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.

We consider the problem of the existence of uniform interpolants in the modal logic K4. We first prove that all \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\square}$$\end{document}-free formulas have uniform interpolants in this logic. In the general case, we shall prove that given a modal formula \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\phi}$$\end{document} and a sublanguage L of the language of the formula, we can decide whether \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} (...) \setlength{\oddsidemargin}{-69pt} \begin{document}$${\phi}$$\end{document} has a uniform interpolant with respect to L in K4. The \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\square}$$\end{document}-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)

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)

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 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)

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)

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 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)

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.

In this article, I focus on Johann Nikolaus Tetens’s linguistic theory to make three arguments: (a) this linguistic theory endorses a phonological (contra phonetic) approach to the acoustic sphere of language; (b) the phonological approach is based on the idea that sounds can turn into phonemes (of a properly human language) only when a minimally rational reflection on them is made; and (c) the phonological approach allows us to understand the phoneme as a differential unity, as being composed of structure (...) and function, and as giving a theoretical basis for language learning. (shrink)

Quale rapporto c'è tra una noticina di appena 30 righe e un'opera filosofica di quasi 500 pagine? A questa domanda, appassionante per chi s'interessa alla genesi delle opere del pensiero, Simone D'Agostino risponde esaminando il rapporto tra il capolavoro blondeliano del 1893, L'Action, e quella che si suole chiamare la "Première notule" del 5 novembre 1882. Il presente lavoro non è uno studio genetico del pensiero blondeliano, bensì una sua interpretazione sistematica da due punti focali e rispecchiantisi l'uno nell'altro. (...) Sotto questo aspetto, la ricerca di D'Agostino può definirsi unica nel suo genere. (shrink)

This book supports the idea that the ancient conception of philosophy as a way of life does not disappear in early modernity, but is transformed into a search for how to cure, guide, and free the human mind.

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)

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)

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

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)

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)

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)

Discussion of the cognitive division of labor has usually made very little contact with relevant materials from other disciplines, including theoretical biology, management science, and design theory. This article draws on these materials to consider some unavoidable conundrums faced by any attempt to present a particular way of dividing tasks among a labor team as the uniquely rational way of doing this, given the interdependence of the underlying evaluative standards by which the products of a system of division of labor (...) will be judged. Divisions of labor will typically cut across these interdependencies in ways which leave the outcomes of a process of labor hostage to path dependencies and suboptimalities. Some attempts to avoid these results are shown to be unsuccessful. All these difficulties are compounded by the fact that, in many cases, the division of labor has to be constructed over a ground of values that is itself being constructed simultaneously with the products which they are invoked to assess. Key Words: risk spreading • interdependency • NK fitness landscapes • complexity • epistasis • Kuhn • modularity • path dependency. (shrink)

In this paper we provide a detailed proof-theoretical analysis of a natural deduction system for classical propositional logic that (i) represents classical proofs in a more natural way than standard Gentzen-style natural deduction, (ii) admits of a simple normalization procedure such that normal proofs enjoy the Weak Subformula Property, (iii) 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, (...) (iv) 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)

We investigate the semantics of the logical systems obtained by introducing the modalities and into the family of substructural implication logics (including relevant, linear and intuitionistic implication). Then, in the spirit of the LDS (Labelled Deductive Systems) methodology, we "import" this semantics into the classical proof system KE. This leads to the formulation of a uniform labelled refutation system for the new logics which is a natural extension of a system for substructural implication developed by the first two authors in (...) a previous paper. (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)

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)

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.

Normal science involves persistent collective application of an agreed research agenda. Anomaly can threaten normal science, but so too can “undue persistence” in that agenda by a normal science peer group. We consider how “undue persistence” might be a collective effect of the common incentive structure that individual members of the peer group typically face in relation to their careers. To understand how “undue persistence” might be ameliorated, we consider the affordances of a peer’s membership of a departmental collegium, organized (...) on a different basis than the specialist peer group and hence able to supply the individual scholar with kinds of information and critical comment that may occasion off-agenda contributions to the specialty. The idea of brokerage is borrowed from the sociology of innovation to see how a scholar’s departmental colleagues might be able to broker new ways of thinking that can assist in the avoidance of “undue persistence”. (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)

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.

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)