I. Professor William L. Rowe begins an interesting paper on the Cosmological Argument by stating that his ‘purpose …is not to resurrect it’ but ‘to uncover, clarify, and examine some of the philosophical concepts and theses essential to the reasoning exhibited in the argument’. However, in the concluding pages of his paper, Rowe is at some pains to show that his discussion does at least demonstrate that the Cosmological Argument is beyond the reach of criticisms levelled against it in the (...) works of its classical critics, Hume and Kant and their modern-day counterparts. To quote his concluding remark: ‘Like most important philosophical arguments, it appears that the Cosmological Argument is neither as good as its supporters have claimed it to be nor as bad as its critics have believed.’ Now I have long suspected that some such estimate of the Cosmological Argument was correct; however I do not think that Rowe's discussions in his paper establish his contention that the reports of the death of the Cosmological Argument have been premature. Nevertheless, Rowe's uncovering of the reasoning that lies behind the argument does have the merit of laying bare the precise point from which, as well as the direction in which, the defence of the argument should proceed if we are to truly make some headway on this hoary topic. In what follows, I will first try to show why I think Rowe's discussion fails to establish his conclusion and thereafter try to give the outline of a more successful strategy for arriving at the same conclusion. (shrink)

My attention in this paper will be focused almost exclusively on the interpretation of Part III of Hume's Dialogues Concerning Natural Religion suggested by Professor Nelson Pike at the very close of his excellent recent commentary on that enduring classic. 1 As I will show briefly in Section II below, Pike's interpretation of Part III emerges from the wider context of his quarrel with Kemp Smith in regard to the final outcome of these Dialogues . I find much in Pike's (...) commentary to applaud and to agree with, especially in its earlier sections, but on this whole question of Hume's final position in the Dialogues , I find his views less than convincing. But in this paper I will confine myself to showing what I think is wrong with his reading of Part III, and no more. 2 In particular, I will make no effort to challenge Pike's disagreement with Kemp Smith in regard to the final dialogue. Though Pike himself regards Part III as crucial, I am well aware that to make anything like a successful case against the kind of interpretation he favours of the upshot of the Dialogues much more would have to be done than I attempt in this paper. I submit this attempt, then, in partial fulfillment of a larger task to be completed at some future time. (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)

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)

Few people would care to deny, whether within India or without, that Buddha is the greatest Indian of all times. Whether from the standpoint of the purity of his life, the daring originality and novelty of his thought, or the extent of his influence in shaping the culture of the world, it would be hard to beat the record of Buddha. Even making every allowance for the common idea that no man is a prophet in his own land, it is (...) difficult to believe that for the last thousand years Buddha has been practically an exile from the land of his birth. It is nothing short of an irony of history, and nothing short of a tragedy for India, that the great unifying force that he represented in his time and does represent even to-day has been allowed to run waste and the great moral stamina that he sought to supply failed in its historic mission. (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)

Over the past few years we have witnessed a sharp resurgence in feminist activism as young women have become increasingly interested in feminist ideas as a means of making sense of their lives. This resurgence in feminist practice is evidenced by the formation of myriad groups and networks across Britain and the initiation of various feminist projects and campaigns, reported regularly and widely in local and national media. This article examines the renaissance of this new feminism through the example of (...) one of the most active and publicly visible organisations, UK Feminista. It draws on ethnographic research into young women’s feminist activism, exploring the emergence of UK Feminista as part of this new wave of feminist activism in Britain. We argue that the wave metaphor has multifarious meanings and that to understand how the political practice of a small feminist organisation contributes to the amplification of this wave of feminist activism it is helpful to draw on concepts developed within social movement theory such as cycles of protest, repertoires of action and collective identities. We show how UK Feminista mobilises young women and provides resources to existing and new feminist organisations and activist groups, commenting particularly on its use of the Internet and social media as a mobilising device. We explore the role of the Internet in young women’s engagement with feminism, the forms of activism in which they take part and the importance of feminist cultural memory to the construction of a collective feminist identity. Our analysis reveals the significant part played by a small mobilising structure, UK Feminista, in the resurgence of feminist activism in the UK. (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.

IN the final section of his chapter on ‘Perception’ in The Problem of Knowledge, Ayer makes the statement that ‘The failure of phenomenalism does not mean, however, that there is no logical connection of any kind between the way physical objects appear to us and the way they really are’. To prove his contention, he sets out ‘a pair of limiting cases’ of conditions in which the truth of premises referring exclusively to ‘appearance’ would allegedly afford logical guarantees for the (...) existence, or the non-existence, of physical objects. In this paper I wish to raise some serious reservations concerning the viability of the second of these two limiting conditions, which reads. (shrink)

IT has been some time now since anyone professing himself to be a phenomenalist has characterized physical objects as ‘logical constructions out of sense-data’ in the strict sense of this expression. If he is to be justified in applying the expression in the strict sense, the phenomenalist must demonstrate that there exists a relation of mutual entailment between a statement implying the existence of a physical object and a statement referring exclusively to our ‘sense-experiences’. As a matter of historical fact, (...) no phenomenalist has ever succeeded in doing this; and it is now generally acknowledged, even by those who still wish to adhere to some form of a phenomenalistic analysis of physical object statements, that there are difficulties in principle why the reductivist programme of the early phenomenalists cannot be carried out. The modern-day phenomenalist, in an attempt to avoid the pitfalls in his predecessors’ position, tends to regard physical objects as fruitful and convenient theoretical constructions for interpreting and predicting our sense-experiences. Thus, although still committed to the view that physical objects are ‘constructed’ out of sense-data, he is no longer committed to the thesis of a strict logical equivalence between statements about objects and statements about experiences. Professor A J Ayer, perhaps the foremost exponent of this type of view today, put the matter this way not so long ago. (shrink)

A Student of ethics cannot but be struck by some fundamental difference of outlook in the ethics of the East and the West. This has a particular bearing on the problem of the relation of the individual to society. In practice this has given rise to a question of supreme importance to every thinking man: is an individual completely subservient to society, or is society completely subservient to the demands of individuality? I.e., is the moksha of any individual impossible till (...) all the needs of society have been completely satisfied, or is an individual free to use or not to use society as a field for his own spiritual expansion? There is no doubt a fundamental discrepancy in these outlooks as thus conceived, and the problem is : can we reconcile them? (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.

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 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.

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)

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

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)

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{*}} $.