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  1. The Fundamental Theorem of Central Element Theory.Mariana Vanesa Badano & Diego Jose Vaggione - 2020 - Journal of Symbolic Logic 85 (4):1599-1606.
    We give a short proof of the fundamental theorem of central element theory. The original proof is constructive and very involved and relies strongly on the fact that the class be a variety. Here we give a more direct nonconstructive proof which applies for the more general case of a first-order class which is both closed under the formation of direct products and direct factors.
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  2. On the Correspondence Between Nested Calculi and Semantic Systems for Intuitionistic Logics.Tim Lyon - 2021 - Journal of Logic and Computation 31 (1):213-265.
    This paper studies the relationship between labelled and nested calculi for propositional intuitionistic logic, first-order intuitionistic logic with non-constant domains and first-order intuitionistic logic with constant domains. It is shown that Fitting’s nested calculi naturally arise from their corresponding labelled calculi—for each of the aforementioned logics—via the elimination of structural rules in labelled derivations. The translational correspondence between the two types of systems is leveraged to show that the nested calculi inherit proof-theoretic properties from their associated labelled calculi, such as (...)
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  3. G. Priest's An Introduction to Non-Classical Logic (2001). [REVIEW]Hans-Peter Leeb - 2003 - History and Philosophy of Logic 24:65-66.
    The review gives a short description of the content of the book and discusses the treatment of conditionals in it.
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  4. El lugar de la lógica en el razonamiento jurídico.Miguel Garcia-Godinez - 2019 - In Gerardo Ramirez & Manuel Jimenez (eds.), Ensayos de retórica jurídica. Mexico City, CDMX, Mexico: pp. 171-180.
  5. Extended Syllogistics in Calculus CL.Jens Lemanski - 2021 - Journal of Applied Logics 8 (2):557-577.
    Extensions of traditional syllogistics have been increasingly researched in philosophy, linguistics, and areas such as artificial intelligence and computer science in recent decades. This is mainly due to the fact that syllogistics is seen as a logic that comes very close to natural language abilities. Various forms of extended syllogistics have become established. This paper deals with the question to what extent a syllogistic representation in CL diagrams can be seen as a form of extended syllogistics. It will be shown (...)
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  6. An Algebraic Study of Tense Operators on Nelson Algebras.A. V. Figallo, G. Pelaitay & J. Sarmiento - 2021 - Studia Logica 109 (2):285-312.
    Ewald considered tense operators G, H, F and P on intuitionistic propositional calculus and constructed an intuitionistic tense logic system called IKt. In 2014, Figallo and Pelaitay introduced the variety IKt of IKt-algebras and proved that the IKt system has IKt-algebras as algebraic counterpart. In this paper, we introduce and study the variety of tense Nelson algebras. First, we give some examples and we prove some properties. Next, we associate an IKt-algebra to each tense Nelson algebras. This result allowed us (...)
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  7. Double Negation Semantics for Generalisations of Heyting Algebras.Rob Arthan & Paulo Oliva - 2020 - Studia Logica 109 (2):341-365.
    This paper presents an algebraic framework for investigating proposed translations of classical logic into intuitionistic logic, such as the four negative translations introduced by Kolmogorov, Gödel, Gentzen and Glivenko. We view these as variant semantics and present a semantic formulation of Troelstra’s syntactic criteria for a satisfactory negative translation. We consider how each of the above-mentioned translation schemes behaves on two generalisations of Heyting algebras: bounded pocrims and bounded hoops. When a translation fails for a particular class of algebras, we (...)
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  8. Kripke Semantics for Intuitionistic Łukasiewicz Logic.A. Lewis-Smith, P. Oliva & E. Robinson - 2021 - Studia Logica 109 (2):313-339.
