mathematica. The focus of the discussion is not the philosophical role that section *9 plays in Principia's full ramified type-theory. Rather, the paper assesses the system of *9 as a quantificational theory for the ordinary predicate calculus. The quantifier-free part of the system of *9 is examined and some misunderstandings of it are corrected. A flaw in the system of *9 is discovered, but it is (...)
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  20.  8
    The Theory of Logical Types: Monographs in Modern Logic.Irving M. Copi - 2011 - Routledge.
    This reissue, first published in 1971, provides a brief historical account of the Theory of Logical Types; and describes the problems that gave rise to it, its various different formulations (Simple and Ramified), the difficulties connected with each, and the criticisms that have been directed against it. Professor Copi seeks to make the subject accessible to the non-specialist and yet provide a sufficiently rigorous exposition for the serious student to see exactly what the theory is and how (...)
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  21. Deduction in TIL: From Simple to Ramified Hierarchy of Types.Marie Duží - 2013 - Organon F: Medzinárodný Časopis Pre Analytickú Filozofiu 20 (2):5-36.
    Tichý’s Transparent Intensional Logic (TIL) is an overarching logical framework apt for the analysis of all sorts of discourse, whether colloquial, scientific, mathematical or logical. The theory is a procedural (as opposed to denotational) one, according to which the meaning of an expression is an abstract, extra-linguistic procedure detailing what operations to apply to what procedural constituents to arrive at the product (if any) of the procedure that is the object denoted by the expression. Such procedures are rigorously defined (...)
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  22. Axiomatic Theories of Partial Ground II: Partial Ground and Hierarchies of Typed Truth.Johannes Korbmacher - 2018 - Journal of Philosophical Logic 47 (2):193-226.
    This is part two of a two-part paper in which we develop an axiomatic theory of the relation of partial ground. The main novelty of the paper is the of use of a binary ground predicate rather than an operator to formalize ground. In this part of the paper, we extend the base theory of the first part of the paper with hierarchically typed truth-predicates and principles about the interaction of partial ground and truth. We show that our (...)
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  23.  61
    Maehara Shôji. Cut-elimination theorem concerning a formal system for ramified theory of types which admits quantifications on types. Annals of the Japan Association for Philosophy of Science, vol. 2 no. 2 , pp. 55–64. [REVIEW]Moto-O. Takahashi - 1970 - Journal of Symbolic Logic 35 (2):325-325.
  24.  53
    Types in logic and mathematics before 1940.Fairouz Kamareddine, Twan Laan & Rob Nederpelt - 2002 - Bulletin of Symbolic Logic 8 (2):185-245.
    In this article, we study the prehistory of type theory up to 1910 and its development between Russell and Whitehead's Principia Mathematica ([71], 1910-1912) and Church's simply typed λ-calculus of 1940. We first argue that the concept of types has always been present in mathematics, though nobody was incorporating them explicitly as such, before the end of the 19th century. Then we proceed by describing how the logical paradoxes entered the formal systems of Frege, Cantor and Peano concentrating (...)
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  25.  28
    Paul Lorenzen -- Mathematician and Logician.Gerhard Heinzmann & Gereon Wolters (eds.) - 2021 - Springer Verlag.
    This open access book examines the many contributions of Paul Lorenzen, an outstanding philosopher from the latter half of the 20th century. It features papers focused on integrating Lorenzen's original approach into the history of logic and mathematics. The papers also explore how practitioners can implement Lorenzen’s systematical ideas in today’s debates on proof-theoretic semantics, databank management, and stochastics. Coverage details key contributions of Lorenzen to constructive mathematics, Lorenzen’s work on lattice-groups and divisibility theory, and modern set theory (...)
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  26.  75
    The Axiom of Reducibility.Russell Wahl - 2011 - Russell: The Journal of Bertrand Russell Studies 31 (1).
    The axiom of reducibility plays an important role in the logic of Principia Mathematica, but has generally been condemned as an ad hoc non-logical axiom which was added simply because the ramified type theory without it would not yield all the required theorems. In this paper I examine the status of the axiom of reducibility. Whether the axiom can plausibly be included as a logical axiom will depend in no small part on the understanding of propositional functions. (...)
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  27. Set-Theoretic Foundations.Stewart Shapiro - 2000 - The Proceedings of the Twentieth World Congress of Philosophy 6:183-196.
    Since virtually every mathematical theory can be interpreted in Zermelo-Fraenkel set theory, it is a foundation for mathematics. There are other foundations, such as alternate set theories, higher-order logic, ramified type theory, and category theory. Whether set theory is the right foundation for mathematics depends on what a foundation is for. One purpose is to provide the ultimate metaphysical basis for mathematics. A second is to assure the basic epistemological coherence of all mathematical (...)
