Tarski [5] showed that for any set X, its set w(X) of well-orderable subsets has cardinality strictly greater than that of X, even in the absence of the axiom of choice. We construct a Fraenkel-Mostowski model in which there is an infinite strictly descending sequence under the relation |w (X)| = |Y|. This contrasts with the corresponding situation for power sets, where use of Hartogs' ℵ-function easily establishes that there can be no infinite descending sequence under the relation |P(X)| (...) = |Y|. (shrink)

A foundational algebra ( , f, ) consists of a hemimorphism f on a Boolean algebra with a greatest solution to the condition f(x). The quasi-variety of foundational algebras has a decidable equational theory, and generates the same variety as the complex algebras of structures (X, R), where f is given by R-images and is the non-wellfounded part of binary relation R.The corresponding results hold for algebras satisfying =0, with respect to complex algebras of wellfounded binary relations. These algebras, however, (...) generate the variety of all ( ,f) with f a hemimorphism on ). (shrink)

We construct by diagonalization a non-well-founded primitive recursive tree, which is well-founded for co-r.e. sets, provable in Σ1 0. It follows that the supremum of order-types of primitive recursive well-orderings, whose well-foundedness on co-r.e. sets is provable in Σ1 0, equals the limit of all recursive ordinals ω1 ck . RID=""ID="" Mathematics Subject Classification (2000): 03B30, 03F15 RID=""ID="" Supported by the Deutschen Akademie der Naturforscher Leopoldina grant #BMBF-LPD 9801-7 with funds from the Bundesministerium für Bildung, (...) Wissenschaft, Forschung und Technologie. RID=""ID="" I would like to thank A. SETZER for his hospitality during my stay in Uppsala in December 1998 – these investigations are inspired by a discussion with him; S. BUSS for his hospitality during my stay at UCSD and for valuable remarks on a previous version of this paper; and M. MÖLLERFELD for remarks on a previous title. (shrink)