    This paper proposes a generalization of the Kripke semantics of intuitionistic logic IL appropriate for intuitionistic Łukasiewicz logic IŁL — a logic in the intersection between IL and (classical) Łukasiewicz logic. This generalised Kripke semantics is based on the poset sum construction, used in Bova and Montagna (Theoret Comput Sci 410(12):1143–1158, 2009) to show the decidability (and PSPACE completeness) of the quasiequational theory of commutative, integral and bounded GBL algebras. The main idea is that w \Vdash \sigma—which for IL is (...)
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  9. A Conservative Negation Extension of Positive Semilattice Logic Without the Finite Model Property.Yale Weiss - 2021 - Studia Logica 109 (1):125-136.
    In this article, I present a semantically natural conservative extension of Urquhart’s positive semilattice logic with a sort of constructive negation. A subscripted sequent calculus is given for this logic and proofs of its soundness and completeness are sketched. It is shown that the logic lacks the finite model property. I discuss certain questions Urquhart has raised concerning the decision problem for the positive semilattice logic in the context of this logic and pose some problems for further research.
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  10. Semi De Morgan Logic Properly Displayed.Giuseppe Greco, Fei Liang, M. Andrew Moshier & Alessandra Palmigiano - 2021 - Studia Logica 109 (1):1-45.
    In the present paper, we endow semi De Morgan logic and a family of its axiomatic extensions with proper multi-type display calculi which are sound, complete, conservative, and enjoy cut elimination and subformula property. Our proposal builds on an algebraic analysis of the variety of semi De Morgan algebras, and applies the guidelines of the multi-type methodology in the design of display calculi.
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  11. Routley Star and Hyperintensionality.Sergei Odintsov & Heinrich Wansing - 2021 - Journal of Philosophical Logic 50 (1):33-56.
    We compare the logic HYPE recently suggested by H. Leitgeb as a basic propositional logic to deal with hyperintensional contexts and Heyting-Ockham logic introduced in the course of studying logical aspects of the well-founded semantics for logic programs with negation. The semantics of Heyting-Ockham logic makes use of the so-called Routley star negation. It is shown how the Routley star negation can be obtained from Dimiter Vakarelov’s theory of negation and that propositional HYPE coincides with the logic characterized by the (...)
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  12. Brouwer’s Weak Counterexamples and the Creative Subject: A Critical Survey.Peter Fletcher - 2020 - Journal of Philosophical Logic 49 (6):1111-1157.
    I survey Brouwer’s weak counterexamples to classical theorems, with a view to discovering what useful mathematical work is done by weak counterexamples; whether they are rigorous mathematical proofs or just plausibility arguments; the role of Brouwer’s notion of the creative subject in them, and whether the creative subject is really necessary for them; what axioms for the creative subject are needed; what relation there is between these arguments and Brouwer’s theory of choice sequences. I refute one of Brouwer’s claims with (...)
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  13. Transfinite Meta-inferences.Chris Scambler - 2020 - Journal of Philosophical Logic 49 (6):1079-1089.
    In Barrio et al. Barrio Pailos and Szmuc prove that there are systems of logic that agree with classical logic up to any finite meta-inferential level, and disagree with it thereafter. This article presents a generalized sense of meta-inference that extends into the transfinite, and proves analogous results to all transfinite orders.
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  14. A General Framework for $$ {FDE}$$ FDE -Based Modal Logics.Sergey Drobyshevich - 2020 - Studia Logica 108 (6):1281-1306.
    We develop a general theory of FDE-based modal logics. Our framework takes into account the four-valued nature of FDE by considering four partially defined modal operators corresponding to conditions for verifying and falsifying modal necessity and possibility operators. The theory comes with a uniform characterization for all obtained systems in terms of FDE-style formula-formula sequents. We also develop some correspondence theory and show how Hilbert-style axiom systems can be obtained in appropriate cases. Finally, we outline how different systems from the (...)
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  15. Reasoning Continuously: A Formal Construction of Continuous Proofs.T. D. P. Brunet & E. Fisher - 2020 - Studia Logica 108 (6):1145-1160.
