In Whitehead’s theory of “events”, the primary focus is on the critical assessment of abstraction. Modern science’s heavy reliance on abstraction has resulted in what Whitehead calls “the fallacy of misplaced concreteness”, where the abstract is mistaken for the actual. To address this issue, Whitehead replaces the traditional category of abstraction with eternal objects and defines the abstractive hierarchy. He aims to clarify the metaphysical status of abstraction and concreteness while dissolving their binary opposition. It is important to (...) note that there are two possible approaches for interpreting an abstractive hierarchy. The differences between these approaches stem from the question of whether abstraction and complexity should be considered identical concepts. Based on their respective understandings, the abstractive hierarchy will exhibit different structural features, as well as varying positions for the concepts of abstraction and eternal objects within Whitehead’s metaphysical thought. (shrink)

The aim of this paper is to enrich the algebraic-axiomatic approach to recursion theory developed in [1] by an analogue to the classical arithmetical hierarchy and an abstract notion of degree. The results presented here are rather initial and elementary; indeed, the main problem was the very choice of right abstract concepts.

Descriptive abstraction means omission of information from descriptions of phenomena. In this paper, I introduce a distinction between vertical and horizontal descriptive abstraction. Vertical abstracts away levels of mechanism or organization, while horizontal abstracts away details within one level of organization. The distinction is implicit in parts of the literature, but it has received insufficient attention and gone mainly unnoticed. I suggest that the distinction can be used to clarify how computational descriptions are formed in some variants of the mechanistic (...) account of physical computation. Furthermore, I suggest that, if this suggestion is adopted, it can be used to resolve what I call abstraction, hierarchy, and generality problems raised against mechanistic account of physical computation. According to the abstraction problem, the mechanistic account of physical computation is conceptually confused in claiming that physical systems process computational, abstract properties. An existing solution distinguishes between descriptive and metaphysical abstraction, suggesting that the abstraction problem unnecessarily postulates metaphysically abstract entities. The solution has been criticized for leading to what I call hierarchy and generality problems: it results in two separate hierarchies, one physical and one computational, making it problematic both to account for the generality of computational descriptions and to specify how the two hierarchies are related to each other. Adopting the vertical-horizontal distinction and the view that computational descriptions are achieved by horizontal abstraction allows one to account for the generality of computational descriptions, and to form a single hierarchy in which there are no separate hierarchies in need of integration. (shrink)

In this article we investigate infinitary propositional logics from the perspective of their completeness properties in abstract algebraic logic. It is well-known that every finitary logic is complete with respect to its relatively subdirectly irreducible models. We identify two syntactical notions formulated in terms of intersection-prime theories that follow from finitarity and are sufficient conditions for the aforementioned completeness properties. We construct all the necessary counterexamples to show that all these properties define pairwise different classes of logics. Consequently, we obtain (...) a new hierarchy of logics going beyond the scope of finitarity. (shrink)

The paper is a study of the logic of existence, negation, and order in the Neoplatonic tradition. The central idea is that Neoplatonists assume a logic in which the existence predicate is a comparative adjective and in which monadic predicates function as scalar adjectives that nest the background order. Various scalar predicate negations are then identifiable with various Neoplatonic negations, including a privative negation appropriate for the lower orders of reality and a hyper-negation appropriate for the higher. Reversion to the (...) One can then be explained as the logical inference of hyper-negations from mundane knowledge. Part I develops the relevant linguistic and logical theory, and Part II defends Wolfson and the scalar interpretation against the more traditional Aristotelian understanding of Whittaker and others of reversion as intensional abstraction. (shrink)

ABSTRACT:We discuss workers’ dignity in hierarchical organizations. First, we explain why a conflict exists between high-ranking individuals’ authority and low-ranking individuals’ dignity. Then, we ask whether there is any justification that reconciles hierarchical authority with the dignity of workers. We advance a communitarian justification for hierarchical authority, drawing upon Confucianism, which provides that workers can justifiably accept hierarchical authority when it enables a certain type of social functioning critical for the good life of workers and other involved parties. The Confucian (...) communitarian perspective shows that promoting workers’ good life or well-being is an important condition for protecting their dignity. (shrink)

Abstract.This article takes on a perhaps impossible task: not only to reconstruct the core argument of Arthur Peacocke's program in science and religion but also to evaluate it in two major areas where it would seem to be vulnerable, namely, more recent developments in systems biology and the philosophy of mind. If his theory of hierarchies is to be successful, it must stand up to developments in these two areas and then be able to apply the results in a productive (...) way to Christian theological reflection. Peacocke recognized that one's model of the mind‐body relation is crucial for one's position on the God‐world relation and divine action. Of the three models that he constructed, it turns out that only the third can serve as a viable model for theology if it is to be more than purely deistic or metaphorical. (shrink)

