ABSTRACT We present a new method for proving theorems in the equational theory of partial maps over toposes introduced in the papers [C'089] and [086], The method is given by a system of rules of formation of proofs. The proofs of f is defined' and the proofs of correctness ‘φ)' formed by application of the rules of the system are such that they contain a computation of the value f, where f is a partial function valued in natural numbers and (...) is a vector of natural numbers. We show that the system is complete, i.e. if an equation holds in every model then it has a proof formed by application of the rules of the system. The system is equipped with a method of visual presentation of proofs by nested commutative triangles, i.e. commutative triangles which contain in their interiors other commutative triangles which may be also nested. To provide formal foundations for the method of visual presentation of proofs we give a mathematical description of nested commutative triangles in terms of directed graphs and graph homomorphisms. AMS Subject Classification 1995: Primary: 68. Secondary: 18. (shrink)

The first volume of the Brandeis University Summer Institute lecture series of 1970 on theories of interacting elementary particles, consisting of four sets of lectures. Every summer since 1959 Brandeis University has conducted a lecture series centered on various areas of theoretical physics. The areas are sufficiently broad to interest a large number of physicists and the lecturers are among the original explorers of these areas. The 1970 lectures, presented in two volumes, are on theories of interacting elementary particles. The (...) four lecturers of Volume 1, and the range of the topics they cover, are as follows: Stephen L. Adler on "Perturbation Theory Anomalies": introduction and review of perturbation theory; the VVA triangle anomaly; absence of radiative corrections; generalizations of our results; connection between Ward identity anomalies and commutator anomalies; applications of the Bjorken limit; and breakdown of the Bjorken limit in perturbation theory. Stanley Mandelstam on "Dynamical Applications of the Veneziano formula for the four-point scalar amplitude; factorization; the operator formalism; Veneziano-type quark models; and higher-order Feynman-like diagrams. Steven Weinberg on "Dynamic and Algebraic Symmetries": Introduction; hadron electrodynamics; local symmetries; and chirality. Wolfhart Zimmermann on "Local Operator Products and Renormalization in Quantum Field Theory": introduction; renormalization; operator product expansions; and local field equations. The second volume contains lectures by Rudolf Haag on observables and fields, by Maurice Jacob on duality, by Michael Reed on non-Fock representations, and by Bruno Zumino on effective Lagrangians and broken symmetries. (shrink)

A new form of the Hyperbolic Pythagorean Theorem, which has a striking intuitive appeal and offers a strong contrast to its standard form, is presented. It expresses the square of the hyperbolic length of the hypotenuse of a hyperbolic right-angled triangle as the “Einstein sum” of the squares of the hyperbolic lengths of the other two sides, Fig. 1, thus completing the long path from Pythagoras to Einstein. Following the pioneering work of Varičak it is well known that relativistic (...) velocities are governed by hyperbolic geometry in the same way that prerelativistic velocities are governed by Euclidean geometry. Unlike prerelativistic velocity composition, given by the ordinary vector addition, the composition of relativistic velocities, given by the Einstein addition, is neither commutative nor associative due to the presence of Thomas precession. Following the discovery of the mathematical regularity that Thomas precession stores, it is now possible to extend Thomas precession by abstraction, (i) allowing hyperbolic geometry to be studied by means of analogies that it shares with Euclidean geometry; and, similarly (ii) allowing velocities and accelerations in relativistic mechanics to be studied by means of analogies that they share with velocities and accelerations in classical mechanics. The abstract Thomas precession, called the Thomas gyration, gives rise to gyrovector space theory in which the prefix gyro is used extensively in terms like gyrogroups and gyrovector spaces, gyroassociative and gyrocommutative laws, gyroautomorphisms, gyrotranslations, etc. We demonstrate the superiority of our gyrovector space formalism in capturing analogies by deriving the Hyperbolic Pythagorean Theorem in a form fully analogous to its Euclidean counterpart, thus contrasting it with the standard form in which the Hyperbolic Pythagorean Theorem is known in the literature. The hyperbolic metric, which supports the Hyperbolic Pythagorean Theorem, has a dual metric. We show that the dual metric does not support a Pythagorean theorem but, instead, it supports the π-Theorem according to which the sum of the three dual angles of a hyperbolic triangle is π. (shrink)

