We discuss the interpolation property on some important families of non classical logics, such as intuitionistic, modal, fuzzy, and linear logics. A special paragraph is devoted to a generalization of the interpolation property, uniform interpolation.

This paper studies the topological duality between diagonalizable algebras and bi-topological spaces. In particular, the correspondence between algebraic properties of a diagonalizable algebra and topological properties of its dual space is investigated. Since the main example of a diagonalizable algebra is the Lindenbaum algebra of an r.e. theory extending Peano Arithmetic, endowed with an operator defined by means of the provability predicate of the theory, this duality gives the possibility to study arithmetical properties of theories from a topological point of (...) view. We find topological characterization of $\Sigma_1$-sound theories and of sentences that are $\Sigma_1$-conservative over such a theory. (shrink)

In this paper we study iterated circular multisets in a coalgebraic framework. We will produce two essentially different universes of such sets. The unisets of the first universe will be shown to be precisely the sets of the Scott universe. The unisets of the second universe will be precisely the sets of the AFA-universe. We will have a closer look into the connection of the iterated circular multisets and arbitrary trees. RID=""ID="" Mathematics Subject Classification (2000): 03B45, 03E65, 03E70, 18A15, 18A22, (...) 18B05, 68Q85 RID=""ID="" Key words or phrases: Multiset – Non-wellfounded set – Scott-universe – AFA – Coalgebra – Modal logic – Graded modalities. (shrink)

We consider the relation between the uniform interpolation property and the elimination of non-standard quantifiers (the bisimulation quantifiers) in the context of the ?-calculus. In particular, we isolate classes of frames where the correspondence between these two properties is nicely smooth.

In Zwicker (1987) the hypergame paradox is introduced and studied. In this paper we continue this investigation, comparing the hypergame argument with the diagonal one, in order to find a proof schema. In particular, in Theorems 9 and 10 we discuss the complexity of the set of founded elements in a recursively enumerable relation on the set N of natural numbers, in the framework of reduction between relations. We also find an application in the theory of diagonalizable algebras and construct (...) an undecidable formula. (shrink)

We consider the problem of the existence of uniform interpolants in the modal logic K4. We first prove that all ${\square}$ -free formulas have uniform interpolants in this logic. In the general case, we shall prove that given a modal formula ${\phi}$ and a sublanguage L of the language of the formula, we can decide whether ${\phi}$ has a uniform interpolant with respect to L in K4. The ${\square}$ -free case is proved using a reduction to the Gödel Löb Logic (...) GL, while in the general case we prove that the question of whether a modal formula has uniform interpolants over transitive frames can be reduced to a decidable expressivity problem on the μ-calculus. (shrink)

Using only propositional connectives and the provability predicate of a Σ1-sound theory T containing Peano Arithmetic we define recursively enumerable relations that are complete for specific natural classes of relations, as the class of all r. e. relations, and the class of all strict partial orders. We apply these results to give representations of these classes in T by means of formulas.

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)

Free Public Reason examines the idea of public justification, stressing its importance but also questioning the coherence of the concept itself. Although public justification is employed in the work of theorists such as John Rawls, Jeremy Waldron, Thomas Nagel, and others, it has received little attention on its own as a philosophical concept. In this book Fred D'Agostino shows that the concept is composed of various values, interests, and notions of the good, and that no ranking of these is possible. (...) The notion of public justification itself is thus shown to be contestable. In demonstrating this, D'Agostino undermines many current political theories that rely on this concept. Having broken down the foundations of public justification, D'Agostino then offers an alternative model of how a workable consensus on its meaning might be reached through the interactions of a community of interpreters or delegates at a constitutional convention. (shrink)

