Results for 'Intuitionistic logic'

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  1. On an Intuitionistic Logic for Pragmatics.Gianluigi Bellin, Massimiliano Carrara & Daniele Chiffi - 2018 - Journal of Logic and Computation 50 (28):935–966..
    We reconsider the pragmatic interpretation of intuitionistic logic [21] regarded as a logic of assertions and their justi cations and its relations with classical logic. We recall an extension of this approach to a logic dealing with assertions and obligations, related by a notion of causal implication [14, 45]. We focus on the extension to co-intuitionistic logic, seen as a logic of hypotheses [8, 9, 13] and on polarized bi-intuitionistic logic (...)
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  2. Intuitionistic logic and its philosophy.Panu Raatikainen - 2013 - Al-Mukhatabat. A Trilingual Journal For Logic, Epistemology and Analytical Philosophy (6):114-127.
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  3. Vagueness and Intuitionistic Logic.Ian Rumfitt - forthcoming - In Alexander Miller (ed.), Language, Logic,and Mathematics: Themes from the Philosophy of Crispin Wright. Oxford University Press.
    In his essay ‘“Wang’s Paradox”’, Crispin Wright proposes a solution to the Sorites Paradox (in particular, the form of it he calls the ‘Paradox of Sharp Boundaries’) that involves adopting intuitionistic logic when reasoning with vague predicates. He does not give a semantic theory which accounts for the validity of intuitionistic logic (and the invalidity of stronger logics) in that area. The present essay tentatively makes good the deficiency. By applying a theorem of Tarski, it shows (...)
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  4.  87
    Syntactic Interpolation for Tense Logics and Bi-Intuitionistic Logic via Nested Sequents.Tim Lyon, Alwen Tiu, Rajeev Gore & Ranald Clouston - 2020 - In Maribel Fernandez & Anca Muscholl (eds.), 28th EACSL Annual Conference on Computer Science Logic (CSL 2020). Dagstuhl, Germany: pp. 1-16.
    We provide a direct method for proving Craig interpolation for a range of modal and intuitionistic logics, including those containing a "converse" modality. We demonstrate this method for classical tense logic, its extensions with path axioms, and for bi-intuitionistic logic. These logics do not have straightforward formalisations in the traditional Gentzen-style sequent calculus, but have all been shown to have cut-free nested sequent calculi. The proof of the interpolation theorem uses these calculi and is purely syntactic, (...)
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  5.  77
    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 (...)
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  6. On Deriving Nested Calculi for Intuitionistic Logics from Semantic Systems.Tim Lyon - 2020 - In Sergei Artemov & Anil Nerode (eds.), Logical Foundations of Computer Science. Cham: pp. 177-194.
    This paper shows how to derive nested calculi from labelled calculi for propositional intuitionistic logic and first-order intuitionistic logic with constant domains, thus connecting the general results for labelled calculi with the more refined formalism of nested sequents. The extraction of nested calculi from labelled calculi obtains via considerations pertaining to the elimination of structural rules in labelled derivations. Each aspect of the extraction process is motivated and detailed, showing that each nested calculus inherits favorable proof-theoretic (...)
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  7. Normalisation and subformula property for a system of intuitionistic logic with general introduction and elimination rules.Nils Kürbis - 2021 - Synthese 199 (5-6):14223-14248.
    This paper studies a formalisation of intuitionistic logic by Negri and von Plato which has general introduction and elimination rules. The philosophical importance of the system is expounded. Definitions of ‘maximal formula’, ‘segment’ and ‘maximal segment’ suitable to the system are formulated and corresponding reduction procedures for maximal formulas and permutative reduction procedures for maximal segments given. Alternatives to the main method used are also considered. It is shown that deductions in the system convert into normal form and (...)
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  8. Buying Logical Principles with Ontological Coin: The Metaphysical Lessons of Adding epsilon to Intuitionistic Logic.David DeVidi & Corey Mulvihill - 2017 - IfCoLog Journal of Logics and Their Applications 4 (2):287-312.