    We begin with the idea that lines of reasoning are continuous mental processes and develop a notion of continuity in proof. This requires abstracting the notion of a proof as a set of sentences ordered by provability. We can then distinguish between discrete steps of a proof and possibly continuous stages, defining indexing functions to pick these out. Proof stages can be associated with the application of continuously variable rules, connecting continuity in lines of reasoning with continuously variable reasons. Some (...)
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  16. Definable Operators on Stable Set Lattices.Robert Goldblatt - 2020 - Studia Logica 108 (6):1263-1280.
    A fundamental result from Boolean modal logic states that a first-order definable class of Kripke frames defines a logic that is validated by all of its canonical frames. We generalise this to the level of non-distributive logics that have a relational semantics provided by structures based on polarities. Such structures have associated complete lattices of stable subsets, and these have been used to construct canonical extensions of lattice-based algebras. We study classes of structures that are closed under ultraproducts and whose (...)
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  17. A Canonical Model for Constant Domain Basic First-Order Logic.Ben Middleton - 2020 - Studia Logica 108 (6):1307-1323.
    I build a canonical model for constant domain basic first-order logic (BQLCD), the constant domain first-order extension of Visser’s basic propositional logic, and use the canonical model to verify that BQLCD satisfies the disjunction and existence properties.
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  18. Supervaluationism, Subvaluationism and the Sorites Paradox.Pablo Cobreros & Luca Tranchini - 2019 - In Elia Zardini & Sergi Oms (eds.), The Sorites Paradox. Cambridge, Reino Unido: pp. 38-62.
    One way in which we might approach the challenge posed by the Sorites Paradox is considering that Sorites-susceptible predicates have several candidate extensions, or several ways in which these expressions can be made precise. For example, a candidate extension for the predicate ‘is a baby’ is the set of humans of less than two years, but also the set of those less than two years and one second, and of those less than two years and two seconds. In this chapter (...)
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  19. Semantics for Pure Theories of Connexive Implication.Yale Weiss - forthcoming - Review of Symbolic Logic:1-16.
    In this article, I provide Urquhart-style semilattice semantics for three connexive logics in an implication-negation language (I call these “pure theories of connexive implication”). The systems semantically characterized include the implication-negation fragment of a connexive logic of Wansing, a relevant connexive logic recently developed proof-theoretically by Francez, and an intermediate system that is novel to this article. Simple proofs of soundness and completeness are given and the semantics is used to establish various facts about the systems (e.g., that two of (...)
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  20. The Quantified Argument Calculus and Natural Logic.Hanoch Ben-Yami - forthcoming - Dialectica.
    The formalisation of Natural Language arguments in a formal language close to it in syntax has been a central aim of Moss’s Natural Logic. I examine how the Quantified Argument Calculus (Quarc) can handle the inferences Moss has considered. I show that they can be incorporated in existing versions of Quarc or in straightforward extensions of it, all within sound and complete systems. Moreover, Quarc is closer in some respects to Natural Language than are Moss’s systems – for instance, is (...)
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  21. Intuitionistic Non-normal Modal Logics: A General Framework.Tiziano Dalmonte, Charles Grellois & Nicola Olivetti - 2020 - Journal of Philosophical Logic 49 (5):833-882.
    We define a family of intuitionistic non-normal modal logics; they can be seen as intuitionistic counterparts of classical ones. We first consider monomodal logics, which contain only Necessity or Possibility. We then consider the more important case of bimodal logics, which contain both modal operators. In this case we define several interactions between Necessity and Possibility of increasing strength, although weaker than duality. We thereby obtain a lattice of 24 distinct bimodal logics. For all logics we provide both a Hilbert (...)
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  22. A Paraconsistent Conditional Logic.Minghui Ma & Chun-Ting Wong - 2020 - Journal of Philosophical Logic 49 (5):883-903.
    We develop a paraconsistent logic by introducing new models for conditionals with acceptive and rejective selection functions which are variants of Chellas’ conditional models. The acceptance and rejection conditions are substituted for truth conditions of conditionals. The paraconsistent conditional logic is axiomatized by a sequent system \ which is an extension of the Belnap-Dunn four-valued logic with a conditional operator. Some acceptive extensions of \ are shown to be sound and complete. We also show the finite acceptive model property and (...)