The notion of “hierarchy” is one of the most commonly posited organizational principles in systems neuroscience. To this date, however, it has received little philosophical analysis. This is unfortunate, because the general concept of hierarchy ranges over two approaches with distinct empirical commitments, and whose conceptual relations remain unclear. We call the first approach the “representational hierarchy” view, which posits that an anatomical hierarchy of feed-forward, feed-back, and lateral connections underlies a signal processing hierarchy of (...) input-output relations. Because the representational hierarchy view holds that unimodal sensory representations are subsequently elaborated into more categorical and rule-based ones, it is committed to an increasing degree of abstraction along the hierarchy. The second view, which we call “topological hierarchy", is not committed to different representational functions or degrees of abstraction at different levels. Topological approaches instead posit that the hierarchical level of a part of the brain depends on how central it is to the pattern of connections in the system. Based on the current evidence, we argue that three conceptual relations between the two approaches are possible: topological hierarchies could substantiate the traditional representational hierarchy, conflict with it, or contribute to a plurality of approaches needed to understand the organization of the brain. By articulating each of these possibilities, our analysis attempts to open a conceptual space in which further neuroscientific and philosophical reasoning about neural hierarchy can proceed. (shrink)

In lieu of an abstract, here is a brief excerpt of the content:THE HIERARCHY OF TRUTHS IN THE CATECHISM AVERY DULLES, S.J. Fordham University Bronx, New Yark IN ORDER to throw light on the question of the hierarchy of truths in The Catechism of the Catholic Church, the topic here being addressed, it may be best to move by stages. I shall begin by saying something about the nature and purpose of the Catechism, then turn to the meaning (...) of the expression, " hierarchy of truths," and thirdly, in the main part of this article, discuss how the hierarchy of truths is or is not respected in the Catechism. Before concluding I shall add some remarks, fourthly, on the question of the different levels of authority of different doctrines. The Nature of the Catechism The Catechism of the Catholic Church was composed in response to the request of the Extraordinary Synod of 1985, which declared in its Final Report: " Very many have expressed the desire that a catechism or compendium of all Catholic doctrine regarding both faith and morals be composed, that it might be, as it were, a point of reference for the catechisms or compendiums that are prepared in the various regions. The presentation of doctrine must be biblical and liturgical. It must be sound doctrine suited to the present life of Christians." 1 From the beginning the new Catechism was conceived on the pattern of the Roman Catechism, also known as the Catechism of the Council of Trent. That catechism, published in 1566, was a lFinal Report, II B (a) 4; text in Origins 15 (December 19, 1985): 44450, at 448. 369 370 AVERY DULLES, S.J. rather complete restatement of Catholic doctrine in the light of the teaching of Trent. Like the Council out of which it grew, the Roman Catechism gave special emphasis to the Catholic doctrine of the sacraments, which had been particularly contested by the Protestants. The volume was written not for children or catechumens but for pastors. In conformity with the requests made in a number of the small working groups (" circuli minores ") at the Synod of 1985, the present Catechism was intended to be a full restatement of Catholic doctrine in the light of the Second Vatican Council. As a compendium of all doctrine, it includes the teaching of the Church on points that were not touched by Vatican II. The result is a comprehensive summa incorporating the data of Scripture, the heritage of the fathers and doctors of the Church, the pronouncements of popes, councils, and congregations of the Holy See, testimonies of the liturgy, and memorable reflections of classical theologians, saints, and spiritual writers. I know of no comparable instrument for ascertaining the relevant data from Scripture and tradition regarding the full range of Catholic faith and morals. For readers professionally concerned with theology and religious education, the footnotes alone would be worth the price of the volume. The notes contain some 4,000 biblical references, more than 1,000 references to conciliar texts, more than 250 references to papal statements (more than half of them from John Paul II), over 100 references to the Eastern and Western codes of canon law, more than 100 to liturgical texts, and nearly 500 references to church fathers and ecclesiastical writers. Like the Roman Catechism, this new synthesis of Catholic doctrine is directed not to the simple faithful but to pastors. As is stated in the Prologue, " this Catechism is addressed principally to those responsible for catechesis : in the first instance, to the bishops as doctors of the faith and pastors of the Church. It is offered to them as an instrument for the fulfillment of their task of teaching the people of God. It is addressed, through the bishops, to the editors of catechisms, priests, and catechists. It will THE HIERARCHY OF TRUTHS 371 also be useful reading for all other faithful Christians " ( 12).2 The present Catechism is composed for Catholics all over the world, but the authors are quite aware that it needs to be adapted to meet the needs and capacities of different groups. They insist in the Prologue that those who instruct the faithful, or compose other catechetical materials, should adapt their teaching to the culture, age, spiritual maturity... (shrink)

Hierarchies abound to help us organize our world. A hierarchy places items into a general order, where more ‘general’ is also more ‘abstract’. The etymology of hierarchy is grounded in notions of religious and social rank. This article, after a historical review, focuses on knowledge systems, an interloper of the term hierarchy since at least the 1800s. Hierarchies in knowledge systems include taxonomies, classification systems, or thesauri in information science, and systems for representing information and knowledge to (...) computers, notably ontologies and knowledge representation languages. Hierarchies are the logical underpinning of inference and reasoning in these systems, as well as the scaffolding for classification and inheritance. Hierarchies in knowledge systems express subsumption relations that have flexible variants, which we can represent algorithmically, and thus computationally. This article dissects that variability, leading to a proposed typology of hierarchies useful to knowledge systems. The article argues through a perspective informed by Charles Peirce that natural hierarchies are real, can be logically determined, and are the appropriate basis for knowledge systems. Description logics and semantic language standards reflect this perspective, importantly through their open-world logic and vocabularies for generalized subsumption hierarchies. Recent research suggests possible mechanisms for the emergence of natural hierarchies. (shrink)