In lieu of an abstract, here is a brief excerpt of the content:Objective Reality of Ideas in Descartes, Caterus, and Su irez NORMAN j. WELLS IT HAS LONG BEEN ACKNOWLEDGEDthat Francisco Sufirez's distinction between a formal and an objective concept exercised some influence upon Descartes's teaching on 'idea'.' It would appear, however, that not enough attention has been given to that distinction of Sufirez (and especially to another to be mentioned shordy) to aid in dispelling what I take to be (...) a widespread confusion and misunderstanding with regard to 'idea' in Descartes, especially with respect to 'idea' taken 'objectively'. Much, if not all, of the confusion in question has to do with Descartes's admittedly equivocal use of the term 'idea'. For he had allowed that 'idea' could be taken 'materially' (materialiter), on the one hand, for the knowing activity of the intellect. On the other hand, it could be taken 'objectively' (objective) for the thing (res) represented by that knowing activity.' At this point, the contemporary reader can only be brought up short, See E. Gilson, Index scolastico-cart~sien (New York: Burt Franklin Reprint, 1912), 48-49; Discours de la m~thode: Texte et commentazre (Paris: J. Vrin, 1947), 321; G. Rodis-Lewisin her edition of Meditationes de Prima PhRosoph.m (Paris: J. Vrin, 1953), 41 n. 1; L'oeuure de Descartes (Paris: J. Vrin, 1971), 53~ n. 8; J. Collins, Modern European Philosophy (Milwaukee: Bruce, 1954), 165;T.J. Cronin, S.J., Objecttve Being m Descartes and SuArez (Rome: Gregorian University Press, 1966); "Objective Reality of Ideas in Human Thought: Descartes and Su~rez," Wisdom in Depth: Essays in Honor of Henri Renard, SJ., eds. V. F. Daues, S.J., M. R. Holloway, S.J., L. Sweeney, S.J. (Milwaukee : Bruce, 1966), 68-79; L. Gilen, S.J., "Uber die Beziehungen Descartes zur zeitgenossischen Scholastik," Scholastik 32 (1957): 41-66; R. Dalbiez, "Les sources scolastiques de la th~orie cart6sienne de l'&re objectif," Revue d'histoire de la philosophic 3 (1929): 464-72;J. Owens, C.S.S.R., in The Cambridge HtstoU of Later Medu,val Philosophy, eds. N. Kretzmann, A. Kenny, J. Pinborg (Cambridge University Press, 1982), 459; G. Nuchelmans,Judgment and Propositwnfrom Descartesto Kant (Amsterdam: North Holland Publishing Co., 1983), 36-54. " Oeuvres de Descartes, ed. C. Adam et P. Tannery, 11 vols. (Paris: J. Vrin, 1897-19o9; hereafter cited according to volume, page and line), Praefatio ad Lectorem, 7: 8. 19-251 "Sed respondeo hic subesse aequivocationem in voce ideae: sumi enim potest vel materialiter, pro operatione intellectus, quo sensu me perfectior dici nequit, vel objective, pro re per istam operationem repraesentata, quae res, etsi non supponatur extra intellectum existere, potest tamen me esse perfectior ratione suae essentiae." It is clear that two realities are to be causally accounted for. The [33] 34 JOURNAL OF THE HISTORY OF PHILOSOPHY 28"1 JANUARY 1990 conditioned as we are by an appreciation of 'objectivity' altogether extramental! It makes for a considerable wrench of perspective to do justice to Descartes's acknowledged doctrine of intramental objectivity and an intramental 'thing represented' (resrepraesentata). 'Idea', then, in traditional Cartesian fashion, designates the act of representing as well as the 'thing represented '. S Yet, for all this, and despite repeated expressions of the same doctrine, there is a persistent tendency in the history of Cartesianism and its scholarship to interpret 'idea' taken 'objectively' as in some way representative. Contrary to what I take to be Descartes's constant position, such a rendering unwittingly transfers that function which is proper to the cognitive activity, mind is the efficient cause of its cognitive activity. With respect to ideas taken 'objectively', such as the ideas of God and the triangle on the levels of metaphysical and mathematical intelligibility, an efficient cause, extrinsic to the mind, is certainly at issue. For here we are in the presence of what is independent of the mind, however intramental it may be. On the triangle, see Medit. 5; 7: 64. ]4s7 : "...est tamen profecto determinata quaedam ejus natura, sire essentia, sive forma, immutabUis et aeterna, quae a me non efficta est, nec a mente mea dependet." For the idea of God, see Medit. 5; 7: 68. l o-12: "Nam sane multis modis inteUigo illam [ideam... (shrink)

Du Châtelet holds that mathematical representations play an explanatory role in natural science. Moreover, she writes that things proceed in nature as they do in geometry. How should we square these assertions with Du Châtelet’s idealism about mathematical objects, on which they are ‘fictions’ dependent on acts of abstraction? The question is especially pressing because some of her important interlocutors (Wolff, Maupertuis, and Voltaire) denied that mathematics informs us about the properties of material things. After situating Du Châtelet in this (...) debate, this chapter argues, first, that her account of the origins of mathematical objects is less subjectivist than it might seem. Mathematical objects are non-arbitrary, public entities. While mathematical objects are partly mind-dependent, so are material things. Mathematical objects can approximate the material. Second, it is argued that this moderate metaphysical position underlies Du Châtelet’s persistent claims that mathematics is required for certain kinds of general knowledge, including in natural science. The chapter concludes with an illustrative example: an analysis of Du Châtelet’s argument that matter is continuous. A key premise in the argument is that mathematical representations and material nature correspond. (shrink)

