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Comments regarding: Pereira SA, Lopez M, Lavado N, Abreu JM, Silva H. A clinical risk prediction model of orthodontic-induced external apical root resorption. Revista Portuguesa de Estomatologia, Medicina Dentária e Cirurgia Maxilofacial 2014;55(2):66-72
, Article Revista Portuguesa de Estomatologia, Medicina Dentaria e Cirurgia Maxilofacial ; Volume 56, Issue 3 , 2015 , Pages 141-142 ; 16462890 (ISSN) ; Sharif University of Technology
Elsevier Doyma
2015
A unification of the basic logics of Sambin and Visser
, Article Logic Journal of the IGPL ; Volume 20, Issue 6 , 2012 , Pages 1202-1213 ; 13670751 (ISSN) ; Vaezian, V ; Sharif University of Technology
2012
Abstract
In logical literature, the phrase 'basic logic' refers to at least three different logical systems. The first one, basic propositional logic, BPL was introduced by Albert Visser in 1981. This logic is a subintuitionistic logic that can be obtained from intuitionistic logic by weakening of modus ponens. The second logical system with the name 'basic logic', is the system B that was introduced by G. Sambin and G. Battilotti in 1997. The goal of this logical system is to provide a common foundation for all usual non-modal logics. The third one is called 'basic logic' by P. Hajek in the field of Fuzzy Logic. We show that the two systems BPL and B do not have a direct relationship (i.e. none of...
On the constructive notion of closure maps
, Article Mathematical Logic Quarterly ; Volume 58, Issue 4-5 , 2012 , Pages 348-355 ; 09425616 (ISSN) ; Ramezanian, R ; Sharif University of Technology
Wiley
2012
Abstract
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(0) = A? In this paper, we study the notions of closed sets and closure maps in constructive reverse mathematics
The double negation of the intermediate value theorem
, Article Annals of Pure and Applied Logic ; Volume 161, Issue 6 , 2010 , Pages 737-744 ; 01680072 (ISSN) ; Ramezanian, R ; Sharif University of Technology
2010
Abstract
In the context of intuitionistic analysis, we consider the set F consisting of all continuous functions φ{symbol} from [0, 1] to R such that φ{symbol} (0) = 0 and φ{symbol} (1) = 1, and the set I0 consisting of φ{symbol}'s in F where there exists x ∈ [0, 1] such that φ{symbol} (x) = frac(1, 2). It is well-known that there are weak counterexamples to the intermediate value theorem, and with Brouwer's continuity principle we have I0 ≠ F. However, there exists no satisfying answer to I0¬ ¬ =? F. We try to answer to this question by reducing it to a schema (which we call ED) about intuitionistic decidability that asserts "there exists an intuitionistically enumerable set that is not...
Decidability and Specker sequences in intuitionistic mathematics
, Article Mathematical Logic Quarterly ; Volume 55, Issue 6 , 2009 , Pages 637-648 ; 09425616 (ISSN) ; Ramezanian, R ; Sharif University of Technology
2009
Abstract
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 (which we call ED) about intuitionistic decidability that asserts "there exists an intuitionistic enumerable set that is not intuitionistic decidable" and show that the existence of a Specker sequence is equivalent to ED. We show that...
Latarres, lattices with an arrow
, Article Studia Logica ; 2017 , Pages 1-32 ; 00393215 (ISSN) ; Ruitenburg, W ; Sharif University of Technology
2017
Abstract
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. © 2017 Springer Science+Business Media B.V
Compactness, colocatedness, measurability and ED
, Article Logic Journal of the IGPL ; Volume 26, Issue 2 , January , 2018 , Pages 244-254 ; 13670751 (ISSN) ; Ghafouri, Z ; Sharif University of Technology
Oxford University Press
2018
Abstract
In classical analysis, every compact subset of ℝ is Lebesgue measurable, but it is not true in constructive analysis. In this paper, we prove that the statement 'every compact set K in a locally compact space X is integrable with respect to a positive measure μ' is equivalent to LPO, over Bishop's constructive analysis. We also prove that the existence of a compact subset of ℝ which is not Lebesgue integrable is equivalent to the schema ED, which asserts that 'there exists an intuitionistically enumerable subset of ℕ which is not intuitionistically decidable'. Moreover, classically, every open subset of ℝ is Lebesgue measurable, but it is not true constructively. We show that Lebesgue...