In abstract algebraic logic, the general study of propositional non-classical logics has been traditionally based on the abstraction of the Lindenbaum-Tarski process. In this process one considers the Leibniz relation of indiscernible formulae. Such approach has resulted in a classification of logics partly based on generalizations of equivalence connectives: the Leibniz hierarchy. This paper performs an analogous abstract study of non-classical logics based on the kind of generalized implication connectives they possess. It yields a new classification of logics expanding (...) Leibniz hierarchy: the hierarchy of implicational logics. In this framework the notion of implicational semilinear logic can be naturally introduced as a property of the implication, namely a logic L is an implicational semilinear logic iff it has an implication such that L is complete w.r.t. the matrices where the implication induces a linear order, a property which is typically satisfied by well-known systems of fuzzy logic. The hierarchy of implicational logics is then restricted to the semilinear case obtaining a classification of implicational semilinear logics that encompasses almost all the known examples of fuzzy logics and suggests new directions for research in the field. (shrink)

In this paper I propose a type-hierarchy approach to provide an intersubjective framework for the evaluation of evolutionary analogies. This approach develops David Hull’s and others’ attempts to provide full generalisation for selection processes, in order to show that sociocultural development and, particularly, scientific change can be considered as an instance of Darwinian selection. I argue that the recent work by Eileen Cornell Way on type hierarchies can offer the kind of generalisation needed to solve the main problems that (...) still affect Hull’s theory and to show that the evolutionary analogy is, after all, only a particular way of grouping phenomena together. If Hull’s main objective is a unified theory of selection, which supports the idea that science selection and natural selection obey the same laws, I also argue that the type hierarchy approach to models shows that this objective is unsustainable as it stands, and is in need of further development. I will firstly introduce the general outline of the type hierarchy approach to models. Then, after a brief recapitulation of Hull’s main points and difficulties, I will try and construct a hierarchy for a general abstraction of selection processes. Finally I will introduce the main criticisms that Hull’s work has faced from philosophers and scientists, and show how they compare with my proposal. (shrink)

The use of Extended Logics to replace ordinary second order definability in Kleene’s Ramified Analytical Hierarchy is investigated. This mirrors a similar investigation of Kennedy, Magidor and Väänänen [11] where Gödel’s universe L of constructible sets is subjected to similar variance. Enhancing second order definability allows models to be defined which may or may not coincide with the original Kleene hierarchy in domain. Extending the logic with game quantifiers, and assuming strong axioms of infinity, we obtain minimal correct (...) models of analysis. A wide spectrum of models can be so generated from abstract definability notions: one may take an abstract Spector Class and extract an extended logic for it. The resultant structure is then a minimal model of the given kind of definability. (shrink)

A Leibniz class is a class of logics closed under the formation of term-equivalent logics, compatible expansions, and non-indexed products of sets of logics. We study the complete lattice of all Leibniz classes, called the Leibniz hierarchy. In particular, it is proved that the classes of truth-equational and assertional logics are meet-prime in the Leibniz hierarchy, while the classes of protoalgebraic and equivalential logics are meet-reducible. However, the last two classes are shown to be determined by Leibniz conditions (...) consisting of meet-prime logics only. (shrink)

ABSTRACT In this paper we introduce and investigate various classes of multimodal logics based on frames with relative accessibility relations. We discuss their applicability to representation and analysis of incomplete information. We provide axiom systems for these logics and we prove their completeness.

We show how Abstract Complexity Theory is related to the degrees of unsolvability and develop machinery by which computability theoretic hierarchies with a complexity theoretic flavor can be defined and investigated. This machinery is used to prove results both on hierarchies of Δ 2 0 sets and hierarchies of Δ 2 0 degrees. We prove a near-optimal lower bound on the effectivity of the Low Basis Theorem and a result showing that array computable c.e. degrees are, in some sense, the (...) simplest possible Δ 2 0 degrees. We also examine the growth rates of iterates of m K . Finally, we indicate how complexity theory can be used to analyze notions of genericity intermediate between 1 -genericity and 2 -genericity, and produce a hierarchy of such notions. (shrink)

Feminists, critical race scholars, and other social‐justice theorists sometimes object to “abstraction” in liberal normative theory. Arguing that oppression affects individual agents in powerful yet subtle ways, they contend that allegedly abstract theories often reinforce oppressive power structures. Here I critically examine and ultimately reject Onora O'Neill's “abstraction without idealization” as a solution to this problem. Because O'Neill defines abstraction as simply the “bracketing of certain predicates,” her methodology fails to guide decisions about what to bracket and what to include (...) in the theory. Moreover, it may not be possible to abstract without also relying on some particular ideals. While abstraction is unavoidable, I conclude that it must be employed with greater attention to sociopolitical hierarchies and by using ideals that do not collude with structures of oppression. I discuss the work of Susan Babbitt and Elizabeth Anderson as examples of how nonoppressive ideals might be incorporated into normative theory. (shrink)

