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Extensions of First Order Logic

Cambridge University Press (1996)

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  1. Quantifier Variance Without Collapse.Hans Halvorson - manuscript
    The thesis of quantifier variance is consistent and cannot be refuted via a collapse argument.
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  • Framing the Epistemic Schism of Statistical Mechanics.Javier Anta - 2021 - Proceedings of the X Conference of the Spanish Society of Logic, Methodology and Philosophy of Science.
    In this talk I present the main results from Anta (2021), namely, that the theoretical division between Boltzmannian and Gibbsian statistical mechanics should be understood as a separation in the epistemic capabilities of this physical discipline. In particular, while from the Boltzmannian framework one can generate powerful explanations of thermal processes by appealing to their microdynamics, from the Gibbsian framework one can predict observable values in a computationally effective way. Finally, I argue that this statistical mechanical schism contradicts the Hempelian (...)
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  • A Novel Approach to Equality.Andrzej Indrzejczak - 2021 - Synthese 199 (1-2):4749-4774.
    A new type of formalization of classical first-order logic with equality is introduced on the basis of the sequent calculus. It serves to justify the claim that equality is a logical constant characterised by well-behaved rules satisfying properties usually regarded as essential. The main feature of this approach is the application of sequents built not only from formulae but also from terms. Two variants of sequent calculus are examined, a structural and a logical one. The former is defined in accordance (...)
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  • Foundations of Applied Mathematics I.Jeffrey Ketland - 2021 - Synthese 199 (1-2):4151-4193.
    This paper aims to study the foundations of applied mathematics, using a formalized base theory for applied mathematics: ZFCAσ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \mathsf {ZFCA}_{\sigma }$$\end{document} with atoms, where the subscript used refers to a signature specific to the application. Examples are given, illustrating the following five features of applied mathematics: comprehension principles, application conditionals, representation hypotheses, transfer principles and abstract equivalents.
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  • Proof Verification and Proof Discovery for Relativity.Naveen Sundar Govindarajalulu, Selmer Bringsjord & Joshua Taylor - 2015 - Synthese 192 (7):2077-2094.
    The vision of machines autonomously carrying out substantive conjecture generation, theorem discovery, proof discovery, and proof verification in mathematics and the natural sciences has a long history that reaches back before the development of automatic systems designed for such processes. While there has been considerable progress in proof verification in the formal sciences, for instance the Mizar project’ and the four-color theorem, now machine verified, there has been scant such work carried out in the realm of the natural sciences—until recently. (...)
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  • Ramsey Equivalence.Neil Dewar - 2019 - Erkenntnis 84 (1):77-99.
    In the literature over the Ramsey-sentence approach to structural realism, there is often debate over whether structural realists can legitimately restrict the range of the second-order quantifiers, in order to avoid the Newman problem. In this paper, I argue that even if they are allowed to, it won’t help: even if the Ramsey sentence is interpreted using such restricted quantifiers, it is still an implausible candidate to capture a theory’s structural content. To do so, I use the following observation: if (...)
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  • Scheinprobleme - Ein explikativer Versuch.Moritz Cordes - 2016 - Dissertation, University of Greifswald
    The traditional use of the expression 'pseudoproblem' is analysed in order to clarify the talk of pseudoproblems and related phenomena. The goal is to produce a philosophically serviceable terminology that stays true to its historical roots. This explicative study is inspired by and makes use of the method of logical reconstruction. Since pseudoproblems are usually expressed by pseudoquestions a formal language of questions is presented as a possible reconstruction language for alleged pseudoproblems. The study yields an informal theory of pseudoproblems (...)
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  • Logic, Essence, and Modality — Review of Bob Hale's Necessary Beings. [REVIEW]Christopher Menzel - 2015 - Philosophia Mathematica 23 (3):407-428.
    Bob Hale’s distinguished record of research places him among the most important and influential contemporary analytic metaphysicians. In his deep, wide ranging, yet highly readable book Necessary Beings, Hale draws upon, but substantially integrates and extends, a good deal his past research to produce a sustained and richly textured essay on — as promised in the subtitle — ontology, modality, and the relations between them. I’ve set myself two tasks in this review: first, to provide a reasonably thorough (if not (...)