Conditionalization and Jeffrey Conditionalization cannot simultaneously satisfy two widely held desiderata on rules for empirical learning. The first desideratum is confirmational holism, which says that the evidential import of an experience is always sensitive to our background assumptions. The second desideratum is commutativity, which says that the order in which one acquires evidence shouldn't affect what conclusions one draws, provided the same total evidence is gathered in the end. (Jeffrey) Conditionalization cannot satisfy either of these desiderata without violating the other. (...) This is a surprising problem, and I offer a diagnosis of its source. I argue that (Jeffrey) Conditionalization is inherently anti-holistic in a way that is just exacerbated by the requirement of commutativity. The dilemma is thus a superficial manifestation of (Jeffrey) Conditionalization's fundamentally anti-holistic nature. (shrink)

Moving deftly among literary and visual arts, as well as the modern critical canon, Christopher Prendergast's book explores the meaning and value of representation as both a philosophical challenge (What does it mean to create an image that "stands for" something absent?) and a political issue (Who has the right to represent whom?). _The Triangle of Representation_ raises a range of theoretical, historical, and aesthetic questions, and offers subtle readings of such cultural critics as Raymond Williams, Paul de Man, (...) Edward Said, Walter Benjamin, and Hélène Cixous, in addition to penetrating investigations of visual artists like Gros, Ingres, and Matisse and significant insights into Proust and the onus of translating him. Above all, Prendergast's work is a striking display of how a firm grounding in theory is essential for the exploration of art and literature. (shrink)

We introduce proof nets and sequent calculus for the multiplicative fragment of non-commutative logic, which is an extension of both linear logic and cyclic linear logic. The two main technical novelties are a third switching position for the non-commutative disjunction, and the structure of order variety.

In this paper we review some properties of fuzzy observables, mainly as realized by commutative positive operator valued measures. In this context we discuss two representation theorems for commutative positive operator valued measures in terms of projection valued measures and describe, in some detail, the general notion of fuzzification. We also make some related observations on joint measurements.

This work empirically tests the concept of the ‘whistleblowing triangle,’ which is modeled on the three factors encapsulated by the fraud triangle, in the Indonesian context. Anchored in the proposition of an original research framework on the whistleblowing triangle and derived hypotheses, this work aims to expand the body of knowledge on this topic by providing empirical evidence. The sample used is taken from audit firms affiliated with both the big 4 and non-big 4 companies operating in (...) Indonesia. The results of analysis using the PLS-PM method found a significant relationship between the components of the whistleblowing triangle and the intention of blowing the whistle. We found that financial incentives are the most significant predictor of auditors’ intention to blow the whistle in Indonesia. Other components, such as opportunity and rationalization, also play an important role in supporting auditors’ intention to blow the whistle. Our findings also suggest that related pressures are the top priority for audit firms in Indonesia to consider in increasing whistleblowing intention. We expand the previous literature on whistleblowing which has been derived from the components of the fraud triangle :28–56, 2016; Smaili and Arroyo in J Bus Ethics, https://doi.org/10.1007/s10551-017-3663-7, 2017) by adding empirical evidence. (shrink)

A standard interpretation of Plato’s metaphysics holds that sensible particulars are images of Forms. Such particulars are fairly independent, like Aristotelian substances. I argue that this is incorrect: Platonic particulars are not Form images but aggregates of Form images, which are property-instances. Timaeus 49e-50a focuses on “this-suches” and even goes so far as to claim that they compose other things. I argue that Form images are this-suches, which are tropes. I also examine the geometrical account, showing that the geometrical constituents (...) of the elements are also Form images. Thus everything in the sensible world is composed of tropes. (shrink)

