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 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)

Plato was the first philosopher in the Western tradition to reflect systematically on rhetoric. In this book, Tushar Irani presents a comprehensive and innovative reading of the Gorgias and the Phaedrus, the only two Platonic dialogues to focus on what an art of argument should look like, treating each of the texts individually, yet ultimately demonstrating how each can best be understood in light of the other. For Plato, the way in which we approach argument typically reveals something about (...) our deeper desires and motivations, particularly with respect to other people, and so the key to understanding his views on the proper practice of argument lies in his understanding of human psychology. According to this reading, rhetoric done well is simply the practice of philosophy, the pursuit of which has far-reaching implications for how we should relate to others and how we ought to live. (shrink)

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)

This paper focuses on a neglected portion of Plato’s Gorgias from 506c to 513d during Socrates’s discussion with Callicles. I claim that Callicles adopts the view that virtue lies in self-preservation in this part of the dialogue. Such a position allows him to assert the value of rhetoric in civic life by appealing not to the goodness of acting unjustly with impunity, but to the badness of suffering unjustly without remedy. On this view, the benefits of the life of rhetoric (...) depend on the idea that virtue consists in the power to protect oneself from the predations of others. I argue that by challenging this understanding of virtue as self-preservation, Socrates both deprives Callicles of any remaining justification for the rhetorical life in the Gorgias and, at the same time, makes room for his own defense of the life of philosophy. (shrink)

Today the halls of Technology, Entertainment, and Design and Davos reverberate with optimism that hacking, brainstorming, and crowdsourcing can transform citizenship, development, and education alike. This article examines these claims ethnographically and historically with an eye toward the kinds of social orders such practices produce. This article focuses on a hackathon, one emblematic site of social practice where techniques from information technology production become ways of remaking culture. Hackathons sometimes produce technologies, and they always, however, produce subjects. This article argues (...) that the hackathon rehearses an entrepreneurial citizenship celebrated in transnational cultures that orient toward Silicon Valley for models of social change. Such optimistic, high-velocity practice aligns, in India, with middle-class politics that favor quick and forceful action with socially similar collaborators over the contestations of mass democracy or the slow construction of coalition across difference. (shrink)

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)

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.

This paper reevaluates the role of Adeimantus in Book 2 of Plato's Republic, arguing that his challenge to Socrates' view of justice—specifically, his interest in the influence of the outer world on our inner lives—serves a crucial yet underappreciated purpose in initiating the political project of the work. I suggest that it's due to Adeimantus' contribution in the Republic that Plato's wide-ranging inquiry into issues in ethics, politics, psychology, epistemology, and metaphysics hangs together as an integrated whole. A further benefit (...) of this reevaluation, I argue, is the way it illuminates Plato's assessment of poetry in the dialogue. This paper was delivered this past year as part of the Boston Area Colloquium in Ancient Philosophy. Comments are welcome but please do not cite without permission. (shrink)

This collection of original essays examines the controversy over and attacks on rationality in the methodologies of the humanities and the physical and social sciences. These essays represent the thinking of a wide variety of philosophers, psychologists, historians, classicists, and economists about the role of rationality in thought and action. Reflecting the differing perspectives of their authors' disciplines, as well as the centrality of rationality to those disciplines, they are important additions to a debate that has been going on for (...) some twenty years. Beginning with an introductory essay in which K.D. Irani covers the various ways in which rationality can be approached, the body of the book is divided into five sections dealing with various aspects of the issue. Respectively, they are concerned with rationality as it relates to ethical and social thought and action; general scientific thought and the particular disciplines of economics, history, and law; the analytic and hermenutic approaches to communications and learning; and the contrasting classical traditions of ancient Greece and China. In the final section, two differing theories concerning the nature of rationality itself are presented. A list of suggested further readings completes the volume. (shrink)

