We prove that the two-variable fragment of first-order logic has the weak Beth definability property. This makes the two-variable fragment a natural logic separating the weak and the strong Beth properties since it does not have the strong Beth definability property.

In this paper it will be shown that the Beth definability property corresponds to surjectiveness of epimorphisms in abstract algebraic logic. This generalizes a result by I. Németi (cf. [11, Theorem 5.6.10]). Moreover, an equally general characterization of the weak Beth property will be given. This gives a solution to Problem 14 in [20]. Finally, the characterization of the projective Beth property for varieties of modal algebras by L. Maksimova (see [15]) will be shown to (...) hold for the larger class of semantically algebraizable logics. (shrink)

A series of high-profile ethical lapses by senior military professionals has generated calls from levels as high as the commander in chief for a renewed emphasis on military ethics. Leaders engaged in professional military education (PME) across the joint force have worked to ensure their programs support this call. This paper explores and assesses the ethics education programs at the service senior leader colleges (war colleges). There are three fundamental questions facing those charged with teaching ethics to senior military officers. (...) What are the desired outcomes of ethics education? How should the curriculum be structured to achieve those outcomes? And, finally, what is the correct faculty composition to develop and employ that curriculum? Using the answers to those questions to produce a rough framework for a model war college ethics education program, this paper then compares the current war college programs to this model form in order to determine areas of strength and weakness. This analysis reveals that the existing ethics education programs at the war colleges compare favorably to the model program structure. However, leaders at these institutions could further strengthen their programs by creating and empowering an "ethics team" that includes trained ethicists and military practitioners and by conducting more robust faculty development programs for nonethicists."--Provided by publisher. (shrink)

There is increasing interest in artifacts among philosophers. The leading edge is the metaphysics of artifacts and artifact kinds. However, an important question has been neglected. What is the ontological status of the category ‘artifact’ itself? Dan Sperber (2007) argues against its theoretical integrity for the purposes of naturalistic social sciences. In Section 2, I lay out Sperber’s argument, which is based on the observed continuum between natural objects and artifacts. I also review the implicit support for this continuum argument (...) in the philosophical literature on artifacts. In Section 3, I diagnose the fatal weakness affecting continuum-based arguments about categories. They fail to take into account recent thinking about classification in philosophy of science, which has been forced to accommodate continua in biology. In Section 4, I focus specifically on the species problem in philosophy of biology, and on pluralism about species concepts as a response to it. I then identify an analogous artifact problem, and argue that pluralism about artifact concepts is an appropriate and defensible response here as well. Finally, in Section 5, I show that just as species pluralism leads to skepticism about the species category, so artifact pluralism leads to skepticism about the artifact category. I thus arrive at Sperber’s conclusion by an alternate route that does not run afoul of the weakness affecting continuum-based arguments about categories. (shrink)

The Krüppel‐like factors (KLFs) comprise a family of evolutionarily conserved zinc finger transcription factors that regulate numerous biological processes including proliferation, differentiation, development and apoptosis. KLF4 and KLF5 are two closely related members of this family and are both highly expressed in epithelial tissues. In the intestinal epithelium, KLF4 is expressed in terminally differentiated epithelial cells at the villus borders of the mucosa and inhibits cell growth, while KLF5 is expressed in proliferating epithelial cells at the base of the intestinal (...) crypts and promotes cell growth. KLF4 and KLF5 respond to a myriad of external stress stimuli and are likely involved in restoring cellular homeostasis following exposure to stressors. Confirming their importance in maintaining tissue integrity, KLF4 and KLF5 are both dysregulated in various types of cancer. Here we review the recent advances in defining the physiological and pathobiological roles of KLF4 and KLF5, focusing on their functions in the intestinal epithelium. BioEssays 29:549–557, 2007. © 2007 Wiley Periodicals, Inc. (shrink)

In the face of an ensuing environmental crisis, this paper suggests that currently accepted modes of environmentalist thought have not been effective enough in enacting positive change. Anthropocentrism provides something that environmental philosophy needs – wide acceptance and public appeal. This paper argues that an environmental ethic that is weakly anthropocentric, in that it finds value in the environment via human values, can be both internally consistent and highly pragmatic. It goes on to examine some pitfalls of Deep Ecological environmental (...) philosophy, which could be avoided if a weakly anthropocentric environmental ethic were adopted now. (shrink)