In model theory, a branch of mathematical logic, we can classify mathematical structures based on their logical complexity. This yields the so-called stability hierarchy. Independence relations play an important role in this stability hierarchy. An independence relation tells us which subsets of a structure contain information about each other, for example, linear independence in vector spaces yields such a relation.Some important classes in the stability hierarchy are stable, simple, and NSOP $_1$, each being contained in the next. (...) For each of these classes there exists a so-called Kim-Pillay style theorem. Such a theorem describes the interaction between independence relations and the stability hierarchy. For example, simplicity is equivalent to admitting a certain independence relation, which must then be unique.All of the above classically takes place in full first-order logic. Parts of it have already been generalised to other frameworks, such as continuous logic, positive logic, and even a very general category-theoretic framework. In this thesis we continue this work.We introduce the framework of AECats, which are a specific kind of accessible category. We prove that there can be at most one stable, simple, or NSOP $_1$ -like independence relation in an AECat. We thus recover (part of) the original stability hierarchy. For this we introduce the notions of long dividing, isi-dividing, and long Kim-dividing, which are based on the classical notions of dividing and Kim-dividing but are such that they work well without compactness.Switching frameworks, we generalise Kim-dividing in NSOP $_1$ theories to positive logic. We prove that Kim-dividing over existentially closed models has all the nice properties that it is known to have in full first-order logic. We also provide a full Kim-Pillay style theorem: a positive theory is NSOP $_1$ if and only if there is a nice enough independence relation, which then must be given by Kim-dividing.Abstract prepared by Mark Kamsma.E-mail:[email protected]:https://markkamsma.nl/phd-thesis. (shrink)

Protoalgebraic logics are characterized by the monotonicity of the Leibniz operator on their theory lattices and are at the lower end of the Leibniz hierarchy of abstract algebraic logic. They have been shown to be the most primitive among those logics with a strong enough algebraic character to be amenable to algebraic study techniques. Protoalgebraic π-institutions were introduced recently as an analog of protoalgebraic sentential logics with the goal of extending the Leibniz hierarchy from the sentential framework to (...) the π-institution framework. Many properties of protoalgebraic logics, studied in the sentential logic framework by Blok and Pigozzi, Czelakowski, and Font and Jansana, among others, have already been adapted in previous work by the author to the categorical level. This work aims at further advancing that study by exploring in this new level some more properties of protoalgebraic sentential logics. (shrink)

The notion of an ℐ -matrix as a model of a given π -institution ℐ is introduced. The main difference from the approach followed so far in CategoricalAlgebraic Logic and the one adopted here is that an ℐ -matrix is considered modulo the entire class of morphisms from the underlying N -algebraic system of ℐ into its own underlying algebraic system, rather than modulo a single fixed -logical morphism. The motivation for introducing ℐ -matrices comes from a desire to formulate (...) a correspondence property for N -protoalgebraic π -institutions closer in spirit to the one for sentential logics than that considered in CAAL before. As a result, in the previously established hierarchy of syntactically protoalgebraic π -institutions, i. e., those with an implication system, and of protoalgebraic π -institutions, i. e., those with a monotone Leibniz operator, the present paper interjects the class of those π -institutions with the correspondence property, as applied to ℐ -matrices. Moreover, this work on ℐ -matrices enables us to prove many results pertaining to the local deduction-detachment theorems, paralleling classical results in Abstract Algebraic Logic formulated, first, by Czelakowski and Blok and Pigozzi. Those results will appear in a sequel to this paper. (shrink)

