Born in 1945, John Lane Bell is not only one of the greatest logicians of our time, but he is also one the most gifted in the art of writing, as witnessed by the success of his introductory texts, many of which were originally written as lecture notes: their essential clarity is an exemplar of the Attic style. More generally, Bell’s works are a rare example of how rigour and sophisticated elegance can coexist.

AbstractIntegrity is considered an important corporate value. Yet recent global events have highlighted the challenges firms face at living up to their stated values, especially when extended supply chain partners are involved. The concept of Supply Chain Integrity (SCI) can help firms shift focus beyond internal corporate integrity, toward supply chain integrity. Researchers and managers will benefit from an understanding of the SCI concept toward implementing SCI to better align supply chain partners with stated corporate values. This research fully develops (...) and empirically grounds the firm-level, inter-firm-oriented SCI concept. The thematic analysis of six firms’ archival and website content elaborated empirical descriptions of SCI themes and enabled the development of a process model for SCI, presenting a novel view of the underlying process by which firms can assess, develop, and maintain SCI across their supply chains. We propose the SCI model as an evolutionary process to improve a firm’s supply chain sustainability, rather than a dichotomous end state where firms either “have” integrity or they don’t. The SCI model could be used as a tool to help leaders create necessary change to better align values and supporting statements with culture, while influencing and affecting stakeholders across the supply chain. This is particularly important in today’s world, where business leaders must consider all stakeholders and address important stakeholder-driven issues such as supply chain sustainability, resilience, and security, which are now at the forefront in the ever-changing environment. (shrink)

Previous research has demonstrated electroencephalogram (EEG) changes in response to low-odor concentrations, resulting in near-chance detection. Such findings have been taken as evidence for olfaction without awareness. We replicated and extended previous work by examining EEG responses to water-water control, 0.0001, 0.001, 0.01, and 1 ppm isoamyl acetate (IAA) in water paired with water only. Detection was above chance (>50%) for .001 and above, and alpha decreased only to those concentrations, suggesting that EEG changes corresponded to IAA awareness. However, when (...) correct trial EEGs were compared to incorrect trial EEGs during .001 ppm, right posterior/central alpha decreased during incorrect trials and alpha decreased more globally (including frontal sites) during correct trials. These data may not reflect awareness or unawareness per se. Instead, results are discussed regarding activation of perceptual systems in the posterior region during incorrect trials and the activation of frontal action systems during a subset of correct trials. (shrink)

ABSTRACT Notwithstanding recent controversies involving echo chambers and social media, “post-truth” has always been central to philosophical investigations of what is knowable and good. The internal tension of the term offers a choice: to gasp in feigned astonishment at the hell-in-a-handbasket state of public discourse, or to reflect critically on what is beyond, after, or other than the truth. In this essay, we approach post-truth via elements of narrative, biography, and myth, portraying Friedrich Nietzsche's polytropic figure, Zarathustra, as he might (...) have spoken to the contemporary moment. We demonstrate how Zarathustra affords access to the idea that truth and life are inextricable in the human condition. To temper this tension, we depict a character whose disposition toward post-truth spans from certainty and doubt to exuberance and despair. Our hope is to indicate how, for the humans of Motley Cow, post-truth is ubiquitous, institutional, and infrastructural. (shrink)

Traditionally, expressions in formal systems have been regarded as signifying finite inscriptions which are—at least in principle—capable of actually being written out in primitive notation. However, the fact that (first-order) formulas may be identified with natural numbers (via "Gödel numbering") and hence with finite sets makes it no longer necessary to regard formulas as inscriptions, and suggests the possibility of fashioning "languages" some of whose formulas would be naturally identified as infinite sets . A "language" of this kind is called (...) an infinitary language : in this article I discuss those infinitary languages which can be obtained in a straightforward manner from first-order languages by allowing conjunctions, disjunctions and, possibly, quantifier sequences, to be of infinite length. In the course of the discussion it will be seen that, while the expressive power of such languages far exceeds that of their finitary (first-order) counterparts, very few of them possess the "attractive" features (e.g., compactness and completeness) of the latter. Accordingly, the infinitary languages that do in fact possess these features merit special attention. (shrink)

