25 found
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  1.  12
    Boolean Algebras in Visser Algebras.Majid Alizadeh, Mohammad Ardeshir & Wim Ruitenburg - 2016 - Notre Dame Journal of Formal Logic 57 (1):141-150.
    We generalize the double negation construction of Boolean algebras in Heyting algebras to a double negation construction of the same in Visser algebras. This result allows us to generalize Glivenko’s theorem from intuitionistic propositional logic and Heyting algebras to Visser’s basic propositional logic and Visser algebras.
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  2.  3
    Decidability and Specker Sequences in Intuitionistic Mathematics.Mohammad Ardeshir & Rasoul Ramezanian - 2009 - Mathematical Logic Quarterly 55 (6):637-648.
    A bounded monotone sequence of reals without a limit is called a Specker sequence. In Russian constructive analysis, Church's Thesis permits the existence of a Specker sequence. In intuitionistic mathematics, Brouwer's Continuity Principle implies it is false that every bounded monotone sequence of real numbers has a limit. We claim that the existence of Specker sequences crucially depends on the properties of intuitionistic decidable sets. We propose a schema about intuitionistic decidability that asserts “there exists an intuitionistic enumerable set that (...)
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  3.  5
    Basic Propositional Calculus I.Mohammad Ardeshir & Wim Ruitenburg - 1998 - Mathematical Logic Quarterly 44 (3):317-343.
    We present an axiomatization for Basic Propositional Calculus BPC and give a completeness theorem for the class of transitive Kripke structures. We present several refinements, including a completeness theorem for irreflexive trees. The class of intermediate logics includes two maximal nodes, one being Classical Propositional Calculus CPC, the other being E1, a theory axiomatized by T → ⊥. The intersection CPC ∩ E1 is axiomatizable by the Principle of the Excluded Middle A V ∨ ⌝A. If B is a formula (...)
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  4.  3
    Basic Propositional Logic and the Weak Excluded Middle.Majid Alizadeh & Mohammad Ardeshir - forthcoming - Logic Journal of the IGPL.
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  5.  5
    Reduction of Provability Logics to Σ1-Provability Logics.Mohammad Ardeshir & S. Mojtaba Mojtahedi - 2015 - Logic Journal of the IGPL 23 (5):842-847.
  6.  15
    Avicenna on the Primary Propositions.Seyed N. Mousavian & Mohammad Ardeshir - 2018 - History and Philosophy of Logic 39 (3):201-231.
    Avicenna introduces the primary propositions as the most fundamental principles of knowledge. However, as far as we are aware, Avicenna’s primaries have not yet been independently studied. Nor do Avicenna scholars agree on how to characterize them in the language of contemporary philosophy. It is well-known that the primaries are indemonstrable; nonetheless, it is not clear what the genealogy of the primaries is, how, epistemologically speaking, they can be distinguished from other principles, what their phenomenology is, what the cause of (...)
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  7.  2
    The de Jongh Property for Basic Arithmetic.Mohammad Ardeshir & S. Mojtaba Mojtahedi - 2014 - Archive for Mathematical Logic 53 (7-8):881-895.
  8.  9
    The Double Negation of the Intermediate Value Theorem.Mohammad Ardeshir & Rasoul Ramezanian - 2010 - Annals of Pure and Applied Logic 161 (6):737-744.
    In the context of intuitionistic analysis, we consider the set consisting of all continuous functions from [0,1] to such that =0 and =1, and the set consisting of ’s in where there exists x[0,1] such that . It is well-known that there are weak counterexamples to the intermediate value theorem, and with Brouwer’s continuity principle we have . However, there exists no satisfying answer to . We try to answer to this question by reducing it to a schema about intuitionistic (...)
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  9.  47
    Gentzen-Style Axiomatizations for Some Conservative Extensions of Basic Propositional Logic.Mojtaba Aghaei & Mohammad Ardeshir - 2001 - Studia Logica 68 (2):263-285.
    We introduce two Gentzen-style sequent calculus axiomatizations for conservative extensions of basic propositional logic. Our first axiomatization is an ipmrovement of, in the sense that it has a kind of the subformula property and is a slight modification of. In this system the cut rule is eliminated. The second axiomatization is a classical conservative extension of basic propositional logic. Using these axiomatizations, we prove interpolation theorems for basic propositional logic.
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  10.  8
    On Löb Algebras.Majid Alizadeh & Mohammad Ardeshir - 2006 - Mathematical Logic Quarterly 52 (1):95-105.
    We study the variety of Löb algebras , the algebraic structures associated with formal propositional calculus. Among other things, we prove a completeness theorem for formal propositional logic with respect to the variety of Löb algebras. We show that the variety of Löb algebras has the weak amalgamation property. Some interesting subclasses of the variety of Löb algebras, e.g. linear, faithful and strongly linear Löb algebras are introduced.
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  11.  16
    Basic Propositional Calculus II. Interpolation.Mohammad Ardeshir & Wim Ruitenburg - 2001 - Archive for Mathematical Logic 40 (5):349-364.
    Let ℒ and ? be propositional languages over Basic Propositional Calculus, and ℳ = ℒ∩?. Weprove two different but interrelated interpolation theorems. First, suppose that Π is a sequent theory over ℒ, and Σ∪ {C⇒C′} is a set of sequents over ?, such that Π,Σ⊢C⇒C′. Then there is a sequent theory Φ over ℳ such that Π⊢Φ and Φ, Σ⊢C⇒C′. Second, let A be a formula over ℒ, and C 1, C 2 be formulas over ?, such that A∧C 1⊢C (...)
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  12.  41
    A Solution to the Surprise Exam Paradox in Constructive Mathematics.Mohammad Ardeshir & Rasoul Ramezanian - 2012 - Review of Symbolic Logic 5 (4):679-686.