This article analyzes positions on pornography using Q-methodology. Eighty-five respondents sorted a sample of 86 opinion statements on definitions of pornography, personal reactions to it, its causes and effects, and social policy recommendations. Factor analysis was used to identify clusters of individuals in the United States who share common subjectively defined points of view on pornography. The three patterns of responses that emerged from the analysis were labeled Religious-Conservative, Liberal, and Antipornography Feminist. Using the empirical data, we examine the logical (...) and ethical structures of these points of view and their political and legal implications. We conclude that the viewpoints are too incompatible to sustain stable and effective political alignments among the adherents. (shrink)

Much of the work coming out of Women’s Studies spaces narrowly defines what it means to be a woman. Antagonizing White Feminism pushes back against this exclusive discourse by invoking intersectionality and centering the experiences of Trans Women, Femmes, Women of Color, Queer Women, and Gender Variant and Gender Non- Conforming people.

In the deviance literature, sexual assaults at work have not been given the sustained attention that harm to property or violation of production guidelines has received. This omission suggests that sexual harassment is considered normative and that when women fail to accommodate this reality, it is the survivor rather than the perpetrator who is considered deviant. This article reports on 64 cases of attempted or completed rape in a sample of heterosexual and lesbian women workers in a wide range of (...) occupations. Two atypical responses, quitting the job or filing a complaint as a result of a workplace sexual assault, highlight the process by which informal deviance defining occurs in everyday interactions at work. (shrink)

Inwardness and interiority are concepts that have a multifaceted currency within many areas of medieval studies. These fields include, but are not limited to, historical studies, theology and religious studies, literary studies, and art history. Studies on inwardness, interiority, and selfhood intersect with an interest in what has often been called “popular religion” and in devotional behavior, both clerical and lay, to produce an engagement, across many fields, with inward or private aspects of religious belief and practice. “Popular religion” has (...) sometimes been presented as generally distinct or separate from official, ecclesiastical, or institutional ritual, and, as such, it is associated with other concepts like “private devotion” and even “interior piety.” Unhelpful binary oppositions are implied by qualifying terms like “private” and “popular,” because those qualifiers invoke the unsaid “public” or “official” or “outward.” More recently, terms like “vernacular theology” and Frömmigkeitstheologie have been preferred for their attempt to break down the “high-low” implications of terms like “popular religion” and “private devotion.” Though these terms themselves are not without potential problems or implications, the consonance or dissonance between individual religiosity and official or corporate expressions of religious belief can, and should, be examined in a more subtle manner than has often been the case. Medieval religious devotion was not a universally or solely private activity that was in some way opposite to the public, structured religiosity of the church's liturgy. It has been increasingly acknowledged in recent years that liturgy and devotion should not be opposed in this way, as private versus public, free and expressionist versus structured and defined. That having been said, it is nevertheless the case that the kind of piety that we are accustomed to call “devotional,” or contemplative, or meditative, presupposes and requires a certain inwardness and self-awareness, even if such activity is carried out in the company of others, or even during liturgical services. (shrink)

