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  1. Erik Aarts (1994). Proving Theorems of the Second Order Lambek Calculus in Polynomial Time. Studia Logica 53 (3):373 - 387.
    In the Lambek calculus of order 2 we allow only sequents in which the depth of nesting of implications is limited to 2. We prove that the decision problem of provability in the calculus can be solved in time polynomial in the length of the sequent. A normal form for proofs of second order sequents is defined. It is shown that for every proof there is a normal form proof with the same axioms. With this normal form we can give (...)
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  2. Klaus Aehlig (2005). Induction and Inductive Definitions in Fragments of Second Order Arithmetic. Journal of Symbolic Logic 70 (4):1087 - 1107.
    A fragment with the same provably recursive functions as n iterated inductive definitions is obtained by restricting second order arithmetic in the following way. The underlying language allows only up to n + 1 nested second order quantifications and those are in such a way, that no second order variable occurs free in the scope of another second order quantifier. The amount of induction on arithmetical formulae only affects the arithmetical consequences of these theories, whereas adding induction for arbitrary formulae (...)
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  3. Thomas Ågotnes, Wiebe van der Hoek & Michael Wooldridge (2008). Quantified Coalition Logic. Synthese 165 (2):269 - 294.
    We add a limited but useful form of quantification to Coalition Logic, a popular formalism for reasoning about cooperation in game-like multi-agent systems. The basic constructs of Quantified Coalition Logic (QCL) allow us to express such properties as “every coalition satisfying property P can achieve φ” and “there exists a coalition C satisfying property P such that C can achieve φ”. We give an axiomatisation of QCL, and show that while it is no more expressive than Coalition Logic, it is (...)
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  4. John Bacon (1989). A Single Primitive Trope Relation. Journal of Philosophical Logic 18 (2):141 - 154.
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  5. David Ballard (1985). Independence in Higher-Order Subclassical Logic. Notre Dame Journal of Formal Logic 26 (4):444-454.
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  6. Jesús Alcolea Banegas (1992). Instituciones y heterogeindad. Theoria 7 (1/2/3):65-85.
    The paper presents and discusses an example, namely a version of heterogeneous frrst-order logic and uses the classical theorem of Herbrand-Schmidt-Wang about the reduction of heterogeneous first-order logic to homogeneous first-order logic, in order to obtain two transformations between heterogeneous and homogeneous frrst-order logic which are different from the institution morphisms defined by Goguen and Burstall. Moreover, by considering a type of 2-cell among institution morphisms it is obtained a 2-category and also a 2-functor from this to another 2-category.
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  7. Jon Barwise (1977). First-Order Logic. In Jon Barwise & H. Jerome Keisler (eds.), Handbook of Mathematical Logic. North-Holland Pub. Co.
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  8. R. Beneyto (1971). First-Order Logic, de RM Smullyan. Teorema: International Journal of Philosophy 1 (3):136-138.
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  9. Kim Bruce (1989). Review: Johan van Benthem, Kees Doets, Higher-Order Logic. [REVIEW] Journal of Symbolic Logic 54 (3):1090-1092.
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  10. Otavio Bueno & Scott A. Shalkowski (2013). On Second-Order Logic. Noûs 47 (1).
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  11. John Corcoran (1995). Information Recovery Problems. Theoria 10 (3):55-78.
    An information recovery problem is the problem of constructing a proposition containing the information dropped in going from a given premise to a given conclusion that folIows. The proposition(s) to beconstructed can be required to satisfy other conditions as well, e.g. being independent of the conclusion, or being “informationally unconnected” with the conclusion, or some other condition dictated by the context. This paper discusses various types of such problems, it presents techniques and principles useful in solving them, and it develops (...)
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  12. Ciro de Florio, N-Th Order Logic.
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  13. Ciro de Florio, N-Order, Logic.
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  14. Roger Fellows (1996). First‐Order Logic. Philosophical Books 37 (4):284-286.
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  15. Joanna Golińska & Konrad Zdanowski (2003). Spectra of Formulae with Henkin Quantifiers. In A. Rojszczak, J. Cachro & G. Kurczewski (eds.), Philosophical Dimensions of Logic and Science. Kluwer Academic Publishers 29--45.
    It is known that various complexity-theoretical problems can be translated into some special spectra problems (see e.g. Fagin [Fa74] or Blass and Gurevich, [Bl-Gu86]). So 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 quanti fiers in the empty vocabulary. This problem has been investigated fi rstly by Krynicki and Mostowski in [Kr-Mo 92] and [Kr- Mo 95]. All presented results can be (...)
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  16. P. K. H. (1968). First Order Mathematical Logic. [REVIEW] Review of Metaphysics 21 (3):556-556.