This paper explores Plato’s views on the purpose of rhetorical form by surveying the way in which Socrates engages in speechmaking at several points in the Gorgias. I argue that Socrates has nothing in principle against the use of a long speech as part of the practice of philosophical inquiry and argument, provided that the speech is geared toward understanding. This reflects a key and relatively unremarked distinction that Socrates makes in the Gorgias between persuasion that comes from being convinced (...) and persuasion that comes from being taught. The kind of long speeches that Socrates objects to are those that have a conviction-based purpose. However, this leaves open the use of a wide variety of rhetorical techniques—pieces of argument, speechifying, analogy, myth, and exhortation—that have a teaching-based purpose, which is precisely the sort of rhetoric that Socrates licenses in the dialogue. (shrink)

We prove that Basic Arithmetic, BA, has the de Jongh property, i.e., for any propositional formula A(p 1,..., p n ) built up of atoms p 1,..., p n, BPC $${\vdash}$$ A(p 1,..., p n ) if and only if for all arithmetical sentences B 1,..., B n, BA $${\vdash}$$ A(B 1,..., B n ). The technique used in our proof can easily be applied to some known extensions of BA.

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 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 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.

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)

O envelhecimento populacional é uma preocupação mundial e exige medidas de prevenção de saúde a serem adotadas com a maior brevidade possível. Esse processo é, muitas vezes, acompanhado pelo declínio das habilidades cognitivas, como a memória e as funções executivas. O objetivo do presente estudo é ..

This paper examines how Plato’s rejection of the friends of the forms at 248a–249b in the Sophist is continuous with the arguments that he develops shortly after this part of the dialogue for the interrelatedness of the forms. I claim that the interrelatedness of the forms implies that they are changed, and that this explains Plato’s rejection of the friends of the forms. Much here turns on the kind of change that Plato wants to attribute to the forms. I distinguish (...) my view of the sort of change that the forms experience from other kinds of change—such as ‘Cambridge change’—that scholars have believed Plato has in mind in rejecting the friends of the forms. On the view that I advance, a form experiences a change (which I call ‘perfect change’) in its association with another form that distinguishes it as the distinctive being that it is—that is, through its possession of its distinctive properties. (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.

I begin with a puzzle. According to some scholars, Plato’s view that the forms possess value as objects of desire gives rise to a problem in his metaphysics: how can forms of injustice and ugliness be considered desirable? To resolve this puzzle, I focus on Plato’s views on eros and argue that the philosopher’s love of forms is best understood as a kind of rational compulsion. Approaching the puzzle from this direction gives us an idea of how Plato’s forms might (...) be viewed as value laden. It also suggests an understanding of his metaphysics that’s less otherworldly than is often thought. (shrink)

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

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{*}} $.

This edited collection focuses on the problem of social justice, or, more particularly, how the demand for social justice was articulated and implemented in ...

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.

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.

The authors suggest that postcolonial science studies can do more than expand answers to questions already posed; it can generate different questions and different ways of looking at the world. To illustrate, the authors draw on existing histories and anthropologies and critical theories of colonial and postcolonial technoscience. To move forward together, rather than remaining mired in regretful contemplation of past biases, the authors offer some analytical and practical suggestions. In reading hegemonic forms of postcolonial computing, this article offers tactics (...) for rereading, rewriting, or reimagining those scripts. (shrink)

_Telling Stories to Change the World_ is a powerful collection of essays about community-based and interest-based projects where storytelling is used as a strategy for speaking out for justice. Contributors from locations across the globe—including Uganda, Darfur, China, Afghanistan, South Africa, New Orleans, and Chicago—describe grassroots projects in which communities use narrative as a way of exploring what a more just society might look like and what civic engagement means. These compelling accounts of resistance, hope, and vision showcase the power (...) of the storytelling form to generate critique and collective action. Together, these projects demonstrate the contemporary power of stories to stimulate engagement, active citizenship, the pride of identity, and the humility of human connectedness. (shrink)

This is an Introduction to the special issue of Metaphilosophy entitled Philosophy as a Way of Life, giving a brief account of the genesis of the project, an overview of the topic, and a summary of the topics covered in the issue.