In lieu of an abstract, here is a brief excerpt of the content:Reviewed by:Christian Ethics: A Very Short Introduction, and: Christian Ethics: A Brief History, and: Behaving in Public: How to Do Christian EthicsBeth K. HaileChristian Ethics: A Very Short Introduction D. Stephen Long Oxford: Oxford University Press, 2010. 144 pp. $11.95Christian Ethics: A Brief History Michael Banner West Sussex, UK: Wiley-Blackwell, 2009. 160 pp. $24.95Behaving in Public: How to Do Christian Ethics Nigel Biggar Grand Rapids, Mich.: Eerdmans, 2011. 142 (...) pp. $16.00In calling ethics “Christian,” the question of distinctiveness, or the normative force of theological claims, inevitably arises. Three new books explore what Christian ethics looks like if it is to take seriously its theological basis for engaging the world.Stephen D. Long, professor of systematic theology at Marquette University, attends to the difficulties in predicating “Christian” of “ethics,” yet he is concerned more with explaining how Christianity can be ethical rather than vice versa. His Christian Ethics: A Very Short Introduction opens with a quote from Christopher Hitchens and later dedicates almost an entire chapter to analyzing the failures of Christian ethics. From the Crusades to colonialism, the Galileo affair to slavery, Long provocatively asks whether modern ethics emerges as a result of the failure of Christian moral claims. While not sidestepping the difficulties in Christian history, Long is careful to note how this history is often distorted to serve the goals of contemporary secular politics, goals which he points out have resulted in the bloody failures of the twentieth century. Perhaps a renewed interest in Christian ethics, Long notes, is a result of the failures of secularism.Long manages to cover a surprising breadth of material in his brief book, from a sweeping historical overview of Christian theology to a final chapter on [End Page 195] the practical matters of sex, money, and political power. Long also summarizes some important theological differences among Christians that lead to practical disagreements about the nature of Christian ethics. His overview of the Catholic-Lutheran convergence on matters of faith and works is particularly helpful in establishing the theological basis for an ecumenical Christian ethics. This historical survey serves to illustrate that ethics is both an integral part of and a contradiction to Christianity. Still, argues Long, while Christianity may claim to be more than ethical, it may never claim to be less.While commendable in breadth, the genre of the “very short introduction” precludes depth and must necessarily omit important information. In his summary of Catholic ethics, for example, Long addresses nominalism and probabilism but omits the Catholic social tradition, a particularly glaring omission given his topic. In the end, the target audience is the educated and curious layperson, more suitable for a church group than a classroom, though this is reflective of the nature of “the very short introduction” series and does not indicate any limitation in Long’s skill as a scholar.In the spirit of brevity, Michael Banner, Fellow of Trinity College in Cambridge, offers Christian Ethics: A Brief History, which, like any brief history, paints with broad and admittedly selective strokes. The aim of this work is not to evaluate the normative value of Christian claims but rather to establish the ethical implications of Christian belief. Unlike Long, who directly addresses the ethical failures of the church, Banner is more apologetic. In clarifying the difficulties of writing a text such as this one, he observes that Christian history has not been guided by texts as much as by action. In this vein, Banner begins not chronologically but with the Rule of St. Benedict and its call to live in a relationship with God and neighbor characterized by love.Banner grounds the theoretical dimension of Christian ethics in this emphasis on practice. Given the way of life Christians are called to, what are its limitations and possibilities? Banner turns to Augustine’s notion of the weakness of the will and the human creature’s need for grace, which he then places in creative tension with Aquinas’s emphasis on the harmony between faith and reason. In somewhat breathtaking conciseness, Banner describes how the Scholastic emphasis on natural law, casuistry, and the categorizing of sins led to what he calls... (shrink)

Our goal in producing this special issue is to encourage our colleagues to incorporate topics related to LGBT populations into bioethics curricula and scholarship. Bioethics has only rarely examined the ways in which law and medicine have defined, regulated, and often oppressed sexual minorities. This is an error on the part of bioethics. Medicine and law have served in the past as society's enforcement arm toward sexual minorities, in ways that robbed many people of their dignity. We feel that bioethics (...) has an obligation to discuss that history and to help us as a society take responsibility for it.We can address only a small number of topics in this special issue of the Hastings Center Report, and we selected topics we believe will stimulate discourse. Andrew Solomon offers an elegant overview of the challenges that bioethics faces in articulating a solid basis for LGBT rights. Timothy F. Murphy asks whether bioethics still faces issues related to lesbian, gay, and bisexual people, given the deletion of homosexuality as a disease and the progress toward same‐sex marriage. Jamie Lindemann Nelson's essay addresses the search for identity for transgender persons and the role of science in that search. Two articles, those by Brendan S. Abel and by Jack Drescher and Jack Pula, take up the complex issue of medical treatment for children who reject their assigned birth gender. Celia B. Fisher and Brian Mustanski address the special challenges of engaging LGBT youth in research, balancing the need for better information about this vulnerable group against the existing restrictions on research involving children. Tia Powell and Edward Stein consider the merits of legal bans on psychotherapies intended to change sexual orientation, particularly in the light of current research on orientation. Mary Beth Foglia and Karen I. Fredricksen‐Goldsen highlight health disparities and resilience among LGBT older adults and then discuss the role of nonconscious bias in perpetuating disparities. Stephan Davis and Nancy Berlinger assess the challenges of access to care and health policy for transgender persons. Edward J. Callahan et al. tackle the ways in which diverse aspects of medicine should change to better incorporate the needs of LGBT patients, including through use of the electronic medical record, education of health professionals, and recruitment efforts for LGBT health professionals. Virginia Ashby Sharpe and Uchenna S. Uchendu describe multifaceted efforts within Veterans Administration facilities to create change for LGBT veterans across the largest integrated health care network in the United States. Lance Wahlert and Autumn Fiester find a mixed record in the use of case studies in teaching about LGBT issues. (shrink)