Excerpt: IT will lead into my subject most conveniently to contrast and separate two divergent types of mind, types which are to be distinguished chiefly by their attitude toward time, and more particularly by the relative importance they attach and the relative amount of thought they give to the future. The first of these two types of mind, and it is, I think, the predominant type, the type of the majority of living people, is that which seems scarcely to think (...) of the future at all, which regards it as a sort of blank non-existence upon which the advancing present will presently write events. The second type, which is, I think, a more modern and much less abundant type of mind, thinks constantly and by preference of things to come, and of present things mainly in relation to the results that must arise from them. The former type of mind, when one gets it in its purity, is retrospective in habit, and it interprets the things of the present, and gives value to this and denies it to that, entirely with relation to the past. The latter type of mind is constructive in habit, it interprets the things of the present and gives value to this or that, entirely in relation to things designed or foreseen. While from that former point of view our life is simply to reap the consequences of the past, from this our life is to prepare the future. The former type one might speak of as the legal or submissive type of mind, because the business, the practice, and the training of a lawyer dispose him toward it; he of all men must constantly refer to the law made, the right established, the precedent set, and consistently ignore or condemn the thing that is only seeking to establish itself. The latter type of mind I might for contrast call the legislative, creative, organizing, or masterful type, because it is perpetually attacking and altering the established order of things, perpetually falling away from respect for what the past has given us. It sees the world as one great workshop, and the present is no more than material for the future, for the thing that is yet destined to be. It is in the active mood of thought, while the former is in the passive; it is the mind of youth, it is the mind more manifest among the western nations, while the former is the mind of age, the mind of the oriental. Things have been, says the legal mind, and so we are here. The creative mind says we are here because things have yet to be. Now I do not wish to suggest that the great mass of people belong to either of these two types. Indeed, I speak of them as two distinct and distinguishable types mainly for convenience and in order to accentuate their distinction. There are probably very few people who brood constantly upon the past without any thought of the future at all, and there are probably scarcely any who live and think consistently in relation to the future. The great mass of people occupy an intermediate position between these extremes, they pass daily and hourly from the passive mood to the active, they see this thing in relation to its associations and that thing in relation to its consequences, and they do not even suspect that they are using two distinct methods in their minds. As well known as HG Wells is for his science fiction, this is simply HG Wells discussing our view of the present and the future and various nuances related to both. Very interesting. (shrink)

This conceptual paper explores the theoretical possibilities of posthumanism and presents ecologies of repair as a heuristic device to explore the association modes of different entities, which, when confronted with the effects of human-induced destructive events, seek to repair the damage and transform the conditions of coexistence of various life forms. The central idea is that severe socio-environmental crisis caused by an intensification of industrial activity are conducive to observing new sociomaterial configurations and affective dispositions that, through the reorganization of (...) practices of resistance, remediation, and mutual care, are oriented to generating reparative and/or transformative processes from damaged ecologies and communities. Crises constitute true ontological experimentation processes where the presence of other-than-human natures, and of artifacts or devices that participate in reparative actions, become visible. A post-human approach to nature allows us to use languages and methodologies that do not restrict the emergence of assemblages under the assumption of theira prioriontological separation, but rather examine their reparative potential based on the efficacy of situated relationships. Methodologically, transdisciplinarity is relevant, with ethnography and other engaged methods applied over units of observation and experience called socio-geo-ecologies. The relevant attributes of these socio-geo-ecologies, beyond the individual, community, or institutional aspects, are the specific geological characteristics that make possible an entanglement of interdependent relationships between human and non-human agents. The conceptual analysis is illustrated with empirical examples stemming from socio-geo-ecologies researched in Southern Chile. (shrink)

To take the idea of a non-anthropocentric ethic of nature seriously is to abandon morality itself. The idea of humanity is not an optional extra for moral seriousness. Non-anthropocentric environmental ethicists mistake the kind of value non-human entities may bear. It is not moral value, but aesthetic value.

I examine the link between extensionality principles of classical mereology and the anti‐symmetry of parthood. Varzi's most recent defence of extensionality depends crucially on assuming anti‐symmetry. I examine the notions of proper parthood, weak supplementation and non‐well‐foundedness. By rejecting anti‐symmetry, the anti‐extensionalist has a unified, independently grounded response to Varzi's arguments. I give a formal construction of a non‐extensional mereology in which anti‐symmetry fails. If the notion of ‘mereological equivalence’ is made explicit, this non‐anti‐symmetric mereology recaptures all of (...) the structure of classical mereology. (shrink)

