This book presents a set of historical recollections on the work of Martin Davis and his role in advancing our understanding of the connections between logic, computing, and unsolvability. The individual contributions touch on most of the core aspects of Davis’ work and set it in a contemporary context. They analyse, discuss and develop many of the ideas and concepts that Davis put forward, including such issues as contemporary satisfiability solvers, essential unification, quantum computing and generalisations of Hilbert’s tenth problem. (...) The book starts out with a scientific autobiography by Davis, and ends with his responses to comments included in the contributions. In addition, it includes two previously unpublished original historical papers in which Davis and Putnam investigate the decidable and the undecidable side of Logic, as well as a full bibliography of Davis’ work. As a whole, this book shows how Davis’ scientific work lies at the intersection of computability, theoretical computer science, foundations of mathematics, and philosophy, and draws its unifying vision from his deep involvement in Logic. (shrink)

As proved recently, the satisfaction problem for all prenex formulae in the set-theoretic Bernays-Shönfinkel-Ramsey class is semi-decidable over von Neumann's cumulative hierarchy. Here that semi-decidability result is strengthened into a decidability result for the same collection of formulae.

We study the decidability problem for metric and layered temporal logics. The logics we consider are suitable to model time granularity in various contexts, and they allow one to build granular temporal models by referring to the "natural scale" in any component of the model and by properly constraining the interactions between differently-grained components. A monadic second-order language combining operators such as temporal contextualization and projection, together with the usual displacement operator of metric temporal logics, is considered, and the theory (...) of finitely-layered metric temporal structures is shown to be decidable. (shrink)

The axiom of infinity can be expressed by stating the existence of sets satisfying a formula which involves restricted universal quantifiers only, even if the axiom of foundation is not assumed.

The still unsettled decision problem for the restricted purely universal formulae 0-formulae) of the first order set-theoretic language based over =, ∈ is discussed in relation with the adoption or rejection of the axiom of foundation. Assuming the axiom of foundation, the related finite set-satisfiability problem for the very significant subclass of the 0-formulae consisting of the formulae involving only nested variables of level 1 is proved to be semidecidable on the ground of a reflection property over the hereditarily finite (...) sets, and various extensions of this result are obtained. When variables are restricted to range only over sets, in universes with infinitely many urelements the set-satisfiability problem is shown to be solvable provided the axiom of foundation is assumed; if it is not, then the decidability of a related derivability problem still holds. That, in turn, suggests the alternative adoption of an antifoundation axiom under which the set-satisfiability problem is also solvable . Turning to set theory without urelements, assuming a form of Boffa's antifoundation axiom, the complement of the set-satisfiability problem for the full class of Δ0-formulae is shown to be semidecidable; a result that is known not to hold, for the set-satisfiability problem itself, even for a very restricted subclass of the Δ0-formulae. (shrink)

As is well-known, the Bernays-Schönfinkel-Ramsey class of all prenex ∃*∀* -sentences which are valid in classical first-order logic is decidable. This paper paves the way to an analogous result which the authors deem to hold when the only available predicate symbols are ∈ and =, no constants or function symbols are present, and one moves inside a (rather generic) Set Theory whose axioms yield the well-foundedness of membership and the existence of infinite sets. Here semi-decidability of the satisfiability problem for (...) the BSR class is proved by following a purely semantic approach, the remaining part of the decidability result being postponed to a forthcoming paper. (shrink)

In this paper, we generalize the set-theoretic translation method for poly-modal logic introduced in [11] to extended modal logics. Instead of devising an ad-hoc translation for each logic, we develop a general framework within which a number of extended modal logics can be dealt with. We first extend the basic set-theoretic translation method to weak monadic second-order logic through a suitable change in the underlying set theory that connects up in interesting ways with constructibility; then, we show how to tailor (...) such a translation to work with specific cases of extended modal logics. (shrink)

