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  1. Proof Theory and Meaning: On Second Order Logic.Greg Restall - manuscript
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  2. Overgeneration in the Higher Infinite.Salvatore Florio & Luca Incurvati - forthcoming - In Gil Sagi & Jack Woods (eds.), The Semantic Conception of Logic: Essays on Consequence, Invariance, and Meaning. Cambridge: Cambridge University Press.
    The Overgeneration Argument is a prominent objection against the model-theoretic account of logical consequence for second-order languages. In previous work we have offered a reconstruction of this argument which locates its source in the conflict between the neutrality of second-order logic and its alleged entanglement with mathematics. Some cases of this conflict concern small large cardinals. In this article, we show that in these cases the conflict can be resolved by moving from a set-theoretic implementation of the model-theoretic account to (...)
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  3. The Proper Treatment of Identity in Dialetheic Metaphysics.Nicholas K. Jones - forthcoming - The Philosophical Quarterly.
    According to one prominent strand of mainstream logic and metaphysics, identity is indistinguishability. Priest has recently argued that this permits counterexamples to the transitivity and substitutivity of identity within dialetheic metaphysics, even in paradigmatically extensional contexts. This paper investigates two alternative regimentations of indistinguishability. Although classically equivalent to the standard regimentation on which Priest focuses, these alternatives are strictly stronger than it in dialetheic settings. Both regimentations are transitive, and one satisfies substitutivity. It is argued that both regimentations provide better (...)
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  4. Themes From Barcan Marcus.Timothy Williamson - forthcoming - Lauener Library of Analytical Philosophy, Vol. 3.
  5. Unrestricted Quantification and the Structure of Type Theory.Nicholas K. Jones & Salvatore Florio - 2021 - Philosophy and Phenomenological Research 102 (1):44-64.
    Semantic theories based on a hierarchy of types have prominently been used to defend the possibility of unrestricted quantification. However, they also pose a prima facie problem for it: each quantifier ranges over at most one level of the hierarchy and is therefore not unrestricted. It is difficult to evaluate this problem without a principled account of what it is for a quantifier to be unrestricted. Drawing on an insight of Russell’s about the relationship between quantification and the structure of (...)
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  6. A Patch to the Possibility Part of Gödel’s Ontological Proof.Johan E. Gustafsson - 2020 - Analysis 80 (2):229-240.
    Kurt Gödel’s version of the Ontological Proof derives rather than assumes the crucial Possibility Claim: the claim that it is possible that something God-like exists. Gödel’s derivation starts off with a proof of the Possible Instantiation of the Positive: the principle that, if a property is positive, it is possible that there exists something that has that property. I argue that Gödel’s proof of this principle relies on some implausible axiological assumptions but it can be patched so that it only (...)
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  7. Neo-Logicism and Its Logic.Panu Raatikainen - 2020 - History and Philosophy of Logic 41 (1):82-95.
    The rather unrestrained use of second-order logic in the neo-logicist program is critically examined. It is argued in some detail that it brings with it genuine set-theoretical existence assumptions and that the mathematical power that Hume’s Principle seems to provide, in the derivation of Frege’s Theorem, comes largely from the ‘logic’ assumed rather than from Hume’s Principle. It is shown that Hume’s Principle is in reality not stronger than the very weak Robinson Arithmetic Q. Consequently, only a few rudimentary facts (...)
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  8. Metalogic and the Overgeneration Argument.Salvatore Florio & Luca Incurvati - 2019 - Mind 128 (511):761-793.
    A prominent objection against the logicality of second-order logic is the so-called Overgeneration Argument. However, it is far from clear how this argument is to be understood. In the first part of the article, we examine the argument and locate its main source, namely, the alleged entanglement of second-order logic and mathematics. We then identify various reasons why the entanglement may be thought to be problematic. In the second part of the article, we take a metatheoretic perspective on the matter. (...)
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  9. The Innocence of Truth in Semantic Paradox.Eric Guindon - 2019 - Erkenntnis 86 (1):71-93.
