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1274 found
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1 — 50 / 1274
  1. Extension, Translation, and the Cantor-Bernstein Property.Thomas William Barrett & Hans Halvorson - manuscript
    The purpose of this paper is to examine in detail a particularly interesting pair of first-order theories. In addition to clarifying the overall geography of notions of equivalence between theories, this simple example yields two surprising conclusions about the relationships that theories might bear to one another. In brief, we see that theories lack both the Cantor-Bernstein and co-Cantor-Bernstein properties.
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  2. A BIBLIOGRAPHY: JOHN CORCORAN's PUBLICATIONS ON ARISTOTLE 1972–2015.John Corcoran - manuscript
    This presentation includes a complete bibliography of John Corcoran’s publications devoted at least in part to Aristotle’s logic. Sections I–IV list 20 articles, 43 abstracts, 3 books, and 10 reviews. It starts with two watershed articles published in 1972: the Philosophy & Phenomenological Research article that antedates Corcoran’s Aristotle’s studies and the Journal of Symbolic Logic article first reporting his original results; it ends with works published in 2015. A few of the items are annotated with endnotes connecting them with (...)
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  3. Zeno Paradox, Unexpected Hanging Paradox (Modeling of Reality & Physical Reality, A Historical-Philosophical View).Farzad Didehvar - manuscript
    . In our research about Fuzzy Time and modeling time, "Unexpected Hanging Paradox" plays a major role. Here, we compare this paradox to the Zeno Paradox and the relations of them with our standard models of continuum and Fuzzy numbers. To do this, we review the project "Fuzzy Time and Possible Impacts of It on Science" and introduce a new way in order to approach the solutions for these paradoxes. Additionally, we have a more general discussion about paradoxes, as Philosophical (...)
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  4. Computational Reverse Mathematics and Foundational Analysis.Benedict Eastaugh - manuscript
    Reverse mathematics studies which subsystems of second order arithmetic are equivalent to key theorems of ordinary, non-set-theoretic mathematics. The main philosophical application of reverse mathematics proposed thus far is foundational analysis, which explores the limits of different foundations for mathematics in a formally precise manner. This paper gives a detailed account of the motivations and methodology of foundational analysis, which have heretofore been largely left implicit in the practice. It then shows how this account can be fruitfully applied in the (...)
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  5. Partitions and Objective Indefiniteness.David Ellerman - manuscript
    Classical physics and quantum physics suggest two meta-physical types of reality: the classical notion of a objectively definite reality with properties "all the way down," and the quantum notion of an objectively indefinite type of reality. The problem of interpreting quantum mechanics (QM) is essentially the problem of making sense out of an objectively indefinite reality. These two types of reality can be respectively associated with the two mathematical concepts of subsets and quotient sets (or partitions) which are category-theoretically dual (...)
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  6. The Impacts of Logic, Paradoxes in One Side and Theory of Computation in the Other Side.Didehvar Farzad - manuscript
    This is a presentation about the impacts of Logic and Theory of Computation. It starts by some explanations about Theory of Computation and its relations with the other subjects in science. Then we have some explanations about paradoxes and some historical points. In continuation, we present some of the most important paradoxes. Forthcoming, Five subjects around the relations between Logic and Theory of computation is introduced. Finally, we present a new approach to solve P vs NP problem via Paradoxes (Presentation (...)
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  7. Algunos tópicos de Lógica matemática y los Fundamentos de la matemática.Franklin Galindo - manuscript
    En este trabajo matemático-filosófico se estudian cuatro tópicos de la Lógica matemática: El método de construcción de modelos llamado Ultraproductos, la Propiedad de Interpolación de Craig, las Álgebras booleanas y los Órdenes parciales separativos. El objetivo principal del mismo es analizar la importancia que tienen dichos tópicos para el estudio de los fundamentos de la matemática, desde el punto de vista del platonismo matemático. Para cumplir con tal objetivo se trabajará en el ámbito de la Matemática, de la Metamatemática y (...)