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  23. A Relevant Logic of Questions.Vít Punčochář - 2020 - Journal of Philosophical Logic 49 (5):905-939.
    This paper introduces the inquisitive extension of R, denoted as InqR, which is a relevant logic of questions based on the logic R as the background logic of declaratives. A semantics for InqR is developed, and it is shown that this semantics is, in a precisely defined sense, dual to Routley-Meyer semantics for R. Moreover, InqR is axiomatized and completeness of the axiomatic system is established. The philosophical interpretation of the duality between Routley-Meyer semantics and the semantics for InqR is (...)
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  24. A Characteristic Frame for Positive Intuitionistic and Relevance Logic.Yale Weiss - forthcoming - Studia Logica:1-13.
    I show that the lattice of the positive integers ordered by division is characteristic for Urquhart’s positive semilattice relevance logic; that is, a formula is valid in positive semilattice relevance logic if and only if it is valid in all models over the positive integers ordered by division. I show that the same frame is characteristic for positive intuitionistic logic, where the class of models over it is restricted to those satisfying a heredity condition. The results of this article highlight (...)
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  25. Impossible Worlds, by Francesco Berto and Mark Jago. [REVIEW]Koji Tanaka - forthcoming - Mind.
    Book Review of Impossible Worlds, by Francesco Berto and Mark Jago. Oxford: Oxford University Press, 2019.
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  26. Structural Proof Theory for First-Order Weak Kleene Logics.Andreas Fjellstad - 2020 - Journal of Applied Non-Classical Logics 30 (3):272-289.
    This paper presents a sound and complete five-sided sequent calculus for first-order weak Kleene valuations which permits not only elegant representations of four logics definable on first-order weak Kleene valuations, but also admissibility of five cut rules by proof analysis.
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  27. The Normal and Self-Extensional Extension of Dunn–Belnap Logic.Arnon Avron - 2020 - Logica Universalis 14 (3):281-296.
    A logic \ is called self-extensional if it allows to replace occurrences of a formula by occurrences of an \-equivalent one in the context of claims about logical consequence and logical validity. It is known that no three-valued paraconsistent logic which has an implication can be self-extensional. In this paper we show that in contrast, the famous Dunn–Belnap four-valued logic has exactly one self-extensional four-valued extension which has an implication. We also investigate the main properties of this logic, determine the (...)
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  28. Exactly True and Non-Falsity Logics Meeting Infectious Ones.Alex Belikov & Yaroslav Petrukhin - 2020 - Journal of Applied Non-Classical Logics 30 (2):93-122.
    In this paper, we study logical systems which represent entailment relations of two kinds. We extend the approach of finding ‘exactly true’ and ‘non-falsity’ versions of four-valued logics that emerged in series of recent works [Pietz & Rivieccio (2013). Nothing but the truth. Journal of Philosophical Logic, 42(1), 125–135; Shramko (2019). Dual-Belnap logic and anything but falsehood. Journal of Logics and their Applications, 6, 413–433; Shramko et al. (2017). First-degree entailment and its relatives. Studia Logica, 105(6), 1291–1317] to the case (...)
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  29. The Inextricable Link Between Conditionals and Logical Consequence.Matheus Silva - manuscript
    There is a profound, but frequently ignored, relationship between the classical conception of logical consequence and the material implication. The first repeats the patterns of the latter, but with a wider modal reach. This relationship suggests that there should be also a connection between the notion of logical consequence and the conditional connective of any given logical system. This implies, among other things, that it is incoherent to propose alternatives to the material implication while maintaining the classical conception of logical (...)
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  30. Proof Theory of Paraconsistent Weak Kleene Logic.Francesco Paoli & Michele Pra Baldi - 2020 - Studia Logica 108 (4):779-802.
