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  1. Arithmetic is Necessary.Zachary Goodsell - 2024 - Journal of Philosophical Logic 53 (4).
    (Goodsell, Journal of Philosophical Logic, 51(1), 127-150 2022) establishes the noncontingency of sentences of first-order arithmetic, in a plausible higher-order modal logic. Here, the same result is derived using significantly weaker assumptions. Most notably, the assumption of rigid comprehension—that every property is coextensive with a modally rigid one—is weakened to the assumption that the Boolean algebra of properties under necessitation is countably complete. The results are generalized to extensions of the language of arithmetic, and are applied to answer a question (...)
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  2. Higher‐Order Being and Time.Lukas Skiba - forthcoming - Noûs.
    Higher‐order metaphysicians take facts to be higher‐order beings, i.e., entities in the range of irreducibly higher‐order quantifiers. In this paper, I investigate the impact of this conception of facts on the debate about the reality of tense. I identify two major repercussions. The first concerns the logical space of tense realism: on a higher‐order conception of facts, a prominent version of tense realism, dynamic absolutism, turns out to conflict with the laws of (higher‐order tense) logic. The second concerns our understanding (...)
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  3. Should Theories of Logical Validity Self-Apply?Marco Grossi - forthcoming - Erkenntnis.
    Some philosophers argue that a theory of logical validity should not interpret its own language, because a Russellian argument shows that self-applicability is inconsistent with the ability to capture all the interpretations of its own language. First, I set up a formal system to examine the Russellian argument. I then defend the need for self-applicability. I argue that self-applicability seems to be implied by generality, and that the Russellian argument rests on a test for meaning that is biased against self-applicability. (...)
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  4. Logic in mathematics and computer science.Richard Zach - forthcoming - In Filippo Ferrari, Elke Brendel, Massimiliano Carrara, Ole Hjortland, Gil Sagi, Gila Sher & Florian Steinberger (eds.), Oxford Handbook of Philosophy of Logic. Oxford, UK: Oxford University Press.
    Logic has pride of place in mathematics and its 20th century offshoot, computer science. Modern symbolic logic was developed, in part, as a way to provide a formal framework for mathematics: Frege, Peano, Whitehead and Russell, as well as Hilbert developed systems of logic to formalize mathematics. These systems were meant to serve either as themselves foundational, or at least as formal analogs of mathematical reasoning amenable to mathematical study, e.g., in Hilbert’s consistency program. Similar efforts continue, but have been (...)
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  5. The Logic of Hyperlogic. Part A: Foundations.Alexander W. Kocurek - 2024 - Review of Symbolic Logic 17 (1):244-271.
    Hyperlogic is a hyperintensional system designed to regiment metalogical claims (e.g., “Intuitionistic logic is correct” or “The law of excluded middle holds”) into the object language, including within embedded environments such as attitude reports and counterfactuals. This paper is the first of a two-part series exploring the logic of hyperlogic. This part presents a minimal logic of hyperlogic and proves its completeness. It consists of two interdefined axiomatic systems: one for classical consequence (truth preservation under a classical interpretation of the (...)
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  6. A Philosophical Introduction to Higher-order Logics.Andrew Bacon - 2023 - Routledge.
    This is the first comprehensive textbook on higher order logic that is written specifically to introduce the subject matter to graduate students in philosophy. The book covers both the formal aspects of higher-order languages -- their model theory and proof theory, the theory of λ-abstraction and its generalizations -- and their philosophical applications, especially to the topics of modality and propositional granularity. The book has a strong focus on non-extensional higher-order logics, making it more appropriate for foundational metaphysics than other (...)
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  7. LF: a Foundational Higher-Order Logic.Zachary Goodsell & Juhani Yli-Vakkuri - manuscript
    This paper presents a new system of logic, LF, that is intended to be used as the foundation of the formalization of science. That is, deductive validity according to LF is to be used as the criterion for assessing what follows from the verdicts, hypotheses, or conjectures of any science. In work currently in progress, we argue for the unique suitability of LF for the formalization of logic, mathematics, syntax, and semantics. The present document specifies the language and rules of (...)