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  8. What is Mathematics: Gödel's Theorem and Around (Edition 2015).Karlis Podnieks - manuscript
    Introduction to mathematical logic, part 2.Textbook for students in mathematical logic and foundations of mathematics. Platonism, Intuition, Formalism. Axiomatic set theory. Around the Continuum Problem. Axiom of Determinacy. Large Cardinal Axioms. Ackermann's Set Theory. First order arithmetic. Hilbert's 10th problem. Incompleteness theorems. Consequences. Connected results: double incompleteness theorem, unsolvability of reasoning, theorem on the size of proofs, diophantine incompleteness, Loeb's theorem, consistent universal statements are provable, Berry's paradox, incompleteness and Chaitin's theorem. Around Ramsey's theorem.
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  9. Random Formula Generators.Ariel Jonathan Roffé & Joaquín Toranzo Calderón - manuscript
    In this article, we provide three generators of propositional formulae for arbitrary languages, which uniformly sample three different formulae spaces. They take the same three parameters as input, namely, a desired depth, a set of atomics and a set of logical constants (with specified arities). The first generator returns formulae of exactly the given depth, using all or some of the propositional letters. The second does the same but samples up-to the given depth. The third generator outputs formulae with exactly (...)
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  10. Category Theory: A Gentle Introduction.Peter Smith - manuscript
    This Gentle Introduction is very much still work in progress. Roughly aimed at those who want something a bit more discursive, slower-moving, than Awodey's or Leinster's excellent books. -/- The current [Jan 2018] version is 291pp.
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  11. Informal and Formal Proofs, Metalogic, and the Groundedness Problem.Mario Bacelar Valente - manuscript
    When modeling informal proofs like that of Euclid’s Elements using a sound logical system, we go from proofs seen as somewhat unrigorous – even having gaps to be filled – to rigorous proofs. However, metalogic grounds the soundness of our logical system, and proofs in metalogic are not like formal proofs and look suspiciously like the informal proofs. This brings about what I am calling here the groundedness problem: how can we decide with certainty that our metalogical proofs are rigorous (...)
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  12. Provably Games.J. P. Aguilera & D. W. Blue - forthcoming - Journal of Symbolic Logic:1-22.
    We isolate two abstract determinacy theorems for games of length $\omega_1$ from work of Neeman and use them to conclude, from large-cardinal assumptions and an iterability hypothesis in the region of measurable Woodin cardinals thatif the Continuum Hypothesis holds, then all games of length $\omega_1$ which are provably $\Delta_1$ -definable from a universally Baire parameter are determined;all games of length $\omega_1$ with payoff constructible relative to the play are determined; andif the Continuum Hypothesis holds, then there is a model of (...)
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  13. Formal Differential Variables and an Abstract Chain Rule.Samuel Alexander - forthcoming - Proceedings of the ACMS.
    One shortcoming of the chain rule is that it does not iterate: it gives the derivative of f(g(x)), but not (directly) the second or higher-order derivatives. We present iterated differentials and a version of the multivariable chain rule which iterates to any desired level of derivative. We first present this material informally, and later discuss how to make it rigorous (a discussion which touches on formal foundations of calculus). We also suggest a finite calculus chain rule (contrary to Graham, Knuth (...)
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  14. Metrically Homogeneous Graphs of Diameter Three.Daniela Amato, Gregory Cherlin & H. Dugald Macpherson - forthcoming - Journal of Mathematical Logic.
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  15. Expanding the Reals by Continuous Functions Adds No Computational Power.Uri Andrews, Julia F. Knight, Rutger Kuyper, Joseph S. Miller & Mariya I. Soskova - forthcoming - Journal of Symbolic Logic:1-19.
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  16. Strong Compactness, Square, GCH, and Woodin Cardinals.Arthur W. Apter - forthcoming - Journal of Symbolic Logic:1-14.
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  17. Dependent Choice, Properness, and Generic Absoluteness.David Asperó & Asaf Karagila - forthcoming - Review of Symbolic Logic:1-25.