In lieu of an abstract, here is a brief excerpt of the content:Renaissance Thought on the Celestial Hierarchy:The Decline of a Tradition?Feisal G. MohamedThe Dionysian arrangement of the angels was dismantled on the one hand because its author was increasingly regarded as a "counterfait," and on the other hand because Protestants upheld the Bible's supremacy over all the "vain babblings of idle men." In consequence, those who like Spenser celebrated the "trinall triplicities," look back upon a great past that (...) had provided a vision hallowed by time and graced by an impressive array of intellects.—C. A. Patrides1It has been some forty years now since C. A. Patrides first argued in these pages that English Renaissance views on the celestial hierarchy constitute "the decline of a tradition," an argument made so convincing by his encyclopedic articles on the subject that scholars have been content to accept it. Indeed there is a good deal of evidence suggesting that the tradition of the ninefold hierarchy of angels associated with the Pseudo-Dionysius suffered serious injury at the hands of the Renaissance and Reformation. Subjected to the scrutiny of Renaissance scholarship and in light of the rediscovery of several Neoplatonists, particularly Proclus, it became quite clear that the Corpus Dionysiacum had been written centuries later than its author pretended. In the mid-fifteenth century Lorenzo Valla argued that because the Areopagus was a legal rather than a philosophical assembly, because none of the Greek or Latin fathers make mention of Dionysius, and because the claim in the letter to Polycarp to have witnessed [End Page 559] in Heliopolis the solar eclipse at the moment of Christ's death is a clear fabrication, placement of the Corpus in the first century should be doubted.2 These suggestions went largely unnoticed until Desiderius Erasmus, returning from England (where William Grocyn had begun murmurs regarding the Areopagitic origin of the Corpus), discovered in Louvain a manuscript of Valla's commentary on the book of Acts and incorporated its conclusions in his influential annotations on the New Testament.3 The discovery of Dionysius's pseudonymity and Erasmus's suggestion that Dionysian ceremonies could no longer be connected to the apostolic church would damage permanently the now Pseudo-Areopagite's influence. In a classic case of complicity between Renaissance humanism and Reformation theology—both Protestant and Catholic, it must be stressed—the sixteenth century turned away from Dionysian thought in general and from The Celestial Hierarchy in particular. The interrogation of apostolic origin begun by prominent Catholic thinkers like Erasmus and Cardinal Cajetan was most convenient to reformers like Martin Luther and John Calvin, both of whom described the author of the Corpus as "Dionysius, whoever he was."4Such an open-and-shut presentation of sixteenth-century Dionysiana, however, ignores several factors. One of them is that Renaissance scholars were not the first to question the dating of the Corpus: Peter Abelard fueled general perception of his heretical tendencies by suggesting that Dionysius may not have been the Pauline convert mentioned in the Acts of the Apostles.5 Much more important, however, is the number of Renaissance scholars for whom Dionysius was a formative influence. Despite his stance on the authenticity of the Corpus, Lorenzo Valla describes Dionysius in the peroration to his Encomium Sancti Thomae Aquinatis as playing the flute in a celestial symphony before the throne of God.6 For Nicholas Cusanus, as F. Edward Cranz describes [End Page 560] it, "Dionysius is the most important Christian theologian with no runner-up who is even close."7 Giovanni Pico della Mirandola saw Dionysius's writings as following directly from the teachings of Paul.8 Marsilio Ficino employed Dionysius in his comments on angels in the Theologia Platonica, translated and commented on The Divine Names and The Mystical Theology, and consistently endorsed first-century dating of the Corpus Dionysiacum.9 Ambrogio Traversari, who described Dionysius as "Apostoli disciplus," would also produce a fresh translation acceptable to humanist tastes in 1437, which would receive wide dissemination in the 1499 edition produced by Jacques Lefèvre d'Étaples.10 Lefèvre saw the Corpus as apostolic in origin and authority and went so far as to endorse the traditional conflation of Dionysius with Saint... (shrink)

This research was partially supported by Grant-in-Aid for Scientific Research (No. 06640178 and No. 06640336), Ministry of Education, Science and Culture of Japan Mathematics Subject Classification: 03E05 --> Abstract. Following Carr's study on diagonal operations and normal filters on \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} ${\cal P}_{\kappa}\lambda$\end{document} in [2], several weakenings of normality have been investigated. One of them is to consider normal filters without \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $\kappa$\end{document}-completeness, for (...) example, see DiPrisco-Uzcategui [3]. The other is weakening normality itself while keeping \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $\kappa$\end{document}-completeness such as in Mignone [10] and Shioya [11]. We take the second one so that all filters are assumed to be \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $\kappa$\end{document}-complete. In Sect. 1 a hierarchy of filters on \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} ${\cal P}_{\kappa}\lambda$\end{document} is presented which corresponds to the length of diagonal intersections under which the filters are closed. It turns out that many ranks exist between \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $FSF_{\kappa\lambda}$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $CF_{\kappa\lambda}$\end{document}. We consider seminormal ideals in Sect. 2 and determine the minimal seminormal ideal extending Johnson's result in [6]. Its precise descripti on changes according to \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $cf(\lambda )$\end{document} although we can write it in a single form as well. We also prove that a nonnormal seminormal ideal \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $I\supset NS_{\kappa\lambda}$\end{document} exists if and only if \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $\lambda$\end{document} is regular. (shrink)

Czelakowski introduced the Suszko operator as a basis for the development of a hierarchy of non-protoalgebraic logics, paralleling the well-known abstract algebraic hierarchy of protoalgebraic logics based on the Leibniz operator of Blok and Pigozzi. The scope of the theory of the Leibniz operator was recently extended to cover the case of, the so-called, protoalgebraic π-institutions. In the present work, following the lead of Czelakowski, an attempt is made at lifting parts of the theory of the Suszko operator (...) to the π-institution framework. (shrink)

This paper provides a semantic analysis of admissible rules and associated completeness conditions for arbitrary deductive systems, using the framework of abstract algebraic logic. Algebraizability is not assumed, so the meaning and significance of the principal notions vary with the level of the Leibniz hierarchy at which they are presented. As a case study of the resulting theory, the nonalgebraizable fragments of relevance logic are considered.