Different approaches analyze the strength of a natural language argument in different ways. This paper contrasts the dialectical, structural, probabilistic (or Bayesian), computational, and empirical approaches by exemplarily applying them to a single argumentative text (Epicureans on Squandering Life; Aikin & Talisse, 2019). Rather than pitching these approaches against one another, our main goal is to show the room for fruitful interaction. Our focus is on a dialectical analysis of the squandering argument as an argumentative response that voids an interlocutor’s (...) right to assertion. This analysis addresses the pragmatic dimensions of arguing and implies an argument structure that is consistent with empirical evidence of perceived argument strength. Results show that the squandering argument can be evaluated as a (non-fallacious) ad hominem argument, which however is not necessarily stronger than possible arguments attacking it. (shrink)

It is argued that the \ cancellation model is a good proxy for discussions of finetuned cancellations in physical theories. XYCM is then analyzed from a statistical perspective, where it is argued that a finetuned point in the parameter space is not abnormal, with any such point being just as probable as any other point. However, landing inside a standardly defined finetuned region has a much lower probability than landing outside the region, and that probability is invariant under assumed ranges (...) of parameters. This proposition requires asserting also that the finetuned target region is a priori established. Therefore, it is surmised that a highly finetuned theory is generally expected to be highly improbable. An actionable implication of this moderate naturalness position is that the search for a non-finetuned explanation to supplant an apparently finetuned theory is likely to be a valid pursuit, but not guaranteed to be. A statistical characterization of this moderate position is presented, as well as those of the extreme pro-naturalness and anti-naturalness positions. (shrink)

McLuhan’s status as the patriarch of Canadian media studies is explored as a troublesome nomological principle in light of Derrida’s Archive Fever. The trouble with archives, for Derrida, is the trouble of the original source. Linking McLuhan’s exploration of typographical and electronic communication systems to Derrida’s deconstruction of the archive as a technology of exteriorization, this paper investigates the nature of subjectivity and objectivity in Western epistemology. Can the archive, conceived of as a medium, allow for an escape from the (...) specters of the past? Derrida’s proposed ‘archive of the future’ is theorized in terms of McLuhan’s sense of acoustic space and Foucault’s non-subjective discourse, and as a spatial and cultural construct reflecting the technological uncertainty in an age of mixed-media. (shrink)

In a recent paper, Lyngzeidetson [1990] has claimed that a type of parallel computer called the ‘Connection Machine’ instantiates architectural principles which will ‘revolutionize which "functions" of the human mind can and cannot be modelled by (non-human) computational automata.’ In particular, he claims that the Connection Machine architecture shows the anti-mechanist argument from Gödel's theorem to be false for at least one kind of parallel computer. In the first part of this paper, I argue that Lyngzeidetson's claims are not supported (...) by his arguments; in the second part I consider some other aspects of parallel computation which may be of theoretical significance in cognitive science. (shrink)

Forensic psychiatry is a particular subspecialty within psychiatry, dedicated in applying psychiatric knowledge and psychiatric training for particular legal purposes. Given that within the scope of forensic psychiatry, a third party usually intervenes in the patient-doctor relationship, an amendment of the traditional ethical principles seems justified. Thus, 47 articles, two book chapters and the guidelines produced by the World Psychiatric Association, the American Association of Psychiatry and the Law, as well as by the Royal Australian and New Zealand College (...) of psychiatrists, were analyzed. The review revealed that the ethics of correctional forensic psychiatry and those of legal forensic psychiatry do not markedly differ from each other, but they are incongruent in terms of implementation. In an effort to better understand which ethical principles apply to forensic psychiatry, a chronological review of the literature published from 1950 to 2015 was carried out. The ethics of correctional forensic psychiatry are primarily deontological. The principle of justice translates into the principle of health care equivalence, the principle of beneficence into providing the best possible care to patients, and the principle of respect of autonomy into ensuring confidentiality and informed consent. The ethics of legal forensic psychiatry are rather consequentialist. In this latter setting, the principle of justice is mainly characterized by professionalism, the principle of beneficence by objectivity and impartiality, and the principle of respect of autonomy by informed consent. However, these two distinct fields of forensic psychiatry share in common the principle of non maleficence, defined as the non collaboration of the psychiatrist in any activity leading to inhuman and degrading treatment or to the death penalty. (shrink)