Multi-method research is any research that uses more than one research technique or strategy to study one or several closely related phenomena. This method is described by triangulation. This article examines the multi-method tradition in empirical legal research, defines basic concepts, discusses when and why multi-method research is useful, and how the different actions of research can provide unique approaches to the same questions. It explores examples of projects to demonstrate how research that employs multiple tactics for observing and understanding (...) is more reliable than a single study if the studies are of comparable quality. At the same time, multi-method research comes with great costs and risks that must be considered at the outset of such a project as well as during the project itself. However, it is speculated that multi-method research is perhaps the most effective way to understand the relationship between law and society. (shrink)

In the early twentieth century, viruses had yet to be defined in a material way. Instead, they were known better by what they were not – not bacteria, not culturable, and not visible with a light microscope. As with the ill-defined “gene” of genetics, viruses were microbes whose nature had not been revealed. Some clarity arrived in 1929 when Francis O. Holmes, a scientist at the Boyce Thompson Institute for Plant Research reported that Tobacco mosaic virus could produce local necrotic (...) lesions on tobacco plants and that these lesions were in proportion to dilutions of the inoculum. Holmes’ method, the local lesion assay, provided the first evidence that viruses were discrete infectious particles, thus setting the stage for physicochemical studies of plant viruses. In a field where there are few eponymous methods or diseases, Holmes’ assay continues to be a useful tool for the study of plant viruses. TMV was a success because the local lesion assay “made the virus visible” and standardized the work of virology towards determining the nature of the virus. (shrink)

A semantical definition of abstract logics is given. It is shown that the Craig interpolation property implies the Beth definability property, and that the Souslin-Kleene interpolation property implies the weak Beth definability property. An example is given, showing that Beth does not imply Souslin-Kleene.

Both the Beth definability theorem and Craig's lemma (interpolation theorem from now on) deal with the issue of the entanglement of one language L1 with another language L2, that is to say, information transfer—or the lack of such transfer—between the two languages. The notion of splitting we study below looks into this issue. We briefly relate our own results in this area as well as the results of other researchers like Kourousias and Makinson, and Peppas, Chopra and Foo.Section (...) 3 does contain one apparently new theorem. (shrink)

The Stone-Weierstrass Theorem for compact Hausdorff spaces is a basic result of functional analysis with far-reaching consequences. We introduce an equational logic ⊨Δ associated with an infinitary variety Δ and show that the Stone-Weierstrass Theorem is a consequence of the Beth definability property of ⊨Δ, stating that every implicit definition can be made explicit. Further, we define an infinitary propositional logic ⊢Δ by means of a Hilbert-style calculus and prove a strong completeness result whereby the semantic notion of (...) consequence associated with ⊢Δ coincides with ⊨Δ. (shrink)

In a previous paper [ 21 ] all extensions of Johansson’s minimal logic J with the weak interpolation property WIP were described. It was proved that WIP is decidable over J. It turned out that the weak interpolation problem in extensions of J is reducible to the same problem over a logic Gl, which arises from J by adding tertium non datur. In this paper we consider extensions of the logic Gl. We prove that only finitely many logics (...) over Gl have the Craig interpolation property CIP, the restricted interpolation property IPR or the projective Beth property PBP. The full list of Gl-logics with the mentioned properties is found, and their description is given. We note that IPR and PBP are equivalent over Gl. It is proved that CIP, IPR and PBP are decidable over the logic Gl. (shrink)

This paper studies definability within the theory of institutions, a version of abstract model theory that emerged in computing science studies of software specification and semantics. We generalise the concept of definability to arbitrary logics, formalised as institutions, and we develop three general definability results. One generalises the classical Beth theorem by relying on the interpolation properties of the institution. Another relies on a meta Birkhoff axiomatizability property of the institution and constitutes a source for many (...) new actual definability results, including definability in (fragments of) classical model theory. The third one gives a set of sufficient conditions for 'borrowing' definability properties from another institution via an 'adequate' encoding between institutions. The power of our general definability results is illustrated with several applications to (many-sorted) classical model theory and partial algebra, leading for example to definability results for (quasi-)varieties of models or partial algebras. Many other applications are expected for the multitude of logical systems formalised as institutions from computing science and logic. (shrink)

A few results generalizing well-known classical model theory ones have been obtained in institution theory these last two decades (e.g. Craig interpolation, ultraproduct, elementary diagrams). In this paper, we propose a generalized institution-independent version of the Beth definability theorem.