    According to some philosophers, the Liar paradox arises because of a mistaken theory of truth. Its lesson is that we must reject some instances of the naive propositional truth-schema \It is true that \ if and only if \\. In this paper, I construct a novel semantic paradox in which no principle even analogous to the truth-schema plays any role. I argue that this undermines the claim that we ought to respond to the Liar by revising our theory of truth.
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  10. A Note on Algebraic Semantics for S5 with Propositional Quantifiers.Wesley H. Holliday - 2019 - Notre Dame Journal of Formal Logic 60 (2):311-332.
    In two of the earliest papers on extending modal logic with propositional quantifiers, R. A. Bull and K. Fine studied a modal logic S5Π extending S5 with axioms and rules for propositional quantification. Surprisingly, there seems to have been no proof in the literature of the completeness of S5Π with respect to its most natural algebraic semantics, with propositional quantifiers interpreted by meets and joins over all elements in a complete Boolean algebra. In this note, we give such a proof. (...)
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  11. A Poetics of Designing.Claudia Westermann - 2019 - In Thomas Fischer & Christiane M. Herr (eds.), Design Cybernetics: Navigating the New. Basel, Switzerland: Springer. pp. 233-245.
    The chapter provides an overview on what it means to be in a world that is uncertain, e.g., how under conditions of limited understanding any activity is an activity that designs and constructs, and how designing objects, spaces, and situations relates to the (designed) meta-world of second-order cybernetics. Designers require a framework that is open, but one that supplies ethical guidance when ‘constructing’ something new. Relating second-order design thinking to insights in philosophy and aesthetics, the chapter argues that second-order cybernetics (...)
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  12. Logicism, Ontology, and the Epistemology of Second-Order Logic.Richard Kimberly Heck - 2018 - In Ivette Fred-Rivera & Jessica Leach (eds.), Being Necessary: Themes of Ontology and Modality from the Work of Bob Hale. Oxford: Oxford University Press. pp. 140-169.
    In two recent papers, Bob Hale has attempted to free second-order logic of the 'staggering existential assumptions' with which Quine famously attempted to saddle it. I argue, first, that the ontological issue is at best secondary: the crucial issue about second-order logic, at least for a neo-logicist, is epistemological. I then argue that neither Crispin Wright's attempt to characterize a `neutralist' conception of quantification that is wholly independent of existential commitment, nor Hale's attempt to characterize the second-order domain in terms (...)
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  13. On Delight: Thoughts for Tomorrow.Claudia Westermann - 2018 - Technoetic Arts 16 (1):43-51.
    The article introduces the problematics of the classical two-valued logic on which Western thought is generally based, outlining that under the conditions of its logical assumptions the subject I is situated in a world that it cannot address. In this context, the article outlines a short history of cybernetics and the shift from first- to second-order cybernetics. The basic principles of Gordon Pask’s 1976 Conversation Theory are introduced. It is argued that this second-order theory grants agency to others through a (...)
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  14. What Russell Should Have Said to Burali–Forti.Salvatore Florio & Graham Leach-Krouse - 2017 - Review of Symbolic Logic 10 (4):682-718.
    The paradox that appears under Burali-Forti’s name in many textbooks of set theory is a clever piece of reasoning leading to an unproblematic theorem. The theorem asserts that the ordinals do not form a set. For such a set would be—absurdly—an ordinal greater than any ordinal in the set of all ordinals. In this article, we argue that the paradox of Burali-Forti is first and foremost a problem about concept formation by abstraction, not about sets. We contend, furthermore, that some (...)
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  15. Second-Order Modal Logic.Andrew Parisi - 2017 - Dissertation, University of Connecticut
    This dissertation develops an inferentialist theory of meaning. It takes as a starting point that the sense of a sentence is determined by the rules governing its use. In particular, there are two features of the use of a sentence that jointly determine its sense, the conditions under which it is coherent to assert that sentence and the conditions under which it is coherent to deny that sentence. From this starting point the dissertation develops a theory of quantification as marking (...)