Among business ethicists, Adam Smith is widely viewed as the defender of an amoral if not anti-moral economics in which individuals’ pursuit of their private self-interest is converted by an ‘invisible hand’ into shared economic prosperity. This is often justified by reference to a select few quotations from The Wealth of Nations. We use new empirical methods to investigate what Smith actually had to say, firstly about the relationship between free market institutions and individuals’ moral virtues, and secondly about the (...) further relationship between virtues and societal flourishing. We show with more quantitative precision than traditional scholarship that the invisible hand reading dramatically misrepresents both the nuance and the sum of Smith’s analysis. Smith paid a great deal of attention to a flourishing society’s dependence on virtues, including the non-self-regarding virtues of justice and benevolence, and he worried also about their fragility in the face of the changed incentives and social conditions of commercial society. (shrink)

The two center time dependent Dirac equation, for an electron in the external field of two colliding ultrarelativistic heavy ions is considered. In the ultrarelativistic limit, the ions are practically moving at the speed of light and the electromagnetic fields of the ions are confined to the light fronts by the extreme Lorentz contraction and by the choice of gauge, designed to remove the long-range Coulomb effects. An exact solution to the ultrarelativistic limit of the two-center Dirac equation is found (...) by using light-front variables and a light-fronts representation. Previously unexplained experimental results obtained at CERN's SPS are explained in this way and predictions are made as to where one should look, in momentum space, and in space-time, if one wants to study and observe non-perturbative electromagnetic pair-production effects in extremely relativistic heavy-ion collisions. (shrink)

To take the idea of a non-anthropocentric ethic of nature seriously is to abandon morality itself. The idea of humanity is not an optional extra for moral seriousness. Non-anthropocentric environmental ethicists mistake the kind of value non-human entities may bear. It is not moral value, but aesthetic value.

This paper investigates provability and non-provability of well-foundedness of ordinal notations in weak theories of bounded arithmetic. We define a notion of well-foundedness on bounded domains. We show that T21 and S22 can prove the well-foundedness on bounded domains of the ordinal notations below 0 and Γ0. As a corollary, the class of polynomial local search problems, PLS, can be augmented with cost functions that take ordinal values below 0 and Γ0 without increasing the (...) class PLS. (shrink)

This paper investigates provability and non-provability of well-foundedness of ordinal notations in weak theories of bounded arithmetic. We define a notion of well-foundedness on bounded domains. We show that T21 and S22 can prove the well-foundedness on bounded domains of the ordinal notations below 0 and Γ0. As a corollary, the class of polynomial local search problems, PLS, can be augmented with cost functions that take ordinal values below 0 and Γ0 without increasing the (...) class PLS. (shrink)

It has recently been argued that foundationalists, those who take grounding to be well-founded, should not understand the well-foundedness of grounding as the condition that every grounding chain terminates in the downward direction, because this interpretation of well-foundedness fails to correctly classify certain complex grounding structures. Some structures that plausibly would be acceptable to the foundationalist are classified as not well-founded and others that plausibly would not be acceptable to the foundationalist are classified as (...) well-founded. In this paper I provide a characterisation of well-foundedness in terms of termination that correctly characterises all these difficult cases. This acts as a response to these recent arguments. Furthermore, it allows us to better evaluate their importance: these arguments have not shown that foundationalists are wrong to harbour the intuition that grounding chains must terminate in the downward direction, rather, they have shown that foundationalists need to be more clear about this intuition and how it is born out in more complex grounding structures. (shrink)

ABSTRACT The technological revolution has created enormous opportunities for the provision of affordable, accessible, and flexible mental healthcare. Yet it also creates complexities and ethical challenges. While some of these challenges may be similar to face-to-face care, their nuance in the online milieu is different, as relationships, identities and boundaries in this setting are fluid, and there is an absence of physical presence. In this paper we consider the specific ethical complexities involved in the provision of a social networking intervention (...) for depression and problematic alcohol use involving providers and peers. We present a case in which clinicians encountered a participant who repeatedly expressed suicidal thoughts in this online forum, and reflect on how this situation was managed. In analyzing this case, we use relational ethics—which proposes that interpersonal relationships are the fulcrum for ethical decision-making and action—together with actor-network theory—which highlights the power of technology to transform and mediate social and ethical relationships. Our analysis demonstrates the need to move beyond human-centric approaches, to recognize the role of both human and non-human actors in shaping moral decisions and behavior in the provision of e-mental healthcare. (shrink)