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  28.  12
    Lorenzen Between Gentzen and Schütte.Reinhard Kahle & Isabel Oitavem - 2021 - In Gerhard Heinzmann & Gereon Wolters, Paul Lorenzen -- Mathematician and Logician. Springer Verlag. pp. 63-76.
    We discuss Lorenzen’s consistency proof for ramified type theory without reducibility, published in 1951, in its historical context and highlight Lorenzen’s contribution to the development of modern proof theory, notably by the introduction of the ω\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}ω\omega \end{document}-rule.
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  29.  70
    A Normative Model of Classical Reasoning in Higher Order Languages.Peter Zahn - 2006 - Synthese 148 (2):309-343.
    The present paper is concerned with a ramified type theory (cf. (Lorenzen 1955), (Russell), (Schütte), (Weyl), e.g.,) in a cumulative version. §0 deals with reasoning in first order languages. is introduced as a first order set.
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  30.  44
    The Versatility of Universality in Principia Mathematica.Brice Halimi - 2011 - History and Philosophy of Logic 32 (3):241-264.
    In this article, I examine the ramified-type theory set out in the first edition of Russell and Whitehead's Principia Mathematica. My starting point is the ‘no loss of generality’ problem: Russell, in the Introduction (Russell, B. and Whitehead, A. N. 1910. Principia Mathematica, Volume I, 1st ed., Cambridge: Cambridge University Press, pp. 53–54), says that one can account for all propositional functions using predicative variables only, that is, dismissing non-predicative variables. That claim is not self-evident at all, (...)
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  31.  54
    Structured lexical concepts, property modifiers, and Transparent Intensional Logic.Bjørn Jespersen - 2015 - Philosophical Studies 172 (2):321-345.
    In a 2010 paper Daley argues, contra Fodor, that several syntactically simple predicates express structured concepts. Daley develops his theory of structured concepts within Tichý’s Transparent Intensional Logic . I rectify various misconceptions of Daley’s concerning TIL. I then develop within TIL an improved theory of how structured concepts are structured and how syntactically simple predicates are related to structured concepts.
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  32.  79
    Wittgenstein's apprenticeship with Russell (review). [REVIEW]Thomas J. Brommage - 2008 - Journal of the History of Philosophy 46 (3):pp. 493-494.
    Although everyone knows that Russell had an immense influence upon Wittgenstein's early philosophy, the degree to which Wittgenstein is either adopting or renouncing Russell's views is still largely a matter of dispute. Recent commentators have been in nearly univocal agreement that the Tractatus should be understood as a rejection of Russell's philosophy, and that Wittgenstein was instead more influenced by the "great works of Frege." In his earlier work, Gregory Landini has proposed a more nuanced way to understand Russell than (...)
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  33.  21
    A Neglected Interpretation of Das Kontinuum.Michele Contente - forthcoming - History and Philosophy of Logic:1-25.
    Hermann's Weyl Das Kontinuum has inspired several studies in logic and foundations of mathematics over the last century. The book provides a remarkable reconstruction of a large portion of classical mathematics on a predicative basis. However, diverging interpretations of the predicative system formulated by Weyl have been proposed in the literature. In the present work, I analyze an early formalization of Weyl's ideas proposed by [Casari, E. 1964. Questioni di Filosofia Della Matematica, Milano: Feltrinelli] and compare it with other, more (...)
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  34. A new interpretation of russell's multiple-relation theory of judgment.Gregory Landini - 1991 - History and Philosophy of Logic 12 (1):37-69.
    This paper offers an interpretation of Russell's multiple-relation theory of judgment which characterizes it as direct application of the 1905 theory of definite descriptions. The paper maintains that it was by regarding propositional symbols (when occurring as subordinate clauses) as disguised descriptions of complexes, that Russell generated the philosophical explanation of the hierarchy of orders and the ramified theory of types of _Principia mathematica (1910). The interpretation provides a new understanding of Russell's abandoned book _Theory of (...)
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  35. Reply to Bacon, Hawthorne and Uzquiano.Timothy Williamson - 2016 - Canadian Journal of Philosophy 46 (4-5):542-547.
  36.  2
    A Neglected Interpretation of Das Kontinuum.Michele Contente Czech Academy of Sciences, Prague & Czech Republic - forthcoming - History and Philosophy of Logic:1-25.