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  17. David Harel (1979). Characterizing Second Order Logic with First Order Quantifiers. Mathematical Logic Quarterly 25 (25‐29):419-422.
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  18. John Heil (1994). First-Order Logic a Concise Introduction. Monograph Collection (Matt - Pseudo).
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  19. Stefan Hetzl, Alexander Leitsch & Daniel Weller (2011). CERES in Higher-Order Logic. Annals of Pure and Applied Logic 162 (12):1001-1034.
    We define a generalization of the first-order cut-elimination method CERES to higher-order logic. At the core of lies the computation of an set of sequents from a proof π of a sequent S. A refutation of in a higher-order resolution calculus can be used to transform cut-free parts of π into a cut-free proof of S. An example illustrates the method and shows that can produce meaningful cut-free proofs in mathematics that traditional cut-elimination methods cannot reach.
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  20. R. I. G. Hughes (1993). A Philosophical Companion to First-Order Logic. Monograph Collection (Matt - Pseudo).
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  21. Shmuel Lifsches & Saharon Shelah (1999). Random Graphs in the Monadic Theory of Order. Archive for Mathematical Logic 38 (4-5):273-312.
    We continue the works of Gurevich-Shelah and Lifsches-Shelah by showing that it is consistent with ZFC that the first-order theory of random graphs is not interpretable in the monadic theory of all chains. It is provable from ZFC that the theory of random graphs is not interpretable in the monadic second order theory of short chains (hence, in the monadic theory of the real line).
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  22. Shmuel Lifsches & Saharon Shelah (1992). The Monadic Theory of (Ω 2, <) May Be Complicated. Archive for Mathematical Logic 31 (3):207-213.
    Assume ZFC is consistent then for everyB⫅ω there is a generic extension of the ground world whereB is recursive in the monadic theory ofω 2.
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  23. A. H. Lighstone (1972). Review: Angelo Margaris, First Order Mathematical Logic. [REVIEW] Journal of Symbolic Logic 37 (3):616-616.
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  24. M. A. Mcbeth (1994). Combinatorial Number Theory a Treatise on Growth, Based on the Goodstein-Skolem Hierarchy, Including a Critique of Non-Constructive or 1st Order Logic. Monograph Collection (Matt - Pseudo).
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  25. Robert K. Meyer (1976). Ackermann, Takeuti, and Schnitt: For Higher-Order Relevant Logic. Bulletin of the Section of Logic 5 (4):138-142.
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  26. Malika More & Frédéric Olive (1997). Rudimentary Languages and Second‐Order Logic. Mathematical Logic Quarterly 43 (3):419-426.
    The aim of this paper is to point out the equivalence between three notions respectively issued from recursion theory, computational complexity and finite model theory. One the one hand, the rudimentary languages are known to be characterized by the linear hierarchy. On the other hand, this complexity class can be proved to correspond to monadic second-order logic with addition. Our viewpoint sheds some new light on the close connection between these domains: We bring together the two extremal notions by providing (...)
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  27. Hirokazu Nishimura (1983). Hauptsatz for Higher-Order Modal Logic. Journal of Symbolic Logic 48 (3):744-751.
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  28. Niels Öffenberger & Albert Menne (1982). [Standing Order : Staff Use Only].
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  29. A. C. Paseau (2012). James Robert Brown. Platonism, Naturalism, and Mathematical Knowledge. New York and London: Routledge, 2012. Isbn 978-0-415-87266-9. Pp. X + 182. [REVIEW] Philosophia Mathematica 20 (3):359-364.
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  30. Peter Roeper (2004). First- and Second-Order Logic of Mass Terms. Journal of Philosophical Logic 33 (3):261-297.
    Provided here is an account, both syntactic and semantic, of first-order and monadic second-order quantification theory for domains that may be non-atomic. Although the rules of inference largely parallel those of classical logic, there are important differences in connection with the identification of argument places and the significance of the identity relation.
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  31. Marcus Rossberg, Second-Order Logic : Ontological and Epistemological Problems.
    In this thesis I provide a survey over different approaches to second-order logic and its interpretation, and introduce a novel approach. Of special interest are the questions whether second-order logic can count as logic in some proper sense of logic, and what epistemic status it occupies. More specifically, second-order logic is sometimes taken to be mathematical, a mere notational variant of some fragment of set theory. If this is the case, it might be argued that it does not have the (...)
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  32. Gábor Sági (2000). A Completeness Theorem for Higher Order Logics. Journal of Symbolic Logic 65 (2):857-884.