In lieu of an abstract, here is a brief excerpt of the content:A THEORY OF BASIC GOODS: STRUCTURE AND HIERARCHY* I. FTEN, PERHAPS ALWAYS, moral theory emerges from particular problems. Just how is obscure. The logic of discovery is elusive; and it is harder to explain how we have come to see matters rightly than to recognize that we do, in fact, see them rightly. What counts as a theory, moreover, calls for explication as much as does a theory's (...) emergence. When we have in mind a theory that shapes our social vision, there is a progression from a central set of propositions, keyed to a range of paradigm cases and seen as a logical structure, to a " way of thinking " widely enough embraced to have political significance, at least for a given oommunity. In this essay I want to explore some fundamental questions raised by Germain Grisez's theory of natural law ethics.1 These questions are chiefly conceptual. But neither they nor Grisez's theory should be seen in a vacuum. Rather, I would begin by suggesting that the importance of his work, its emergence in some Catholic circles, is linked with two specific moral and political problems. The first is abortion; the second is the superpowers' policy of nuclear deterrence. *I thank David Blake, Robert Gordh, Carroll Kearley, Gary Mar. Martin Woods, and Linda Zagzebski for their comments on an earlier version of this essay. 1 For the theory's fullest presentation, see Germain Grisez, The Way of the Lord Jesus, vol. one, Ohristian Moral Principles, with the help of Joseph M. Boyle, Jr., Basil Cole, O.P., John Finnis, John A. Geinzer, Robert G. Kennedy, Patrick Lee, William E. May, and Russell Shaw (Chicago: Franciscan Herald Press, 1983). Especially critical are chapters two through twelve. 221 JAMES G. HANINK As a point of chronology, Grisez's early study of rubortion was the context for one of the first formulations of his natural law ethics.2 But why should this have been so? And why should abortion so engage Catholic thinkers, some of whom had felt a distance from explicitly Catholic positions, that Grisez's work should become pivotal? Answers to such questions must be tentative, but they include the following considerations. In a brief period, no more than a decade, legal abortion in the West became entrenched social policy.8 It was a policy, too, supported by dominant intellectual forces. But the new policy, defended in terms of individual freedom and a right to privacy, struck many Catholicsand non-Catholics-as vicious and tragic. It was vicious in that it made expendable the weakest and most vulnerable human beings. Human rights somehow became restricted to the strong and self-sufficient. The policy was tragic in its denial of human community. We are not, however much radical individualism supposes, merely a collection of separate egos. l/Ve are, rather, a community tied to a past and pledged to a future, so pledged, in part, by the children we bear and nurture.4 For in hearing and nurturing children we carry on the love of parents who have done as much for us; and we carry ourselves into a future where we hope our own love will be thus extended. But abortion, and surely unrestricted abortion, betrays this community of trust. Still, a policy so vicious and tragic can make us look for a fresh moral vision. And, of course, to the extent that we have contributed to,a climate in which abortion could become entrenched -either by our failures to support women in need or 2 Germain Grisez, Abortion: The Myths, the Realities, and the Arguments (New York and Cleveland: Corpus Books, 1970). a A milepost, of course, was the United States Supreme Court decision, Roe v. Wade, in 1973. Recent U.S. Government figures indicate that there are now well over a million legal abortions annually in this country. 4 For a provocative discussion of this theme, see Stanley Hauerwas's " The Moral Value of the Family " in his A Commimity of Character (Notre Dame: University of Notre Dame Press, 1981). A THEORY OF BASIC GOODS: STRUCTURE & HIERARCHY 223 by an acquiescence in our culture's individualism-the call for theoretical reconstruction is all the more urgent... (shrink)

It is generally accepted that, in the cognitive and neural sciences, there are both computational and mechanistic explanations. We ask how computational explanations can integrate into the mechanistic hierarchy. The problem stems from the fact that implementation and mechanistic relations have different forms. The implementation relation, from the states of an abstract computational system to the physical, implementing states is a homomorphism mapping relation. The mechanistic relation, however, is that of part/whole; the explaining features in a mechanistic explanation are (...) the components of the explanandum phenomenon and their causal organization. Moreover, each component in one level of mechanism is constituted and explained by components of an underlying level of mechanism. Hence, it seems, computational variables and functions cannot be mechanistically explained by the medium-dependent states and properties that implement them. How then, do the computational and the implementational integrate to create the mechanistic hierarchy? After explicating the general problem, we further demonstrate it through a concrete example, of reinforcement learning, in the cognitive and neural sciences. We then examine two possible solutions. On one solution, the mechanistic hierarchy embeds at the same levels computational and implementational properties. This picture fits with the view that computational explanations are mechanistic sketches. On the other solution, there are two separate hierarchies, one computational and another implementational, which are related by the implementation relation. This picture fits with the view that computational explanations are functional and autonomous explanations. It is less clear how these solutions fit with the view that computational explanations are full-fledged mechanistic explanations. Finally, we argue that both pictures are consistent with the reinforcement learning example, but that scientific practice does not align with the view that computational models are merely mechanistic sketches. (shrink)

We examine a variety of dialogue protocols, taking inspiration from two fields: natural language dialogue modelling and multiagent systems. In communicative interaction, one can identify different features that may increase the complexity of the dialogue structure. This motivates a hierarchy of abstract models for protocols that takes as a starting point protocols based on deterministic finite automata. From there, we proceed by looking at particular examples that justify either an enrichment or a restriction of the initial model.