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  16. 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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  17. Plurals and Modals.Øystein Linnebo - 2016 - Canadian Journal of Philosophy 46 (4-5):654-676.
    Consider one of several things. Is the one thing necessarily one of the several? This key question in the modal logic of plurals is clarified. Some defenses of an affirmative answer are developed and compared. Various remarks are made about the broader philosophical significance of the question.
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  18. Reply to Fritz.Timothy Williamson - 2016 - Canadian Journal of Philosophy 46 (4-5):610-612.
  19. Reply to Goodman.Timothy Williamson - 2016 - Canadian Journal of Philosophy 46 (4-5):640-653.
  20. 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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  21. Somehow Things Do Not Relate: On the Interpretation of Polyadic Second-Order Logic.Marcus Rossberg - 2015 - Journal of Philosophical Logic 44 (3):341-350.
    Boolos has suggested a plural interpretation of second-order logic for two purposes: to escape Quine’s allegation that second-order logic is set theory in disguise, and to avoid the paradoxes arising if the second-order variables are given a set-theoretic interpretation in second-order set theory. Since the plural interpretation accounts only for monadic second-order logic, Rayo and Yablo suggest an new interpretation for polyadic second-order logic in a Boolosian spirit. The present paper argues that Rayo and Yablo’s interpretation does not achieve the (...)
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  22. Barcan Formulas in Second-Order Modal Logic.Timothy Williamson - 2015 - In Michael Frauchiger (ed.), Themes From Barcan Marcus. Ontos Verlag. pp. 51-74.
    Second-order logic and modal logic are both, separately, major topics of philosophical discussion. Although both have been criticized by Quine and others, increasingly many philosophers find their strictures uncompelling, and regard both branches of logic as valuable resources for the articulation and investigation of significant issues in logical metaphysics and elsewhere. One might therefore expect some combination of the two sorts of logic to constitute a natural and more comprehensive background logic for metaphysics. So it is somewhat surprising to find (...)
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  23. Second-Order Arithmetic Sans Sets.L. Berk - 2013 - Philosophia Mathematica 21 (3):339-350.
    This paper examines the ontological commitments of the second-order language of arithmetic and argues that they do not extend beyond the first-order language. Then, building on an argument by George Boolos, we develop a Tarski-style definition of a truth predicate for the second-order language of arithmetic that does not involve the assignment of sets to second-order variables but rather uses the same class of assignments standardly used in a definition for the first-order language.
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  24. Modalising Plurals.Simon Thomas Hewitt - 2012 - Journal of Philosophical Logic 41 (5):853-875.
    There has been very little discussion of the appropriate principles to govern a modal logic of plurals. What debate there has been has accepted a principle I call (Necinc); informally if this is one of those then, necessarily: this is one of those. On this basis Williamson has criticised the Boolosian plural interpretation of monadic second-order logic. I argue against (Necinc), noting that it isn't a theorem of any logic resulting from adding modal axioms to the plural logic PFO+, and (...)
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  25. Second Order Logic or Set Theory?Jouko Väänänen - 2012 - Bulletin of Symbolic Logic 18 (1):91-121.
    We try to answer the question which is the “right” foundation of mathematics, second order logic or set theory. Since the former is usually thought of as a formal language and the latter as a first order theory, we have to rephrase the question. We formulate what we call the second order view and a competing set theory view, and then discuss the merits of both views. On the surface these two views seem to be in manifest conflict with each (...)
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  26. A Logic for Frege's Theorem.Richard Heck - 2011 - In Frege’s Theorem: An Introduction. Oxford University Press.
    It has been known for a few years that no more than Pi-1-1 comprehension is needed for the proof of "Frege's Theorem". One can at least imagine a view that would regard Pi-1-1 comprehension axioms as logical truths but deny that status to any that are more complex—a view that would, in particular, deny that full second-order logic deserves the name. Such a view would serve the purposes of neo-logicists. It is, in fact, no part of my view that, say, (...)