Two operations, e.g. measurements, successively applied to the state of a system are said to be non-commutative if the sequence of their application makes a difference for the final result. Non-commuting operations play a crucial role in quantum theory, where they are intimately related to concepts as central as those of complementarity and entanglement. However, their significance is not restricted to the small dimensions of the microworld. For reasons easy to understand, non-commuting operations must be expected to be the (...) rule rather than the exception for operations on mental systems. We demonstrate this with a few examples from our own work, and hope this may contribute to the increasing attention that this exciting new area in consciousness studies is currently attracting worldwide. (shrink)

Conciliatory views of peer disagreement hold that when an agent encounters peer disagreement she should conciliate by adjusting her doxastic attitude towards that of her peer. In this paper I distinguish different ways conciliation can be understood and argue that the way conciliationism is typically understood violates the principle of commutativity of evidence. Commutativity of evidence holds that the order in which evidence is acquired should not influence what it is reasonable to believe based on that evidence. I argue that (...) when an agent encounters more than one peer, and applies the process of conciliation serially, the order she encounters the peers influences the resulting credence. I argue this is a problem for conciliatory views of disagreement, and suggest some responses available to advocates of conciliation. (shrink)

The notions of a commutative (∈, ∈)-neutrosophic ideal and a commutative falling neutrosophic ideal are introduced, and several properties are investigated. Characterizations of a commutative (∈, ∈)-neutrosophic ideal are obtained. Relations between commutative (∈, ∈)-neutrosophic ideal and (∈, ∈)-neutrosophic ideal are discussed. Conditions for an (∈, ∈)-neutrosophic ideal to be a commutative (∈, ∈)-neutrosophic ideal are established. Relations between commutative (∈, ∈)-neutrosophic ideal, falling neutrosophic ideal and commutative falling neutrosophic ideal are considered. Conditions (...) for a falling neutrosophic ideal to be commutative are provided. (shrink)

In this work we introduce a class of commutative rings whose defining condition is that its lattice of ideals, augmented with the ideal product, the semi-ring of ideals, is isomorphic to an MV-algebra. This class of rings coincides with the class of commutative rings which are direct sums of local Artinian chain rings with unit.

This work empirically tests the concept of the ‘whistleblowing triangle,’ which is modeled on the three factors encapsulated by the fraud triangle (pressure or financial incentives, opportunity and rationalization), in the Indonesian context. Anchored in the proposition of an original research framework on the whistleblowing triangle and derived hypotheses, this work aims to expand the body of knowledge on this topic by providing empirical evidence. The sample used is taken from audit firms affiliated with both the big (...) 4 and non-big 4 companies operating in Indonesia. The results of analysis using the PLS-PM method found a significant relationship between the components of the whistleblowing triangle and the intention of blowing the whistle. We found that financial incentives are the most significant predictor of auditors’ intention to blow the whistle in Indonesia. Other components, such as opportunity and rationalization, also play an important role in supporting auditors’ intention to blow the whistle. Our findings also suggest that related pressures are the top priority for audit firms in Indonesia to consider in increasing whistleblowing intention. We expand the previous literature on whistleblowing which has been derived from the components of the fraud triangle (Brown et al. in Account Public Interest 16(1):28–56, 2016; Smaili and Arroyo in J Bus Ethics, 10.1007/s10551-017-3663-7, 2017) by adding empirical evidence. (shrink)