    Paraconsistent Weak Kleene Logic is the 3-valued propositional logic defined on the weak Kleene tables and with two designated values. Most of the existing proof systems for PWK are characterised by the presence of linguistic restrictions on some of their rules. This feature can be seen as a shortcoming. We provide a cut-free calculus for PWK that is devoid of such provisos. Moreover, we introduce a Priest-style tableaux calculus for PWK.
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  31. Intuitionistic Conditional Logics.Ivano Ciardelli & Xinghan Liu - 2020 - Journal of Philosophical Logic 49 (4):807-832.
    Building on recent work by Yale Weiss, we study conditional logics in the intuitionistic setting. We consider a number of semantic conditions which give rise, among others, to intuitionistic counterparts of Lewis’s logic VC and Stalnaker’s C2. We show how to obtain a sound and complete axiomatization of each logic arising from a combination of these conditions. On the way, we remark how, in the intuitionistic setting, certain classically equivalent principles of conditional logic come apart, and how certain logical connections (...)
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  32. Kurt Gödel's Anticipation of the Turing Machine: A Vitalistic Approach.Tim Lethen - 2020 - History and Philosophy of Logic 41 (3):252-264.
    In 1935/1936 Kurt Gödel wrote three notebooks on the foundations of quantum mechanics, which have now been entirely transcribed for the first time. Whereas a lot of the material is rather technical in character, many of Gödel's remarks have a philosophical background and concentrate on Leibnizian monadology as well as on vitalism. Obviously influenced by the vitalistic writings of Hans Driesch and his ‘proofs’ for the existence of an entelechy in every living organism, Gödel briefly develops the idea of a (...)
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  33. Epimorphism Surjectivity in Varieties of Heyting Algebras.T. Moraschini & J. J. Wannenburg - 2020 - Annals of Pure and Applied Logic 171 (9):102824.
    It was shown recently that epimorphisms need not be surjective in a variety K of Heyting algebras, but only one counter-example was exhibited in the literature until now. Here, a continuum of such examples is identified, viz. the variety generated by the Rieger-Nishimura lattice, and all of its (locally finite) subvarieties that contain the original counter-example K . It is known that, whenever a variety of Heyting algebras has finite depth, then it has surjective epimorphisms. In contrast, we show that (...)
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  34. Eigenlogic in the Spirit of George Boole.Zeno Toffano - 2020 - Logica Universalis 14 (2):175-207.
    This work presents an operational and geometric approach to logic. It starts from the multilinear elective decomposition of binary logical functions in the original form introduced by George Boole. A justification on historical grounds is presented bridging Boole’s theory and the use of his arithmetical logical functions with the axioms of Boolean algebra using sets and quantum logic. It is shown that this algebraic polynomial formulation can be naturally extended to operators in finite vector spaces. Logical operators will appear as (...)
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  35. Positive Amalgamation.Mohammed Belkasmi - 2020 - Logica Universalis 14 (2):243-258.
    We study the amalgamation property in positive logic, where we shed light on some connections between the amalgamation property, Robinson theories, model-complete theories and the Hausdorff property.
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  36. Modal Extension of Ideal Paraconsistent Four-Valued Logic and its Subsystem.Norihiro Kamide & Yoni Zohar - 2020 - Annals of Pure and Applied Logic 171 (10):102830.
    This study aims to introduce a modal extension M4CC of Arieli, Avron, and Zamansky's ideal paraconsistent four-valued logic 4CC as a Gentzen-type sequent calculus and prove the Kripke-completeness and cut-elimination theorems for M4CC. The logic M4CC is also shown to be decidable and embeddable into the normal modal logic S4. Furthermore, a subsystem of M4CC, which has some characteristic properties that do not hold for M4CC, is introduced and the Kripke-completeness and cut-elimination theorems for this subsystem are proved. This subsystem (...)
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  37. Algebraically Closed Structures in Positive Logic.Mohammed Belkasmi - 2020 - Annals of Pure and Applied Logic 171 (9):102822.