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  8. Internalism and the Determinacy of Mathematics.Lavinia Picollo & Daniel Waxman - 2023 - Mind 132 (528):1028-1052.
    A major challenge in the philosophy of mathematics is to explain how mathematical language can pick out unique structures and acquire determinate content. In recent work, Button and Walsh have introduced a view they call ‘internalism’, according to which mathematical content is explained by internal categoricity results formulated and proven in second-order logic. In this paper, we critically examine the internalist response to the challenge and discuss the philosophical significance of internal categoricity results. Surprisingly, as we argue, while internalism arguably (...)
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  9. John MacFarlane, Philosophical Logic: A Contemporary Introduction, Routledge Contemporary Introductions to Philosophy, Routledge, New York, and London, 2021, xx + 238 pp. [REVIEW]Bruno Bentzen - 2023 - Bulletin of Symbolic Logic 29 (3):456-457.
  10. Against Instantiation.Christopher Frugé - forthcoming - Australasian Journal of Philosophy.
    According to traditional universalism, properties are instantiated by objects, where instantiation is a ‘tie’ that binds objects and properties into facts. I offer two arguments against this view. I then develop an alternative higher-order account which holds that properties are primitively predicated of objects yet, unlike traditional nominalism, are nevertheless genuinely real. When it’s a fact that Fo, it’s not because object o instantiates F-ness, but just that Fo – where F still exists. Against orthodox higher-order approaches, however, my arguments (...)
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  11. On Type Distinctions and Expressivity.Salvatore Florio - 2023 - Proceedings of the Aristotelian Society 123 (2):150-172.
    Quine maintained that philosophical and scientific theorizing should be conducted in an untyped language, which has just one style of variables and quantifiers. By contrast, typed languages, such as those advocated by Frege and Russell, include multiple styles of variables and matching kinds of quantification. Which form should our theories take? In this article, I argue that expressivity does not favour typed languages over untyped ones.
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  12. Peano, Frege and Russell’s Logical Influences.Kevin C. Klement - forthcoming - Forthcoming.
    This chapter clarifies that it was the works Giuseppe Peano and his school that first led Russell to embrace symbolic logic as a tool for understanding the foundations of mathematics, not those of Frege, who undertook a similar project starting earlier on. It also discusses Russell’s reaction to Peano’s logic and its influence on his own. However, the chapter also seeks to clarify how and in what ways Frege was influential on Russell’s views regarding such topics as classes, functions, meaning (...)
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  13. Russell's Logicism.Kevin C. Klement - 2018 - In Russell Wahl (ed.), The Bloomsbury Companion to Bertrand Russell. London, UK: BloomsburyAcademic. pp. 151-178.
    Bertrand Russell was one of the best-known proponents of logicism: the theory that mathematics reduces to, or is an extension of, logic. Russell argued for this thesis in his 1903 The Principles of Mathematics and attempted to demonstrate it formally in Principia Mathematica (PM 1910–1913; with A. N. Whitehead). Russell later described his work as a further “regressive” step in understanding the foundations of mathematics made possible by the late 19th century “arithmetization” of mathematics and Frege’s logical definitions of arithmetical (...)
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  14. Two conceptions of absolute generality.Salvatore Florio & Nicholas K. Jones - 2023 - Philosophical Studies 180 (5-6):1601-1621.
    What is absolutely unrestricted quantification? We distinguish two theoretical roles and identify two conceptions of absolute generality: maximally strong generality and maximally inclusive generality. We also distinguish two corresponding kinds of absolute domain. A maximally strong domain contains every potential counterexample to a generalisation. A maximally inclusive domain is such that no domain extends it. We argue that both conceptions of absolute generality are legitimate and investigate the relations between them. Although these conceptions coincide in standard settings, we show how (...)
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  15. Type Polymorphism, Natural Language Semantics, and TIL.Ivo Pezlar - 2023 - Journal of Logic, Language and Information 32 (2):275-295.
