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  1. Mohamed A. Amer (1989). First Order Logic with Empty Structures. Studia Logica 48 (2):169 - 177.
    For first order languages with no individual constants, empty structures and truth values (for sentences) in them are defined. The first order theories of the empty structures and of all structures (the empty ones included) are axiomatized with modus ponens as the only rule of inference. Compactness is proved and decidability is discussed. Furthermore, some well known theorems of model theory are reconsidered under this new situation. Finally, a word is said on other approaches to the whole problem.
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  2. Colin G. Bailey (2013). Some Jump-Like Operations in $\Mathbf \Beta $-Recursion Theory. Journal of Symbolic Logic 78 (1):57-71.
    In this paper we show that there are various pseudo-jump operators definable over inadmissible $J_{\beta}$ that relate to the failure of admissiblity and to non-regularity. We will use these ideas to construct some intermediate degrees.
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  3. Dwight R. Bean (1976). Effective Coloration. Journal of Symbolic Logic 41 (2):469-480.
    We are concerned here with recursive function theory analogs of certain problems in chromatic graph theory. The motivating question for our work is: Does there exist a recursive (countably infinite) planar graph with no recursive 4-coloring? We obtain the following results: There is a 3-colorable, recursive planar graph which, for all k, has no recursive k-coloring; every decidable graph of genus p ≥ 0 has a recursive 2(χ(p) - 1)-coloring, where χ(p) is the least number of colors which will suffice (...)
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  4. Arnold Beckmann (2002). Proving Consistency of Equational Theories in Bounded Arithmetic. Journal of Symbolic Logic 67 (1):279-296.
    We consider equational theories for functions defined via recursion involving equations between closed terms with natural rules based on recursive definitions of the function symbols. We show that consistency of such equational theories can be proved in the weak fragment of arithmetic S 1 2 . In particular this solves an open problem formulated by TAKEUTI (c.f. [5, p.5 problem 9.]).
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  5. Michael Beeson (1976). The Unprovability in Intuitionistic Formal Systems of the Continuity of Effective Operations on the Reals. Journal of Symbolic Logic 41 (1):18-24.
  6. Lev D. Beklemishev (2003). On the Induction Schema for Decidable Predicates. Journal of Symbolic Logic 68 (1):17-34.
    We study the fragment of Peano arithmetic formalizing the induction principle for the class of decidable predicates, $I\Delta_1$ . We show that $I\Delta_1$ is independent from the set of all true arithmetical $\Pi_2-sentences$ . Moreover, we establish the connections between this theory and some classes of oracle computable functions with restrictions on the allowed number of queries. We also obtain some conservation and independence results for parameter free and inference rule forms of $\Delta_1-induction$ . An open problem formulated by J. (...)
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  7. Patrick Blackburn, Maarten de Rijke & Yde Venema (2002). Modal Logic. Cambridge University Press.
    This modern, advanced textbook reviews modal logic, a field which caught the attention of computer scientists in the late 1970's.
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  8. D. Bollman & M. Tapia (1972). On the Recursive Unsolvability of the Provability of the Deduction Theorem in Partial Propositional Calculi. Notre Dame Journal of Formal Logic 13 (1):124-128.
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  9. Carlos Caleiro, Luca Viganò & Marco Volpe (2013). On the Mosaic Method for Many-Dimensional Modal Logics: A Case Study Combining Tense and Modal Operators. [REVIEW] Logica Universalis 7 (1):33-69.
    We present an extension of the mosaic method aimed at capturing many-dimensional modal logics. As a proof-of-concept, we define the method for logics arising from the combination of linear tense operators with an “orthogonal” S5-like modality. We show that the existence of a model for a given set of formulas is equivalent to the existence of a suitable set of partial models, called mosaics, and apply the technique not only in obtaining a proof of decidability and a proof of completeness (...)
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  10. William J. Collins & Paul Young (1983). Discontinuities of Provably Correct Operators on the Provably Recursive Real Numbers. Journal of Symbolic Logic 48 (4):913-920.
    In this paper we continue, from [2], the development of provably recursive analysis, that is, the study of real numbers defined by programs which can be proven to be correct in some fixed axiom system S. In particular we develop the provable analogue of an effective operator on the set C of recursive real numbers, namely, a provably correct operator on the set P of provably recursive real numbers. In Theorems 1 and 2 we exhibit a provably correct operator on (...)
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  11. René David & Karim Nour (1995). Storage Operators and Directed Lambda-Calculus. Journal of Symbolic Logic 60 (4):1054-1086.