It has been acknowledged that education includes ‘a love of what one teaches and a love of those whom one teaches’ (Hogan 2010: 81), but two traditions of writing in philosophy of education—concerning love for student and love for subject—have rarely been brought together. This paper considers the extent to which the ‘triangular’ relationship of teacher, student and subject matter runs the risk of the rivalry, jealousy and strife that are characteristic of ‘tragic’ love triangles, or entails undesirable consequences such (...) as transference from one intended object of love to another. It argues that this faultline in the literature of educational love corresponds to education's ‘divided heart’. The implication of this exploration of education's triangular relationship is that we cannot ignore the ‘dark of love’, nor can we address it simply by asserting that educational love must be of a more honourable sort than romantic love. These tensions can be reconciled through the loving recourse of ‘ceasing to strive’ and the possibility of sublimating the two originary loves into a higher ‘love of truth’. (shrink)

We characterize the generalized quantifiers Q which satisfy the scheme $QxQy\phi \leftrightarrow QyQx\phi$ , the so-called self-commuting quantifiers, or quantifiers with the Fubini property.

This book develops a liberal theory of justice in exchange. It identifies the conditions that market exchanges need to fulfill to be just. It also addresses head-on a consequentialist challenge to existing theories of exchange, namely that, in light of new harms faced at the global level, we need to consider the combined consequences of millions of market exchanges to reach a final judgment about whether some individual exchange is just. The author argues that, even if we accept this challenge, (...) the effect of it is minimal. For different reasons, normatively problematic collective market outcomes like externalities, monopolies, violations of the Lockean proviso, inequality, and commodification do not pose particular problems to the justice of market exchanges. He outlines the various conditions a market exchange needs to fulfill to be considered just from a liberal background and in light of the new harms. Ultimately, he shows, it is not the market which is to blame; if we want to tackle issues like global warming or global economic injustice, we should not blindly follow the intuition that we best restrain and regulate markets. Commutative Justice is unique in its focus on justice in exchange rather than on end-state distributive justice, and the way in which it addresses the new harms we are facing today. It will be of interest to researchers and advanced students in philosophy, politics, and economics who are working on questions of economic justice. (shrink)

We investigate the connection between $\triangle^1_3$-stability for random and Cohen forcing notions and the measurability and categoricity of the $\triangle^1_3$-sets. We show that Shelah's model for $\triangle^1_3$-measurability and categoricity satisfies $\triangle^1_3$-random-stability while it does not satisfy $\triangle^1_3$-Cohen-stability. This gives an example of measure-category asymmetry. We also present a result concerning finite support iterations of Suslin forcing.

Quantum B-algebras, the partially ordered implicational algebras arising as subreducts of quantales, are introduced axiomatically. It is shown that they provide a unified semantic for non-commutative algebraic logic. Specifically, they cover the vast majority of implicational algebras like BCK-algebras, residuated lattices, partially ordered groups, BL- and MV-algebras, effect algebras, and their non-commutative extensions. The opposite of the category of quantum B-algebras is shown to be equivalent to the category of logical quantales, in the way that every quantum B-algebra (...) admits a natural embedding into a logical quantale, the enveloping quantale. Partially defined products of algebras related to effect algebras are handled efficiently in this way. The unit group of the enveloping quantale of a quantum B-algebra X is shown to be always contained in X, which gives a functorial subgroup X× of X. Similar subfunctors are obtained for the non-commutative extensions of BCK-algebras and effect algebras. The results of Galatos, Jónsson, and Tsinakis on the splitting of generalized BL-algebras into a semidirect product of a partially ordered group operating on an integral residuated poset are extended to a characterization of twisted semidirect products of a po-group by a quantum B-algebra. (shrink)

A symmetric residuated lattice is an algebra such that is a commutative integral bounded residuated lattice and the equations x=x and =xy are satisfied. The aim of the paper is to investigate the properties of the unary operation ε defined by the prescription εx=x→0. We give necessary and sufficient conditions for ε being an interior operator. Since these conditions are rather restrictive →0)=1 is satisfied) we consider when an iteration of ε is an interior operator. In particular we consider (...) the chain of varieties of symmetric residuated lattices such that the n iteration of ε is a boolean interior operator. For instance, we show that these varieties are semisimple. When n=1, we obtain the variety of symmetric stonean residuated lattices. We also characterize the subvarieties admitting representations as subdirect products of chains. These results generalize and in many cases also simplify, results existing in the literature. (shrink)