This paper treats some of the issues raised by Putnam's discussion of, and claims for, quantum logic, specifically: that its proposal is a response to experimental difficulties; that it is a reasonable replacement for classical logic because its connectives retain their classical meanings, and because it can be derived as a logic of tests. We argue that the first claim is wrong (1), and that while conjunction and disjunction can be considered to retain their classical meanings, negation crucially does not. (...) The argument is conducted via a thorough analysis of how the meet, join and complementation operations are defined in the relevant logical structures, respectively Boolean- and ortholattices (3). Since Putnam wishes to reinstate a realist interpretation of quantum mechanics, we ask how quantum logic can be a logic of realism. We show that it certainly cannot be a logic of bivalence realism (i.e., of truth and falsity), although it is consistent with some form of ontological realism (4). Finally, we show that while a reasonable explication of the idealized notion of test yields interesting mathematical structure, it by no means yields the rich ortholattice structure which Putnam (following Finkelstein) seeks. (shrink)

then E has a subset which is the domain of a model of Peano's axioms for the natural numbers. (This result is proved explicitly, using classical reasoning, in section 3 of [1].) My purpose in this note is to strengthen this result in two directions: first, the premise will be weakened so as to require only that the map ν be defined on the family of (Kuratowski) finite subsets of the set E, and secondly, the argument will be constructive, i.e., (...) will involve no use of the law of excluded middle. To be precise, we will prove, in constructive (or intuitionistic) set theory3, the following.. (shrink)

If we imagine a chess-board with alternate blue and red squares, then this is something in which the individual red and blue areas allow themselves to be distinguished from each other in juxtaposition, and something similar holds also if we imagine each of the squares divided into four smaller squares also alternating between these two colours. If, however, we were to continue with such divisions until we had exceeded the boundary of noticeability for the individual small squares which result, then (...) it would no longer be possible to apprehend the individual red and blue areas in their respective positions. But would we then see nothing at all? Not in the least; rather we would see the whole chessboard as violet, i.e. apprehend it as something that participates simultaneously in red and blue. (shrink)

Call a family F of subsets of a set E inductive if ∅ ∈ F and F is closed under unions with disjoint singletons, that is, if ∀X∈F ∀x∈E–X(X ∪ {x} ∈ F]. A Frege structure is a pair (E.

Full proofs of the Gödel incompleteness theorems are highly intricate affairs. Much of the intricacy lies in the details of setting up and checking the properties of a coding system representing the syntax of an object language (typically, that of arithmetic) within that same language. These details are seldom illuminating and tend to obscure the core of the argument. For this reason a number of efforts have been made to present the essentials of the proofs of Gödel’s theorems without getting (...) mired in syntactic or computational details. One of the most important of these efforts was made by Löb [8] in connection with his analysis of sentences asserting their own provability. Löb formulated three conditions (now known as the Hilbert-Bernays-Löb derivability conditions), on the provability predicate in a formal system which are jointly sufficient to yield the Gödel’s second incompleteness theorem for it. A key role in Löb’s analysis is played by (a special case of) what later became known as the diagonalization or fixed point property of formal systems, a property which had already, in essence, been exploited by Gödel in his original proofs of the incompleteness theorems. The fixed point property plays a central role in Lawvere’s [7] category-theoretic account of incompleteness phenomena (see also [10]). Incompleteness theorems have also been subjected to intensive investigation within the framework of modal logic (see, e.g.[4], [5]). In this formulation the modal operator takes up the role previously played by the provability predicate, and the derivability conditions on the latter are translated into algebraic conditions (the so-called GL, i.e., Gödel–Löb, conditions) on the former. My purpose here is to present a framework for incompleteness phenomena, fully compatible with intuitionistic or constructive principles, in which the idea of a coding system is retained, only in a 2 simple, but very general form, a form wholly free of syntactical notions. As codes we shall take the elements of an arbitrary given nonempty set, possibly, but not necessarily, the set of natural numbers.. (shrink)