    We show that Dependent Choice is a sufficient choice principle for developing the basic theory of proper forcing, and for deriving generic absoluteness for the Chang model in the presence of large cardinals, even with respect to $\mathsf {DC}$ -preserving symmetric submodels of forcing extensions. Hence, $\mathsf {ZF}+\mathsf {DC}$ not only provides the right framework for developing classical analysis, but is also the right base theory over which to safeguard truth in analysis from the independence phenomenon in the presence of (...)
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  18. How Much Propositional Logic Suffices for Rosser's Essential Undecidability Theorem?Guillermo Badia, Petr Cintula, Petr Hajek & Andrew Tedder - forthcoming - Review of Symbolic Logic.
    In this paper we explore the following question: how weak can a logic be for Rosser’s essential undecidability result to be provable for a weak arithmetical theory? It is well known that Robinson’s Q is essentially undecidable in intuitionistic logic, and P. Hájek proved it in the fuzzy logic BL for Grzegorczyk’s variant of Q which interprets the arithmetic operations as nontotal nonfunctional relations. We present a proof of essential undecidability in a much weaker substructural logic and for a much (...)
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  19. How Much Propositional Logic Suffices for Rosser’s Essential Undecidability Theorem?Guillermo Badia, Petr Cintula, Petr Hajek & Andrew Tedder - forthcoming - Review of Symbolic Logic:1-18.
    In this paper we explore the following question: how weak can a logic be for Rosser's essential undecidability result to be provable for a weak arithmetical theory? It is well known that Robinson's Q is essentially undecidable in intuitionistic logic, and P. Hajek proved it in the fuzzy logic BL for Grzegorczyk's variant of Q which interprets the arithmetic operations as non-total non-functional relations. We present a proof of essential undecidability in a much weaker substructural logic and for a much (...)
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  20. More on the Preservation of Large Cardinals Under Class Forcing.Joan Bagaria & Alejandro Poveda - forthcoming - Journal of Symbolic Logic:1-34.
    We prove two general results about the preservation of extendible and $C^{}$ -extendible cardinals under a wide class of forcing iterations. As applications we give new proofs of the preservation of Vopěnka’s Principle and $C^{}$ -extendible cardinals under Jensen’s iteration for forcing the GCH [17], previously obtained in [8, 27], respectively. We prove that $C^{}$ -extendible cardinals are preserved by forcing with standard Easton-support iterations for any possible $\Delta _2$ -definable behaviour of the power-set function on regular cardinals. We show (...)
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  21. Defining Measures in a Mereological Space.Giuseppina Barbieri & Giangiacomo Gerla - forthcoming - Logic and Logical Philosophy:1.
    We explore the notion of a measure in a mereological structure and we deal with the difficulties arising. We show that measure theory on connection spaces is closely related to measure theory on the class of ortholattices and we present an approach akin to Dempster’s and Shafer’s. Finally, the paper contains some suggestions for further research.
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  22. Structural Relativity and Informal Rigour.Neil Barton - forthcoming - In Objects, Structures, and Logics, FilMat Studies in the Philosophy of Mathematics.
    Informal rigour is the process by which we come to understand particular mathematical structures and then manifest this rigour through axiomatisations. Structural relativity is the idea that the kinds of structures we isolate are dependent upon the logic we employ. We bring together these ideas by considering the level of informal rigour exhibited by our set-theoretic discourse, and argue that different foundational programmes should countenance different underlying logics (intermediate between first- and second-order) for formulating set theory. By bringing considerations of (...)
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  23. Bolzano’s Mathematical Infinite.Anna Bellomo & Guillaume Massas - forthcoming - Review of Symbolic Logic:1-80.
    Bernard Bolzano (1781-1848) is commonly thought to have attempted to develop a theory of size for infinite collections that follows the so-called part-whole principle, according to which the whole is always greater than any of its proper parts. In this paper, we develop a novel interpretation of Bolzano's mature theory of the infinite and show that, contrary to mainstream interpretations, it is best understood as a theory of infinite sums. Our formal results show that Bolzano's infinite sums can be equipped (...)
