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)

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

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.

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.

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)

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)

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.

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 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 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.

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)

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

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.

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.

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)

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.

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 $1$.

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)

In lieu of an abstract, here is a brief excerpt of the content:Answer to Catherine König-Pralong, Eun-Jeung Lee, and Jyoti MohanSelusi Ambrogio (bio)I want to start my reply by expressing my deep gratitude to the three reviewers who devoted their energy and time to reading and commenting on my book. Their wise comments and criticisms helped in shaping my upcoming research plans, as well as in refining my understanding of this historiographical topic. The eventual readers of this research will certainly (...) benefit from their arguments. I will reply to each review separately, one after the other. Also I could never thank the editors of Philosophy East and West enough for the honor they bestowed on me in hosting this debate.Mapping the ExclusionCatherine König-Pralong provides an extremely knowledgeable and refined review of the book in three sections. The first section is a theoretical evaluation and discussion of the outputs of the monograph through the lens of Reinhard Koselleck's theory of the "temporalization of history" (Verzeitlichung [End Page 230] der Geschichte)1 and her unequalled expertise on the imaginary maps of civilizations. I completely agree with her statement that "from the doxography of the seventeenth century to Brucker's eclecticism, philosophical historiography seemed to be much less attached to historical accuracy than to delimiting regions and identifying peoples and their minds—in a word, to territorializing." The territorialization of philosophy into maps that presented first the lost seeds of divine wisdom (perennialism) and afterwards the exclusive regions of rationality (eclecticism) shows more the Western necessity for a self-definition rather than an effective understanding of the thought of other civilizations. As Metha perfectly expresses it, "Long ago 'Asia' and the 'Orient' became almost existential categories in the European mind."2 It is in this context of 'spatialization of identities' that philosophy became one of the leading characteristics—when not the leading one—of European identity. As the reviewer lucidly remarks, this happened at the expense of philosophy itself. Practical philosophy was put under the superior control of theoretical and logical faculties. It is true that this process was achieved in German pragmatism and Kantian historiography, but this was already evident in Heumann and Brucker's historiographical productions.In her dense analysis, König-Pralong provides bibliographical suggestions that could be integrated within the monograph. I have to admit that, unluckily, I've never come across the two outstanding essays by Wimmer and Rüdiger on Chinese philosophy as a problem of historiography in the eighteenth century, both published in 2017.3 On the contrary, I eagerly read the reviewer's book La colonie philosophique, which was published in March 2019. However, at that time, my monograph was already in the hands of the editor, undergoing the process of double-blind peer review and editing, and hence my argumentation could not benefit from her discussion on temporalization versus historicization and the mapping of civilizations. To quote a sentence from her brilliant book: "L'histoire de la philosophie des Lumières programme en effet une rationalisation de l'histoire et une temporalisation de la philosophie."4 This sentence is a perfect explanation of the "de-territorialization" of India and China from the land of historicity and rationality, namely philosophy. For Brucker, India and China did not possess reliable historical sources, showed childish theoretical capacities, and presented oppressive norms that were not based on moral principles and reasoning.The reviewer agrees with the fact that, at the time of Brucker, we can already find reasons for the exclusion of Asian philosophies, but she contests the thesis that the character of this exclusion was already completely displayed. She suggests that Brucker's Historia critica is mostly a destructive history, showing human mistakes, and that the real targets were Neoplatonism and mystical syncretism, not Asian thought. She writes that "Brucker excluded non-European thoughts from philosophy, but not from the history of philosophy." In my view, Brucker did not insert "barbarian" [End Page 231] and "exotic" civilizations in the history of philosophy, but rather in a critical history—that is, historia critica—of human mistakes. He clearly states that nothing philosophical could be detected in these lands and that his own use of the term "philosophy" about them is... (shrink)

Two cut-free sequent calculi which are conservative extensions of Visser's Formal Propositional Logic are introduced. These satisfy a kind of subformula property and by this property the interpolation theorem for FPL are proved. These are analogies to Aghaei-Ardeshir's calculi for Visser's Basic Propositional Logic.

It is a well-known fact that although the poset of open sets of a topological space is a Heyting algebra, its Heyting implication is not necessarily stable under the inverse image of continuous functions and hence is not a geometric concept. This leaves us wondering if there is any stable family of implications that can be safely called geometric. In this paper, we will first recall the abstract notion of implication as a binary modality introduced in Akbar Tabatabai (Implication via (...) spacetime. In: Mathematics, logic, and their philosophies: essays in honour of Mohammad Ardeshir, pp 161–216, 2021). Then, we will use a weaker version of categorical fibrations to define the geometricity of a category of pairs of spaces and implications over a given category of spaces. We will identify the greatest geometric category over the subcategories of open-irreducible (closed-irreducible) maps as a generalization of the usual injective open (closed) maps. Using this identification, we will then characterize all geometric categories over a given category $${\mathcal {S}}$$ S, provided that $${\mathcal {S}}$$ S has some basic closure properties. Specially, we will show that there is no non-trivial geometric category over the full category of spaces. Finally, as the implications we identified are also interesting in their own right, we will spend some time to investigate their algebraic properties. We will first use a Yoneda-type argument to provide a representation theorem, making the implications a part of an adjunction-style pair. Then, we will use this result to provide a Kripke-style representation for any arbitrary implication. (shrink)

