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 (...) result due to Wehmeier then implies . We prove is not -conservative over . If , then is not closed under MR or Friedman's translation, so range (). Both and are closed under the negative translation. (shrink)

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 (...) such that → B is not derivable, then the lattice of formulas built from one propositional variable p using only the binary connectives, is isomorphically preserved if B is substituted for p. A formula → B is derivable exactly when B is provably equivalent to a formula of the form → A) →. (shrink)

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.

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 (...) 2. Then there exists a formula B over ℳ such that A⊢B and B∧C 1⊢C 2. (shrink)

We prove that Basic Arithmetic, BA, has the de Jongh property, i.e., for any propositional formula A built up of atoms p1,..., pn, BPC⊢\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\vdash}$$\end{document}A if and only if for all arithmetical sentences B1,..., Bn, BA⊢\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\vdash}$$\end{document}A. The technique used in our proof can easily be applied to some known extensions of BA.

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 (...) is not intuitionistic decidable” and show that the existence of a Specker sequence is equivalent to ED. We show that ED is consistent with some certain well known axioms of intuitionistic analysis as Weak Continuity Principle, bar induction, and Kripke Schema. Thus, the assumption of the existence of a Specker sequence is conceivable in intuitionistic analysis. We will also introduce the notion of double Specker sequence and study the existence of them. (shrink)

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 (...) the assent to them is, how to explain the relationship between the ‘innate [nature] of the intellect’ and the primaries and, finally, back to their indemonstrability, in what sense they are ‘indemonstrable’. We will try to fill this gap. As a corollary, we will explain why Gutas’s view [Gutas, Dimitri.... (shrink)

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.

We prove that every infinite rooted narrow tree Kripke model of HA is locally PA.

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.

We give a counterexample to the claim that every provably total function of Basic Arithmetic is a polynomially bounded primitive recursive function.

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.

Let denote a first‐order logic in a language that contains infinitely many constant symbols and also containing intuitionistic logic. By, we mean the associated logic axiomatized by the double negation of the universal closure of the axioms of plus. We shall show that if is strongly complete for a class of Kripke models, then is strongly complete for the class of Kripke models that are ultimately in.

Kolmogorov established the principle of the double negation translation by which to embed Classical Predicate Logic |${\operatorname {CQC}}$| into Intuitionistic Predicate Logic |${\operatorname {IQC}}$|⁠. We show that the obvious generalizations to the Basic Predicate Logic of [3] and to |${\operatorname {BQC}}$| of [12], a proper subsystem of |${\operatorname {IQC}}$|⁠, go through as well. The obvious generalizations of Kuroda’s embedding are shown to be equivalent to the Kolmogorov variant. In our proofs novel nontrivial techniques are needed to overcome the absence of (...) full modus ponens in Basic Predicate Logic. In [3] we argued that |${\operatorname {IQC}}$| is not the logic of constructive mathematics. Our doubts were far from new. New was that we put forward an alternative, |${\operatorname {BQC}}$|⁠. One concern is that |${\operatorname {BQC}}$| is too weak for serious mathematics, or even trivial. This paper is one step to alleviate such concerns. (shrink)

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.

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 (...) decidability that asserts “there exists an intuitionistically enumerable set that is not intuitionistically decidable”. We also introduce the notion of strong Specker double sequence, and prove that the existence of such a double sequence is equivalent to the existence of a function where. (shrink)

For the Heyting Arithmetic HA, $HA^{\text{*}} $ is defined [14, 15] as the theory $\left\{ {A|HA \vdash A^\square } \right\}$, where $A^\square $ is called the box translation of A. We characterize the ${\text{\Sigma }}_1 $-provability logic of $HA^{\text{*}} $ as a modal theory $iH_\sigma ^{\text{*}} $.

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.

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.

We study the linear Lindenbaum algebra of Basic Propositional Calculus, called linear basic algebra.

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.

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.

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.

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.

We prove that $\textbf {K}5$ and some of its extensions that do not contain $\textbf {K}4$ are of unification type $1$.

Employing Constructive Type Theory Constructive Type Theory, we provide a logical analysis of[aut]Ibn SīnāIbn Sīnā’sIbn Sīnā descriptional propositions. Compared to its rivals, our analysis is more faithful to the grammatical subject-predicate structure of propositions and can better reflect the morphological features of the verbs that extend time to intervals. We also study briefly the logical structure of some fallacious inferences that are discussed by Ibn Sīnā. The CTT-framework makes the fallacious nature of these inferences apparent.

