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  1. Combining Montague Semantics and Discourse Representation.Reinhard Muskens - 1996 - Linguistics and Philosophy 19 (2):143 - 186.
    This paper embeds the core part of Discourse Representation Theory in the classical theory of types plus a few simple axioms that allow the theory to express key facts about variables and assignments on the object level of the logic. It is shown how the embedding can be used to combine core analyses of natural language phenomena in Discourse Representation Theory with analyses that can be obtained in Montague Semantics.
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  • Intuitionist Type Theory and Foundations.J. Lambek & P. J. Scott - 1981 - Journal of Philosophical Logic 10 (1):101 - 115.
    A version of intuitionistic type theory is presented here in which all logical symbols are defined in terms of equality. This language is used to construct the so-called free topos with natural number object. It is argued that the free topos may be regarded as the universe of mathematics from an intuitionist's point of view.
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  • Extending Montague's System: A Three Valued Intensional Logic.E. H. Alves & J. A. D. Guerzoni - 1990 - Studia Logica 49 (1):127 - 132.
    In this note we present a three-valued intensional logic, which is an extension of both Montague's intensional logic and ukasiewicz three-valued logic. Our system is obtained by adapting Gallin's version of intensional logic (see Gallin, D., Intensional and Higher-order Modal Logic). Here we give only the necessary modifications to the latter. An acquaintance with Gallin's work is pressuposed.
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  • Completeness in Equational Hybrid Propositional Type Theory.Maria Manzano, Manuel Martins & Antonia Huertas - 2019 - Studia Logica 107 (6):1159-1198.
    Equational hybrid propositional type theory ) is a combination of propositional type theory, equational logic and hybrid modal logic. The structures used to interpret the language contain a hierarchy of propositional types, an algebra and a Kripke frame. The main result in this paper is the proof of completeness of a calculus specifically defined for this logic. The completeness proof is based on the three proofs Henkin published last century: Completeness in type theory, The completeness of the first-order functional calculus (...)
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  • Completeness in Equational Hybrid Propositional Type Theory.Maria Manzano, Manuel Martins & Antonia Huertas - 2019 - Studia Logica 107 (6):1159-1198.
    Equational hybrid propositional type theory ) is a combination of propositional type theory, equational logic and hybrid modal logic. The structures used to interpret the language contain a hierarchy of propositional types, an algebra and a Kripke frame. The main result in this paper is the proof of completeness of a calculus specifically defined for this logic. The completeness proof is based on the three proofs Henkin published last century: Completeness in type theory, The completeness of the first-order functional calculus (...)
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  • A Brief History of Fuzzy Logic in the Czech Republic and Significance of P. Hájek for Its Development.Vilém Novák - 2017 - Archives for the Philosophy and History of Soft Computing 2017 (1).
    In this paper, we will briefly look at the history of mathematical fuzzy logic in Czechoslovakia starting from the 1970s and extending until 2009. The role of P. Ha ́jek in the development of fuzzy logic is especially emphasized.
     
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  • Anaphora and the Logic of Change.Reinhard Muskens - 1991 - In Jan Van Eijck (ed.), Logics in AI, Proceedings of JELIA '90, volume 478 of Lecture Notes in Computer Science. Berlin: Springer-Verlag. pp. 412-427.
    This paper shows how the dynamic interpretation of natural language introduced in work by Hans Kamp and Irene Heim can be modeled in classical type logic. This provides a synthesis between Richard Montague's theory of natural language semantics and the work by Kamp and Heim.
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  • Ins and Outs of Russell's Theory of Types.Ali Bora Enderer - unknown
    The thesis examines A.N. Whitehead and B. Russell’s Ramified Theory of Types. It consists of three parts. The first part is devoted to understanding the source of impredicativity implicit in the induction principle. The question I raise here is whether second-order explicit definitions are responsible for cases when impredicativity turns pathological. The second part considers the interplay between the vicious-circle principle and the no-class theory. The main goal is to give an explanation for the predicative restrictions entailed by the vicious-circle (...)
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  • The Seven Virtues of Simple Type Theory.William M. Farmer - 2008 - Journal of Applied Logic 6 (3):267-286.
  • Lesniewski and Russell's Paradox: Some Problems.Rafal Urbaniak - 2008 - History and Philosophy of Logic 29 (2):115-146.
    Sobocinski in his paper on Leśniewski's solution to Russell's paradox (1949b) argued that Leśniewski has succeeded in explaining it away. The general strategy of this alleged explanation is presented. The key element of this attempt is the distinction between the collective (mereological) and the distributive (set-theoretic) understanding of the set. The mereological part of the solution, although correct, is likely to fall short of providing foundations of mathematics. I argue that the remaining part of the solution which suggests a specific (...)
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  • Description of All Functions Definable by Formulæ of the 2nd Order Intuitionistic Propositional Calculus on Some Linear Heyting Algebras.Dimitri Pataraia - 2006 - Journal of Applied Non-Classical Logics 16 (3-4):457-483.