In this paper, we generalize the set-theoretic translation method for polymodal logic introduced in [11] to extended modal logics. Instead of devising an ad-hoc translation for each logic, we develop a general framework within which a number of extended modal logics can be dealt with. We first extend the basic set-theoretic translation method to weak monadic second-order logic through a suitable change in the underlying set theory that connects up in interesting ways with constructibility; then, we show how to tailor (...) such a translation to work with specific cases of extended modal logics. (shrink)

Being able to state the principles which lie deepest in the foundations of mathematics by sentences in three variables is crucially important for a satisfactory equational rendering of set theories along the lines proposed by Alfred Tarski and Steven Givant in their monograph of 1987. The main achievement of this paper is the proof that the 'kernel' set theory whose postulates are extensionality. (E), and single-element adjunction and removal. (W) and (L), cannot be axiomatized by means of three-variable sentences. This (...) highlights a sharp edge to be crossed in order to attain an 'algebraization' of Set Theory. Indeed, one easily shows that the theory which results from the said kernel by addition of the null set axiom, (N), is in its entirety expressible in three variables. (shrink)

The problem is addressed of establishing the satisfiability of prenex formulas involving a single universal quantifier, in diversified axiomatic set theories. A rather general decision method for solving this problem is illustrated through the treatment of membership theories of increasing strength, ending with a subtheory of Zermelo-Fraenkel which is already complete with respect to the ∀*∀ class of sentences. NP-hardness and NP-completeness results concerning the problems under study are achieved and a technique for restricting the universal quantifier is presented.

In lieu of an abstract, here is a brief excerpt of the content:A Bridge From Analysis to Action:Psychodynamic Analyses of Religion and Michael S. Hogue's American ImmanenceAJ Turner (bio)I. IntroductionThe purpose of this essay is to work constructively with Michael S. Hogue's groundbreaking American Immanence: Democracy for an Uncertain World to demonstrate how psychodynamic analyses of religion are essential theoretical allies in the fight for resilient democracy. The "revolution in mind"1 that psychodynamic approaches contribute, especially in their analyses of religion, (...) both complements and complicates socio-political theory as a whole. In the improvisational spirit of "yes-and," the goal here is not to identify and decry the failures and shortcomings of Hogue's theopolitical vision but rather to inspire the recovery of a sibling discourse—a kind of "rebirth of a forgotten alternative"2—that also draws deeply from the American immanental tradition in order to add psychodynamic depth to Hogue's timely and important analysis. Psychodynamic theories of religion shed light on how symbolic identifications, including and especially ultimate concerns, structure personal and collective experience. By engaging religion and theology sympathetically, constructively, and creatively, theorists such as Rollo May, Gordon Allport, Karen Horney, Erich Fromm, and Ernest Becker departed from the skepticism, reductionism, and dismissal that characterized psychoanalytic analyses of religion in the wake of Freud.3 Each of these figures, and especially Becker whose work will be a focus in this essay, are noteworthy interlocutors for extending Hogue's theopolitical theorizing of resilient democracy, particularly in how they drew unique inspiration from the American immanental tradition [End Page 44] in their social analyses. In this respect, they provide a more robust theorization of the psychosocial conditions of the associational ethos needed to connect Hogue's conception of resilient democracy with forms of agonistic organization and democratic activism. Briefly stated, psychodynamic approaches are a helpful bridge for connecting these proposals to strategies of organization, their impediments, and collective action.In this way, both psychodynamic analyses of religion and Hogue's theopolitical innovation insist that no revolutionary political change is possible, and certainly not of the kind or scale needed to counter our wickedly entangled global crises, without a deep engagement with theological imaginaries. They differ, however, in their theoretical aims. Whereas Hogue compellingly argues for the political upshot of a theology deriving from the American immanental tradition, psychodynamic analysts of religion focus on the function of ultimate concerns in individual and collective life. In other words, American Immanence rests on a methodology of analyzing the philosophical and theological assumptions that have contributed to our climate crises and then presenting clear and compelling articulation of an alternative paradigm meant to open up new ways of being. In contrast psychodynamic analyses leverage conceptual clarity of psychic reality for the fundamentally therapeutic aim of transformation. At their best, both approaches share what could be called an etiological orientation to ontological questions in order to provide liberative alternatives through rational understanding and clarification, but the terrain, and thus the problems they hope to mitigate in their analyses are focused differently. Extending the visual metaphor, they are different lenses highlighting different areas of the same landscape. Furthermore, it must be said that neither are sufficient in themselves, but the psychodynamic theorists of religion to be highlighted in this essay share deeply with Hogue's heading in American Immanence.II. A Problematic AssumptionIn emphasizing and attempting to explain4 the role of non-rational processes in the both individual and social processes of perception and organization of experience, psychodynamic theories confound a widely held and problematic assumption that the triumph of rational analysis is the hallmark for building the society to which we should aspire. Political theory in general, [End Page 45] especially as it has taken its cues from a rationalist legacy about the inherently liberating force of Reason, has been weighted toward an assumption that its task of exposing the mystifications of power would suffice in inspiring more "rational" and therefore necessarily equitable social organization. This speaks to the longstanding socio-political interest in "consciousness raising" tactics, which are often meant to dispel "false consciousness." It even comes across in the emphases of Deweyan pragmatists on education. Psychologists of all stripes point out that this approach... (shrink)