We begin with the following quotation from Karp [1964]: My interest in infinitary logic dates back to a February day in 1956 when I remarked to my thesis supervisor, Professor Leon Henkin, that a particularly vexing problem would be so simple if only I could write a formula which would say x = 0 or x = 1 or x = 2 etc. To my surprise, he replied, "Well, go ahead." Traditionally, expressions in formal systems have been regarded as signifying (...) finite inscriptions which are—at least in principle—capable of actually being written out in primitive notation. However, the fact that (first-order) formulas may be identified with natural numbers (via "Gödel numbering") and hence with finite sets makes it no longer necessary to regard formulas as inscriptions, and suggests the possibility of fashioning "languages" some of whose formulas— such as that in the above quotation—would be naturally identified as infinite sets. A "language" of this kind is called an infinitary language: in this article we discuss those infinitary languages which can be obtained in a straightforward manner from first-order languages by allowing conjunctions, disjunctions and, possibly, quantifier sequences, to be of infinite length. In the course of the discussion we shall see that, while the expressive power of such languages far exceeds that of their finitary (first-order) counterparts, very few of them possess the "attractive" features (e.g., compactness and completeness) of the latter. Accordingly, the infinitary languages that do in fact possess these features merit special attention. In §1 we lay down the basic syntax and semantics of infinitary languages and demonstrate their expressive power by means of examples. §2 is devoted to those infinitary languages which permit only finite quantifier sequences: these languages turn out to be relatively well-behaved. In §3 we discuss the compactness problem for infinitary languages and its.. (shrink)

One of the most familiar uses of the Russell paradox, or, at least, of the idea underlying it, is in proving Cantor's theorem that the cardinality of any set is strictly less than that of its power set. The other method of proving Cantor's theorem — employed by Cantor himself in showing that the set of real numbers is uncountable — is that of diagonalization. Typically, diagonalization arguments are used to show that function spaces are "large" in a suitable sense. (...) Classically, these two methods are equivalent. But constructively they are not: while the argument for Russell's paradox is perfectly constructive, (i.e., employs intuitionistically acceptable principles of logic) the method of diagonalization fails to be so. I describe the ways in which these two methods.. (shrink)

A structure is a triple A = (A, {Ri: i ∈ I}, {ej: j ∈ J}), where A, the domain or universe of A, is a nonempty set, {Ri: i ∈ I} is an indexed family of relations on A and {ej: j ∈ J}) is an indexed set of elements —the designated elements of A. For each i ∈ I there is then a natural number λ(i) —the degree of Ri —such that Ri is a λ(i)-place relation on A, (...) i.e., Ri ⊆ Aλ(i). This λ may be regarded as a function from I to the set ω of natural numbers; the pair (λ, J) is called the type of A. Structures of the same type are said to be similar. Note that since an n-place operation f: An → A can be regarded as an (n+1)-place relation on A, algebraic structures containing operations such as groups, rings, vector spaces, etc. may be construed as structures in the above sense. (shrink)

Axioms for the continuum, or smooth real line R. These include the usual axioms for a commutative ring with unit expressed in terms of two operations + and i , and two distinguished elements 0 ≠ 1. In addition we stipulate that R is a local ring, i.e., the following axiom: ∃y x i y = 1 ∨ ∃y (1 – x) i y = 1. Axioms for the strict order relation < on R. These are: 1. a < b (...) and b < c implies a < c. 2. ¬(a < a) 3. a < b implies a + c < b + c for any c. ≤ 4. a < b and 0 < c implies acbc.. (shrink)

Theorem 3.1. can be strengthened. Let S be a (well-termed) local set theory and (E, ≤) a partially ordered S-set. An element m of E is internally maximal if it satisfies S ∀x∈E [m ≤ x → m = x]. We can then prove the..

By a lattice we shall always mean a distributive lattice which is bounded, i.e. has both a bottom element 0 and a top element 1. Lattice homomorphisms will always be assumed to preserve 0 and 1.

In this paper I reflect on the nature of mathematical beauty, and examine the connections between mathematics and the arts. I employ Plutarch’s distinction between the intelligible and the sensible, to compare the beauty of mathematics with the beauties of music, poetry and painting. While the beauty of mathematics is almost exclusively intelligible, and the beauties of these arts primarily sensible, it is pointed out that the latter share with mathematics a certain kind of intelligible beauty. The paper also contains (...) reflections on the formal beauty and timelessness of mathematics, beauty as richness flowing from simplicity, form and content in mathematics, and mathematics and fiction. It concludes with some remarks on the question of why mathematical beauty is so little appreciated by non-mathematicians. (shrink)

Curriculum 2000 has meant significant change for the post-16 sector. New qualifications have been introduced (e.g. the new Advanced Subsidiary examination) and the number of students involved in education and training post-16 has increased. In this scenario how can the standards of new qualifications, particularly the new Advanced Subsidiary examinations, be compared with those of previous qualifications? One method is to use the prior achievement of candidates (i.e. GCSE results) as a basis for comparison of their results on subsequent qualifications (...) (i.e. A levels and AS). This method of comparability and its limitations will be explored using examples with actual data. (shrink)