Deductive inference is usually regarded as being “tautological” or “analytical”: the information conveyed by the conclusion is contained in the information conveyed by the premises. This idea, however, clashes with the undecidability of first-order logic and with the (likely) intractability of Boolean logic. In this article, we address the problem both from the semantic and the proof-theoretical point of view. We propose a hierarchy of propositional logics that are all tractable (i.e. decidable in polynomial time), although by means of growing (...) computational resources, and converge towards classical propositional logic. The underlying claim is that this hierarchy can be used to represent increasing levels of “depth” or “informativeness” of Boolean reasoning. Special attention is paid to the most basic logic in this hierarchy, the pure “intelim logic”, which satisfies all the requirements of a natural deduction system (allowing both introduction and elimination rules for each logical operator) while admitting of a feasible (quadratic) decision procedure. We argue that this logic is “analytic” in a particularly strict sense, in that it rules out any use of “virtual information”, which is chiefly responsible for the combinatorial explosion of standard classical systems. As a result, analyticity and tractability are reconciled and growing degrees of computational complexity are associated with the depth at which the use of virtual information is allowed. (shrink)

Decision-making assisted by algorithms developed by machine learning is increasingly determining our lives. Unfortunately, full opacity about the process is the norm. Would transparency contribute to restoring accountability for such systems as is often maintained? Several objections to full transparency are examined: the loss of privacy when datasets become public, the perverse effects of disclosure of the very algorithms themselves, the potential loss of companies’ competitive edge, and the limited gains in answerability to be expected since sophisticated algorithms usually are (...) inherently opaque. It is concluded that, at least presently, full transparency for oversight bodies alone is the only feasible option; extending it to the public at large is normally not advisable. Moreover, it is argued that algorithmic decisions preferably should become more understandable; to that effect, the models of machine learning to be employed should either be interpreted ex post or be interpretable by design ex ante. (shrink)

In our information societies, tasks and decisions are increasingly outsourced to automated systems, machines, and artificial agents that mediate human relationships, by taking decisions and acting on the basis of algorithms. This raises a critical issue: how are algorithmic procedures and applications to be appraised and governed? This question needs to be investigated, if one wishes to avoid the traps of ICTs ending up in isolating humans behind their screens and digital delegates, or harnessing them in a passive role, by (...) curtailing their freedom and autonomy. (shrink)

In this paper we provide a detailed proof-theoretical analysis of a natural deduction system for classical propositional logic that represents classical proofs in a more natural way than standard Gentzen-style natural deduction, admits of a simple normalization procedure such that normal proofs enjoy the Weak Subformula Property, provides the means to prove a Non-contamination Property of normal proofs that is not satisfied by normal proofs in the Gentzen tradition and is useful for applications, especially in formal argumentation, naturally leads to (...) defining a notion of depth of a proof, to the effect that, for every fixed natural k, normal k-depth deducibility is a tractable problem and converges to classical deducibility as k tends to infinity. (shrink)

x philosophy when he inaugurated a debate about the principle of methodologi cal individualism, a debate which continues to this day, and which has inspired a literature as great as any in contemporary philosophy. Few collections of material in the general area of philosophy of social science would be considered complete unless they contained at least one of Watkins's many contributions to the discussion of this issue. In 1957 Watkins published the flrst of a series of three papers in which (...) he tried to codify and rehabilitate metaphysics within the Popperian philosophy, placing it somewhere between the analytic and the empirical. He thus signalled the emergence of an important implica tion of Popper's thought that had not to that point been stressed by Sir Karl himself, and which marked off his followers from the antimetaphysical ideas of the regnant logical positivists. In 1965 years of work in political philosophy and in the history of philosophy in the seventeenth century were brought to fruition in Watkins's widely cited and admired Hobbes's System of Ideas. This book is an important contribution not just to our understanding of Hobbes's political thinking, but, perhaps more importantly, to our understanding of the way in which a system of ideas is constituted and applied. Watkins built on earlier work in developing an account of Hobbes's ideas in which was revealed and clarifled the unity of Hobbes's metaphysical, epistemological and political ideas. (shrink)

To understand the continuing importance of John Rawls’s work, we need to understand the background, the object and the method of his fifty-year quest as a political thinker. The background to Rawls’s investigation was a (carefully circumscribed) acknowledgement of a certain kind of evaluative pluralism. The object of Rawls’s work was to develop a method of commensuration that would enable us, the free and equal citizens of a democratic society, to identify a common basis for our dealings, in search of (...) mutual benefit, with one another. Finally, the method used by Rawls in his work was broadly, though tacitly rather than explicitly, pragmatist in character, especially in the sense that Rawls was aiming, now explicitly, to develop principles of political association that generate their own support in the attitudes and actions of those whose behavior they are intended to guide. (shrink)