    Transparent intensional logic (TIL) is a well-explored type-theoretical framework for semantics of natural language. However, its treatment of polymorphic functions, which are essential for the analysis of various natural language phenomena, is still underdeveloped. In this paper, we address this issue and propose an extension of TIL that introduces polymorphism via type variables ranging over types and generalized variables ranging over constructions and types. Furthermore, we offer an analysis of sentences involving non-specific notional attitudes of the general form ‘_A_ considers (...)
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  16. Svojstva klasične logike [Properties of Classical Logic].Srećko Kovač - 2013 - Zagreb: Hrvatski studiji Sveučilišta u Zagrebu.
    The content for an advanced logic course is presented, which includes the properties of first-order logic language, soundness and completeness of the first-order logic deductive system, Peano arithmetic, Gödel's incompleteness theorems, higher-order logic and its properties. As a reminder, a brief description of first-order logic is included.
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  17. Going Nowhere and Back: Is Trivialization the Same as Zero Execution?Ivo Pezlar - 2022 - In Pavel Materna & Bjørn Jespersen (eds.), Logically Speaking. A Festschrift for Marie Duží. College Publications. pp. 187-202.
    In this paper I will explore the question whether the Trivialization construction of transparent intensional logic (TIL) can be understood in terms of the Execution construction, specifically, in terms of its degenerate case known as the 0-Execution. My answer will be positive and the apparent contrast between the intuitive understanding of Trivialization and 0-Execution will be explained as a matter of distinct yet related informal perspectives, not as a matter of technical or conceptual differences.
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  18. Pure Logic and Higher-order Metaphysics.Christopher Menzel - 2024 - In Peter Fritz & Nicholas K. Jones (eds.), Higher-Order Metaphysics. Oxford University Press.
    W. V. Quine famously defended two theses that have fallen rather dramatically out of fashion. The first is that intensions are “creatures of darkness” that ultimately have no place in respectable philosophical circles, owing primarily to their lack of rigorous identity conditions. However, although he was thoroughly familiar with Carnap’s foundational studies in what would become known as possible world semantics, it likely wouldn’t yet have been apparent to Quine that he was fighting a losing battle against intensions, due in (...)
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  19. A note on fragments of uniform reflection in second order arithmetic.Emanuele Frittaion - 2022 - Bulletin of Symbolic Logic 28 (3):451-465.
    We consider fragments of uniform reflection for formulas in the analytic hierarchy over theories of second order arithmetic. The main result is that for any second order arithmetic theory $T_0$ extending $\mathsf {RCA}_0$ and axiomatizable by a $\Pi ^1_{k+2}$ sentence, and for any $n\geq k+1$, $$\begin{align*}T_0+ \mathrm{RFN}_{\varPi^1_{n+2}} \ = \ T_0 + \mathrm{TI}_{\varPi^1_n}, \end{align*}$$ $$\begin{align*}T_0+ \mathrm{RFN}_{\varSigma^1_{n+1}} \ = \ T_0+ \mathrm{TI}_{\varPi^1_n}^{-}, \end{align*}$$ where T is $T_0$ augmented with full induction, and $\mathrm {TI}_{\varPi ^1_n}^{-}$ denotes the schema of transfinite induction up (...)
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  20. The Logic of Hyperlogic. Part B: Extensions and Restrictions.Alexander W. Kocurek - 2022 - Review of Symbolic Logic:1-28.
    This is the second part of a two-part series on the logic of hyperlogic, a formal system for regimenting metalogical claims in the object language (even within embedded environments). Part A provided a minimal logic for hyperlogic that is sound and complete over the class of all models. In this part, we extend these completeness results to stronger logics that are sound and complete over restricted classes of models. We also investigate the logic of hyperlogic when the language is enriched (...)
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  21. MASS SURVEILLANCE, BEHAVIOURAL CONTROL, AND PSYCHOLOGICAL COERCION THE MORAL ETHICAL RISKS IN COMMERCIAL DEVICES.Yang Immanuel Pachankis - 2022 - In David C. Wyld & Dhinaharan Nagamalai (eds.), Computer Science and Information Technology. pp. 151-168.