A number of philosophers think that grounding is, in some sense, well-founded. This thesis, however, is not always articulated precisely, nor is there a consensus in the literature as to how it should be characterized. In what follows, I consider several principles that one might have in mind when asserting that grounding is well-founded, and I argue that one of these principles, which I call ‘full foundations’, best captures the relevant claim. My argument is by the process of (...) elimination. For each of the inadequate principles, I illustrate its inadequacy by showing either that it excludes cases that should not be ruled out by a well-foundedness axiom for grounding, or that it admits cases that should be ruled out. (shrink)

According to the traditional bundle theory, particulars are bundles of compresent universals. I think we should reject the bundle theory for a variety of reasons. But I will argue for the thesis at the core of the bundle theory: that all the facts about particulars are grounded in facts about universals. I begin by showing how to meet the main objection to this thesis (which is also the main objection to the bundle theory): that it is inconsistent with the possibility (...) of distinct qualitative indiscernibles. Here, the key idea appeals to a non-standard theory of haecceities as non-well-founded properties of a certain sort. I will then defend this theory from a number of objections, and finally argue that we should accept it on the basis of considerations of parsimony about the fundamental. (shrink)

We highlight that the connection of well-foundedness and recursive definitions is more than just convenience. While the consequences of making well-foundedness a sufficient condition for the existence of hierarchies have been extensively studied, we point out that well-foundedness is a necessary condition for the existence of hierarchies e.g. that even in an intuitionistic setting α⊢wfwhereα\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${_\alpha \vdash \mathsf{wf}\, {\rm where}\, _\alpha}$$\end{document} stands for the iteration of Π10\documentclass[12pt]{minimal} (...) \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\Pi^0_1}$$\end{document} comprehension along some ordinal α\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\alpha}$$\end{document} and wf\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathsf{wf}}$$\end{document} stands for the well-foundedness of α\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\alpha}$$\end{document}. (shrink)

We present a sequent calculus for the Grzegorczyk modal logic $\mathsf {Grz}$ allowing cyclic and other non-well-founded proofs and obtain the cut-elimination theorem for it by constructing a continuous cut-elimination mapping acting on these proofs. As an application, we establish the Lyndon interpolation property for the logic $\mathsf {Grz}$ proof-theoretically.

We consider Hilbert-style non–well-founded derivations in the Gödel-Löb provability logic GL and establish that GL with the obtained derivability relation is globally complete for algebraic and neighbourhood semantics.

Non-well-founded trees are used in mathematics and computer science, for modelling non-well-founded sets, as well as non-terminating processes or infinite data structures. Categorically, they arise as final coalgebras for polynomial endofunctors, which we call M-types. We derive existence results for M-types in locally cartesian closed pretoposes with a natural numbers object, using their internal logic. These are then used to prove stability of such categories with M-types under various topos-theoretic constructions; namely, slicing, formation of coalgebras , and (...) sheaves for an internal site. (shrink)

George Boolos has described an interpretation of a fragment of ZFC in a consistent second-order theory whose only axiom is a modification of Frege's inconsistent Axiom V. We build on Boolos's interpretation and study the models of a variety of such theories obtained by amending Axiom V in the spirit of a limitation of size principle. After providing a complete structural description of all well-founded models, we turn to the non-well-founded ones. We show how to build models in (...) which foundation fails in prescribed ways. In particular, we obtain models in which every relation is isomorphic to the membership relation on some set as well as models of Aczel's anti-foundation axiom (AFA). We suggest that Fregean extensions provide a natural way to envisage non-well-founded membership. (shrink)

Typed feature structures are used extensively for the specification of linguistic information in many formalisms. The subsumption relation orders TFSs by their information content. We prove that subsumption of acyclic TFSs is well founded, whereas in the presence of cycles general TFS subsumption is not well founded. We show an application of this result for parsing, where the well-foundedness of subsumption is used to guarantee termination for grammars that are off-line parsable. We define a new version (...) of off-line parsability that is less strict than the existing one; thus termination is guaranteed for parsing with a larger set of grammars. (shrink)