In a recent pair of publications, Richard Bradley has offered two novel no-go theorems involving the principle of Preservation for conditionals, which guarantees that one’s prior conditional beliefs will exhibit a certain degree of inertia in the face of a change in one’s non-conditional beliefs. We first note that Bradley’s original discussions of these results—in which he finds motivation for rejecting Preservation, first in a principle of Commutativity, then in a doxastic analogue of the rule of modus ponens —are problematic (...) in a significant number of respects. We then turn to a recent U-turn on his part, in which he winds up rescinding his commitment to modus ponens, on the grounds of a tension with the rule of Import-Export for conditionals. Here we offer an important positive contribution to the literature, settling the following crucial question that Bradley leaves unanswered: assuming that one gives up on full-blown modus ponens on the grounds of its incompatibility with Import-Export, what weakened version of the principle should one be settling for instead? Our discussion of the issue turns out to unearth an interesting connection between epistemic undermining and the apparent failures of modus ponens in McGee’s famous counterexamples. (shrink)

Commuting conversions were introduced in the natural deduction calculus as ad hoc devices for the purpose of guaranteeing the subformula property in normal proofs. In a well known book, Jean-Yves Girard commented harshly on these conversions, saying that ‘one tends to think that natural deduction should be modified to correct such atrocities.’ We present an embedding of the intuitionistic predicate calculus into a second-order predicative system for which there is no need for commuting conversions. Furthermore, we show that the redex (...) and the conversum of a commuting conversion of the original calculus translate into equivalent derivations by means of a series of bidirectional applications of standard conversions. (shrink)

"C'est un palimpseste : une succession de textes tirés des oeuvres complètes des auteurs à peu près complets. Le choix effectué par la lecture se trouve incarné dans l'ordre de l'écriture. Et cet ordre est porteur d'un message, il véhicule une thèse, il articule une idée dont le sens ne devient transparent que par le tout. Désormais les fragments extraits des texte ne représentent que le maillons d'une unique chaîne connexe de paroles, segmentss d'un entier : le texte du Palimpseste. (...) Derrière ce choix un axiome est caché, une assertion qui stimule et justifie la composition et, en dernière instance, garantit l'existence et légitime la subsistance autonome du texte. C'est ce que, par un terme emprunté à la théorie des ensembles de Georg Cantor, on pourra appeler "axiome du choix" ou encore, en une formulation quelque peu différente mais peut-être plus adéquate pour le traitement des textes, "axiome de sélection". L'axiome du choix permet donc l'assemblage d'un texte uniquement constitué de citations. Il permet aussi de citer des textes inexistants : ce sont alors des textes écrits exprès et pour l'unique raison d'être cités.... Le parler, comme l'a fait remarquer à plusieurs reprises Platon, est une praxis et l'axiome du choix confère au sujet le privilège d'exercer la fonction de choix : de sélectionner, de découper et d'assembler en un polymère verbal des fragments hétérogènes appartenant initialement à des textes préexistants. Le choix qui a présidé et guidé la navigation dans la métagalaxie des textes préexistants relève de cette liberté : sélectionner des "Propos" concernant le "Triangle", euclidien ou non, et de les assembler en un unique lieu et dans le présent indivisible d'une unique écriture étalée sur la surface du papier.". (shrink)