We give a level-by-level analysis of the Weak Vopěnka Principle for definable classes of relational structures ( $\mathrm {WVP}$ ), in accordance with the complexity of their definition, and we determine the large-cardinal strength of each level. Thus, in particular, we show that $\mathrm {WVP}$ for $\Sigma _2$ -definable classes is equivalent to the existence of a strong cardinal. The main theorem (Theorem 5.11) shows, more generally, that $\mathrm {WVP}$ for $\Sigma _n$ -definable classes is equivalent to the existence (...) of a $\Sigma _n$ -strong cardinal (Definition 5.1). Hence, $\mathrm {WVP}$ is equivalent to the existence of a $\Sigma _n$ -strong cardinal for all $n<\omega $. (shrink)

In this paper we introduce the modelwise interpolation property of a logic that states that whenever \(\models\phi\to\psi\) holds for two formulas \(\phi\) and \(\psi\), then for every model \(\mathfrak{M}\) there is an interpolant formula \(\chi\) formulated in the intersection of the vocabularies of \(\phi\) and \(\psi\), such that \(\mathfrak{M}\models\phi\to\chi\) and \(\mathfrak{M}\models\chi\to\psi\), that is, the interpolant formula in Craig interpolation may vary from model to model. We compare the modelwise interpolation property with the standard Craig interpolation and with the local interpolation (...) property by discussing examples, most notably the finite variable fragments of first order logic, and difference logic. As an application we connect the modelwise interpolation property with the local Beth definability, and we prove that the modelwise interpolation property of an algebraizable logic can be characterized by a weak form of the superamalgamation property of the class of algebras corresponding to the models of the logic. (shrink)

For each completion of Peano Arithmetic there is a weakly definable type which is not definable.

We shall prove that if a model of cardinality κ can be expanded to a model of a sentence ψ of Lλ+,ω by adding a suitable predicate in more than κ ways, then, it has a submodel of power μ which can be expanded to a model of ψ in $> \mu$ ways provided that λ,κ,μ satisfy suitable conditions.

The paper is aimed at studying the topological dimension for sets definable in weakly o-minimal structures in order to prepare background for further investigation of groups, group actions and fields definable in the weakly o-minimal context. We prove that the topological dimension of a set definable in a weakly o-minimal structure is invariant under definable injective maps, strengthening an analogous result from [2] for sets and functions definable in models of weakly o-minimal theories. We pay special attention to large subsets (...) of Cartesian products of definable sets, showing that if X, Y and S are non-empty definable sets and S is a large subset of X × Y, then for a large set of tuples $\langle \overline{a}_{1},\ldots,\overline{a}_{2^{k}}\rangle \in X^{2^{k}}$ , where k = dim(Y), the union of fibers $S_{\overline{a}_{1}}\cup \cdots \cup S_{\overline{a}_{2^{k}}}$ is large in Y. Finally, given a weakly o-minimal structure ������, we find various conditions equivalent to the fact that the topological dimension in ������ enjoys the addition property. (shrink)

We consider the extent to which certain properties of definably complete structures may persist in structures which are not definably complete, particularly in the weakly o-minimal structures.

We develop a notion of cell decomposition suitable for studying weak p-adic structures definable). As an example, we consider a structure with restricted addition.

Modifying the descriptive and theoretical generalizations of Relativized Minimality, we argue that a significant subset of weak island violations arise when an extracted phrase should scope over some intervener but is unable to. Harmless interveners seem harmless because they can support an alternative reading. This paper focuses on why certain wh-phrases are poor wide scope takers, and offers an algebraic perspective on scope interaction. Each scopal element SE is associated with certain operations (e.g., not with complements). When a wh-phrase (...) scopes over some SE, the operations associated with that SE are performed in its denotation domain. The requisite operations may or may not be available in a domain, however. We present an empirical analysis of a variety of wh-phrases. It is argued that the wh-phrases that escape all weak islands (i.e., can scope over any intervener) are those that range over individuals, the reason being that all Boolean operations are defined for their domain. Collectives, manners, amounts, numbers, etc. all denote in domains with fewer operations and are thus selectively sensitive to scopal interveners—a “semantic relativized minimality effect”. (shrink)