    We discuss the philosophical implications of formal results showing the con- sequences of adding the epsilon operator to intuitionistic predicate logic. These results are related to Diaconescu’s theorem, a result originating in topos theory that, translated to constructive set theory, says that the axiom of choice (an “existence principle”) implies the law of excluded middle (which purports to be a logical principle). As a logical choice principle, epsilon allows us to translate that result to a logical setting, where (...)
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  9.  52
    Set theory INC# based on intuitionistic logic with restricted modus ponens rule.Jaykov Foukzon (ed.) - 2021 - AP LAMBERT Academic Publishing (June 23, 2021).
    In this book set theory INC# based on intuitionistic logic with restricted modus ponens rule is proposed. It proved that intuitionistic logic with restricted modus ponens rule can to safe Cantor naive set theory from a triviality. Similar results for paraconsistent set theories were obtained in author papers [13]-[16].
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  10. Set Theory INC# Based on Intuitionistic Logic with Restricted Modus Ponens Rule (Part. I).Jaykov Foukzon - 2021 - Journal of Advances in Mathematics and Computer Science 36 (2):73-88.
    In this article Russell’s paradox and Cantor’s paradox resolved successfully using intuitionistic logic with restricted modus ponens rule.
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  11. Five Observations Concerning the Intended Meaning of the Intuitionistic Logical Constants.Gustavo Fernández Díez - 2000 - Journal of Philosophical Logic 29 (4):409-424.
    This paper contains five observations concerning the intended meaning of the intuitionistic logical constants: (1) if the explanations of this meaning are to be based on a non-decidable concept, that concept should not be that of 'proof'; (2) Kreisel's explanations using extra clauses can be significantly simplified; (3) the impredicativity of the definition of → can be easily and safely ameliorated; (4) the definition of → in terms of 'proofs from premises' results in a loss of the inductive character (...)
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  12.  66
    Set Theory INC# Based on Infinitary Intuitionistic Logic with Restricted Modus Ponens Rule (Part.II) Hyper inductive definitions.Jaykov Foukzon - 2021 - Journal of Advances in Mathematics and Computer Science 36 (4):22.
    In this paper intuitionistic set theory INC# in infinitary set theoretical language is considered. External induction principle in nonstandard intuitionistic arithmetic were derived. Non trivial application in number theory is considered.
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  13. Existence Assumptions and Logical Principles: Choice Operators in Intuitionistic Logic.Corey Edward Mulvihill - 2015 - Dissertation, University of Waterloo
    Hilbert’s choice operators τ and ε, when added to intuitionistic logic, strengthen it. In the presence of certain extensionality axioms they produce classical logic, while in the presence of weaker decidability conditions for terms they produce various superintuitionistic intermediate logics. In this thesis, I argue that there are important philosophical lessons to be learned from these results. To make the case, I begin with a historical discussion situating the development of Hilbert’s operators in relation to his evolving (...)
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  14.  42
    Book "Set theory INC^# based on intuitionistic logic with restricted modus ponens rule".Jaykov Foukzon - 2021 - LAP LAMBERT Academic Publishing.
    In this book set theory INC# based on intuitionistic logic with restricted modus ponens rule is proposed. It proved that intuitionistic logic with restricted modus ponens rule can to safe Cantor naive set theory from a triviality.
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  15. Intuitionism and the Modal Logic of Vagueness.Susanne Bobzien & Ian Rumfitt - 2020 - Journal of Philosophical Logic 49 (2):221-248.
    Intuitionistic logic provides an elegant solution to the Sorites Paradox. Its acceptance has been hampered by two factors. First, the lack of an accepted semantics for languages containing vague terms has led even philosophers sympathetic to intuitionism to complain that no explanation has been given of why intuitionistic logic is the correct logic for such languages. Second, switching from classical to intuitionistic logic, while it may help with the Sorites, does not appear to (...)
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  16.  42
    Set Theory INC_{∞^{#}}^{#} Based on Infinitary Intuitionistic Logic with Restricted Modus Ponens Rule (Part III).Hyper inductive definitions. Application in transcendental number theory.Jaykov Foukzon - 2021 - Journal of Advances in Mathematics and Computer Science 36 (8):43.