This volume is a collection of essays in honour of Professor Mohammad Ardeshir. It examines topics which, in one way or another, are connected to the various aspects of his multidisciplinary research interests. Based on this criterion, the book is divided into three general categories. The first category includes papers on non-classical logics, including intuitionistic logic, constructive logic, basic logic, and substructural logic. The second category is made up of papers discussing issues in the contemporary philosophy of mathematics (...) and logic. The third category contains papers on Avicenna’s logic and philosophy. Mohammad Ardeshir is a full professor of mathematical logic at the Department of Mathematical Sciences, Sharif University of Technology, Tehran, Iran, where he has taught generations of students for around a quarter century. Mohammad Ardeshir is known in the first place for his prominent works in basic logic and constructive mathematics. His areas of interest are however much broader and include topics in intuitionistic philosophy of mathematics and Arabic philosophy of logic and mathematics. In addition to numerous research articles in leading international journals, Ardeshir is the author of a highly praised Persian textbook in mathematical logic. Partly through his writings and translations, the school of mathematical intuitionism was introduced to the Iranian academic community. (shrink)

This intriguing work offers a new perspective on Islamic Peripatetic philosophy, critiquing modern receptions of such thought and highlighting the contribution it can make to contemporary Western philosophy. Mohammad Azadpur focuses on the thought of Alfarabi and Avicenna, who, like ancient Greek philosophers and some of their successors, viewed philosophy as a series of spiritual exercises. However, Muslim Peripatetics differed from their Greek counterparts in assigning importance to prophecy. The Islamic philosophical account of the cultivation of the soul to (...) the point of prophecy unfolds new vistas of intellectual and imaginative experience and accords the philosopher an exceptional dignity and freedom. With reference to both Islamic and Western philosophers, Azadpur discusses how Islamic Peripatetic thought can provide an antidote to some of modernity’s philosophical problems. A discussion of the development of later Islamic Peripatetic thought is also included. (shrink)

Biographies of Imam Ghazali, Socrates, Plato and other philosophers.

Autonomous vehicles (AV) offer promising benefits to society in terms of safety, environmental impact and increased mobility. However, acute challenges persist with any novel technology, inlcuding the perceived risks and trust underlying public acceptance. While research examining the current state of AV public perceptions and future challenges related to both societal and individual barriers to trust and risk perceptions is emerging, it is highly fragmented across disciplines. To address this research gap, by using the Web of Science database, our study (...) undertakes a bibliometric and performance analysis to identify the conceptual and intellectual structures of trust and risk narratives within the AV research field by investigating engineering, social sciences, marketing, and business and infrastructure domains to offer an interdisciplinary approach. Our analysis provides an overview of the key research area across the search categories of ‘trust’ and ‘risk’. Our results show three main clusters with regard to trust and risk, namely, behavioural aspects of AV interaction; uptake and acceptance; and modelling human–automation interaction. The synthesis of the literature allows a better understanding of the public perception of AV and its historical conception and development. It further offers a robust model of public perception in AV, outlining the key themes found in the literature and, in turn, offers critical directions for future research. (shrink)

The connection between the infinite nature of Quantum Cosmology and the infinite nature of God is presented here. At the beginning of the creation process, there was a single God/Void that was divided into many Gods/Voids all filled with Dark Energy consisting of photons which were responsible for creating the Multiverses made of matter, antimatter, space, time, charge, and multiple dimensions of space. The one God initially had no material existence which along with the laws of science was a creation (...) of the powers of His infinite mind. Since the process of creation started with the Dark Energy of the Voids, the Conservation of Energy Principle applied to all of them. If the Dark Energy of each Void were to be finite, then its energy would begin to decrease as Multiverses were created, and the only solution for not violating the Conservation of Energy Principle was that the Dark Energy of each of the Voids must be infinite implying that the Voids themselves must be infinite, and with infinite amount of energy be able to create an infinite number of Multiverses. According to Mathematics, all infinite numbers are not equal, some infinities are greater than other infinities. Hence of the many infinite Voids that were initially created, some Voids were greater than others. Each Void has a Supreme Leader that we refer to as God, and therefore there exist many Gods, some Gods more powerful than other Gods. (shrink)

The discourse on the integration of science and Islam is being realised through the establishment of various Islamic religious universities in Indonesia. One of the Islamic universities that accommodates this discourse is the Al-Qur’an Science University, Central Java, Indonesia (UNSIQ). This study aims to examine the basic concept of scientific integration at the UNSIQ and critically analyses the academic tradition and research development patterns based on the Lakatos research development pattern, both of which are hard-core and auxiliary hypotheses. This research (...) shows that the scientific base at the UNSIQ has been structured in an awareness that supports the establishment of a distinctive academic tradition and scientific culture in the framework of al-Qur’an-based academic knowledge. However, research as auxiliary hypotheses in building scientific paradigm is still limited in certain disciplines and needs to be improved. This needs to be a common concern in order to increase the scientific base of the Syajarah al-Qur’an into a solid scientific paradigm.Contribution: This research figured out that the scientific base at the UNSIQ has been structured in an awareness that supports the establishment of a distinctive academic tradition and scientific culture, but this scientific base needs to be supported by research plan and researches in various disciplines. Thus, those researches become the auxiliary hypotheses to build a particular scientific paradigm of this university. (shrink)