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  • Structure and Categoricity: Determinacy of Reference and Truth Value in the Philosophy of Mathematics.Tim Button & Sean Walsh - 2016 - Philosophia Mathematica 24 (3):283-307.
    This article surveys recent literature by Parsons, McGee, Shapiro and others on the significance of categoricity arguments in the philosophy of mathematics. After discussing whether categoricity arguments are sufficient to secure reference to mathematical structures up to isomorphism, we assess what exactly is achieved by recent ‘internal’ renditions of the famous categoricity arguments for arithmetic and set theory.
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  • Ontology, Quantification, and Fundamentality.Jason Theodore Turner - unknown
    The structuralist conception of metaphysics holds that it aims to uncover the ultimate structure of reality and explain how the world's richness and variety are accounted for by that ultimate structure. On this conception, metaphysicians produce fundamental theories, the primitive, undefined expressions of which are supposed to 'carve reality at its joints', as it were. On this conception, ontological questions are understood as questions about what there is, where the existential quantifier 'there is' has a fundamental, joint-carving interpretation. Structuralist orthodoxy (...)
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  • Quine’s Conjecture on Many-Sorted Logic.Thomas William Barrett & Hans Halvorson - 2017 - Synthese 194 (9):3563-3582.
    Quine often argued for a simple, untyped system of logic rather than the typed systems that were championed by Russell and Carnap, among others. He claimed that nothing important would be lost by eliminating sorts, and the result would be additional simplicity and elegance. In support of this claim, Quine conjectured that every many-sorted theory is equivalent to a single-sorted theory. We make this conjecture precise, and prove that it is true, at least according to one reasonable notion of theoretical (...)
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  • Hybrid Type Theory: A Quartet in Four Movements.Carlos Areces, Patrick Blackburn, Antonia Huertas & María Manzano - 2011 - Principia: An International Journal of Epistemology 15 (2):225.
    Este artigo canta uma canção — uma canção criada ao unir o trabalho de quatro grandes nomes na história da lógica: Hans Reichenbach, Arthur Prior, Richard Montague, e Leon Henkin. Embora a obra dos primeiros três desses autores tenha sido previamente combinada, acrescentar as ideias de Leon Henkin é o acréscimo requerido para fazer com que essa combinação funcione no nível lógico. Mas o presente trabalho não se concentra nas tecnicalidades subjacentes (que podem ser encontradas em Areces, Blackburn, Huertas, e (...)
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  • Empirical Adequacy and Ramsification.Jeffrey Ketland - 2004 - British Journal for the Philosophy of Science 55 (2):287-300.
    Structural realism has been proposed as an epistemological position interpolating between realism and sceptical anti-realism about scientific theories. The structural realist who accepts a scientific theory thinks that is empirically correct, and furthermore is a realist about the ‘structural content’ of . But what exactly is ‘structural content’? One proposal is that the ‘structural content’ of a scientific theory may be associated with its Ramsey sentence (). However, Demopoulos and Friedman have argued, using ideas drawn from Newman's earlier criticism of (...)
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  • El pensamiento incorporado percepcional-lingüístico-lógico/The embodied, perceptional, linguistic and logic thougth.Rómulo Sanmartin - 2012 - Sophia. Colección de Filosofía de la Educación 13:26-72.
    El pensamiento es incorporado por la articulación de los dos algoritmos: la estructura interna para conocer y la realidad, externa, a ser reconocida. La realidad interna está dada desde la estructura de la lengua, la cual más adelante dará lugar a un formato lógica-matemática. La realidad externa, que es también antropológica, está mediada por las áreas somatosensoriales cerebrales, que protológicamente acercan a lo distinto del humano. La filosofía y la ciencia tienen la tarea de acercarse a estas realidades para enhebrarlas, (...)
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  • Some Uses of Logic in Rigorous Philosophy.Guillermo E. Rosado Haddock - 2010 - Axiomathes 20 (2-3):385-398.
    This paper is concerned with the use of logic to solve philosophical problems. Such use of logic goes counter to the prevailing empiricist tradition in analytic circles. Specifically, model-theoretic tools are applied to three fundamental issues in the philosophy of logic and mathematics, namely, to the issue of the existence of mathematical entities, to the dispute between first- and second-order logic and to the definition of analyticity.