    Here we investigate the classes RCA $^\uparrow_\alpha$ of representable directed cylindric algebras of dimension α introduced by Nemeti[12]. RCA $^\uparrow_\alpha$ can be seen in two different ways: first, as an algebraic counterpart of higher order logics and second, as a cylindric algebraic analogue of Quasi-Projective Relation Algebras. We will give a new, "purely cylindric algebraic" proof for the following theorems of Nemeti: (i) RCA $^\uparrow_\alpha$ is a finitely axiomatizable variety whenever α ≥ 3 is finite and (ii) one can obtain (...)
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  33. Charles Sayward (1983). What is a Second Order Theory Committed To? Erkenntnis 20 (1):79 - 91.
    The paper argues that no second order theory is ontologically commited to anything beyond what its individual variables range over.
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  34. George F. Schumm (1995). Review: R. I. G. Hughes, A Philosophical Companion to First-Order Logic. [REVIEW] Journal of Symbolic Logic 60 (2):684-685.
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  35. K. Schutte (1974). Review: Dag Prawitz, Hauptsatz for Higher Order Logic; Dag Prawitz, Completeness and Hauptsatz for Second Order Logic; Moto-o Takahashi, A Proof of Cut-Elimination in Simple Type-Theory. [REVIEW] Journal of Symbolic Logic 39 (3):607-607.
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  36. I. A. Stewart (1991). On the Expressibility of Extensions of First-Order Logic. University of Newcastle Upon Tyne, Computing Laboratory.
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  37. Iain A. Stewart (1997). Regular Subgraphs in Graphs and Rooted Graphs and Definability in Monadic Second‐Order Logic. Mathematical Logic Quarterly 43 (1):1-21.
    We investigate the definability in monadic ∑11 and monadic Π11 of the problems REGk, of whether there is a regular subgraph of degree k in some given graph, and XREGk, of whether, for a given rooted graph, there is a regular subgraph of degree k in which the root has degree k, and their restrictions to graphs in which every vertex has degree at most k, namely REGkk and XREGkk, respectively, for k ≥ 2 . Our motivation partly stems from (...)
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  38. Sean Walsh & Sean Ebels-Duggan (2015). Relative Categoricity and Abstraction Principles. Review of Symbolic Logic 8 (3):572-606.
    Many recent writers in the philosophy of mathematics have put great weight on the relative categoricity of the traditional axiomatizations of our foundational theories of arithmetic and set theory. Another great enterprise in contemporary philosophy of mathematics has been Wright's and Hale's project of founding mathematics on abstraction principles. In earlier work, it was noted that one traditional abstraction principle, namely Hume's Principle, had a certain relative categoricity property, which here we term natural relative categoricity. In this paper, we show (...)
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  39. John J. Wellmuth (1941). Philosophy and Order in Logic. Proceedings of the American Catholic Philosophical Association 17:12-18.
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  40. J. Wolenski (2004). First-Order Logic:(Philosophical) Pro and Contra. In Vincent F. Hendricks (ed.), First-Order Logic Revisited. Logos 369--398.
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  41. G. H. Wright (1983). Norms of Higher Order. Studia Logica 42 (2-3):119-127.
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Second-Order Logic
  1. S. Awodey & C. Butz (2000). Topological Completeness for Higher-Order Logic. Journal of Symbolic Logic 65 (3):1168-1182.
    Using recent results in topos theory, two systems of higher-order logic are shown to be complete with respect to sheaf models over topological spaces- so -called "topological semantics." The first is classical higher-order logic, with relational quantification of finitely high type; the second system is a predicative fragment thereof with quantification over functions between types, but not over arbitrary relations. The second theorem applies to intuitionistic as well as classical logic.
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  2. Ruth C. Barcan (1947). The Identity of Individuals in a Strict Functional Calculus of Second Order. Journal of Symbolic Logic 12 (1):12-15.
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  3. K. Jon Barwise (1972). The Hanf Number of Second Order Logic. Journal of Symbolic Logic 37 (3):588-594.
    We prove, among other things, that the number mentioned above cannot be shown to exist without using some $\Pi_1(\mathscr{P})$ instance of the axiom of replacement.
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  4. Anne Bauval (1985). Polynomial Rings and Weak Second-Order Logic. Journal of Symbolic Logic 50 (4):953-972.
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  5. L. Berk (2013). Second-Order Arithmetic Sans Sets. 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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  6. George Boolos (1985). Nominalist Platonism. Philosophical Review 94 (3):327-344.
  7. George S. Boolos (1975). On Second-Order Logic. Journal of Philosophy 72 (16):509-527.
  8. James Robert Brown (1996). Foundations Without Foundationalism: A Case for Second-Order Logic Stewart Shapiro Oxford: Oxford University Press, 1991, Xx + 277 Pp. [REVIEW] Dialogue 35 (03):624-.
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  9. Otávio Bueno (2010). A Defense of Second-Order Logic. 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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