A consequence relation is strongly classical if it has all the theorems and entailments of classical logic as well as the usual meta-rules (such as Conditional Proof). A consequence relation is weakly classical if it has all the theorems and entailments of classical logic but lacks the usual meta-rules. The most familiar example of a weakly classical consequence relation comes from a simple supervaluational approach to modelling vague language. This approach is formally equivalent to an account of logical consequence according (...) to which α1, ..., αn entails β just in case □α1, ..., □αn entails □β in the modal logic S5. This raises a natural question: If we start with a different underlying modal logic, can we generate a strongly classical logic? This paper explores this question. In particular, it discusses four related technical issues: (1) Which base modal logics generate strongly classical logics and which generate weakly classical logics? (2) Which base logics generate themselves? (3) How can we directly characterize the logic generated from a given base logic? (4) Given a logic that can be generated, which base logics generate it? The answers to these questions have philosophical interest. They can help us to determine whether there is a plausible supervaluational approach to modelling vague language that yields the usual meta-rules. They can also help us to determine the feasibility of other philosophical projects that rely on an analogous formalism, such as the project of defining logical consequence in terms of the preservation of an epistemic status. (shrink)

We consider the sets definable in the countable models of a weakly o-minimal theory T of totally ordered structures. We investigate under which conditions their Boolean algebras are isomorphic , in other words when each of these definable sets admits, if infinite, an infinite coinfinite definable subset. We show that this is true if and only if T has no infinite definable discrete subset. We examine the same problem among arbitrary theories of mere linear orders. Finally we prove that, within (...) expansions of Boolean lattices, every weakly o-minimal theory is p-ω-categorical. (shrink)

Is the restricted, consistent, version of the T-scheme sufficient for an ‘implicit definition’ of truth? In a sense, the answer is yes (Haack 1978 , Quine 1953 ). Section 4 of Ketland 1999 mentions this but gives a result saying that the T-scheme does not implicitly define truth in the stronger sense relevant for Beth’s Definability Theorem. This insinuates that the T-scheme fares worse than the compositional truth theory as an implicit definition. However, the insinuation is mistaken. For, (...) as Bays rightly points out, the result given extends to the compositional truth theory also. So, as regards implicit definability, both kinds of truth theory are equivalent. Some further discussion of this topic is mentioned (Gupta 2008 , Ketland 2003 , McGee 1991 ), all in agreement with Bays’s analysis. (shrink)

Let M=\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\mathcal{M}}=}$$\end{document} be a weakly o-minimal expansion of a dense linear order without endpoints. Some tame properties of sets and functions definable in M\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\mathcal{M}}}$$\end{document} which hold in o-minimal structures, are examined. One of them is the intermediate value property, say IVP. It is shown that strongly continuous definable functions in M\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\mathcal{M}}}$$\end{document} satisfy an extended (...) version of IVP. After introducing a weak version of definable connectedness in M\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\mathcal{M}}}$$\end{document}, we prove that strong cells in M\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\mathcal{M}}}$$\end{document} are weakly definably connected, so every set definable in M\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\mathcal{M}}}$$\end{document} is a finite union of its weakly definably connected components, provided that M\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\mathcal{M}}}$$\end{document} has the strong cell decomposition property. Then, we consider a local continuity property for definable functions in M\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\mathcal{M}}}$$\end{document} and conclude some results on cell decomposition regarding that property. Finally, we extend the notion of having no dense graph which was examined for definable functions in and related to uniform finiteness, definable completeness, and others. We show that every weakly o-minimal structure M\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\mathcal{M}}}$$\end{document} having cell decomposition, satisfies NDG, i.e. every definable function in M\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\mathcal{M}}}$$\end{document} has no dense graph. (shrink)

The logic BKc1 is the basic constructive logic for weak consistency in the ternary relational semantics without a set of designated points. In this paper, a number of extensions of B Kc1 defined with a propositional falsity constant are defined. It is also proved that weak consistency is not equivalent to negation-consistency or absolute consistency in any logic included in positive contractionless intermediate logic LC plus the constructive negation of BKc1 and the contraposition axioms.