    Main results are: (i) number e^{e} is transcendental; (ii) the both numbers e+π and e-π are irrational.
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  17.  19
    The commonly ignored aspects of the history of symmetries. Their link with intuitionist logic.Antonino Drago - manuscript
    The obscure and punctuated history of symmetry is compared with the history of the celebrated and exalting notion of infinitesimal; some considerations about them are derived. A long list of odd and hidden events concerning symmetry in theoretical physics is offered. The last event is the discovery of the nature of the same word “symmetry” which pertains to non-classical logic, and it is linked to the principle of sufficient reason. A comparison of the roles played by the two mathematical (...)
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  18. Bilateralist Detours: From Intuitionist to Classical Logic and Back.Nils Kürbis - 2017 - Logique Et Analyse 60 (239):301-316.
    There is widespread agreement that while on a Dummettian theory of meaning the justified logic is intuitionist, as its constants are governed by harmonious rules of inference, the situation is reversed on Huw Price's bilateralist account, where meanings are specified in terms of primitive speech acts assertion and denial. In bilateral logics, the rules for classical negation are in harmony. However, as it is possible to construct an intuitionist bilateral logic with harmonious rules, there is no formal argument (...)
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  19. Definite Descriptions in Intuitionist Positive Free Logic.Nils Kürbis - 2020 - Logic and Logical Philosophy 30:1.
    This paper presents rules of inference for a binary quantifier I for the formalisation of sentences containing definite descriptions within intuitionist positive free logic. I binds one variable and forms a formula from two formulas. Ix[F, G] means ‘The F is G’. The system is shown to have desirable proof-theoretic properties: it is proved that deductions in it can be brought into normal form. The discussion is rounded up by comparisons between the approach to the formalisation of definite descriptions (...)
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  20. From Intuitionism to Many-Valued Logics Through Kripke Models.Saeed Salehi - 2021 - In Mojtaba Mojtahedi, Shahid Rahman & Mohammad Saleh Zarepour (eds.), Mathematics, Logic, and their Philosophies: Essays in Honour of Mohammad Ardeshir. Springer. pp. 339-348.
    Intuitionistic Propositional Logic is proved to be an infinitely many valued logic by Gödel (Kurt Gödel collected works (Volume I) Publications 1929–1936, Oxford University Press, pp 222–225, 1932), and it is proved by Jaśkowski (Actes du Congrés International de Philosophie Scientifique, VI. Philosophie des Mathématiques, Actualités Scientifiques et Industrielles 393:58–61, 1936) to be a countably many valued logic. In this paper, we provide alternative proofs for these theorems by using models of Kripke (J Symbol Logic (...)
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  21. The ILLTP Library for Intuitionistic Linear Logic.Carlos Olarte, Valeria Correa Vaz De Paiva, Elaine Pimentel & Giselle Reis - manuscript
    Benchmarking automated theorem proving (ATP) systems using standardized problem sets is a well-established method for measuring their performance. However, the availability of such libraries for non-classical logics is very limited. In this work we propose a library for benchmarking Girard's (propositional) intuitionistic linear logic. For a quick bootstrapping of the collection of problems, and for discussing the selection of relevant problems and understanding their meaning as linear logic theorems, we use translations of the collection of Kleene's (...) theorems in the traditional monograph "Introduction to Metamathematics". We analyze four different translations of intuitionistic logic into linear logic and compare their proofs using a linear logic based prover with focusing. In order to enhance the set of problems in our library, we apply the three provability-preserving translations to the propositional benchmarks in the ILTP Library. Finally, we generate a comprehensive set of reachability problems for Petri nets and encode such problems as linear logic sequents, thus enlarging our collection of problems. (shrink)
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  22.  82
    Nested Sequents for Intuitionistic Modal Logics via Structural Refinement.Tim Lyon - 2021 - In Anupam Das & Sara Negri (eds.), Automated Reasoning with Analytic Tableaux and Related Methods: TABLEAUX 2021. 93413 Cham, Germany: pp. 409-427.