    The research observed, in parallel and comparatively, a surveillance state’s use of communication & cyber networks with satellite applications for power political & realpolitik purposes, in contrast to the outer space security & legit scientific purpose driven cybernetics. The research adopted a psychoanalytic & psychosocial method of observation in the organizational behaviors of the surveillance state, and a theoretical physics, astrochemical, & cosmological feedback method in the contrast group of cybernetics. Military sociology and multilateral movements were adopted in the diagnostic (...)
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  22. Higher-order logic as metaphysics.Jeremy Goodman - 2024 - In Peter Fritz & Nicholas K. Jones (eds.), Higher-Order Metaphysics. Oxford University Press.
    This chapter offers an opinionated introduction to higher-order formal languages with an eye towards their applications in metaphysics. A simply relationally typed higher-order language is introduced in four stages: starting with first-order logic, adding first-order predicate abstraction, generalizing to higher-order predicate abstraction, and finally adding higher-order quantification. It is argued that both β-conversion and Universal Instantiation are valid on the intended interpretation of this language. Given these two principles, it is then shown how we can use pure higher-order logic to (...)
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  23. A Case For Higher-Order Metaphysics.Andrew Bacon - 2024 - In Peter Fritz & Nicholas K. Jones (eds.), Higher-Order Metaphysics. Oxford University Press.
    Higher-order logic augments first-order logic with devices that let us generalize into grammatical positions other than that of a singular term. Some recent metaphysicians have advocated for using these devices to raise and answer questions that bear on many traditional issues in philosophy. In contrast to these 'higher-order metaphysicians', traditional metaphysics has often focused on parallel, but importantly different, questions concerning special sorts of abstract objects: propositions, properties and relations. The answers to the higher-order and the property-theoretic questions may coincide (...)
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  24. Neo-Fregeanism and the Burali-Forti Paradox.Ian Rumfitt - 2018 - In Ivette Fred Rivera & Jessica Leech (eds.), Being Necessary: Themes of Ontology and Modality from the Work of Bob Hale. Oxford, England: Oxford University Press. pp. 188-223.
    Philip Jourdain put this question to Frege in a letter of 28 January 1909. Frege had, indeed, next to nothing to say about ordinals, and in this respect Bob Hale has followed the master. As I hope this chapter will show, though, the topic is worth addressing. The natural abstraction principle for ordinals combines with full, impredicative second-order logic to engender a contradiction, the so-called Burali-Forti Paradox. I shall contend that the best solution involves a retreat to a predicative logic. (...)
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  25. Higher-Order Logic and Disquotational Truth.Lavinia Picollo & Thomas Schindler - 2022 - Journal of Philosophical Logic 51 (4):879-918.
    Truth predicates are widely believed to be capable of serving a certain logical or quasi-logical function. There is little consensus, however, on the exact nature of this function. We offer a series of formal results in support of the thesis that disquotational truth is a device to simulate higher-order resources in a first-order setting. More specifically, we show that any theory formulated in a higher-order language can be naturally and conservatively interpreted in a first-order theory with a disquotational truth or (...)
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  26. Essence and Necessity.Andreas Ditter - 2022 - Journal of Philosophical Logic 51 (3):653-690.
    What is the relation between metaphysical necessity and essence? This paper defends the view that the relation is one of identity: metaphysical necessity is a special case of essence. My argument consists in showing that the best joint theory of essence and metaphysical necessity is one in which metaphysical necessity is just a special case of essence. The argument is made against the backdrop of a novel, higher-order logic of essence, whose core features are introduced in the first part of (...)
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  27. A Theory of Necessities.Andrew Bacon & Jin Zeng - 2022 - Journal of Philosophical Logic 51 (1):151-199.
    We develop a theory of necessity operators within a version of higher-order logic that is neutral about how fine-grained reality is. The theory is axiomatized in terms of the primitive of *being a necessity*, and we show how the central notions in the philosophy of modality can be recovered from it. Various questions are formulated and settled within the framework, including questions about the ordering of necessities under strength, the existence of broadest necessities satisfying various logical conditions, and questions about (...)