In this paper we present an equivalence between the category of commutative regular rings and the category of Boolean-valued fields, i.e., Boolean-valued sets for which the field axioms are true. The author used this equivalence in [12] to develop a Galois theory for commutative regular rings. Here we apply the equivalence to give an alternative construction of an algebraic closure for any commutative regular ring.Boolean-valued sets were developed in 1965 by Scott and Solovay [10] to simplify independence (...) proofs in set theory. They later were applied by Takeuti [13] to obtain results on Hilbert and Banach spaces. Ellentuck [3] and Weispfenning [14] considered Boolean-valued rings which consisted of rings and associated Boolean-valued relations. To the author's knowledge, the present work is the first to employ the Boolean-valued sets of Scott and Solovay to obtain results in algebra.The idea that commutative regular rings can be studied by examining the properties of related fields is not new. For several years algebraists and logicians have investigated commutative regular rings by representing a commutative regular ring as a subdirect product of fields or as the ring of global sections of a sheaf of fields over a Boolean space. These representations depend, as does the work presented here, on the fact that the set of central idempotents of any ring with identity forms a Boolean algebra. The advantage of the Boolean-valued set approach is that the axioms of classical logic and set theory are true in the Boolean universe. Therefore, if the axioms for a field are true for a Boolean-valued set, then other properties of the set can be deduced immediately from field theory. (shrink)

The non-commutative counterpart of the well-known Łukasiewicz propositional logic is developed, in strong connection with the algebraic theory of psMV-algebras. An extension by a new unary logical connective is also considered and a stronger completeness result is proved for this system.

Lange (2000) famously argues that although Jeffrey Conditionalization is non-commutative over evidence, it’s not defective in virtue of this feature. Since reversing the order of the evidence in a sequence of updates that don’t commute does not reverse the order of the experiences that underwrite these revisions, the conditions required to generate commutativity failure at the level of experience will fail to hold in cases where we get commutativity failure at the level of evidence. If our interest in commutativity (...) is, fundamentally, an interest in the order-invariance of information, an updating sequence that does not violate such a principle at the more fundamental level of experiential information should not be deemed defective. This paper claims that Lange’s argument fails as a general defense of the Jeffrey framework. Lange’s argument entails that the inputs to the Jeffrey framework differ from those of classical Bayesian Conditionalization in a way that makes them defective. Therefore, either the Jeffrey framework is defective in virtue of not commuting its inputs, or else it is defective in virtue of commuting the wrong kinds of ones. (shrink)

A large-scale field experiment in Rotterdam, Netherlands, tested whether nudging could increase public transport use. During one work week, 4000 commuters on six bus lines, received a free travel card holder. On the three bus lines in the experimental condition, the card holders displayed a social label that branded bus passengers as sustainable travelers because of their bus use. On the three bus lines in the control condition, there was no such message on the card holders. Analysis of the number (...) of rides per hour showed that the intervention led to a change from pre-intervention to post-intervention period that was estimated to be 1.18 rides per day greater on experimental lines than on control lines. This experiment shows that public transport operators can increase public transport use by incorporating messages that positively label passengers as sustainable travelers in their communication strategies. (shrink)

Several investigations in probability theory and the theory of expert systems show that it is important to search for some reasonable generalizations of fuzzy logics (e.g. Łukasiewicz, Gödel or product logic) having a non-associative conjunction. In the present paper, we offer a non-associative fuzzy logic L CBA having as an equivalent algebraic semantics lattices with section antitone involutions satisfying the contraposition law, so-called commutative basic algebras. The class (variety) CBA of commutative basic algebras was intensively studied in several (...) recent papers and includes the class of MV-algebras. We show that the logic L CBA is very close to the Łukasiewicz one, both having the same finite models, and can be understood as its non-associative generalization. (shrink)

How should we respond to cases of disagreement where two epistemic agents have the same evidence but come to different conclusions? Adam Elga has provided a Bayesian framework for addressing this question. In this paper, I shall highlight two unfortunate consequences of this framework, which Elga does not anticipate. Both problems derive from a failure of commutativity between application of the equal weight view and updating in the light of other evidence.