Emergency care providers regularly deal with ethical dilemmas that must be addressed. In comparison with in-hospital nurses, emergency medical service personnel are faced with more problems such as distance to resources including personnel, medico-technical aids, and information; the unpredictable atmosphere at the scene; arriving at the crime scene and providing emergency care for accident victims and patients at home. As a result of stressfulness, unpredictability, and often the life threatening nature of tasks that ambulance professionals have to deal with every (...) day, ethical decision-making has become an inevitable challenge. The content analysis approach was used to conduct the present qualitative study in Iran. The participants consisted of 14 EMS personnel selected through purposive sampling, which continued until the data became saturated. Data were collected using semi-structured interviews and analyzed concurrently with their collection through the constant comparison method. The process of data analysis resulted in the emergence of 3 main categories “respecting client’s values”, “performing tasks within the professional manner”, “personal characteristic”, and the emergence of eight sub-categories signifying participants’ experiences with regard to EDM. According to the results, when EMS personnel are faced with ethical dilemmas, they consider the client’s values and professional dignity, and perform the assigned tasks within the framework of the regulation. The findings also suggest that pre-hospital care providers assess legal consequences before making any decision. Further studies should be conducted regarding the experiences of the subordinates and other related parties. (shrink)

There has recently been a surge of development in augmented reality (AR) technologies that has led to an ecosystem of hardware and software for AR, including tools for artists and designers to accelerate the design of AR content and experiences without requiring complex programming. AR is viewed as a key “disruptive technology” and future display technologies (such as digital eyewear) will provide seamless continuity between reality and the digitally augmented. This article will argue that the technologization of human perception and (...) experience of reality, coupled with the development of artificial intelligence (AI)–based natural language assistants, may lead to a secular re‐enchantment of the world, in the sense outlined by Charles Taylor, where human existence is shaped through AR inhabited by advanced personal and social AI agents in the form of digital avatars and daemons, and that enchantment has been persistent throughout the formation of modernity and is being rekindled by the integration of AI in the plane of AR. (shrink)

Background:Experiencing moral distress is traumatic for nurses. Ignoring moral distress can lead to job dissatisfaction, improper handling in the care of patients, or even leaving the job. Thus, it is crucial to use valid and reliable instruments to measure moral distress.Objective:The purpose of this study was to determine the reliability and the validity of the Persian version of the Moral Distress Scale–Revised among a sample of Iranian nurses.Research design:In this methodological study, 310 nurses were recruited from all hospitals affiliated with (...) the Qazvin University of Medical Sciences from February 2014 to April 2015. Data were collected using a demographic questionnaire and the Moral Distress Scale–Revised. The construct validity of the Moral Distress Scale–Revised was evaluated using principal component analysis and confirmatory factor analysis. Internal consistency reliability was assessed with Cronbach’s alpha.Ethical considerations:This study was approved by the Regional Committee of Medical Research Ethics. The ethical principles of voluntary participation, anonymity, and confidentiality were considered.Findings:The construct validity of the scale showed four factors with eigenvalues greater than one. The model had a good fit ( χ2(162) = 307.561, χ2/ df = 1.899, goodness-of-fit index =.904, comparative fit index =.927, incremental fit index =.929, and root mean square error of approximation (90% confidence interval) =.049 (.040–.057)) with all factor loadings greater than.5 and statistically significant. Cronbach’s alpha coefficients were.853,.686,.685, and.711for the four factors. Moreover, the model structure was invariant across different income groups.Discussion and conclusion:The Persian version of the Moral Distress Scale–Revised demonstrated suitable validity and reliability among nurses. The factor analysis also revealed that the Moral Distress Scale–Revised has a multidimensional structure. Regarding the proper psychometric characteristics, the validated scale can be used to further research about moral distress in this population. (shrink)

The rapid advancement of artificial intelligence (AI) has led to renewed ambitions of developing artificial general intelligence. Alongside this has been a resurgence in the development of virtual and augmented reality (V/AR) technologies, which are viewed as “disruptive” technologies and the computing platforms of the future. V/AR effectively bring the digital world of machines, robots, and artificial agents to our senses while entailing the transposition of human activity and presence into the digital world of artificial agents and machine forms of (...) intelligence. The intersection of humans and machines in this shared space brings humans and machines into ontological continuity as informational entities in a totalizing informational environment, which subsumes both cyber and physical space in an artificially constructed virtual world. The reconstruction of mind (through AI) and world (through V/AR) thus has significant epistemological, ontological, and anthropological implications, which constitute the underlying features in the artificialization of mind and world. (shrink)

Recent literature on Islam and the digital covers a wide range of topics and themes; however, what is yet to be developed from an Islamic perspective is a broader philosophical framework that accounts for the nature, exigencies and affordances of contemporary digital technologies. In advance of such a framework, this article is an attempt to open the way to philosophical engagement with issues of digital ethics from an Islamic perspective. After a brief review of recent literature on Islam and the (...) digital and a significantly earlier work by Ziauddin Sardar, in which he proposed an information strategy for the Muslim world, this paper provides some background to Islamic ethics and the wide field of ethical theories in the Islamic tradition. This paper then proceeds by identifying and outlining examples of contemporary themes in Islam and the digital which conceal underlying philosophical and theological issues. Finally, this paper considers the significant scale and scope of the transformations involved in the ongoing transposition to the ‘onlife’ and highlights several areas where Islamic perspectives may be seen to converge or diverge with other strands of scholarship on digital ethics. (shrink)