The principle of open induction and specker sequences
, Article Logic Journal of the IGPL ; Volume 25, Issue 2 , 2017 , Pages 232-238 ; 13670751 (ISSN) ; Ghafouri, Z ; Sharif University of Technology
Oxford University Press
2017
Abstract
The schema ED asserts that 'there exists an intuitionistically enumerable subset of N which is not intuitionistically decidable.' In this article, we prove that in the presence of Markov's Principle over Bishop's constructive analysis, ¬ED is equivalent to the principle of open induction on [0,1], via Specker sequences. © 2016. Oxford University Press. All rights reserved
The Σ1-provability logic of HA
, Article Annals of Pure and Applied Logic ; Volume 169, Issue 10 , 2018 , Pages 997-1043 ; 01680072 (ISSN) ; Mojtahedi, M ; Sharif University of Technology
Elsevier B.V
2018
Abstract
In this paper we introduce a modal theory iHσ which is sound and complete for arithmetical Σ1-interpretations in HA, in other words, we will show that iHσ is the Σ1-provability logic of HA. Moreover we will show that iHσ is decidable. As a by-product of these results, we show that HA+□⊥ has de Jongh property. © 2018 Elsevier B.V
The Σ1-Provability Logic of HA
, Article Journal of Symbolic Logic ; Volume 84, Issue 3 , 2019 , Pages 1118-1135 ; 00224812 (ISSN) ; Mojtahedi, M ; Sharif University of Technology
Cambridge University Press
2019
Abstract
For the Heyting Arithmetic HA,HA is defined [14, 15] as the theory {A | HA-A}, where is called the box translation of A (Definition 2.4). We characterize the Σ1-provability logic of HA as a modal theory (Definition 3.17). © 2019 The Association for Symbolic Logic
Latarres, Lattices with an Arrow
, Article Studia Logica ; Volume 106, Issue 4 , 2018 , Pages 757-788 ; 00393215 (ISSN) ; Ruitenburg, W ; Sharif University of Technology
Springer Netherlands
2018
Abstract
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. © 2017, Springer Science+Business Media B.V
An introduction to basic arithmetic
, Article Logic Journal of the IGPL ; Volume 16, Issue 1 , 2008 , Pages 1-13 ; 13670751 (ISSN) ; Hesaam, B ; Sharif University of Technology
Oxford University Press
2008
Abstract
We study Basic Arithmetic BA, which is the basic logic BQC equivalent of Heyting Arithmetic HA over intuitionistic logic IQC, and of Peano Arithmetic PA over classical logic CQC. It turns out that The Friedman translation is applicable to BA. Using this translation, we prove that BA is closed under a restricted form of the Markov rule. Moreover, it is proved that all nodes of a finite Kripke model of BA are classical models of Ι∃1+, a fragment of PA with Induction restricted to the formulas made up of ∃, ∧ and/or ∨. We also study an interesting extension of BQC, called EBQC, which is the extension by the axiom schema ⊤ → →. We show that this extension behaves very like to IQC, and the...