This review of Martin Jay’s recent published collection of essays examines his ongoing rethinking, supplementation, and revision of central themes—the negative and positive dialectics of historical totalization, the varieties and uses of conceptions of experience, the nature of visual cultures and scopic regimes, and the ambiguities of truth-construction in the public realm—that have been the focus of his major works since the 1970s. It argues that his more recent work indicates a gradual shift toward an affirmation of the kinds of (...) paratactic and deconstructive thinking of Adorno and Derrida as models for producing appropriate forms of historical consciousness and historical critique in the present, and it raises the question of how the issues of historical truth-telling, consensual collective identity, ethical action, and the cultural role of the critical intellectual are reformulated in this process. (shrink)

The popular belief that religion is the same everywhere or that all religions are ‘at bottom’ identical in essentials is a widespread falsehood that is saved from being completely worthless by the fact that religion does exhibit a universal or common structure wherever it appears. This structure is intimately related to the structure of human life in the world. The enduring pattern that enables us to understand religions widely separated in both time and space depends largely on the fact that (...) man and the process of human life in the world have their own structures which remain, despite the undeniable variety introduced by vast differences of culture, ethnic features, geographical location, climate etc. Structure means pattern or form; it is reality significantly organised. It can be grasped as that which endures above and beyond changing historical details. Because human life has a structure, we are able to understand the wrath of Achilles or sympathise with the love of Abélard for Héloïse although we are separated from both by centuries of time. (shrink)

The dissertation's primary task is to discern to what extent the investigations contained in Aristotle's Metaphysics conform to the model of science developed in the Posterior Analytics. It concludes that the Metaphysics substantially follows the model of the Analytics in studying the causes and attributes of a specific nature, although it makes significant departures especially in its conception of the principles of being and substance. ;Two introductory chapters discuss respectively Aristotle's conception of science in the Analytics and the problems one (...) is likely to face in attempting to apply this conception to the Metaphysics . Chapter 3 clarifies the meaning of the phrase "science of being qua being" by reference to the Posterior Analytics, and introduces the concept of pro&d13;v 3&d3;n equivocity. Chapter 4 considers the role of dialectic in metaphysics with particular reference to the principle of noncontradiction. Chapter 5 argues that the second part of Meta. 4.2 introduces a demonstrative science of the per se attributes of being and unity. ;Chapter 6 introduces the investigation into the principles and causes of being and reviews the aporias about the principles . Chapter 7 offers a hypothesis to explain Aristotle's identification of the science of being with first philosophy. The last two chapters argue that Aristotle is pursuing this investigation in the central books of Melaphysics, but his pursuit of the investigation there is incomplete. ;This interpretation provides an alternative to those of T. H. Irwin and Walter Leszl, who argue that metaphysics is a second-order discipline and not a science after the model of the Analytics. It finds explicit support in recent work by Robert Bolton and Alan Code, while widening the range of issues to include the attributes of being and the importance of first philosophy for understanding the nature of being. It supports many of the conclusions of Joseph Owens's, Urbain Dhondt's, and Frede and Patzig's work on the object of metaphysics. There are also substantial discussions of the contributions of G. E. L. Owen, J. G. Stevenson, Theodore Scaltsas, and John Thorp. (shrink)

Partant du substrat intellectuel et scientifique sur lequel repose, en dernière analyse, l’Atlas linguistique de la France (ALF : 1902-1910) de Jules Gilliéron (1854-1926), cet article envisage l’évolution ultérieure de la géographie linguistique « atlantiste », surtout sous l’égide de Karl Jaberg (1877-1958) et Jakob Jud (1882-1952), auteurs de l’atlas italo-suisse AIS, et des promoteurs des Nouveaux atlas linguistiques de la France (NALFs) à partir de 1950. Il en appert que l’orientation théorique de base et la recherche inconditionnée de l’intercomparabilité (...) des données d’atlas collectées ont fini par céder devant une « faim empirique » générale qui risque de saper la raison d’être de la géographie linguistique à proprement parler. A la fin, il est fait mention des deux parties de l’atlas linguistique ladin ALD, publiées – chaque fois en 7 volumes – en 1998 et en 2012, qui ont été élaborées dans le plus strict respect des principes « atlantistes » de Jules Gilliéron. (shrink)