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  27. The Reality of Mathematics and the Case of Set Theory.Daniel Isaacson - 2011 - In Zsolt Novak & Andras Simonyi (eds.), Truth, Reference and Realism. Budapest: Central European University Press. pp. 1-76.
  28. Resonances of the Unknown.Claudia Westermann - 2011 - Kybernetes 40 (7/8):1189-1195.
    Purpose – The purpose of this paper is to discuss the relevance of second-order cybernetics for a theory of architectural design and related discourse. -/- Design/methodology/approach – First, the relation of architectural design to the concept of “poiesis” is clarified. Subsequently, selected findings of Gotthard Günther are revisited and related to an architectural poetics. The last part of the paper consists of revisiting ideas mentioned previously, however, on the level of a discourse that has incorporated the ideas and offers a (...)
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  29. A Defense of Second-Order Logic.Otávio Bueno - 2010 - Axiomathes 20 (2-3):365-383.
    Second-order logic has a number of attractive features, in particular the strong expressive resources it offers, and the possibility of articulating categorical mathematical theories (such as arithmetic and analysis). But it also has its costs. Five major charges have been launched against second-order logic: (1) It is not axiomatizable; as opposed to first-order logic, it is inherently incomplete. (2) It also has several semantics, and there is no criterion to choose between them (Putnam, J Symbol Logic 45:464–482, 1980 ). Therefore, (...)
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  30. Pure Second-Order Logic with Second-Order Identity.Alexander Paseau - 2010 - Notre Dame Journal of Formal Logic 51 (3):351-360.
    Pure second-order logic is second-order logic without functional or first-order variables. In "Pure Second-Order Logic," Denyer shows that pure second-order logic is compact and that its notion of logical truth is decidable. However, his argument does not extend to pure second-order logic with second-order identity. We give a more general argument, based on elimination of quantifiers, which shows that any formula of pure second-order logic with second-order identity is equivalent to a member of a circumscribed class of formulas. As a (...)
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  31. When Best Theories Go Bad.David Manley - 2009 - Philosophy and Phenomenological Research 78 (2):392-405.
    It is common for contemporary metaphysical realists to adopt Quine's criterion of ontological commitment while at the same time repudiating his ontological pragmatism. 2 Drawing heavily from the work of others—especially Joseph Melia and Stephen Yablo—I will argue that the resulting approach to meta-ontology is unstable. In particular, if we are metaphysical realists, we need not accept ontological commitment to whatever is quantified over by our best first-order theories.
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  32. Second-Order and Higher-Order Logic.Herbert B. Enderton - 2008 - Stanford Encyclopedia of Philosophy.
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  33. Plural Quantification.Ø Linnebo - 2008 - Stanford Encyclopedia of Philosophy.
    Ordinary English contains different forms of quantification over objects. In addition to the usual singular quantification, as in 'There is an apple on the table', there is plural quantification, as in 'There are some apples on the table'. Ever since Frege, formal logic has favored the two singular quantifiers ∀x and ∃x over their plural counterparts ∀xx and ∃xx (to be read as for any things xx and there are some things xx). But in recent decades it has been argued (...)
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  34. Second-Order Quantifier Elimination in Higher-Order Contexts with Applications to the Semantical Analysis of Conditionals.Dov M. Gabbay & Andrzej Szałas - 2007 - Studia Logica 87 (1):37-50.
    Second-order quantifier elimination in the context of classical logic emerged as a powerful technique in many applications, including the correspondence theory, relational databases, deductive and knowledge databases, knowledge representation, commonsense reasoning and approximate reasoning. In the current paper we first generalize the result of Nonnengart and Szałas [17] by allowing second-order variables to appear within higher-order contexts. Then we focus on a semantical analysis of conditionals, using the introduced technique and Gabbay’s semantics provided in [10] and substantially using a third-order (...)
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  35. Burgess on Plural Logic and Set Theory.Øystein Linnebo - 2007 - Philosophia Mathematica 15 (1):79-93.