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  28. Engineering Existence?Lukas Skiba - forthcoming - Inquiry: An Interdisciplinary Journal of Philosophy.
    This paper investigates the connection between two recent trends in philosophy: higher-orderism and conceptual engineering. Higher-orderists use higher-order quantifiers (in particular quantifiers binding variables that occupy the syntactic positions of predicates) to express certain key metaphysical doctrines, such as the claim that there are properties. I argue that, on a natural construal, the higher-orderist approach involves an engineering project concerning, among others, the concept of existence. I distinguish between a modest construal of this project, on which it aims at engineering (...)
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  29. Higher‐order metaphysics.Lukas Skiba - 2021 - Philosophy Compass 16 (10):1-11.
    Subverting a once widely held Quinean paradigm, there is a growing consensus among philosophers of logic that higher-order quantifiers (which bind variables in the syntactic position of predicates and sentences) are a perfectly legitimate and useful instrument in the logico-philosophical toolbox, while neither being reducible to nor fully explicable in terms of first-order quantifiers (which bind variables in singular term position). This article discusses the impact of this quantificational paradigm shift on metaphysics, focussing on theories of properties, propositions, and identity, (...)
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  30. Logic Works: A Rigorous Introduction to Formal Logic.Lorne Falkenstein, Scott Stapleford & Molly Kao - 2021 - New York: Routledge. Edited by Scott Stapleford & Molly Kao.
    Logic Works is a critical and extensive introduction to logic. It asks questions about why systems of logic are as they are, how they relate to ordinary language and ordinary reasoning, and what alternatives there might be to classical logical doctrines. It considers how logical analysis can be applied to carefully represent the reasoning employed in academic and scientific work, better understand that reasoning, and identify its hidden premises. Aiming to be as much a reference work and handbook for further, (...)
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  31. Modes of Truth: The Unified Approach to Truth, Modality, and Paradox.Carlo Nicolai & Johannes Stern (eds.) - 2021 - New York, NY: Routledge.
    The aim of this volume is to open up new perspectives and to raise new research questions about a unified approach to truth, modalities, and propositional attitudes. The volume's essays are grouped thematically around different research questions. The first theme concerns the tension between the theoretical role of the truth predicate in semantics and its expressive function in language. The second theme of the volume concerns the interaction of truth with modal and doxastic notions. The third theme covers higher-order solutions (...)
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  32. Inner models from extended logics: Part 1.Juliette Kennedy, Menachem Magidor & Jouko Väänänen - 2020 - Journal of Mathematical Logic 21 (2):2150012.
    If we replace first-order logic by second-order logic in the original definition of Gödel’s inner model L, we obtain the inner model of hereditarily ordinal definable sets [33]. In this paper...
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  33. A characterization of Σ 1 1 -reflecting ordinals.J. P. Aguilera - 2021 - Annals of Pure and Applied Logic 172 (10):103009.
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  34. The Logic of Logical Necessity.Andrew Bacon & Kit Fine - manuscript
    Prior to Kripke's seminal work on the semantics of modal logic, McKinsey offered an alternative interpretation of the necessity operator, inspired by the Bolzano-Tarski notion of logical truth. According to this interpretation, `it is necessary that A' is true just in case every sentence with the same logical form as A is true. In our paper, we investigate this interpretation of the modal operator, resolving some technical questions, and relating it to the logical interpretation of modality and some views in (...)
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  35. Logical Combinatorialism.Andrew Bacon - 2020 - Philosophical Review 129 (4):537-589.
    In explaining the notion of a fundamental property or relation, metaphysicians will often draw an analogy with languages. The fundamental properties and relations stand to reality as the primitive predicates and relations stand to a language: the smallest set of vocabulary God would need in order to write the “book of the world.” This paper attempts to make good on this metaphor. To that end, a modality is introduced that, put informally, stands to propositions as logical truth stands to sentences. (...)