    Storage operators have been introduced by J. L. Krivine in [5] they are closed λ-terms which, for a data type, allow one to simulate a "call by value" while using the "call by name" strategy. In this paper, we introduce the directed λ-calculus and show that it has the usual properties of the ordinary λ-calculus. With this calculus we get an equivalent--and simple--definition of the storage operators that allows to show some of their properties: $\bullet$ the stability of the set (...)
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  12. Thomas John (1986). Recursion in Kolmogorov's R-Operator and the Ordinal Σ. Journal of Symbolic Logic 51 (1):1 - 11.
  13. N. D. Jones (1997). Computability and Complexity: From a Programming Perspective Vol. 21. Mit Press.
    This makes his book especially valuable." -- Yuri Gurevich, Professor of Computer Science, University of Michigan Computability and complexity theory should be of central concern to practitioners as well as theorists.
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  14. Saul A. Kripke (2013). The Church-Turing ‘Thesis’ as a Special Corollary of Gödel’s Completeness Theorem. In B. J. Copeland, C. Posy & O. Shagrir (eds.), Computability: Turing, Gödel, Church, and Beyond. MIT Press.
    Traditionally, many writers, following Kleene (1952), thought of the Church-Turing thesis as unprovable by its nature but having various strong arguments in its favor, including Turing’s analysis of human computation. More recently, the beauty, power, and obvious fundamental importance of this analysis, what Turing (1936) calls “argument I,” has led some writers to give an almost exclusive emphasis on this argument as the unique justification for the Church-Turing thesis. In this chapter I advocate an alternative justification, essentially presupposed by Turing (...)
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  15. Ian Mason (1985). The Metatheory of the Classical Propositional Calculus is Not Axiomatizable. Journal of Symbolic Logic 50 (2):451-457.
  16. Albert R. Meyer & Patrick C. Fischer (1972). Computational Speed-Up by Effective Operators. Journal of Symbolic Logic 37 (1):55-68.
  17. Antje Nowack (2005). A Guarded Fragment for Abstract State Machines. Journal of Logic, Language and Information 14 (3):345-368.
    Abstract State Machines (ASMs) provide a formal method for transparent design and specification of complex dynamic systems. They combine advantages of informal and formal methods. Applications of this method motivate a number of computability and decidability problems connected to ASMs. Such problems result for example from the area of verifying properties of ASMs. Their high expressive power leads rather directly to undecidability respectively uncomputability results for most interesting problems in the case of unrestricted ASMs. Consequently, it is rather natural to (...)
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  18. Claes Strannegård, Fredrik Engström, Abdul Rahim Nizamani & Lance Rips (2013). Reasoning About Truth in First-Order Logic. Journal of Logic, Language and Information 22 (1):115-137.
    First, we describe a psychological experiment in which the participants were asked to determine whether sentences of first-order logic were true or false in finite graphs. Second, we define two proof systems for reasoning about truth and falsity in first-order logic. These proof systems feature explicit models of cognitive resources such as declarative memory, procedural memory, working memory, and sensory memory. Third, we describe a computer program that is used to find the smallest proofs in the aforementioned proof systems when (...)
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  19. Michał Walicki (2012). Introduction to Mathematical Logic. World Scientific.
    A history of logic -- Patterns of reasoning -- A language and its meaning -- A symbolic language -- 1850-1950 mathematical logic -- Modern symbolic logic -- Elements of set theory -- Sets, functions, relations -- Induction -- Turning machines -- Computability and decidability -- Propositional logic -- Syntax and proof systems -- Semantics of PL -- Soundness and completeness -- First order logic -- Syntax and proof systems of FOL -- Semantics of FOL -- More semantics -- Soundness and (...)
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  20. Jonathan A. Waskan (forthcoming). A Virtual Solution to the Frame Problem. Proceedings of the First Ieee-Ras International Conference on Humanoid Robots.
    We humans often respond effectively when faced with novel circumstances. This is because we are able to predict how particular alterations to the world will play out. Philosophers, psychologists, and computational modelers have long favored an account of this process that takes its inspiration from the truth-preserving powers of formal deduction techniques. There is, however, an alternative hypothesis that is better able to account for the human capacity to predict the consequences worldly alterations. This alternative takes its inspiration from the (...)
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  21. Guohua Wu (2006). Jump Operator and Yates Degrees. Journal of Symbolic Logic 71 (1):252 - 264.
    In [9]. Yates proved the existence of a Turing degree a such that 0. 0′ are the only c.e. degrees comparable with it. By Slaman and Steel [7], every degree below 0′ has a 1-generic complement, and as a consequence. Yates degrees can be 1-generic, and hence can be low. In this paper, we prove that Yates degrees occur in every jump class.
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