    John Burgess in a 2004 paper combined plural logic and a new version of the idea of limitation of size to give an elegant motivation of the axioms of ZFC set theory. His proposal is meant to improve on earlier work by Paul Bernays in two ways. I argue that both attempted improvements fail. I am grateful to Philip Welch, two anonymous referees, and especially Ignacio Jané for written comments on earlier versions of this paper, which have led to substantial (...)
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  36. Neutrosophy in Arabic Philosophy (Arabic version).Salah Osman & Florentin Smarandache - 2007 - Alexandria, Egypt: Al Maaref Establishment Press.
    لأننا نعيش في عالم يكتنفه الغموض من كل جانب؛ عالم تتسم معرفتنا لأحداثه ووقائعه بالتناقض واللاتحديد، وتُفصح قضايانا اللغوية الواصفة له عن الصدق تارة وعن الكذب تارة أخرى، فنحن في حاجة إلى فلسفة جديدة تعكس حقيقة رؤيتنا النسبية لهذا العالم وقصور معرفتنا به؛ ونحن في حاجة إلى نسقٍِ منطقي يُلائم معطياته غير المكتملة ويُشبع معالجاتنا لها، سواء على مستوى ممارسات الحياة اليومية أو على مستوى الممارسة العلمية بمختلف أشكالها. والفلسفة التي يقترحها هذا الكتاب هي «النيوتروسوفيا»؛ تلك النظرية التي قدمها الفيلسوف (...)
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  37. On Quantifying Into Predicate Position: Steps Towards a New (Tralist) Perspective.Crispin Wright - 2007 - In Mary Leng, Alexander Paseau & Michael Potter (eds.), Mathematical Knowledge. Oxford University Press. pp. 150--74.
  38. Expressivity of Second Order Propositional Modal Logic.Balder ten Cate - 2006 - Journal of Philosophical Logic 35 (2):209-223.
    We consider second-order propositional modal logic (SOPML), an extension of the basic modal language with propositional quantifiers introduced by Kit Fine in 1970. We determine the precise expressive power of SOPML by giving analogues of the Van Benthem–Rosen theorem and the Goldblatt Thomason theorem. Furthermore, we show that the basic modal language is the bisimulation invariant fragment of SOPML, and we characterize the bounded fragment of first-order logic as being the intersection of first-order logic and SOPML.
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  39. On Spectra of Sentences of Monadic Second Order Logic with Counting.E. Fischer & J. A. Makowsky - 2004 - Journal of Symbolic Logic 69 (3):617-640.
    We show that the spectrum of a sentence ϕ in Counting Monadic Second Order Logic (CMSOL) using one binary relation symbol and finitely many unary relation symbols, is ultimately periodic, provided all the models of ϕ are of clique width at most k, for some fixed k. We prove a similar statement for arbitrary finite relational vocabularies τ and a variant of clique width for τ-structures. This includes the cases where the models of ϕ are of tree width at most (...)
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  40. First Order Quantifiers in Monadic Second Order Logic.H. Jerome Keisler & Wafik Boulos Lotfallah - 2004 - Journal of Symbolic Logic 69 (1):118-136.
    This paper studies the expressive power that an extra first order quantifier adds to a fragment of monadic second order logic, extending the toolkit of Janin and Marcinkowski [JM01].We introduce an operation existsn on properties S that says "there are n components having S". We use this operation to show that under natural strictness conditions, adding a first order quantifier word u to the beginning of a prefix class V increases the expressive power monotonically in u. As a corollary, if (...)
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  41. Querying Linguistic Treebanks with Monadic Second-Order Logic in Linear Time.Stephan Kepser - 2004 - Journal of Logic, Language and Information 13 (4):457-470.
    In recent years large amounts of electronic texts have become available. While the first of these corpora had only a low level of annotation, the more recent ones are annotated with refined syntactic information. To make these rich annotations accessible for linguists, the development of query systems has become an important goal. One of the main difficulties in this task consists in the choice of the right query language, a language which at the same time should be powerful enough to (...)