Vauzeilles, J., Cut elimination for the Unified Logic, Annals of Pure and Applied Logic 62 1-16. In the paper entitled “On the Unity of Logic” Girard introduced and motivated the system LU. In Girard's article, the cut-elimination result for LU is stated and used as a key lemma, but not supported by any rigourous proof. In the present paper, we prove that LU enjoys cut elimination under minimal hypotheses: a notion of degree for a formula is introduced, which depends only (...) on the number of exponentials of the formula; this refinement yields much better bounds for cut elimination and leaves open many possibilities as to non-well-foundedness of formulae. (shrink)

A finitary characterization for non-well-founded sets with finite transitive closure is established in terms of a greatest fixpoint formula of the modal μ-calculus. This generalizes the standard result in the literature where a finitary modal characterization is provided only for wellfounded sets with finite transitive closure. The proof relies on the concept of automaton, leading then to new interlinks between automata theory and non-well-founded sets.

A complete list of Finsler, Scott and Boffa sets whose transitive closures contain 1, 2 and 3 elements is given. An algorithm for deciding the identity of hereditarily finite Scott sets is presented. Anti-well-founded sets, i. e., non-well-founded sets whose all maximal ∈-paths are circular, are studied. For example they form transitive inner models of ZFC minus foundation and empty set, and they include uncountably many hereditarily finite awf sets. A complete list of Finsler and Boffa awf sets (...) with 2 and 3 elements in their transitive closure is given. Next the existence of infinite descending ∈-sequences in Aczel universes is shown. Finally a theorem of Ballard and Hrbáček concerning nonstandard Boffa universes of sets is considerably extended. (shrink)

We model infinite regress structures -not arguments- by means of ungrounded recursively defined functions in order to show that no such structure can perform the task of providing determination to the items composing it, that is, that no determination process containing an infinite regress structure is successful.

The paper uses the formalism of indexed categories to recover the proof of a standard final coalgebra theorem, thus showing existence of final coalgebras for a special class of functors on finitely complete and cocomplete categories. As an instance of this result, we build the final coalgebra for the powerclass functor, in the context of a Heyting pretopos with a class of small maps. This is then proved to provide models for various non-well-founded set theories, depending on the chosen (...) axiomatisation for the class of small maps. (shrink)

In this paper, we show that non-well-founded sets can be defined constructively by formalizing Hallnäs' limit definition of these within Martin-Löf's theory of types. A system is a type W together with an assignment of ᾱ ∈ U and α̃ ∈ ᾱ → W to each α ∈ W. We show that for any system W we can define an equivalence relation = w such that α = w β ∈ U and = w is the maximal bisimulation. Aczel's (...) proof that CZF can be interpreted in the type V of iterative sets shows that if the system W satisfies an additional condition (*), then we can interpret CZF minus the set induction scheme in W. W is then extended to a complete system W * by taking limits of approximation chains. We show that in W * the antifoundation axiom AFA holds as well as the axioms of CFZ -. (shrink)

In this article, we discuss expertise by considering the activity of reading. Cognitive scientists have traditionally conceptualised reading as a single, well-defined task, namely the decoding of letter sequences into meaningful sequences of speech sounds. This definition captures a core feature of the reading activity at the computational level, but it is an overly narrow model of how reading behaviour occurs in the real world. We propose a more expansive model of expertise. In our view, expertise in general is (...) best conceptualised as a distributed process that takes place within a cultural-cognitive ecosystem. Our model allows for a more inclusive view of expertise in reading. We argue that reading is better understood as a manifold task domain that admits multiple reasonable criteria for evaluating performance. We draw on ethnographic data to show how our model allows for a wider appreciation of what expertise in reading amounts to in educational contexts. (shrink)

We show that there are universes of sets which contain descending ϵ-sequences of length α for every ordinal α, though they do not contain any ϵ-cycle. It is also shown that there is no set universe containing a descending ϵ-sequence of length On. MSC: 03E30; 03E65.