    Hermann's Weyl Das Kontinuum has inspired several studies in logic and foundations of mathematics over the last century. The book provides a remarkable reconstruction of a large portion of classical mathematics on a predicative basis. However, diverging interpretations of the predicative system formulated by Weyl have been proposed in the literature. In the present work, I analyze an early formalization of Weyl's ideas proposed by [Casari, E. 1964. Questioni di Filosofia Della Matematica, Milano: Feltrinelli] and compare it with other, more (...)
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  37.  37
    Subsystems of Quine's "New Foundations" with Predicativity Restrictions.M. Randall Holmes - 1999 - Notre Dame Journal of Formal Logic 40 (2):183-196.
    This paper presents an exposition of subsystems and of Quine's , originally defined and shown to be consistent by Crabbé, along with related systems and of type theory. A proof that (and so ) interpret the ramified theory of types is presented (this is a simplified exposition of a result of Crabbé). The new result that the consistency strength of is the same as that of is demonstrated. It will also be shown that cannot be finitely (...)
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  38. Conceptual realism versus Quine on classes and higher-order logic.Nino B. Cocchiarella - 1992 - Synthese 90 (3):379 - 436.
    The problematic features of Quine's set theories NF and ML are a result of his replacing the higher-order predicate logic of type theory by a first-order logic of membership, and can be resolved by returning to a second-order logic of predication with nominalized predicates as abstract singular terms. We adopt a modified Fregean position called conceptual realism in which the concepts (unsaturated cognitive structures) that predicates stand for are distinguished from the extensions (or intensions) that their nominalizations denote (...)
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  39.  99
    Russell's way out of the paradox of propositions.André Fuhrmann - 2002 - History and Philosophy of Logic 23 (3):197-213.
    In Appendix B of Russell's The Principles of Mathematics occurs a paradox, the paradox of propositions, which a simple theory of types is unable to resolve. This fact is frequently taken to be one of the principal reasons for calling ramification onto the Russellian stage. The paper presents a detaiFled exposition of the paradox and its discussion in the correspondence between Frege and Russell. It is argued that Russell finally adopted a very simple solution to the paradox. This solution (...)
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  40.  68
    Scott Soames: The analytic tradition in philosophy, volume 1: Founding giants: Princeton University Press.Charles R. Pigden - 2015 - Philosophical Studies 172 (6):1671-1680.
    The Analytic Tradition in Philosophy is an excellent successor to an excellent book : It is a fine an example of the necromantic style in the history of philosophy where the object of the exercise is to resurrect the mighty dead in order to get into an argument with them, either because we think them importantly right or instructively wrong. However what was a pardonable a simplification and a reasonable omission in the earlier book has now metamorphosed into a sin (...)
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  41.  19
    Les classes dans les Principia Mathematica sont‐elles des expressions incomplétes?Par Jocelyne Couture - 1983 - Dialectica 37 (4):249-267.
    RésuméLa théorie des expressions incomplétes dans Principia Mathematica, se fonde sur le principe déja appliqué par Russell dans “On Denoting”, selon lequel il est souhaitable dans certains cas, ?on;établir le statut syntaxique des expressions catégorématiques. Grâce à la théorie intensionnelle ramifyée des types, les expressions incomplétes réféientiellement pourront être logiquement caractérisées par un mode de dérivation principalement basé sur la quantification non‐objectuelle. Ľintroduction des classes cependant, n'est en aucune façon reliée à ce mode intensionnel de dérivation; il en résulte qu'elles (...)
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  42. On Hierarchical Propositions.Giorgio Sbardolini - 2020 - Journal of Philosophical Logic 49 (1):1-11.
    There is an apparent dilemma for hierarchical accounts of propositions, raised by Bruno Whittle : either such accounts do not offer adequate treatment of connectives and quantifiers, or they eviscerate the logic. I discuss what a plausible hierarchical conception of propositions might amount to, and show that on that conception, Whittle’s dilemma is not compelling. Thus, there are good reasons why proponents of hierarchical accounts of propositions did not see the difficulty Whittle raises.
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  43. Report on some ramified-type assignment systems and their model-theoretic semantics.Harold Hodes - 2013 - In Nicholas Griffin & Bernard Linsky, The Palgrave Centenary Companion to Principia Mathematica. London and Basingstoke: Palgrave-Macmillan.
     
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  44.  14
    Pedagogical Implication of The Principle of Identity and Russell"s Paradox. 은은숙 - 2023 - Journal of the New Korean Philosophical Association 114:263-294.