Indeterminacy of reference appears to be incompatible with the deflationist conceptions of reference: in deflationismDeflationism, the singular term ‘a’ refers to a, if it exists, and to nothing else, whereas if the term is referentially indeterminate, it has a variety of equally permissible reference-candidates: referential indeterminacyReferential indeterminacy and deflationismDeflationism cannot both be maintained. In this paper, I discuss the incompatibility thesis, critically examine the arguments leading to it, and thereby point towards ways in which the deflationist can explain referential indeterminacyReferential (...) indeterminacy. (shrink)

Recent work on compositional distributional models shows that bialgebras over finite dimensional vector spaces can be applied to treat generalised quantifiersGeneralised quantifiers for natural language. That technique requires one to construct the vector space over powersets, and therefore is computationally costly. In this paper, we overcome this problem by considering fuzzy versions of quantifiers along the lines of ZadehZadeh, L. A., within the category of many valued relationsMany valued relations. We show that this category is a concrete instantiation of the (...) compositional distributional model. We show that the semantics obtained in this model is equivalent to the semantics of the fuzzy quantifiers of ZadehZadeh, L. A.. As a result, we are now able to treat fuzzy quantification without requiring a powerset construction. (shrink)

It has been claimed that contracting connectivesContracting connective are conditionalsConditional. Our modest aim here is to show that the conditional-like features of a contracting connectiveContracting connective depend on the defining features of the conditionalConditional in a particular logic, yes, but they also depend on the underlying notion of logical consequence and the structure of the collection of truth values. More concretely, we will show that under P-consequenceP-consequence and suitable satisfiability conditions for the conditionalConditional, conjunctionsConjunction are contracting connectivesContracting connective for some (...) logics without thereby being conditional-ish. (shrink)

We continue our investigation of paraorthomodular BZ*-lattices PBZ*-lattices, started in Giuntini et al., Mureşan. We shed further light on the structure of the subvariety lattice of the variety PBZL∗\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb {PBZL}^{\mathbb {*}}$$\end{document} of PBZ∗\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$^{*}$$\end{document}–lattices; in particular, we provide axiomatic bases for some of its members. Further, we show that some distributive subvarieties of PBZL∗\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb {PBZL}^{\mathbb (...) {*}}$$\end{document} are term-equivalent to well-known varieties of expanded KleeneKleene, C. lattices or of nonclassical modal algebrasNonclassical modal algebras. By so doing, we somehow help the reader to locate PBZ∗\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$^{*}$$\end{document}–lattices on the atlas of algebraic structures for nonclassical logics. (shrink)

With the aim of providing an empiricist-friendly rational reconstruction of scientists’ modal talk, I represent and defend the following unoriginal idea of relative modalities, focused on natural ones: the assertion of physical necessity of φ can be understood as the logical provability of φ from the background theory of the context of assertion. I elaborate on my conception of the background theory, and reply to a number of objections, among which an objection concerning the failure of factivity.

I will focus on the following question: “Is Avicenna[aut]Avicenna an empiricist?”. I will introduce Avicenna’sAvicenna language of “signification”, “understood content”, “mentalMental impression” and “conception”. Then, following Kenneth P. Winkler[aut]Winkler, K. ~ P., I will distinguish between origin-empiricism and content-empiricism Empiricism and reinterpret the distinction in Avicenna’sAvicenna language as OEA and CEA. I will show that Avicenna’sAvicenna analysis of the relationship between knowledge, on the one hand, and sensation and imagination, on the other hand, includes three empiricist themes. I use these (...) themes to argue that that Avicenna is committed to OEA. Then, I will consider three “possible” limitations to Avicenna’sAvicenna origin-empiricism. I will show that a common empiricist solution, that relies on the compositionality of the “understood contentContent”, quia demonstration and relative conceptions, has significant limitations. I will argue that a careful examination of these limitations, and the epistemology of the primary conceptions, show that Avicenna is not committed to CEA. I will conclude that “Is Avicenna[aut]Avicenna an empiricist?” has no simple yes-or-no answer. This raises a parallel open question: “Is Avicenna a rationalist?”, in some sense. After briefly commenting on this question, I will consider a related, but generally dissociated, question on the reality of abstraction, namely “Is Avicenna an abstractionist?” I will explain how the common replies to both questions rely on different incompatibility principles, according to which emanation from the active intellect is incompatible with apprehension by or abstraction from sense-perception. I will end by outlining the elements of a reading of Avicenna that assumes neither of these incompatibility principles. (shrink)

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.

A gossip protocolGossip protocol is a procedure for sharing secrets in a network. The basic action in a gossip protocolGossip protocol is a telephone call wherein the caller and the callee exchange all the secrets they know. An agent who knows all secrets is an expert. The usual termination condition is that all agents are experts. Instead, we explore some protocols wherein the termination condition is that all agents know that all agents are experts. We call such agents super experts. (...) Additionally, we model that agents who already know that all agents are experts, do not make and do not answer calls. We also model that such protocols are common knowledgeCommon knowledge among the agents. We investigate conditions under which such gossip protocolsGossip protocol terminate, both in the synchronous case, where there is a global clock, and in the asynchronous case, where there is not. We show that a protocol with missed calls can terminate faster than the same protocol without missed calls. (shrink)