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  24. Characterizing Existence of a Measurable Cardinal Via Modal Logic.G. Bezhanishvili, N. Bezhanishvili, J. Lucero-Bryan & J. van Mill - forthcoming - Journal of Symbolic Logic:1-15.
    We prove that the existence of a measurable cardinal is equivalent to the existence of a normal space whose modal logic coincides with the modal logic of the Kripke frame isomorphic to the powerset of a two element set.
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  25. A Negative Solution of Kuznetsov’s Problem for Varieties of Bi-Heyting Algebras.Guram Bezhanishvili, David Gabelaia & Mamuka Jibladze - forthcoming - Journal of Mathematical Logic.
    In this paper, we show that there exist varieties of bi-Heyting algebras that are not generated by their complete members. It follows that there exist extensions o...
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  26. The Additive Groups of ℤ and ℚ with Predicates for Being Square‐Free.Neer Bhardwaj & Minh Chieu Tran - forthcoming - Journal of Symbolic Logic:1-26.
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  27. Frege’s Theory of Real Numbers: A Consistent Rendering.Francesca Boccuni & Marco Panza - forthcoming - Review of Symbolic Logic:1-44.
    Frege's definition of the real numbers, as envisaged in the second volume of Grundgesetze der Arithmetik, is fatally flawed by the inconsistency of Frege's ill-fated Basic Law V. We restate Frege's definition in a consistent logical framework and investigate whether it can provide a logical foundation of real analysis. Our conclusion will deem it doubtful that such a foundation along the lines of Frege's own indications is possible at all.
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  28. On Sequences of Homomorphisms Into Measure Algebras and the Efimov Problem.Piotr Borodulin–Nadzieja & Damian Sobota - forthcoming - Journal of Symbolic Logic:1-28.
    For given Boolean algebras $\mathbb {A}$ and $\mathbb {B}$ we endow the space $\mathcal {H}$ of all Boolean homomorphisms from $\mathbb {A}$ to $\mathbb {B}$ with various topologies and study convergence properties of sequences in $\mathcal {H}$. We are in particular interested in the situation when $\mathbb {B}$ is a measure algebra as in this case we obtain a natural tool for studying topological convergence properties of sequences of ultrafilters on $\mathbb {A}$ in random extensions of the set-theoretical universe. This (...)
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  29. Logics of Formal Inconsistency Enriched with Replacement: An Algebraic and Modal Account.Walter Carnielli, Marcelo E. Coniglio & David Fuenmayor - forthcoming - Review of Symbolic Logic.
    One of the most expected properties of a logical system is that it can be algebraizable, in the sense that an algebraic counterpart of the deductive machinery could be found. Since the inception of da Costa's paraconsistent calculi, an algebraic equivalent for such systems have been searched. It is known that these systems are non self-extensional (i.e., they do not satisfy the replacement property). More than this, they are not algebraizable in the sense of Blok-Pigozzi. The same negative results hold (...)
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  30. Hilbert Algebras with Hilbert–Galois Connections.Sergio A. Celani & Daniela Montangie - forthcoming - Studia Logica:1-26.
    In this paper we introduce Hilbert algebras with Hilbert–Galois connections and we study the Hilbert–Galois connections defined in Heyting algebras, called HGC-algebras. We assign a categorical duality between the category HilGC-algebras with Hilbert homomorphisms that commutes with Hilbert–Galois connections and Hilbert spaces with certain binary relations and whose morphisms are special functional relations. We also prove a categorical duality between the category of Heyting Galois algebras with Heyting homomorphisms that commutes with Hilbert–Galois connections and the category of spectral Heyting spaces (...)
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  31. Invariant Measures in Simple and in Small Theories.Artem Chernikov, Ehud Hrushovski, Alex Kruckman, Krzysztof Krupinski, Slavko Moconja, Anand Pillay & Nicholas Ramsey - forthcoming - Journal of Mathematical Logic.
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  32. The Well-Ordered Society Under Crisis: A Formal Analysis of Public Reason Vs. Convergence Discourse.Hun Chung - forthcoming - American Journal of Political Science:1-20.