    본 연구는 논리학 및 수리논리학의 토대 개념인 동일성 원리에 대한 역사적인 논쟁들의 교육학적 함의를 도출하는 것이다. 이때 필자가 사용할 중심 방법은 구조-구성주의 인식론이다. 따라서 필자는 구조-구성주의 인식론의 관점에서 동일성 원리에 대한 핵심 논쟁들을 역사-비판적으로 재구성함으로써, 필자가 지속적으로 논변해 온 구조-구성주의 교수학습이론의 확고한 토대를 제공하고자 한다. 이를 위해 본고는 동일성 원리에 대한 역사발생학적 탐구와 정신발생학적 탐구를 종합한다. 구체적인 내용은 피아제의 발생학적 인식론의 관점에서 논리적 개념들 및 공리화에 대한 프레게-러셀의 선험주의적 논리주의와 비트겐슈타인의 회의론적 유명론을 동시에 비판하면서, 구조-구성주의 인식론 및 이것의 교육학적 함의를 (...)
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  45.  11
    Apuntes para una introducción al logicismo.Ricardo Da Silva - 2019 - Apuntes Filosóficos 28 (55):181-199.
    The following note has on purpose to introduce interested students to logicism. Our objective is not to show any new interpretation or thesis about logicism or its rebirth between the 60s and 80s of the last century. What we will do is systematically show the evolution of logicism from Frege to Russell-Whitehead, with greater emphasis on this latest development, and approach some problems that arise within that movement, for example: The logical paradoxes and the principle of intuitive comprehension, the impredicative (...)
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  46. Against Cumulative Type Theory.Tim Button & Robert Trueman - 2022 - Review of Symbolic Logic 15 (4):907-49.
    Standard Type Theory, STT, tells us that b^n(a^m) is well-formed iff n=m+1. However, Linnebo and Rayo have advocated the use of Cumulative Type Theory, CTT, has more relaxed type-restrictions: according to CTT, b^β(a^α) is well-formed iff β > α. In this paper, we set ourselves against CTT. We begin our case by arguing against Linnebo and Rayo’s claim that CTT sheds new philosophical light on set theory. We then argue that, while CTT ’s (...)-restrictions are unjustifiable, the type-restrictions imposed by STT are justified by a Fregean semantics. What is more, this Fregean semantics provides us with a principled way to resist Linnebo and Rayo’s Semantic Argument for CTT. We end by examining an alternative approach to cumulative types due to Florio and Jones; we argue that their theory is best seen as a misleadingly formulated version of STT. (shrink)
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  47. Constructive Type Theory, an appetizer.Laura Crosilla - 2024 - In Peter Fritz & Nicholas K. Jones, Higher-Order Metaphysics. Oxford University Press.
    Recent debates in metaphysics have highlighted the significance of type theories, such as Simple Type Theory (STT), for our philosophical analysis. In this chapter, I present the salient features of a constructive type theory in the style of Martin-Löf, termed CTT. My principal aim is to convey the flavour of this rich, flexible and sophisticated theory and compare it with STT. I especially focus on the forms of quantification which are available in CTT. A (...)
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  48.  18
    Type theory and formal proof: an introduction.R. P. Nederpelt & Herman Geuvers - 2014 - New York: Cambridge University Press. Edited by Herman Geuvers.
    Type theory is a fast-evolving field at the crossroads of logic, computer science and mathematics. This gentle step-by-step introduction is ideal for graduate students and researchers who need to understand the ins and outs of the mathematical machinery, the role of logical rules therein, the essential contribution of definitions and the decisive nature of well-structured proofs. The authors begin with untyped lambda calculus and proceed to several fundamental type systems culminating in the well-known and powerful Calculus of (...)
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  49. Selection type theories.Lindley Darden & Joseph A. Cain - 1989 - Philosophy of Science 56 (1):106-129.
    Selection type theories solve adaptation problems. Natural selection, clonal selection for antibody production, and selective theories of higher brain function are examples. An abstract characterization of typical selection processes is generated by analyzing and extending previous work on the nature of natural selection. Once constructed, this abstraction provides a useful tool for analyzing the nature of other selection theories and may be of use in new instances of theory construction. This suggests the potential fruitfulness of research to find (...)
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  50. A contextual type theory with judgemental modalities for reasoning from open assumptions.Giuseppe Primiero - 2012 - Logique and Analyse 220:579-600.
    Contextual type theories are largely explored in their applications to programming languages, but less investigated for knowledge representation purposes. The combination of a constructive language with a modal extension of contexts appears crucial to explore the attractive idea of a type-theoretical calculus of provability from refutable assumptions for non-monotonic reasoning. This paper introduces such a language: the modal operators are meant to internalize two different modes of correctness, respectively with necessity as the standard notion of constructive verification and (...)
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