In this paper, I sketch out the way our bodies are engaged while commuting in order to elucidate several key aspects of the bodily experience of “in-between-ness.” I discover that within the rhythm and movement of the in-between, our bodies can open to a specific kind of conceptual creativity—an insight that I unfold in reference to the unanticipated innovation and transformation that accompanies other bodily experiences of in-between-ness more generally. This sketch, however, also demands that I reflect on phenomenological methodology, (...) and our ontological presuppositions about the “things” that we intend to “go back to.” I conclude by suggesting that while Deleuze and Guattari’s rhizomatic ontology helps us explain the findings of my phenomenological sketch of commuting, Merleau-Ponty’s embodied phenomenology can help to flesh out (quite literally) Deleuze and Guattari’s suggestion that all the work happens in “the logic of the AND” ( A Thousand Plateaus 25). While the method I call on in this paper is decidedly phenomenological, the conclusions I reach invite me to explore further the promise of a more “rhizomatic phenomenological” practice. (shrink)

This work presents a computational interpretation of the construction process for cyclic linear logic and non-commutative logic sequential proofs. We assume a proof construction paradigm, based on a normalisation procedure known as focussing, which efficiently manages the non-determinism of the construction. Similarly to the commutative case, a new formulation of focussing for NL is used to introduce a general constraint-based technique in order to dealwith partial information during proof construction. In particular, the procedure develops through construction steps propagating (...) constraints in intermediate objects called abstract proofs. (shrink)

It is shown that all the assumptions for symmetric monoidal categories follow from a unifying principle involving natural isomorphisms of the type →, called medial commutativity. Medial commutativity in the presence of the unit object enables us to define associativity and commutativity natural isomorphisms. In particular, Mac Lane’s pentagonal and hexagonal coherence conditions for associativity and commutativity are derived from the preservation up to a natural isomorphism of medial commutativity by the biendofunctor . This preservation boils down to an isomorphic (...) representation of the Yang–Baxter equation of symmetric and braid groups. The assumptions of monoidal categories, and in particular Mac Lane’s pentagonal coherence condition, are explained in the absence of commutativity, and also of the unit object, by a similar preservation of associativity by the biendofunctor . In the final section one finds coherence conditions for medial commutativity in the absence of the unit object. These conditions are obtained by taking the direct product of the symmetric groups for 0≤i≤n. (shrink)

A simple rule of probability revision ensures that the final result of a sequence of probability revisions is undisturbed by an alteration in the temporal order of the learning prompting those revisions. This Uniformity Rule dictates that identical learning be reflected in identical ratios of certain new-to-old odds, and is grounded in the old Bayesian idea that such ratios represent what is learned from new experience alone, with prior probabilities factored out. The main theorem of this paper includes as special (...) cases Field's theorem on commuting probability-kinematical revisions and the equivalence of two strategies for generalizing Jeffrey 's solution to the old evidence problem to the case of uncertain old evidence and probabilistic new explanation. (shrink)

We build models where all $\underset{\sim}{\triangle}^1_3$-sets of reals are measurable and (or) have the property of Baire and (or) are Ramsey. We will show that there is no implication between any of these properties for $\underset{\sim}{\triangle}^1_3$-sets of reals.

J.P. Moreland—Christian philosopher, theologian, and apologist—issues a call to recapture the drama and power of kingdom living—to cultivate a revolution of Evangelical life, spirituality, thought, and Spirit-led power. Drawing insights from the early church, he unpacks three essential ingredients of this revolution: Recovery of the Christian mind. Renovation of Christian spirituality. Restoration of the power of the Holy Spirit. Western society is in crisis: the result of our culture's embrace of naturalism and postmodernism, and a biblical worldview has been pushed (...) to the margins. Christians have been strongly influenced by these trends, with the result that their personal lives often reflect the surrounding culture more than the way of Christ, and the church's transforming influence on society has waned as a result. Kingdom Triangle is divided into two major sections: The first examines and provides a critique of secular worldviews and shows how they have ushered in the current societal crisis. The second lays out a strategy for the Christian community to regain the potency of kingdom life and influence in the world. Moreland believes that evangelical Christianity can mature and lead the surrounding society out of the meaningless morass it finds itself in with humility and vision. With clear insight, he puts the thoughtful Christian in a position to understand our current cultural struggle and to return to a responsible presentation of "the way of Christ" as not just a way of right living, but also a way of knowledge and meaningful life. (shrink)