For many Christians, personal experiences of God provide an important ground or justification for accepting the truth of the gospel. But we are sometimes mistaken about our experiences, and followers of other religions also provide impressive testimonies to support their religious beliefs. This book explores from a philosophical and theological perspective the viability of divine encounters as support for belief in God, arguing that some religious experiences can be accepted as genuine experiences of God and can provide evidence for Christian (...) beliefs. (shrink)

A. W. Price explores the varying ways in which context is relevant to our reasoning about what to do.

Publicación compilatoria que recoge los avances que en los últimos tiempos se han producido en materia normativa sobre acceso a los recursos genéticos, en un intento por implementar los principios del Convenio sobre la Diversidad Biológica (1992).

Ludwig Wittgenstein (1889-1951) was an extraordinarily original thinker, whose influence on twentieth-century thinking far outside the bounds of philosophy alone. In this engaging Introduction, A.C. Grayling makes Wittgenstein's thought accessible to the general reader by explaining the nature and impact of Wittgenstein's views. He describes both his early and later philosophy, the differences and connections between them, and gives a fresh assessment of Wittgenstein's continuing influence on contemporary thought.

The elephant in the room -- Crabs and octopuses -- The central intelligence of a frog -- The cerebral cortex and consciousness -- Social consciousness -- Yoda and Darth: how can we find -- Consciousness in the brain? -- The hard problem and other perspectives on consciousness -- Conscious machines -- Uploading minds -- How to build visual consciousness.

The Human A Priori is a collection of essays by A. W. Moore, one of them previously unpublished and the rest all revised. These essays are all concerned, more or less directly, with something ineliminably anthropocentric in our systematic pursuit of a priori sense-making. Part I deals with the nature, scope, and limits of a priori sense-making in general. Parts II, III, and IV deal with what are often thought to be the three great exemplars of the systematic pursuit of (...) such sense-making: philosophy in the case of Part II, ethics in the case of Part III, and mathematics in the case of Part IV. Much of the attention throughout is devoted to the work of other philosophers: Kant and Wittgenstein feature prominently, and five of the essays take the form of reviews or critical notices of recent work in philosophy. But the interest in never purely exegetical. One of the lessons that emerges from the essays, either in opposition to the views of these other philosophers or by invocation of their views, is that we humans achieve nothing of real significance in philosophy, ethics, or mathematics except from a human point of view, and hence that all three of these pursuits can be said to betoken what may reasonably be called 'the human a priori'. (shrink)