Despite the title, I do not intend to launch another expedition into the domain of epistemology. I wish instead to call attention to some problems which have arisen for philosophical theologians and philosophers of religion, as a result of two facts about the development of modern philosophy and its bearing on the analysis and interpretation of religious insight. Following these considerations, I shall propose in brief compass a programme for the future which I believe will prove fruitful for the philosophical (...) treatment of religious concerns. (shrink)

This is the third edition of a book originally published in the 1970s; it provides a systematic and nicely organized presentation of the elegant method of using Boolean-valued models to prove independence results. Four things are new in the third edition: background material on Heyting algebras, a chapter on ‘Boolean-valued analysis’, one on using Heyting algebras to understand intuitionistic set theory, and an appendix explaining how Boolean and Heyting algebras look from the perspective of category theory. The book presents results (...) from a number of set theorists and includes an insightful and informative foreword by Dana Scott. Bell's presentation is lively and pleasant to read, and the material is given in a nicely cohesive way.One obvious reason to be interested in independence proofs is that they concern the important question, what is the set-theoretic hierarchy like? The proofs in Bell's book cover some of the most basic and fundamental independence results, such as those concerning the size of the continuum, the independence of the Axiom of Choice from ZF, cardinal collapsing, Souslin's hypothesis, and Martin's Axiom.Since Gödel's incompleteness theorem, it has been known that for any serious candidate list of set-theoretic axioms, there will be statements neither provable nor disprovable from those axioms. What is really interesting, however, and what the independence proofs discussed here show, is how many of the most natural questions about sets are not decided by the standard axioms of ZFC, and how …. (shrink)

In this paper a number of oppositions which have haunted mathematics and philosophy are described and analyzed. These include the Continuous and the Discrete, the One and the Many, the Finite and the Infinite, the Whole and the Part, and the Constant and the Variable.

BellJohn L.* * _ Oppositions and Paradoxes: Philosophical Perplexities in Science and Mathematics _. Peterborough, Ontario: Broadview Press, 2016. ISBN: 978-1-55481302-5 ; 978-1-77048603-4. Pp. xiv + 202.

Some concepts that are now part and parcel of mathematics used to be, at least until the beginning of the twentieth century, a central preoccupation of mathematicians and philosophers. The concept of continuity, or the continuous, is one of them. Nowadays, many philosophers of mathematics take it for granted that mathematicians of the last quarter of the nineteenth century found an adequate conceptual analysis of the continuous in terms of limits and that serious philosophical thinking is no longer required, except (...) perhaps when the question of the continuum is transferred to the arena of set theory where it takes the form of the infamous continuum hypothesis. As Philip Ehrlich has recently shown, this conviction goes back to the early writings of Russell who, in 1903 and then again in later writings, forcefully and eloquently pushed the view that mathematicians had given the final answer to immemorial conundrums arising from the continuous and infinitesimals . This proclamation of victory came with what was announced as the necessary defeat of the notion of the infinitesimal, despite the fact that mathematicians like Thomae, Du Bois-Reymond, Stolz, Bettazi, Veronese, Levi-Civita, and Hahn were investigating mathematical structures containing infinitesimals in a mathematically rigorous and logically consistent manner. In this respect Russell was merely walking in the footsteps of Cantor, and many of Russell's contemporaries were only too keen to keep infinitesimals out of Cantor's paradise. However, although Cantor certainly wanted to send the notion of infinitesimal to Hell, the notion kept a low profile in the mathematical purgatory, making its way in the study of non-Archimedean ordered algebraic systems. Of course, nowadays everyone has heard of Robinson's attempt at resurrecting infinitesimals in analysis in the form of non-standard analysis, but Robinson's work, despite the fact that it had, in …. (shrink)

Let it be clear at the outset that in reappraising Dewey's thought we have to do with no minute philosopher. In breadth of interest and range of thought he belongs with the great comprehensive thinkers of the past. And in contrast to many thinkers both in his own time and since, he had a constructive program. Philosophy for him meant more than analysis, even though analysis is an important part of the philosophic enterprise. Dewey's constructive philosophy has too often been (...) lost in polemic discussion. I subscribe to the confession made some years ago by Ernest Hocking in which he said that he began to understand Dewey when he started reading him for enjoyment and not for the purpose of showing that he was all wrong! As Dewey's work shapes up in historical perspective, it assumes a great substantiality. One may disagree and one may correct, but in comparison with philosophy of a wholly technical and professional sort, Dewey's large-minded approach to genuinely philosophic questions places him among philosophers of stature. (shrink)