    We employ a recently developed methodology -- called "structural refinement" -- to extract nested sequent systems for a sizable class of intuitionistic modal logics from their respective labelled sequent systems. This method can be seen as a means by which labelled sequent systems can be transformed into nested sequent systems through the introduction of propagation rules and the elimination of structural rules, followed by a notational translation. The nested systems we obtain incorporate propagation rules that are parameterized with formal (...)
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  23.  43
    A Framework for Intuitionistic Grammar Logics.Tim Lyon - 2021 - In Pietro Baroni, Christoph Benzmüller & Yὶ N. Wang (eds.), Lecture Notes in Computer Science. 93413 Cham, Germany: pp. 495-503.
    We generalize intuitionistic tense logics to the multi-modal case by placing grammar logics on an intuitionistic footing. We provide axiomatizations for a class of base intuitionistic grammar logics as well as provide axiomatizations for extensions with combinations of seriality axioms and what we call "intuitionistic path axioms". We show that each axiomatization is sound and complete with completeness being shown via a typical canonical model construction.
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  24. One-step Modal Logics, Intuitionistic and Classical, Part 1.Harold T. Hodes - 2021 - Journal of Philosophical Logic 50 (5):837-872.
    This paper and its sequel “look under the hood” of the usual sorts of proof-theoretic systems for certain well-known intuitionistic and classical propositional modal logics. Section 1 is preliminary. Of most importance: a marked formula will be the result of prefixing a formula in a propositional modal language with a step-marker, for this paper either 0 or 1. Think of 1 as indicating the taking of “one step away from 0.” Deductions will be constructed using marked formulas. Section 2 (...)
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  25.  78
    One-Step Modal Logics, Intuitionistic and Classical, Part 2.Harold T. Hodes - 2021 - Journal of Philosophical Logic 50 (5):873-910.
    Part 1 [Hodes, 2021] “looked under the hood” of the familiar versions of the classical propositional modal logic K and its intuitionistic counterpart. This paper continues that project, addressing some familiar classical strengthenings of K and GL), and their intuitionistic counterparts. Section 9 associates two intuitionistic one-step proof-theoretic systems to each of the just mentioned intuitionistic logics, this by adding for each a new rule to those which generated IK in Part 1. For the systems (...)
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  26. A Binary Quantifier for Definite Descriptions in Intuitionist Negative Free Logic: Natural Deduction and Normalisation.Nils Kürbis - 2019 - Bulletin of the Section of Logic 48 (2):81-97.
    This paper presents a way of formalising definite descriptions with a binary quantifier ι, where ιx[F, G] is read as ‘The F is G’. Introduction and elimination rules for ι in a system of intuitionist negative free logic are formulated. Procedures for removing maximal formulas of the form ιx[F, G] are given, and it is shown that deductions in the system can be brought into normal form.
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  27. From Classical to Intuitionistic Probability.Brian Weatherson - 2003 - Notre Dame Journal of Formal Logic 44 (2):111-123.
    We generalize the Kolmogorov axioms for probability calculus to obtain conditions defining, for any given logic, a class of probability functions relative to that logic, coinciding with the standard probability functions in the special case of classical logic but allowing consideration of other classes of "essentially Kolmogorovian" probability functions relative to other logics. We take a broad view of the Bayesian approach as dictating inter alia that from the perspective of a given logic, rational degrees of (...)
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  28. Two Treatments of Definite Descriptions in Intuitionist Negative Free Logic.Nils Kürbis - 2019 - Bulletin of the Section of Logic 48 (4):299-317.
    Sentences containing definite descriptions, expressions of the form ‘The F’, can be formalised using a binary quantifier ι that forms a formula out of two predicates, where ιx[F, G] is read as ‘The F is G’. This is an innovation over the usual formalisation of definite descriptions with a term forming operator. The present paper compares the two approaches. After a brief overview of the system INFι of intuitionist negative free logic extended by such a quantifier, which was presented (...)