The so-called integration problem concerning mechanistic and computational explanation asks how they are related to each other. One approach is that a computational explanation is a species of mechanistic explanation. According to this view, computational or mathematical descriptions are mechanism sketches or macroscopic descriptions that include computationally relevant and exclude computationally irrelevant physical properties. Some suggest that this results in a so-called single hierarchy view of physical computation, where computational or mathematical properties sit together in the same mechanistic (...) class='Hi'>hierarchy with the implementational properties. This view can be contrasted with a separate hierarchy view, according to which computational and physical descriptions have their own hierarchies which are related to each other via a bridging implementation relation. The single hierarchy view has been criticized for downplaying the explanatory value of computational explanations and not being hospitable to multiple realization of cognitive processes. In this paper, I argue that the aforementioned criticisms fail, and there might be a deeper problem with the single hierarchy view, which is that the single hierarchy view might collapse into a separate hierarchy view. The kind of abstraction used by the single hierarchy view does not seem to grant mathematical or computational descriptions but only more stripped physical or implementational descriptions. (shrink)

In lieu of an abstract, here is a brief excerpt of the content:Beauty, Transcendence, and the Inclusive Hierarchy of Creation1Thomas Joseph White, O.P.Interpreters of Thomas Aquinas have long argued about whether he holds that beauty is a “transcendental,” a feature of reality coextensive with all that exists, like unity, goodness, and truthfulness.2 In the first part of this article, I will argue that Aquinas can [End Page 1215] be read to affirm in an implicit way that beauty is a (...) transcendental. In the second part, I will consider what it might mean from a Thomistic point of view to speak of a transcendent divine beauty, given Aquinas’s metaphysical commitments, particularly with respect to his doctrine of divine simplicity. In the final part, I will treat the question of how the beauty of the creation both manifests and conceals divine beauty.Beauty as a Transcendental Feature of RealityAquinas does not list beauty as a transcendental term in texts on transcendental notions. Perhaps, then, one should simply exclude it from a responsible account of his teaching on this subject. However, at least two well-known texts should give us reason to pause before reaching such a conclusion. One is found in his Commentary on Dionysius’ Divine Names, chapter 4, lectio 5. The other is in his discussion of the beauty of the eternal Son of God in a discussion of the Holy Trinity in the Summa theologiae I, q. 39, a. 8.In the first of these texts, Aquinas is commenting on Dionysius. The extended text is analytically dense. Aquinas is discussing ways in which one might say that God is beautiful and in what ways one might not say so. I will return to his topic below. Here, however, it is pertinent to consider Aquinas’s discussion of the presence of beauty in all that exists. He makes six main points.3 First, all beauty comes [End Page 1216] from God insofar as God is the cause of all that exists. Second, he gives a first definition of beauty: beauty can be defined ontologically as the splendor (claritas) that results from form; everything has a formal determination of some kind insofar as it has existence (esse); therefore, insofar as anything exists (and has some formal ontological content) it has some degree of beauty. Third, the splendor of the form in created things is a participation in the divine splendor from which it originates. The divine nature is the transcendent exemplar of beauty in diverse finite created realities. Fourth, then (and perhaps most importantly), “ex divina pulchritudine esse omnium derivatur”: literally, the existence of everything originates from divine beauty. Fifth, a second definition of beauty is considered: beauty can be defined ontologically as a property of being that emerges from proportion or harmony (consonantia). For example, authentic relationships of personal friendship imply spiritual harmony or concord and are beautiful and noble in this respect. Sixth, then, the concord or beautiful harmonies we find in the created order are expressive of the wisdom of God, who is the author of creation.Evidently, if the existence of everything derives from divine beauty, and if everything that has existence is in some way beautiful by virtue of its intrinsic form, then it would seem to follow logically that beauty, for Aquinas, is a characteristic of being that is coextensive with all that exists. We see a similar idea expressed in the aforementioned passage of ST. Here, however, Aquinas gives a more synthetic definition of beauty in things that combines both the definitions found in our previous discussion, claritas and proportio, but it also adds a third, integritas: ontological integrity or wholeness.Species or beauty has a likeness to the property of the Son. For beauty includes three conditions, “integrity” or “perfection,” [End Page 1217] since those things which are impaired are by the very fact ugly; due “proportion” or “harmony”; and lastly, “brightness” or “clarity,” whence things are called beautiful which have a bright color.4The implication of this point of view is readily apparent. God is essentially beautiful, and God has created all that exists in light of the eternal Word and Wisdom of God, who is the Son. Consequently, all that exists and that... (shrink)

Theoria, EarlyView. -/- In his recent book Thin Objects, Øystein Linnebo (2018) argues for the existence of a hierarchy of abstract objects, sufficient to model ZFC, via a novel and highly interesting argument that relies on a process called dynamic abstraction. This paper presents a way for a nominalist, someone opposed to the existence of abstract objects, to avoid Linnebo's conclusion by rejecting his claim that certain abstraction principles are sufficient for reference (RBA). Section 1 of the paper explains (...) Linnebo's argument for RBA. It offers a reading of Linnebo's work upon which he has two arguments for RBA: one deductive and one abductive, and argues that whilst the deductive argument is unsound the abductive one is prima facie plausible. The nominalist must therefore find a way to respond to the abductive argument. Section 2 outlines just such a response, by offering an alternative explanation of the cases Linnebo wishes to argue from. Most interestingly, it shows that abstraction in Linnebo's most difficult case (the “reference to ordinary bodies” case) can be achieved using mereological means, rather than relying on RBA. (shrink)