    Explicit description of maps definable by formulæ of the second order intuitionistic propositional calculus is given on two classes of linear Heyting algebras?the dense ones and the ones which possess successors. As a consequence, it is shown that over these classes every formula is equivalent to a quantifier free formula in the dense case, and to a formula with quantifiers confined to the applications of the successor in the second case.
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  • Supra-Logic: Using Transfinite Type Theory with Type Variables for Paraconsistency.Jørgen Villadsen - 2005 - Journal of Applied Non-Classical Logics 15 (1):45-58.
    We define the paraconsistent supra-logic Pσ by a type-shift from the booleans o of propositional logic Po to the supra-booleans σ of the propositional type logic P obtained as the propositional fragment of the transfinite type theory Q defined by Peter Andrews as a classical foundation of mathematics. The supra-logic is in a sense a propositional logic only, but since there is an infinite number of supra-booleans and arithmetical operations are available for this and other types, virtually anything can be (...)
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  • Hyperfine-Grained Meanings in Classical Logic.Reinhard Muskens - 1991 - Logique Et Analyse 133:159-176.
    This paper develops a semantics for a fragment of English that is based on the idea of `impossible possible worlds'. This idea has earlier been formulated by authors such as Montague, Cresswell, Hintikka, and Rantala, but the present set-up shows how it can be formalized in a completely unproblematic logic---the ordinary classical theory of types. The theory is put to use in an account of propositional attitudes that is `hyperfine-grained', i.e. that does not suffer from the well-known problems involved with (...)
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  • Types, Sets and Categories.John L. Bell - unknown
    This essay is an attempt to sketch the evolution of type theory from its beginnings early in the last century to the present day. Central to the development of the type concept has been its close relationship with set theory to begin with and later its even more intimate relationship with category theory. Since it is effectively impossible to describe these relationships (especially in regard to the latter) with any pretensions to completeness within the space of a comparatively short article, (...)
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  • Alonzo Church:His Life, His Work and Some of His Miracles.Maía Manzano - 1997 - History and Philosophy of Logic 18 (4):211-232.
    This paper is dedicated to Alonzo Church, who died in August 1995 after a long life devoted to logic. To Church we owe lambda calculus, the thesis bearing his name and the solution to the Entscheidungsproblem.His well-known book Introduction to Mathematical LogicI, defined the subject matter of mathematical logic, the approach to be taken and the basic topics addressed. Church was the creator of the Journal of Symbolic Logicthe best-known journal of the area, which he edited for several decades This (...)
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  • From Sets to Types to Categories to Sets.Steve Awodey - 2009 - Philosophical Explorations.
    Three different styles of foundations of mathematics are now commonplace: set theory, type theory, and category theory. How do they relate, and how do they differ? What advantages and disadvantages does each one have over the others? We pursue these questions by considering interpretations of each system into the others and examining the preservation and loss of mathematical content thereby. In order to stay focused on the “big picture”, we merely sketch the overall form of each construction, referring to the (...)
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  • A Simple Type Theory with Partial Functions and subtypes11Supported by the MITRE-Sponsored Research Program. Presented at the 9th International Congress of Logic, Methodology and Philosophy of Science Held in Uppsala, Sweden, August 7-14, 1991. [REVIEW]William M. Farmer - 1993 - Annals of Pure and Applied Logic 64 (3):211-240.
    Simple type theory is a higher-order predicate logic for reasoning about truth values, individuals, and simply typed total functions. We present in this paper a version of simple type theory, called PF*, in which functions may be partial and types may have subtypes. We define both a Henkin-style general models semantics and an axiomatic system for PF*, and we prove that the axiomatic system is complete with respect to the general models semantics. We also define a notion of an interpretation (...)
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  • Eq-Algebra-Based Fuzzy Type Theory And Its Extensions.Vilém Novák - 2011 - Logic Journal of the IGPL 19 (3):512-542.
    In this paper, we introduce a new algebra called ‘EQ-algebra’, which is an alternative algebra of truth values for formal fuzzy logics. It is specified by replacing implication as the main operation with a fuzzy equality. Namely, EQ-algebra is a semilattice endowed with a binary operation of fuzzy equality and a binary operation of multiplication. Implication is derived from the fuzzy equality and it is not a residuation with respect to multiplication. Consequently, EQ-algebras overlap with residuated lattices but are not (...)
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  • Visions of Henkin.María Manzano & Enrique Alonso - 2015 - Synthese 192 (7):2123-2138.
    Leon Henkin (1921–2006) was not only an extraordinary logician, but also an excellent teacher, a dedicated professor and an exceptional person. The first two sections of this paper are biographical, discussing both his personal and academic life. In the last section we present three aspects of Henkin’s work. First we comment part of his work fruit of his emphasis on teaching. In a personal communication he affirms that On mathematical induction, published in 1969, was the favourite among his articles with (...)
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