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  29. Hypersequents and the proof theory of intuitionistic fuzzy logic.Matthias Baaz & Richard Zach - 2000 - In Peter G. Clote & Helmut Schwichtenberg (eds.), Computer Science Logic. 14th International Workshop, CSL 2000. Berlin: Springer. pp. 187– 201.
    Takeuti and Titani have introduced and investigated a logic they called intuitionistic fuzzy logic. This logic is characterized as the first-order Gödel logic based on the truth value set [0,1]. The logic is known to be axiomatizable, but no deduction system amenable to proof-theoretic, and hence, computational treatment, has been known. Such a system is presented here, based on previous work on hypersequent calculi for propositional Gödel logics by Avron. It is shown that the (...)
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  30.  73
    Intuitionist Reasoning in the Tri-Unitarian Theology of Nicholas of Cues (1401-1464).Antonino Drago - 2019 - Journal of Applied Logic 6 (6):1143-1186.
    The main subject of Cusanus’ investigations was the name of God. He claimed to have achieved the best possible one, Not-Other. Since Cusanus stressed that these two words do not mean the corresponding affirmative word, i.e. the same, they represent the failure of the double negation law and there￾fore belong to non-classical, and above all, intuitionist logic. Some of his books implicitly applied intuitionist reasoning and the corresponding organization of a theory which is governed by intuitionist logic. A (...)
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  31. Conceptions of truth in intuitionism.Panu Raatikainen - 2004 - History and Philosophy of Logic 25 (2):131--45.
    Intuitionism’s disagreement with classical logic is standardly based on its specific understanding of truth. But different intuitionists have actually explicated the notion of truth in fundamentally different ways. These are considered systematically and separately, and evaluated critically. It is argued that each account faces difficult problems. They all either have implausible consequences or are viciously circular.
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  32. The Logic of Hyperlogic. Part A: Foundations.Alexander W. Kocurek - forthcoming - Review of Symbolic Logic:1-27.
    Hyperlogic is a hyperintensional system designed to regiment metalogical claims (e.g., "Intuitionistic logic is correct" or "The law of excluded middle holds") into the object language, including within embedded environments such as attitude reports and counterfactuals. This paper is the first of a two-part series exploring the logic of hyperlogic. This part presents a minimal logic of hyperlogic and proves its completeness. It consists of two interdefined axiomatic systems: one for classical consequence (truth preservation under a (...)
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  33. The logic of partitions: Introduction to the dual of the logic of subsets: The logic of partitions.David Ellerman - 2010 - Review of Symbolic Logic 3 (2):287-350.
    Modern categorical logic as well as the Kripke and topological models of intuitionistic logic suggest that the interpretation of ordinary “propositional” logic should in general be the logic of subsets of a given universe set. Partitions on a set are dual to subsets of a set in the sense of the category-theoretic duality of epimorphisms and monomorphisms—which is reflected in the duality between quotient objects and subobjects throughout algebra. If “propositional” logic is thus seen (...)
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  34. Failures of Categoricity and Compositionality for Intuitionistic Disjunction.Jack Woods - 2012 - Thought: A Journal of Philosophy 1 (4):281-291.
    I show that the model-theoretic meaning that can be read off the natural deduction rules for disjunction fails to have certain desirable properties. I use this result to argue against a modest form of inferentialism which uses natural deduction rules to fix model-theoretic truth-conditions for logical connectives.
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  35.  99
    Refining Labelled Systems for Modal and Constructive Logics with Applications.Tim Lyon - 2021 - Dissertation, Technischen Universität Wien
    This thesis introduces the "method of structural refinement", which serves as a means of transforming the relational semantics of a modal and/or constructive logic into an 'economical' proof system by connecting two proof-theoretic paradigms: labelled and nested sequent calculi. The formalism of labelled sequents has been successful in that cut-free calculi in possession of desirable proof-theoretic properties can be automatically generated for large classes of logics. Despite these qualities, labelled systems make use of a complicated syntax that explicitly incorporates (...)
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  36. Against Reflective Equilibrium for Logical Theorizing.Jack Woods - 2019 - Australasian Journal of Logic 16 (7):319.