    In this paper we extend of the notion of algebraically closed given in the case of groups and skew fields to an arbitrary h-inductive theory. The main subject of this paper is the study of the notion of positive algebraic closedness and its relationship with the notion of positive closedness and the amalgamation property.
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  38. A Refined Interpretation of Intuitionistic Logic by Means of Atomic Polymorphism.José Espírito Santo & Gilda Ferreira - 2020 - Studia Logica 108 (3):477-507.
    We study an alternative embedding of IPC into atomic system F whose translation of proofs is based, not on instantiation overflow, but instead on the admissibility of the elimination rules for disjunction and absurdity. As compared to the embedding based on instantiation overflow, the alternative embedding works equally well at the levels of provability and preservation of proof identity, but it produces shorter derivations and shorter simulations of reduction sequences. Lambda-terms are employed in the technical development so that the algorithmic (...)
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  39. Interpolation in Extensions of First-Order Logic.Guido Gherardi, Paolo Maffezioli & Eugenio Orlandelli - 2020 - Studia Logica 108 (3):619-648.
    We prove a generalization of Maehara’s lemma to show that the extensions of classical and intuitionistic first-order logic with a special type of geometric axioms, called singular geometric axioms, have Craig’s interpolation property. As a corollary, we obtain a direct proof of interpolation for first-order logic with identity, as well as interpolation for several mathematical theories, including the theory of equivalence relations, partial and linear orders, and various intuitionistic order theories such as apartness and positive partial and linear orders.
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  40. Completeness and Cut-Elimination for First-Order Ideal Paraconsistent Four-Valued Logic.Norihiro Kamide & Yoni Zohar - 2020 - Studia Logica 108 (3):549-571.
    In this study, we prove the completeness and cut-elimination theorems for a first-order extension F4CC of Arieli, Avron, and Zamansky’s ideal paraconsistent four-valued logic known as 4CC. These theorems are proved using Schütte’s method, which can simultaneously prove completeness and cut-elimination.
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  41. Rosser Provability and Normal Modal Logics.Taishi Kurahashi - 2020 - Studia Logica 108 (3):597-617.
    In this paper, we investigate Rosser provability predicates whose provability logics are normal modal logics. First, we prove that there exists a Rosser provability predicate whose provability logic is exactly the normal modal logic \. Secondly, we introduce a new normal modal logic \ which is a proper extension of \, and prove that there exists a Rosser provability predicate whose provability logic includes \.
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  42. Simplified Kripke-Style Semantics for Some Normal Modal Logics.Andrzej Pietruszczak, Mateusz Klonowski & Yaroslav Petrukhin - 2020 - Studia Logica 108 (3):451-476.
    Pietruszczak :163–171, 2009. https://doi.org/10.12775/LLP.2009.013) proved that the normal logics \, \ ), \ are determined by suitable classes of simplified Kripke frames of the form \, where \. In this paper, we extend this result. Firstly, we show that a modal logic is determined by a class composed of simplified frames if and only if it is a normal extension of \. Furthermore, a modal logic is a normal extension of \ ; \; \) if and only if it is (...)
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  43. Paraconsistent Logics From a Philosophical Point of View.Diogo Dias - 2012 - Cognitio-Estudos 9 (2):139-148.
    This article begins with a general and abstract definition of logic and, particularly, of paraconsistent logics, to establish a common ground for the discussion. Briefly stating, these kinds of logics have the property of being non-explosive, that is, it is not possible to infer any conclusion from contradictory premises. Using these definitions, it is possible to analyze some of the philosophical aspects of paraconsistent logics, in particular, the relation between the notion of explosion and the law of non-contradiction, as well (...)
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  44. A Note on FDE “All the Way Up”.Jc Beall & Caleb Camrud - 2020 - Notre Dame Journal of Formal Logic 61 (2):283-296.