    We represent the well-known surprise exam paradox in constructive and computable mathematics and offer solutions. One solution is based on Brouwer’s continuity principle in constructive mathematics, and the other involves type 2 Turing computability in classical mathematics. We also discuss the backward induction paradox for extensive form games in constructive logic.
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  13.  9
    Every Rooted Narrow Tree Kripke Model of HA is Locally PA.Mohammad Ardeshir & Bardyaa Hesaam - 2002 - Mathematical Logic Quarterly 48 (3):391-395.
    We prove that every infinite rooted narrow tree Kripke model of HA is locally PA.
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  14.  6
    Intuitionistic Axiomatizations for Bounded Extension Kripke Models.Mohammad Ardeshir, Wim Ruitenburg & Saeed Salehi - 2003 - Annals of Pure and Applied Logic 124 (1-3):267-285.
    We present axiom systems, and provide soundness and strong completeness theorems, for classes of Kripke models with restricted extension rules among the node structures of the model. As examples we present an axiom system for the class of cofinal extension Kripke models, and an axiom system for the class of end-extension Kripke models. We also show that Heyting arithmetic is strongly complete for its class of end-extension models. Cofinal extension models of HA are models of Peano arithmetic.
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  15.  13
    Intuitionistic Open Induction and Least Number Principle and the Buss Operator.Mohammad Ardeshir & Mojtaba Moniri - 1998 - Notre Dame Journal of Formal Logic 39 (2):212-220.
    In "Intuitionistic validity in -normal Kripke structures," Buss asked whether every intuitionistic theory is, for some classical theory , that of all -normal Kripke structures for which he gave an r.e. axiomatization. In the language of arithmetic and denote PA plus Open Induction or Open LNP, and are their intuitionistic deductive closures. We show is recursively axiomatizable and , while . If proves PEM but not totality of a classically provably total Diophantine function of , then and so . A (...)
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  16.  3
    Compactness, Colocatedness, Measurability and ED.Mohammad Ardeshir & Zahra Ghafouri - 2018 - Logic Journal of the IGPL 26 (2):244-254.
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  17.  48
    A Translation of Intuitionistic Predicate Logic Into Basic Predicate Logic.Mohammad Ardeshir - 1999 - Studia Logica 62 (3):341-352.
    Basic Predicate Logic, BQC, is a proper subsystem of Intuitionistic Predicate Logic, IQC. For every formula in the language {, , , , , , }, we associate two sequences of formulas 0,1,... and 0,1,... in the same language. We prove that for every sequent , there are natural numbers m, n, such that IQC , iff BQC n m. Some applications of this translation are mentioned.
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  18.  15
    A Gentzen-Style Axiomatization for Basic Predicate Calculus.Mojtaba Aghaei & Mohammad Ardeshir - 2003 - Archive for Mathematical Logic 42 (3):245-259.
    We introduce a Gentzen-style sequent calculus axiomatization for Basic Predicate Calculus. Our new axiomatization is an improvement of the previous axiomatizations, in the sense that it has the subformula property. In this system the cut rule is eliminated.
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  19.  1
    The Σ 1 -Provability Logic of HA.Mohammad Ardeshir & Mojtaba Mojtahedi - 2018 - Annals of Pure and Applied Logic 169 (10):997-1043.
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  20.  2
    Latarres, Lattices with an Arrow.Mohammad Ardeshir & Wim Ruitenburg - 2018 - Studia Logica 106 (4):757-788.
    A latarre is a lattice with an arrow. Its axiomatization looks natural. Latarres have a nontrivial theory which permits many constructions of latarres. Latarres appear as an end result of a series of generalizations of better known structures. These include Boolean algebras and Heyting algebras. Latarres need not have a distributive lattice.
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  21.  13
    On the Constructive Notion of Closure Maps.Mohammad Ardeshir & Rasoul Ramezanian - 2012 - Mathematical Logic Quarterly 58 (4-5):348-355.
    Let A be a subset of the constructive real line. What are the necessary and sufficient conditions for the set A such that A is continuously separated from other reals, i.e., there exists a continuous function f with f−1 = A? In this paper, we study the notions of closed sets and closure maps in constructive reverse mathematics.
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  22.  3
    The Principle of Open Induction and Specker Sequences.Mohammad Ardeshir & Zahra Ghafouri - 2017 - Logic Journal of the IGPL 25 (2):232-238.
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  23.  10
    A Bounded Translation of Intuitionistic Propositional Logic Into Basic Propositional Logic.Mojtaba Aghaei & Mohammad Ardeshir - 2000 - Mathematical Logic Quarterly 46 (2):195-206.
    In this paper we prove a bounded translation of intuitionistic propositional logic into basic propositional logic. Our new theorem, compared with the translation theorem in [1], has the advantage that it gives an effective bound on the translation, depending on the complexity of formulas.
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  24.  8
    Amalgamation Property for the Class of Basic Algebras and Some of its Natural Subclasses.Majid Alizadeh & Mohammad Ardeshir - 2006 - Archive for Mathematical Logic 45 (8):913-930.
    We study Basic algebra, the algebraic structure associated with basic propositional calculus, and some of its natural extensions. Among other things, we prove the amalgamation property for the class of Basic algebras, faithful Basic algebras and linear faithful Basic algebras. We also show that a faithful theory has the interpolation property if and only if its correspondence class of algebras has the amalgamation property.
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  25.  6
    Completeness of Intermediate Logics with Doubly Negated Axioms.Mohammad Ardeshir & S. Mojtaba Mojtahedi - 2014 - Mathematical Logic Quarterly 60 (1-2):6-11.