This book was published in 2003.This volume presents a detailed examination of incommensurability in the value-theoretical sense. Exploring how choosers deal with problems and constraints of choice, the author draws on work in cognitive psychology, in sociology, in jurisprudence, in economics, and in the theory of value to show how choosers learn to make "trade-offs" when there is potential incommensurability among the options they are considering. The analysis is also informed by recent work in the tradition of Michel Foucault. With (...) so many modern devices and ideals of government dependent on the comparability of options, this book is timely and can inform public debate about de-regulation, user-pays, accountability, and the substitution of market mechanisms for government regulation and supply. (shrink)

In identifying that the 'essential tension' is the balance between conservative and innovative approaches in the development of knowledge - tried-and tested or new directions - Kuhn pointed out that these two attitudes are both appropriate. This study adds to this picture the social and psychological dynamics that underpin any such balancing.

Critical notice of The Order of Public Reason by Gerald Gaus.

It has frequently been suggested that Rawls’s characteristic method of justification, a method crucially involving the notion of reflective equilibrium, is in some sense relativistic in its implications. No sustained development of this suggestion has been undertaken by those who advance it; likewise, no sustained attempt to refute this suggestion has been made by those who are otherwise sympathetic to Rawls’s account of justification. I here attempt to fill these gaps in the already extensive literature associated with the method of (...) reflective equilibrium. At below, I review claims that Rawls’s method has relativistic implications. I also attempt, for purposes of argument, to buttress these claims by citing remarks of Rawls which could be interpreted as relativistic in character. At below, I consider problems of political reasoning as these arise in societies which are ill- and well-ordered in Rawls’s sense. At below, I consider problems of political reasoning as these arise between the members of two or more well-ordered societies, each of which is regulated by a different conception of justice. I also there adapt Williams’s distinction between notional and real confrontations between incompatible systems of beliefs in order to distinguish two kinds of conflict with respect to fundamental political principles. Finally, I show how a notion of broad reflective equilibrium can be deployed in order to solve those problems of political adjudication which may arise between the members of societies which have distinct conceptions of justice. At this point it should be clear that Rawls’s method of justification is not, in intention at least, relativistic in its implications. (shrink)

The Routledge Companion to Social and Political Philosophy is a comprehensive, definitive reference work, providing an up-to-date survey of the field, charting its history and key figures and movements, and addressing enduring questions as well as contemporary research. Features unique to the Companion are: an extensive coverage of the history of social and political thought, including separate chapters on the development of political thought in the Islamic world, India, and China as well in modern Germany, France, and Britain a focus (...) on the core concepts and the normative foundations of social and political theory a seven-chapter section devoted exclusively to distributive justice, the central issue of political philosophy since Rawls' Theory of Justice extensive coverage of global justice and international issues, which recently have emerged as vital topics an eight-chapter section on issues in social and political philosophy. The Companion is divided into eight thematic sections: The History of Social and Political Theory; Political Theories and Ideologies; Normative Foundations; The National State and Beyond; Distributive Justice; Political Concepts; Concepts and Methods in Social Philosophy; Issues in Social and Political Philosophy. Comprised of sixty-nine newly commissioned essays by leading scholars from throughout the world, The Routledge Companion to Social and Political Philosophy is the most comprehensive and authoritative resource in social and political philosophy for students and scholars. (shrink)

Recent years have been blessed with an abundance of logical systems, arising from a multitude of applications. A logic can be characterised in many different ways. Traditionally, a logic is presented via the following three components: 1. an intuitive non-formal motivation, perhaps tie it in to some application area 2. a semantical interpretation 3. a proof theoretical formulation. There are several types of proof theoretical methodologies, Hilbert style, Gentzen style, goal directed style, labelled deductive system style, and so on. The (...) tableau methodology, invented in the 1950s by Beth and Hintikka and later per fected by Smullyan and Fitting, is today one of the most popular, since it appears to bring together the proof-theoretical and the semantical approaches to the pre of a logical system and is also very intuitive. In many universities it is sentation the style first taught to students. Recently interest in tableaux has become more widespread and a community crystallised around the subject. An annual tableaux conference is being held and proceedings are published. The present volume is a Handbook a/Tableaux pre senting to the community a wide coverage of tableaux systems for a variety of logics. It is written by active members of the community and brings the reader up to frontline research. It will be of interest to any formal logician from any area. (shrink)