    A well-ordered society faces a crisis whenever a sufficient number of noncompliers enter into the political system. This has the potential to destabilize liberal democratic political order. This article provides a formal analysis of two competing solutions to the problem of political stability offered in the public reason liberalism literature—namely, using public reason or using convergence discourse to restore liberal democratic political order in the well-ordered society. The formal analyses offered in this article show that using public reason fails completely, (...)
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  33. There are Infinitely Many Mersenne Prime Numbers. Applications of Rasiowa–Sikorski Lemma in Arithmetic.Janusz Czelakowski - forthcoming - Studia Logica:1-9.
    The paper is concerned with the old conjecture that there are infinitely many Mersenne primes. It is shown in the work that this conjecture is true in the standard model of arithmetic. The proof refers to the general approach to first–order logic based on Rasiowa-Sikorski Lemma and the derived notion of a Rasiowa–Sikorski set. This approach was developed in the papers [2,3,4]. This work is a companion piece to [4].
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  34. The Twin Primes Conjecture is True in the Standard Model of Peano Arithmetic: Applications of Rasiowa–Sikorski Lemma in Arithmetic.Janusz Czelakowski - forthcoming - Studia Logica:1-52.
    The paper is concerned with the old conjecture that there are infinitely many twin primes. In the paper we show that this conjecture is true, that is, it is true in the standard model of arithmetic. The proof is based on Rasiowa–Sikorski Lemma. The key role are played by the derived notion of a Rasiowa–Sikorski set and the method of forcing adjusted to arbitrary first–order languages. This approach was developed in the papers Czelakowski [4, 5]. The central idea consists in (...)
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  35. The Ramsey Theory of Henson Graphs.Natasha Dobrinen - forthcoming - Journal of Mathematical Logic.
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  36. Discrete Duality for Nelson Algebras with Tense Operators.Aldo V. Figallo, Gustavo Pelaitay & Jonathan Sarmiento - forthcoming - Studia Logica:1-19.
    In this paper, we continue with the study of tense operators on Nelson algebras :285–312, 2021, Studia Logica 110:241–263, 2022). We define the variety of algebras, which we call tense Nelson D-algebras, as a natural extension of tense De Morgan algebras :255–267, 2014). In particular, we give a discrete duality for these algebras. To do this, we will extend the representation theorems for Nelson algebras given in Sendlewski :257–280, 1984) to the tense case.
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  37. Extensional Realizability and Choice for Dependent Types in Intuitionistic Set Theory.Emanuele Frittaion - forthcoming - Journal of Symbolic Logic:1-31.
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  38. Conservation Theorems on Semi-Classical Arithmetic.Makoto Fujiwara & Taishi Kurahashi - forthcoming - Journal of Symbolic Logic:1-28.
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  39. Models of Martin-Löf Type Theory From Algebraic Weak Factorisation Systems.Nicola Gambino & Marco Federico Larrea - forthcoming - Journal of Symbolic Logic:1-48.
    We introduce type-theoretic algebraic weak factorisation systems and show how they give rise to homotopy-theoretic models of Martin-Löf type theory. This is done by showing that the comprehension category associated with a type-theoretic algebraic weak factorisation system satisfies the assumptions necessary to apply a right adjoint method for splitting comprehension categories. We then provide methods for constructing several examples of type-theoretic algebraic weak factorisation systems, encompassing the existing groupoid and cubical sets models, as well as new models based on normal (...)
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  40. On a Problem of Friedman and its Solution by Rybakov.Jeroen P. Goudsmit - forthcoming - Bulletin of Symbolic Logic:1-48.
    Rybakov (1984a) proved that the admissible rules of IPC are decidable. We give a proof of the same theorem, using the same core idea, but couched in the many notions that have been developed in the mean time. In particular, we illustrate how the argument can be interpreted as using refinements of the notions of exactness and extendibility.
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  41. On the Invariance of Gödel’s Second Theorem with Regard to Numberings.Balthasar Grabmayr - forthcoming - Review of Symbolic Logic:1-34.