    I distinguish two ways of developing anti-exceptionalist approaches to logical revision. The first emphasizes comparing the theoretical virtuousness of developed bodies of logical theories, such as classical and intuitionistic logic. I'll call this whole theory comparison. The second attempts local repairs to problematic bits of our logical theories, such as dropping excluded middle to deal with intuitions about vagueness. I'll call this the piecemeal approach. I then briefly discuss a problem I've developed elsewhere for comparisons of logical theories. (...)
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  37. The philosophy of alternative logics.Andrew Aberdein & Stephen Read - 2009 - In Leila Haaparanta (ed.), The Development of Modern Logic. Oxford University Press. pp. 613-723.
    This chapter focuses on alternative logics. It discusses a hierarchy of logical reform. It presents case studies that illustrate particular aspects of the logical revisionism discussed in the chapter. The first case study is of intuitionistic logic. The second case study turns to quantum logic, a system proposed on empirical grounds as a resolution of the antinomies of quantum mechanics. The third case study is concerned with systems of relevance logic, which have been the subject of (...)
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  38. Epsilon theorems in intermediate logics.Matthias Baaz & Richard Zach - 2022 - Journal of Symbolic Logic 87 (2):682-720.
    Any intermediate propositional logic can be extended to a calculus with epsilon- and tau-operators and critical formulas. For classical logic, this results in Hilbert’s $\varepsilon $ -calculus. The first and second $\varepsilon $ -theorems for classical logic establish conservativity of the $\varepsilon $ -calculus over its classical base logic. It is well known that the second $\varepsilon $ -theorem fails for the intuitionistic $\varepsilon $ -calculus, as prenexation is impossible. The paper investigates the effect of (...)
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  39. Boxes and Diamonds: An Open Introduction to Modal Logic.Richard Zach - 2019 - Open Logic Project.
    A textbook for modal and other intensional logics based on the Open Logic Project. It covers normal modal logics, relational semantics, axiomatic and tableaux proof systems, intuitionistic logic, and counterfactual conditionals.
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  40. Some strongly undecidable natural arithmetical problems, with an application to intuitionistic theories.Panu Raatikainen - 2003 - Journal of Symbolic Logic 68 (1):262-266.
    A natural problem from elementary arithmetic which is so strongly undecidable that it is not even Trial and Error decidable (in other words, not decidable in the limit) is presented. As a corollary, a natural, elementary arithmetical property which makes a difference between intuitionistic and classical theories is isolated.
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  41. The Craig interpolation theorem for prepositional logics with strong negation.Valentin Goranko - 1985 - Studia Logica 44 (3):291 - 317.
    This paper deals with, prepositional calculi with strong negation (N-logics) in which the Craig interpolation theorem holds. N-logics are defined to be axiomatic strengthenings of the intuitionistic calculus enriched with a unary connective called strong negation. There exists continuum of N-logics, but the Craig interpolation theorem holds only in 14 of them.
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  42. Co-constructive logic for proofs and refutations.James Trafford - 2014 - Studia Humana 3 (4):22-40.
    This paper considers logics which are formally dual to intuitionistic logic in order to investigate a co-constructive logic for proofs and refutations. This is philosophically motivated by a set of problems regarding the nature of constructive truth, and its relation to falsity. It is well known both that intuitionism can not deal constructively with negative information, and that defining falsity by means of intuitionistic negation leads, under widely-held assumptions, to a justification of bivalence. For example, we (...)
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  43. Logic and the Concept of God.Stanisław Krajewski & Ricardo Sousa Silvestre - 2019 - Journal of Applied Logics 6 (6):999-1005.
    This paper introduces the special issue on the Concept of God of the Journal of Applied Logics (College Publications). The issue contains the following articles: Logic and the Concept of God, by Stanisław Krajewski and Ricardo Silvestre; Mathematical Models in Theology. A Buber-inspired Model of God and its Application to “Shema Israel”, by Stanisław Krajewski; Gödel’s God-like Essence, by Talia Leven; A Logical Solution to the Paradox of the Stone, by Héctor Hernández Ortiz and Victor Cantero; No New Solutions (...)