    A very natural and philosophically important subclassical logic is FDE. This account of logical consequence can be seen as going beyond the standard two-valued account to a four-valued account. A natural question arises: What account of logical consequence arises from considering further combinations of such values? A partial answer was given by Priest in 2014; Shramko and Wansing had also given a partial result some years earlier, although in a different context. In this note we generalize Priest’s result to show (...)
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  45. Formal Notes on the Substitutional Analysis of Logical Consequence.Volker Halbach - 2020 - Notre Dame Journal of Formal Logic 61 (2):317-339.
    Logical consequence in first-order predicate logic is defined substitutionally in set theory augmented with a primitive satisfaction predicate: an argument is defined to be logically valid if and only if there is no substitution instance with true premises and a false conclusion. Substitution instances are permitted to contain parameters. Variants of this definition of logical consequence are given: logical validity can be defined with or without identity as a logical constant, and quantifiers can be relativized in substitution instances or not. (...)
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  46. The Symbolic Epistemological Implications of the Different Mythological Set Up of the (Egyptian)-Mesopotamian Culture Compared to the Grecian One.Donato Santarcangelo - 2017 - Enkelados 6.
    The Mesopotamian peoples were never really dominated by the reason the way we conceptualize it. It's to the revelation as direct emanation of the divine that they ascribed the appearance of knowledge.
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  47. Belnap–Dunn Modal Logics: Truth Constants Vs. Truth Values.Sergei P. Odintsov & Stanislav O. Speranski - 2020 - Review of Symbolic Logic 13 (2):416-435.
    We shall be concerned with the modal logic BK—which is based on the Belnap–Dunn four-valued matrix, and can be viewed as being obtained from the least normal modal logic K by adding ‘strong negation’. Though all four values ‘truth’, ‘falsity’, ‘neither’ and ‘both’ are employed in its Kripke semantics, only the first two are expressible as terms. We show that expanding the original language of BK to include constants for ‘neither’ or/and ‘both’ leads to quite unexpected results. To be more (...)
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  48. A Fully Classical Truth Theory Characterized by Substructural Means.Federico Matías Pailos - 2020 - Review of Symbolic Logic 13 (2):249-268.
    We will present a three-valued consequence relation for metainferences, called CM, defined through ST and TS, two well known substructural consequence relations for inferences. While ST recovers every classically valid inference, it invalidates some classically valid metainferences. While CM works as ST at the inferential level, it also recovers every classically valid metainference. Moreover, CM can be safely expanded with a transparent truth predicate. Nevertheless, CM cannot recapture every classically valid meta-metainference. We will afterwards develop a hierarchy of consequence relations (...)
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  49. Two New Series of Principles in the Interpretability Logic of All Reasonable Arithmetical Theories.Evan Goris & Joost J. Joosten - 2020 - Journal of Symbolic Logic 85 (1):1-25.
    The provability logic of a theory T captures the structural behavior of formalized provability in T as provable in T itself. Like provability, one can formalize the notion of relative interpretability giving rise to interpretability logics. Where provability logics are the same for all moderately sound theories of some minimal strength, interpretability logics do show variations.The logic IL is defined as the collection of modal principles that are provable in any moderately sound theory of some minimal strength. In this article (...)
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  50. Chaitin’s Ω as a Continuous Function.Rupert Hölzl, Wolfgang Merkle, Joseph Miller, Frank Stephan & Liang Yu - 2020 - Journal of Symbolic Logic 85 (1):486-510.
    We prove that the continuous function${\rm{\hat \Omega }}:2^\omega \to $ that is defined via$X \mapsto \mathop \sum \limits_n 2^{ - K\left} $ for all $X \in {2^\omega }$ is differentiable exactly at the Martin-Löf random reals with the derivative having value 0; that it is nowhere monotonic; and that $\mathop \smallint \nolimits _0^1{\rm{\hat{\Omega }}}\left\,{\rm{d}}X$ is a left-c.e. $wtt$-complete real having effective Hausdorff dimension ${1 / 2}$.We further investigate the algorithmic properties of ${\rm{\hat{\Omega }}}$. For example, we show that the maximal (...)
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