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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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  • Probabilities Defined on Standard and Non-Standard Cylindric Set Algebras.Miklós Ferenczi - 2015 - Synthese 192 (7):2025-2033.
    Cylindric set algebras are algebraizations of certain logical semantics. The topic surveyed here, i.e. probabilities defined on cylindric set algebras, is closely related, on the one hand, to probability logic (to probabilities defined on logical formulas), on the other hand, to measure theory. The set algebras occuring here are associated, in particular, with the semantics of first order logic and with non-standard analysis. The probabilities introduced are partially continous, they are continous with respect to so-called cylindric sums.
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  • Completeness in Hybrid Type Theory.Carlos Areces, Patrick Blackburn, Antonia Huertas & María Manzano - 2013 - Journal of Philosophical Logic (2-3):1-30.
    We show that basic hybridization (adding nominals and @ operators) makes it possible to give straightforward Henkin-style completeness proofs even when the modal logic being hybridized is higher-order. The key ideas are to add nominals as expressions of type t, and to extend to arbitrary types the way we interpret [email protected]_i$ in propositional and first-order hybrid logic. This means: interpret [email protected]_i\alpha _a$ , where $\alpha _a$ is an expression of any type $a$ , as an expression of type $a$ that (...)
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  • INVENTING LOGIC: THE LÖWENHEIM-SKOLEM THEOREM AND FIRST- AND SECOND-ORDER LOGIC.Valérie Lynn Therrien - 2012 - Pensées Canadiennes 10.
  • Löwenheim–Skolem Theorems for Non-Classical First-Order Algebraizable Logics: Table 1.Pilar Dellunde, Àngel García-Cerdaña & Carles Noguera - 2016 - Logic Journal of the IGPL 24 (3):321-345.
  • Relations Vs Functions at the Foundations of Logic: Type-Theoretic Considerations.Paul Oppenheimer & Edward N. Zalta - 2011 - Journal of Logic and Computation 21:351-374.
    Though Frege was interested primarily in reducing mathematics to logic, he succeeded in reducing an important part of logic to mathematics by defining relations in terms of functions. By contrast, Whitehead & Russell reduced an important part of mathematics to logic by defining functions in terms of relations (using the definite description operator). We argue that there is a reason to prefer Whitehead & Russell's reduction of functions to relations over Frege's reduction of relations to functions. There is an interesting (...)
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  • Some New Results on Decidability for Elementary Algebra and Geometry.Robert M. Solovay, R. D. Arthan & John Harrison - 2012 - Annals of Pure and Applied Logic 163 (12):1765-1802.
    We carry out a systematic study of decidability for theories of real vector spaces, inner product spaces, and Hilbert spaces and of normed spaces, Banach spaces and metric spaces, all formalized using a 2-sorted first-order language. The theories for list turn out to be decidable while the theories for list are not even arithmetical: the theory of 2-dimensional Banach spaces, for example, has the same many-one degree as the set of truths of second-order arithmetic.We find that the purely universal and (...)
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  • A Note on Identity and Higher Order Quantification.Rafal Urbaniak - 2009 - Australasian Journal of Logic 7:48--55.
    It is a commonplace remark that the identity relation, even though not expressible in a first-order language without identity with classical set-theoretic semantics, can be defined in a language without identity, as soon as we admit second-order, set-theoretically interpreted quantifiers binding predicate variables that range over all subsets of the domain. However, there are fairly simple and intuitive higher-order languages with set-theoretic semantics in which the identity relation is not definable. The point is that the definability of identity in higher-order (...)
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  • Identity and Indiscernibility.Jeffrey Ketland - 2011 - Review of Symbolic Logic 4 (2):171-185.
    The notion of strict identity is sometimes given an explicit second-order definition: objects with all the same properties are identical. Here, a somewhat different problem is raised: Under what conditions is the identity relation on the domain of a structure first-order definable? A structure may have objects that are distinct, but indiscernible by the strongest means of discerning them given the language (the indiscernibility formula). Here a number of results concerning the indiscernibility formula, and the definability of identity, are collected (...)
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