    The prevalent interpretation of Gödel’s Second Theorem states that a sufficiently adequate and consistent theory does not prove its consistency. It is however not entirely clear how to justify this informal reading, as the formulation of the underlying mathematical theorem depends on several arbitrary formalisation choices. In this paper I examine the theorem’s dependency regarding Gödel numberings. I introduce deviant numberings, yielding provability predicates satisfying Löb’s conditions, which result in provable consistency sentences. According to the main result of this paper (...)
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  42. Compatibility and Accessibility: Lattice Representations for Semantics of Non-Classical and Modal Logics.Wesley Holliday - forthcoming - In David Fernández Duque & Alessandra Palmigiano (eds.), Advances in Modal Logic, Vol. 14. London: College Publications.
    In this paper, we study three representations of lattices by means of a set with a binary relation of compatibility in the tradition of Ploščica. The standard representations of complete ortholattices and complete perfect Heyting algebras drop out as special cases of the first representation, while the second covers arbitrary complete lattices, as well as complete lattices equipped with a negation we call a protocomplementation. The third topological representation is a variant of that of Craig, Haviar, and Priestley. We then (...)
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  43. Free Definite Description Theory – Sequent Calculi and Cut Elimination.Andrzej Indrzejczak - forthcoming - Logic and Logical Philosophy:1.
    We provide an application of a sequent calculus framework to the formalization of definite descriptions. It is a continuation of research undertaken in [20, 22]. In the present paper a so-called free description theory is examined in the context of different kinds of free logic, including systems applied in computer science and constructive mathematics for dealing with partial functions. It is shown that the same theory in different logics may be formalised by means of different rules and gives results of (...)
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  44. The Poset of All Logics II: Leibniz Classes and Hierarchy.R. Jansana & T. Moraschini - forthcoming - Journal of Symbolic Logic:1-39.
    A Leibniz class is a class of logics closed under the formation of term-equivalent logics, compatible expansions, and non-indexed products of sets of logics. We study the complete lattice of all Leibniz classes, called the Leibniz hierarchy. In particular, it is proved that the classes of truth-equational and assertional logics are meet-prime in the Leibniz hierarchy, while the classes of protoalgebraic and equivalential logics are meet-reducible. However, the last two classes are shown to be determined by Leibniz conditions consisting of (...)
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  45. Boolean Types in Dependent Theories.Itay Kaplan, Ori Segel & Saharon Shelah - forthcoming - Journal of Symbolic Logic:1-32.
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  46. Modal Homotopy Type Theory. The Prospect of a New Logic for Philosophy.A. Klev & C. Zwanziger - forthcoming - History and Philosophy of Logic:1-6.
    The theory referred to by the—perhaps intimidating—main title of this book is an extension of Per Martin-Löf's dependent type theory. Much philosophical work pertaining to dependent type theory tak...
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  47. On Shavrukov’s Non-Isomorphism Theorem for Diagonalizable Algebras.Evgeny A. Kolmakov - forthcoming - Review of Symbolic Logic:1-39.
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  48. Approximate Counting and NP Search Problems.Leszek Aleksander Kołodziejczyk & Neil Thapen - forthcoming - Journal of Mathematical Logic.
    We study a new class of NP search problems, those which can be proved total using standard combinatorial reasoning based on approximate counting. Our model for this kind of reasoning is the bounded...
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  49. Tractarian Logicism: Operations, Numbers, Induction.Gregory Landini - forthcoming - Review of Symbolic Logic:1-41.
  50. Most(?) Theories Have Borel Complete Reducts.Michael C. Laskowski & Douglas S. Ulrich - forthcoming - Journal of Symbolic Logic:1-9.
    We prove that many seemingly simple theories have Borel complete reducts. Specifically, if a countable theory has uncountably many complete one-types, then it has a Borel complete reduct. Similarly, if $Th$ is not small, then $M^{eq}$ has a Borel complete reduct, and if a theory T is not $\omega $ -stable, then the elementary diagram of some countable model of T has a Borel complete reduct.
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