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  36. Higher-order metaphysics and the tropes versus universals dispute.Lukas Skiba - 2021 - Philosophical Studies 178 (9):2805-2827.
    Higher-order realists about properties express their view that there are properties with the help of higher-order rather than first-order quantifiers. They claim two types of advantages for this way of formulating property realism. First, certain gridlocked debates about the nature of properties, such as the immanentism versus transcendentalism dispute, are taken to be dissolved. Second, a further such debate, the tropes versus universals dispute, is taken to be resolved. In this paper I first argue that higher-order realism does not in (...)
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  37. The FAN principle and weak König's lemma in herbrandized second-order arithmetic.Fernando Ferreira - 2020 - Annals of Pure and Applied Logic 171 (9):102843.
    We introduce a herbrandized functional interpretation of a first-order semi-intuitionistic extension of Heyting Arithmetic and study its main properties. We then extend the interpretation to a certain system of second-order arithmetic which includes a (classically false) formulation of the FAN principle and weak König's lemma. It is shown that any first-order formula provable in this system is classically true. It is perhaps worthy of note that, in our interpretation, second-order variables are interpreted by finite sets of natural numbers.
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  38. The placeholder view of assumptions and the Curry–Howard correspondence.Ivo Pezlar - 2020 - Synthese (11):1-17.
    Proofs from assumptions are amongst the most fundamental reasoning techniques. Yet the precise nature of assumptions is still an open topic. One of the most prominent conceptions is the placeholder view of assumptions generally associated with natural deduction for intuitionistic propositional logic. It views assumptions essentially as holes in proofs, either to be filled with closed proofs of the corresponding propositions via substitution or withdrawn as a side effect of some rule, thus in effect making them an auxiliary notion subservient (...)
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  39. Randomness notions and reverse mathematics.André Nies & Paul Shafer - 2020 - Journal of Symbolic Logic 85 (1):271-299.
    We investigate the strength of a randomness notion ${\cal R}$ as a set-existence principle in second-order arithmetic: for each Z there is an X that is ${\cal R}$-random relative to Z. We show that the equivalence between 2-randomness and being infinitely often C-incompressible is provable in $RC{A_0}$. We verify that $RC{A_0}$ proves the basic implications among randomness notions: 2-random $\Rightarrow$ weakly 2-random $\Rightarrow$ Martin-Löf random $\Rightarrow$ computably random $\Rightarrow$ Schnorr random. Also, over $RC{A_0}$ the existence of computable randoms is equivalent (...)
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  40. Mechanizing principia logico-metaphysica in functional type-theory.Daniel Kirchner, Christoph Benzmüller & Edward N. Zalta - 2018 - Review of Symbolic Logic 13 (1):206-218.
    Principia Logico-Metaphysica contains a foundational logical theory for metaphysics, mathematics, and the sciences. It includes a canonical development of Abstract Object Theory [AOT], a metaphysical theory that distinguishes between ordinary and abstract objects.This article reports on recent work in which AOT has been successfully represented and partly automated in the proof assistant system Isabelle/HOL. Initial experiments within this framework reveal a crucial but overlooked fact: a deeply-rooted and known paradox is reintroduced in AOT when the logic of complex terms is (...)
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  41. Elementary inductive dichotomy: Separation of open and clopen determinacies with infinite alternatives.Kentaro Sato - 2020 - Annals of Pure and Applied Logic 171 (3):102754.
    We introduce a new axiom called inductive dichotomy, a weak variant of the axiom of inductive definition, and analyze the relationships with other variants of inductive definition and with related axioms, in the general second order framework, including second order arithmetic, second order set theory and higher order arithmetic. By applying these results to the investigations on the determinacy axioms, we show the following. (i) Clopen determinacy is consistency-wise strictly weaker than open determinacy in these frameworks, except second order arithmetic; (...)
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  42. Substitution Structures.Andrew Bacon - 2019 - Journal of Philosophical Logic 48 (6):1017-1075.