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  42. First-Order Logic, Second-Order Logic, and Completeness.Marcus Rossberg - 2004 - In Vincent Hendricks, Fabian Neuhaus, Stig Andur Pedersen, Uwe Scheffler & Heinrich Wansing (eds.), First-Order Logic Revisited. Logos. pp. 303-321.
    This paper investigates the claim that the second-order consequence relation is intractable because of the incompleteness result for SOL. The opponents’ claim is that SOL cannot be proper logic since it does not have a complete deductive system. I argue that the lack of a completeness theorem, despite being an interesting result, cannot be held against the status of SOL as a proper logic.
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  43. Spectra of Formulae with Henkin Quantifiers.Joanna Golinska-Pilarek & Konrad Zdanowski - 2003 - In A. Rojszczak, J. Cachro & G. Kurczewski (eds.), Philosophical Dimensions of Logic and Science. Kluwer Academic Publishers. pp. 29-45.
    It is known that various complexity-theoretical problems can be translated into some special spectra problems. Thus, questions about complexity classes are translated into questions about the expressive power of some languages. In this paper we investigate the spectra of some logics with Henkin quantifiers in the empty vocabulary.
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  44. Plural Quantification Exposed.Øystein Linnebo - 2003 - Noûs 37 (1):71–92.
    This paper criticizes George Boolos's famous use of plural quantification to argue that monadic second-order logic is pure logic. I deny that plural quantification qualifies as pure logic and express serious misgivings about its alleged ontological innocence. My argument is based on an examination of what is involved in our understanding of the impredicative plural comprehension schema.
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  45. Speaking with Shadows: A Study of Neo‐Logicism.Fraser MacBride - 2003 - British Journal for the Philosophy of Science 54 (1):103-163.
    According to the species of neo-logicism advanced by Hale and Wright, mathematical knowledge is essentially logical knowledge. Their view is found to be best understood as a set of related though independent theses: (1) neo-fregeanism-a general conception of the relation between language and reality; (2) the method of abstraction-a particular method for introducing concepts into language; (3) the scope of logic-second-order logic is logic. The criticisms of Boolos, Dummett, Field and Quine (amongst others) of these theses are explicated and assessed. (...)
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  46. Strong Normalization of a Symmetric Lambda Calculus for Second-Order Classical Logic.Yoriyuki Yamagata - 2002 - Archive for Mathematical Logic 41 (1):91-99.
  47. Second-Order Logic.John Corcoran - 2001 - In M. Zeleny (ed.), Logic, Meaning, and Computation: Essays in Memory of Alonzo Church. KLUKER. pp. 61–76.
    “Second-order Logic” in Anderson, C.A. and Zeleny, M., Eds. Logic, Meaning, and Computation: Essays in Memory of Alonzo Church. Dordrecht: Kluwer, 2001. Pp. 61–76. -/- Abstract. This expository article focuses on the fundamental differences between second- order logic and first-order logic. It is written entirely in ordinary English without logical symbols. It employs second-order propositions and second-order reasoning in a natural way to illustrate the fact that second-order logic is actually a familiar part of our traditional intuitive logical framework and (...)
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  48. Nominalism Through de-Nominalization.Agustín Rayo & Stephen Yablo - 2001 - Noûs 35 (1):74–92.
  49. Classical Logic II: Higher-Order Logic.Stewart Shapiro - 2001 - In Lou Goble (ed.), The Blackwell Guide to Philosophical Logic. Blackwell. pp. 33--54.
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  50. Second-Order Logic and Foundations of Mathematics.Jouko Väänänen - 2001 - Bulletin of Symbolic Logic 7 (4):504-520.
    We discuss the differences between first-order set theory and second-order logic as a foundation for mathematics. We analyse these languages in terms of two levels of formalization. The analysis shows that if second-order logic is understood in its full semantics capable of characterizing categorically central mathematical concepts, it relies entirely on informal reasoning. On the other hand, if it is given a weak semantics, it loses its power in expressing concepts categorically. First-order set theory and second-order logic are not radically (...)
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