Ian C. Jarvie developed the idea of situational logic in a subtle and effective way. He was also interested in, as well as a contributor to, the institution of academic publication. This chapter provides a situational analysis of an important recurrent pattern in academic publishing, namely, the concentration of work around particular topics, despite the fact that most such work will be unrewarded in the economy of esteem that is meant to be in play.

We show that Smullyan's analytic tableaux cannot p-simulate the truth-tables. We identify the cause of this computational breakdown and relate it to an underlying semantic difficulty which is common to the whole tradition originating in Gentzen's sequent calculus, namely the dissonance between cut-free proofs and the Principle of Bivalence. Finally we discuss some ways in which this principle can be built into a tableau-like method without affecting its analytic nature.

One of Thomas Kuhn's profoundest arguments is introduced in the 1970 “Postscript” to The Structure of Scientific Revolutions . Kuhn is discussing the idea of a “disciplinary matrix” as a more adequate articulation of the “paradigm” notion he'd introduced in the first, 1962, edition of his famous work . He notes that one “element” of disciplinary matrices is likely to be common to most or even all such matrices, unlike the other elements which serve to distinguish specific disciplines and sub-disciplines (...) from one another. This is the element which he calls “values”, which, as he notes , being common to a number of otherwise distinct disciplinary matrices, “do much to provide a sense of community to natural scientists as a whole”. On the other hand, they also do much, and crucially in Kuhn's view, to promote and sustain a healthy diversity among the practitioners who share any specific disciplinary matrix. In particular, Kuhn claims that “individual variability in the application of shared values may serve functions essential to science.”. (shrink)

Someone is “verballed” in the Anglo-Australian idiom if they have attributed to them statements they did not actually make and indeed have explicitly denied. We will examine the evidence that Kuhn and Feyerabend were verballed in this sense by their critics and that the role of the idea of incommensurability in their argumentation has been systematically misunderstood and -represented. In particular, we will see that neither Kuhn nor Feyerabend, despite what their critics often say about them, held that incommensurability of (...) theories implies the rational incomparability of theories. This is especially clear in the case of Feyerabend, whose argument is NOT that theories in a scientific tradition are on occasion incommensurable, but, rather, that, when the relations of theories in a tradition are represented in a particular way, they may on occasion be incommensurable according to that representation and hence incomparable if that representation is taken as providing the mechanisms of comparison. And the point of this claim is not to establish something about science, but, rather, to establish something about the representations of science which yield this result (i.e. that two theories might be incommensurable). Feyerabend in other words invokes incommensurability (according to the standards of a particular representation) as a reductio of that mode of representation. And this argument in fact depends precisely on the comparability of theories which are, according to the representation, incommensurable. Feyerabend’s argument is about the ways in which we should understand progress in science and he is concerned, in particular, to establish that a historically informed approach is superior to an approach which, if applicable, is applicable only to what he calls “abstract traditions”. Kuhn’s work, especially in the Postscript—1969, provides complementary materials, especially in relation to a collectivised and non-“algorithmic” account of theory choice across formally incommensurable paradigms. (shrink)

I want to consider how the general characteristics of a discipline might facilitate ?social mechanisms for distributing knowledge? that do not depend on uniformity of use, but, in fact, on different uses by different people. Indeed, I want to show that the ways in which a discipline is organized afford the growth of knowledge and do so, in particular, by facilitating an approach to what Thomas Kuhn described as ?the essential tension? between, on the one hand, the traditional or customary (...) elements of disciplined enquiry, which are prerequisites for there being a community of enquiry, and, on the other hand, the innovative elements of disciplined enquiry, which are needed on account of the always already inadequate character of our engagement with the objects of enquiry. (shrink)

INTRODUCTION The editors of this volume - Jarvie and D'Agostino - encountered John Watkins at such different times in his career that they have never ...