    An increasing amount of twenty-first century metaphysics is couched in explicitly hyperintensional terms. A prerequisite of hyperintensional metaphysics is that reality itself be hyperintensional: at the metaphysical level, propositions, properties, operators, and other elements of the type hierarchy, must be more fine-grained than functions from possible worlds to extensions. In this paper I develop, in the setting of type theory, a general framework for reasoning about the granularity of propositions and properties. The theory takes as primitive the notion of a (...)
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  43. Radical Anti‐Disquotationalism.Andrew Bacon - 2018 - Philosophical Perspectives 32 (1):41-107.
    A number of `no-proposition' approaches to the liar paradox find themselves implicitly committed to a moderate disquotational principle: the principle that if an utterance of the sentence `$P$' says anything at all, it says that $P$ (with suitable restrictions). I show that this principle alone is responsible for the revenge paradoxes that plague this view. I instead propose a view in which there are several closely related language-world relations playing the `semantic expressing' role, none of which is more central to (...)
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  44. A model of second-order arithmetic satisfying AC but not DC.Sy-David Friedman, Victoria Gitman & Vladimir Kanovei - 2019 - Journal of Mathematical Logic 19 (1):1850013.
    We show that there is a [Formula: see text]-model of second-order arithmetic in which the choice scheme holds, but the dependent choice scheme fails for a [Formula: see text]-assertion, confirming a conjecture of Stephen Simpson. We obtain as a corollary that the Reflection Principle, stating that every formula reflects to a transitive set, can fail in models of [Formula: see text]. This work is a rediscovery by the first two authors of a result obtained by the third author in [V. (...)
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  45. On the mathematical and foundational significance of the uncountable.Dag Normann & Sam Sanders - 2019 - Journal of Mathematical Logic 19 (1):1950001.
    We study the logical and computational properties of basic theorems of uncountable mathematics, including the Cousin and Lindelöf lemma published in 1895 and 1903. Historically, these lemmas were among the first formulations of open-cover compactness and the Lindelöf property, respectively. These notions are of great conceptual importance: the former is commonly viewed as a way of treating uncountable sets like e.g. [Formula: see text] as “almost finite”, while the latter allows one to treat uncountable sets like e.g. [Formula: see text] (...)
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  46. Minimum models of second-order set theories.Kameryn J. Williams - 2019 - Journal of Symbolic Logic 84 (2):589-620.
    In this article I investigate the phenomenon of minimum and minimal models of second-order set theories, focusing on Kelley–Morse set theory KM, Gödel–Bernays set theory GB, and GB augmented with the principle of Elementary Transfinite Recursion. The main results are the following. (1) A countable model of ZFC has a minimum GBC-realization if and only if it admits a parametrically definable global well order. (2) Countable models of GBC admit minimal extensions with the same sets. (3) There is no minimum (...)
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  47. M. R. Krom. Separation principles in the hierarchy theory of pure first-order logic. The journal of symbolic logic, vol. 28 no. 3 , pp. 222–236.D. A. Clarke - 1966 - Journal of Symbolic Logic 31 (3):503.
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  48. Angelo Margaris. First order mathematical logic. Blaisdell Publishing Company, Waltham, Massachusetts, Toronto, and London, 1967, x + 211 pp. [REVIEW]A. H. Lightstone - 1972 - Journal of Symbolic Logic 37 (3):616.
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  49. Dag Prawitz. Hauptsatz for higher order logic. The journal of symbolic logic, Bd. 33 , S. 452–457. - Dag Prawitz. Completeness and Hauptsatz for second order logic. Theoria , Bd. 33 , S. 246–258. - Moto-o Takahashi. A proof of cut-elimination in simple type-theory. Journal of the Mathematical Society of Japan, Bd. 19 , S. 399–410. [REVIEW]K. Schutte - 1974 - Journal of Symbolic Logic 39 (3):607-607.
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  50. Saharon Shelah. There are just four second-order quantifiers. Israel journal of mathematics, vol. 15 , pp. 282–300.John T. Baldwin - 1986 - Journal of Symbolic Logic 51 (1):234.
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