In 1999, Jeffrey Ketland published a paper which posed a series of technical problems for deflationary theories of truth. Ketland argued that deflationism is incompatible with standard mathematical formalizations of truth, and he claimed that alternate deflationary formalizations are unable to explain some central uses of the truth predicate in mathematics. He also used Beth’s definability theorem to argue that, contrary to deflationists’ claims, the T-schema cannot provide an ‘implicit definition’ of truth. In this article, I want to (...) challenge this final argument. Whatever other faults deflationism may have, the T-schema does provide an implicit definition of the truth predicate. Or so, at any rate, I shall argue. (shrink)

An innocent form of emergence—what I call "weak emergence"—is now a commonplace in a thriving interdisciplinary nexus of scientific activity—sometimes called the "sciences of complexity"—that include connectionist modelling, non-linear dynamics (popularly known as "chaos" theory), and artificial life.1 After defining it, illustrating it in two contexts, and reviewing the available evidence, I conclude that the scientific and philosophical prospects for weak emergence are bright.

Applying an ever more radical hermeneutics, John D. Caputo breaks down the name of God in this irrepressible book. Instead of looking at God as merely a name, Caputo views it as an event, or what the name conjures or promises in the future. For Caputo, the event exposes God as weak, unstable, and barely functional. While this view of God flies in the face of most religions and philosophies, it also puts up a serious challenge to fundamental tenets (...) of theology and ontology. Along the way, Caputo’s readings of the New Testament, especially of Paul’s view of the Kingdom of God, help to support the "weak force" theory. This penetrating work cuts to the core of issues and questions—What is the nature of God? What is the nature of being? What is the relationship between God and being? What is the meaning of forgiveness, faith, piety, or transcendence?—that define the terrain of contemporary philosophy of religion. (shrink)

A weakly o-minimal structure image expanding an ordered group is called nonvaluational iff for every cut left angle bracketC,Dright-pointing angle bracket of definable in image, we have that inf{y−x:xset membership, variantC,yset membership, variantD}=0. The study of nonvaluational weakly o-minimal expansions of real closed fields carried out in [D. Macpherson, D. Marker, C. Steinhorn,Weakly o-minimal structures and real closed fields, Trans. Amer. Math. Soc. 352 5435–5483. MR1781273 (2001i:03079] suggests that this class is very close to the class of o-minimal expansions of (...) real closed fields. Here we further develop this analogy. We establish an o-minimal style cell decomposition for weakly o-minimal non-valuational expansions of ordered groups. For structures enjoying such a strong cell decomposition we construct a canonical o-minimal extension. Finally, we make attempts towards generalizing the o-minimal Euler characteristic to the class of sets definable in weakly o-minimal structures with the strong cell decomposition property. (shrink)

The logic BKc1 is the basic constructive logic in the ternary relational semantics adequate to consistency understood as the absence of the negation of any theorem. Negation is introduced in BKc1 with a negation connective. The aim of this paper is to define the logic BKc1F. In this logic negation is introduced via a propositional falsity constant. We prove that BKc1 and BKc1F are definitionally equivalent.

Multiple-conclusion Hilbert-style systems allow us to finitely axiomatize every logic defined by a finite matrix. Having obtained such axiomatizations for Paraconsistent Weak Kleene and Bochvar–Kleene logics, we modify them by replacing the multiple-conclusion rules with carefully selected single-conclusion ones. In this way we manage to introduce the first finite Hilbert-style single-conclusion axiomatizations for these logics.

We consider a weak version of Schindler’s remarkable cardinals that may fail to be ${{\rm{\Sigma }}_2}$-reflecting. We show that the ${{\rm{\Sigma }}_2}$-reflecting weakly remarkable cardinals are exactly the remarkable cardinals, and that the existence of a non-${{\rm{\Sigma }}_2}$-reflecting weakly remarkable cardinal has higher consistency strength: it is equiconsistent with the existence of an ω-Erdős cardinal. We give an application involving gVP, the generic Vopěnka principle defined by Bagaria, Gitman, and Schindler. Namely, we show that gVP + “Ord is not (...) ${{\rm{\Delta }}_2}$-Mahlo” and ${\text{gVP}}$ + “there is no proper class of remarkable cardinals” are both equiconsistent with the existence of a proper class of ω-Erdős cardinals, extending results of Bagaria, Gitman, Hamkins, and Schindler. (shrink)