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  44. What is Logical Monism?Justin Clarke-Doane - forthcoming - In Christopher Peacocke & Paul Boghossian (eds.), Normative Realism.
    Logical monism is the view that there is ‘One True Logic’. This is the default position, against which pluralists react. If there were not ‘One True Logic’, it is hard to see how there could be one true theory of anything. A theory is closed under a logic! But what is logical monism? In this article, I consider semantic, logical, modal, scientific, and metaphysical proposals. I argue that, on no ‘factualist’ analysis (according to which ‘there is One (...)
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  45. Topics in Philosophical Logic.Jon Erling Litland - 2012 - Dissertation, Harvard
    In “Proof-Theoretic Justification of Logic”, building on work by Dummett and Prawitz, I show how to construct use-based meaning-theories for the logical constants. The assertability-conditional meaning-theory takes the meaning of the logical constants to be given by their introduction rules; the consequence-conditional meaning-theory takes the meaning of the logical constants to be given by their elimination rules. I then consider the question: given a set of introduction rules \, what are the strongest elimination rules that are validated by an (...)
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  46.  9
    Classical Logic and Neutrosophic Logic. Answers to K. Georgiev.Florentin Smarandache - 2016 - Neutrosophic Sets and Systems 13:79-83.
    In this paper, we make distinctions between Classical Logic (where the propositions are 100% true, or 100 false) and the Neutrosophic Logic (where one deals with partially true, partially indeterminate and partially false propositions) in order to respond to K. Georgiev’s criticism [1]. We recall that if an axiom is true in a classical logic system, it is not necessarily that the axiom be valid in a modern (fuzzy, intuitionistic fuzzy, neutrosophic etc.) logic system.
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  47.  42
    Decidable Formulas Of Intuitionistic Primitive Recursive Arithmetic.Saeed Salehi - 2002 - Reports on Mathematical Logic 36 (1):55-61.
    By formalizing some classical facts about provably total functions of intuitionistic primitive recursive arithmetic (iPRA), we prove that the set of decidable formulas of iPRA and of iΣ1+ (intuitionistic Σ1-induction in the language of PRA) coincides with the set of its provably ∆1-formulas and coincides with the set of its provably atomic formulas. By the same methods, we shall give another proof of a theorem of Marković and De Jongh: the decidable formulas of HA are its provably ∆1-formulas.
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  48. Logic of Probability and Conjecture.Harry Crane - unknown
    I introduce a formalization of probability which takes the concept of 'evidence' as primitive. In parallel to the intuitionistic conception of truth, in which 'proof' is primitive and an assertion A is judged to be true just in case there is a proof witnessing it, here 'evidence' is primitive and A is judged to be probable just in case there is evidence supporting it. I formalize this outlook by representing propositions as types in Martin-Lof type theory (MLTT) and defining (...)
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  49. Normalisation and subformula property for a system of classical logic with Tarski’s rule.Nils Kürbis - 2022 - Archive for Mathematical Logic 61 (1):105-129.
    This paper considers a formalisation of classical logic using general introduction rules and general elimination rules. It proposes a definition of ‘maximal formula’, ‘segment’ and ‘maximal segment’ suitable to the system, and gives reduction procedures for them. It is then shown that deductions in the system convert into normal form, i.e. deductions that contain neither maximal formulas nor maximal segments, and that deductions in normal form satisfy the subformula property. Tarski’s Rule is treated as a general introduction rule for (...)
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  50. Proof Theory of Finite-valued Logics.Richard Zach - 1993 - Dissertation, Technische Universität Wien
    The proof theory of many-valued systems has not been investigated to an extent comparable to the work done on axiomatizatbility of many-valued logics. Proof theory requires appropriate formalisms, such as sequent calculus, natural deduction, and tableaux for classical (and intuitionistic) logic. One particular method for systematically obtaining calculi for all finite-valued logics was invented independently by several researchers, with slight variations in design and presentation. The main aim of this report is to develop the proof theory of finite-valued (...)
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