In the present article, we consider the question of the primary elements in Plato’s Timaeus, the components of the whole universe reduced, by an extraordinarily elegant construction, to two right triangles. But how does he reconcile such a model with the infinite diversity of the universe? A large part of this study is devoted to Cornford’s explanation in his commentary of the Timaeus and its shortcomings, in order to finally propose a revised one, which we think to be entirely consistent (...) with Plato’s text. This analysis is an essential step for the understanding of the connection between the sensible world and the mathematical principles that underlies Timaeus’ cosmological account. (shrink)

In [9] it is proved the categorical isomorphism of two varieties: bounded commutative BCK-algebras and MV -algebras. The class of MV -algebras is the algebraic counterpart of the infinite valued propositional calculus L of Lukasiewicz . The main objective of the present paper is to study that isomorphism from the perspective of logic. The B-C-K logic is algebraizable and the quasivariety of BCKalgebras is the equivalent algebraic semantics for that logic . We call commutative B-C-K logic, briefly cBCK, (...) to the extension of B-C-K logic associated to the variety of commutative BCK–algebras. Moreover, we present the extension Boc of cBCK obtained by adding the axiom of “boundness”. We prove that the deductive system Boc is equivalent to L. We observe that cBCK admits two interesting extensions: the logic Boc, treated in this paper, which is equivalent to the system L of Lukasiewicz, and the logic Co that is naturally associated to the system Balo of `-groups . This constructions establish a link between L and Balo , that would be a logical approach to the categorical relationship between MV–algebras and `-groups. (shrink)

Tim Maudlin argues that we should take facts about distance to be analyzed in terms of facts about path lengths. His reason is that if we take distances to be fundamental, we must stipulate that constraints like the triangle inequality hold, but we get these constraints for free if we take path lengths to be prior. I argue that Maudlin is mistaken. Even if we take path lengths as primitive, the triangle inequality follows only if we stipulate that (...) the fundamental properties and relations obey constraints that are just as puzzling as the triangle inequality. (shrink)

The Convention on Biological Diversity and its Nagoya Protocol established an Access and Benefit Sharing system between utilizers and providers of genetic resources. ABS is understood as a tool that should promote commutative justice between the involved parties. This essay discusses what exactly it is that is being exchanged in the ABS process. It critically analyses moral claims to compensation that are implied by the ABS system for genetic resources. It argues that with the exception of cases in which (...) traditional knowledge is involved, states are not automatically entitled to compensation in return for the utilization of genetic resources growing within their territory. However, biodiversity-rich states that make an effort to protect biodiversity must be compensated for complying with the requests set out in the CBD. Although it acknowledges that the NP is a step towards recognizing this claim, this essay argues that ABS is not the appropriate method to compensate for biodivers... (shrink)

Higher education in general, and more specifically in the South African environment, is under pressure to transform. Although learning is often seen as the main focal point, the education process consists of three equally important pillars that form the triangle of effective education that fits within the intersection of the spheres of the community of inquiry framework. The basic pillars expand to student-centred teaching, blended learning and transformative assessment. This study is a short explanation of how these three pillars (...) form a basic framework for effective theological training. This is a qualitative study, using a philosophical and analytical research design and illuminative evaluation as research method. This is a model for effective higher education in Theology. The triangle of effective education is formed by student-centred teaching, blended learning and transformative assessment. The model fits into the intersection of the spheres identified by the community of inquiry framework of Garrison, Anderson and Archer. (shrink)