On some questions of L. Åqvist
, Article Logic Journal of the IGPL ; Volume 14, Issue 1 , 2006 , Pages 1-13 ; 13670751 (ISSN) ; Nabavi, F ; Sharif University of Technology
Oxford University Press
2006
Abstract
We give answers to some questions raised by L. Åqvist in [6] and [7]. The question raised in [7] which we will answer positively is about the representability of Åqvist's system G in a hierarchy of alethic modal logics Hm, m = 1, 2, .... On the other hand, the questions raised in [6] are about the completeness of some monadic alethic denotic logics with respect to their Kripke semantics. © Copyright 2006 Oxford University Press
Every rooted narrow tree Kripke model of HA is locally PA
, Article Mathematical Logic Quarterly ; Volume 48, Issue 3 , 2002 , Pages 391-395 ; 09425616 (ISSN) ; Hesaam, B ; Sharif University of Technology
2002
Abstract
We prove that every infinite rooted narrow tree Kripke model of HA is locally PA
A logical framework for the Islamic law
, Article Logic, Argumentation and Reasoning ; Volume 23 , 2022 , Pages 53-81 ; 22149120 (ISSN) ; Nabavi, F ; Sharif University of Technology
Springer Science and Business Media B.V
2022
Abstract
We introduce a new aspect of the notion of obligation inspired from the Islamic legal system. We then construct a dynamic deontic logic to model this notion of obligation. A semantics for this logic is introduced, and then its soundness and completeness theorems with respect to this semantics are proved. © 2022, Springer Nature Switzerland AG
Basic propositional calculus II. Interpolation
, Article Archive for Mathematical Logic ; Volume 40, Issue 5 , 2001 , Pages 349-364 ; 09335846 (ISSN) ; Ruitenburg, W ; Sharif University of Technology
Springer New York
2001
Abstract
Let ℒ and script N be propositional languages over Basic Propositional Calculus, and script M = ℒ ∩ script N. We prove 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 script N, such that Π, Σ ⊢ C ⇒ C′. Then there is a sequent theory Φ over script N such that Π ⊢ Φ and Φ, Σ ⊢ C ⇒ C′. Second, let A be a formula over ℒ, and C1, C2 be formulas over script N, such that A ∧ C1 ⊢ C2. Then there exists a formula B over script M such that A ⊢ B and B ∧ C1 ⊢ C2
Advanced Pregroup Analysis of Persian Grammar
, M.Sc. Thesis Sharif University of Technology ; Ardeshir, Mohammad (Supervisor)
Abstract
Pregroups as a mathematical structure, are replacement for Lambek's type caregorial grammar which much used in Computational Linguistics. Because of computational and logical properties of pregroups, we can use them as strong tool to analyse the sentence structure of many natural languages. This kind of analysis has been done for English, French, German, Polish, Italian, Arabic and Japanese. In case of Persian language, analysis of simple and compound sentences structure with simple tense verbs and explicit subjects and objects has been studied. In this M.Sc. thesis, we will extent analysis of Persian sentence structure to sentences with compound tense verbs and implicit subjects and objects...
Description Logic and Its Application in Model Checking
,
M.Sc. Thesis
Sharif University of Technology
;
Ardeshir, Mohammad
(Supervisor)
Abstract
Description logic is a family of knowledge representive languages which represents knowledge via propositional logic (first order logic) propositions and constructors and applies its services for reasoning and consistency checking. Nowadays description logic and its popular reasoner FaCT++ which applies tablue reasoning technique are widely used in applications such as semantic web and onthologies. Model checking is a technique for systems and models verification and to guarantee the accuracy of design. Given a model description and a specification formula, the model checker verifies the model against the specification and decides if the model satisfies the description or not. Main model...
Investigating Basic Logics and Their Possible Interrelation
, M.Sc. Thesis Sharif University of Technology ; Ardeshir, Mohammad (Supervisor)
Abstract
In logic literature, the phrase “basic logic” can refer to three different logical systems. First, basic propositional logic (BPL) that was introduced by A. Visser in 1981. This logic is a subintuitionistic logic, that can be obtained from intuitionistic logic by a weakening of modus ponens. One decade later, Wim Ruitenberg regarding philosophical critiques of logical connectives, reintroduced BPL and its first order extension, BQC. From then, different aspects of basic logic have been investigated by logicians all over the world. Another logical system with the name “basic logic” is the system that was introduced by G. Sambin and G. Battilotti in 1997. The goal of this system is to provide...
Application of Logic in Legal Systems
, M.Sc. Thesis Sharif University of Technology ; Ardeshir, Mohammad (Supervisor)
Abstract
Deontic logic is used to formalize legal reasoning. To apply this logic in law, we describe tersely some efforts to improve this logic by relativizing its operations with respect to different people and groups of society. Until now, this logic was restricted to formalize “what must be”. We extend this logic to dyadic logic to formalize “what must be done”.
In practice, legal reasoning leads to non-monotonic logics, the most applicable one in law is defeasible logic. So it is necessary to combine deontic and defeasible logics to formalize legal reasoning in a more appropriate way. To do that, we must adjust possible worlds of these two logics. In this way, we find a method for...
In practice, legal reasoning leads to non-monotonic logics, the most applicable one in law is defeasible logic. So it is necessary to combine deontic and defeasible logics to formalize legal reasoning in a more appropriate way. To do that, we must adjust possible worlds of these two logics. In this way, we find a method for...