Given a π -institution I , a hierarchy of π -institutions I is constructed, for n ≥ 1. We call I the n-th order counterpart of I . The second-order counterpart of a deductive π -institution is a Gentzen π -institution, i.e. a π -institution associated with a structural Gentzen system in a canonical way. So, by analogy, the second order counterpart I of I is also called the “Gentzenization” of I . In the main result of the paper, (...) it is shown that I is strongly Gentzen , i.e. it is deductively equivalent to its Gentzenization via a special deductive equivalence, if and only if it has the deduction-detachment property. (shrink)

It is argued that explanations of shock waves display explanatory emergence in two different ways. Firstly, the use of discontinuities to model jumps in flow variables is an example of “physics avoidance”. This is where microphysical details can be ignored in an abstract model thus allowing us access to modal information which cannot be attained in principle in any other way. Secondly, Whitham’s interleaving criterion for continuous shock structure is an example of the way different characteristic scales interact in shock (...) dynamics. To fully explain the shock structure one must take account of these different scales, and by doing so explanations of shock structure have irreducible aspects. Lastly, the implications of this explanatory irreducibility are examined in the context of explanatory indispensability arguments used in realism debates elsewhere in the philosophy of science. It is concluded that explanatory emergence on its own only supports an epistemic form of emergence. Yet this epistemic emergence is fully objective. (shrink)

Some scholars claim that epistemology of science and machine learning are actually overlapping disciplines studying induction, respectively affected by Hume's problem of induction and its formal machine-learning counterpart, the “no-free-lunch” (NFL) theorems, to which even advanced AI systems such as LLMs are not immune. Extending Kevin Korb's view, this paper envisions a hierarchy of disciplines where the lowermost is a basic science, and, recursively, the metascience at each level inductively learns which methods work best at the immediately lower level. (...) Due to Hume's dictum and NFL theorems, no exact metanorms for the good performance of each object science can be obtained after just a finite number of levels up the hierarchy, and the progressive abstractness of each metadiscipline and consequent ill-definability of its methods and objects makes science—as defined by a minimal standard of scientificity—cease to exist above a certain metalevel, allowing for a still rational style of inquiry into science that can be called “philosophical.” Philosophical levels, transitively reflecting on science, peculiarly manifest a non–empirically learned urge to self-reflection constituting the properly normative aspect of philosophy of science. (shrink)

This paper explores issues of abstraction and space in Sande Zeig's movie The Girl (2001), based on a novella by Monique Wittig, who also co-wrote the script. It argues that, with this movie, Zeig and Wittig strive to re-materialize the lesbian body abstracted by the ‘Straight Mind’ as defined by Wittig in her 1980 essay. The plot revolves around the love affair of two women, the narrator and the Girl (a lesbian painter and a straight B-grade jazz singer), under the (...) oppressive scrutiny of the Man (the Girl's boss). Each part of this paper corresponds to one space where the characters interact: first, it shows how the nightclub stands as a space of transgression for the lesbian subject and how it opens a reflection on temporality as represented by space; then the analysis of the hotel room where the Girl lives and meets the narrator demonstrates how, notably through the use of close-ups in erotic scenes, Zeig redefines abstraction as a new way to envisage the materiality of the lesbian body; finally, in the third part, the discussion on abstraction moves on to the narrator's studio where, through her work as a painter, she also explores abstract representations of the female body. This approach reconciles Wittigian theory with post-modern queer theory, and confirms that the former still offers paths to positive reformulations of modes of relationality through the de-spatialization of gender – de-genderization; the rematerialization of the body outside of heteronormative conventions; and a reconfiguration of human relationships that refuses the inevitability of social and amorous hierarchy, and that shifts the representation of the erotic from possessiveness to disinterested reciprocity. (shrink)

This article argues that the English-language nonsense words “foo,” “bar,” “baz,” and others in a more or less standardized sequence of so-called metasyntactic variables commonly used in computer programming ought to be understood as meta-abstractive, re-representing a linguistically derived code’s abstraction of language and the abstraction of the programming language hierarchy itself, making it legible in a manner that rewards culturally oriented study: for example, of programming as a culture and of cultures of software development or engineering.

This paper presents a unified framework that explains and extends the already successful applications of the Leibniz operator, the Suszko operator, and the Tarski operator in recent developments in abstract algebraic logic. To this end, we refine Czelakowski’s notion of an S-compatibility operator, and introduce the notion of coherent family of S-compatibility operators, for a sentential logic S. The notion of coherence is a restricted property of commutativity with inverse images by surjective homomorphisms, which is satisfied by both the Leibniz (...) and the Suszko operators. We generalize several constructions and results already existing for the mentioned operators; in particular, the well-known classes of algebras associated with a logic through each of them, and the notions of full generalized model of a logic and a special kind of S-filters. We obtain a General Correspondence Theorem, extending the well-known one from the theory of protoalgebraic logics to arbitrary logics and to more general operators, and strengthening its formulation. We apply the general results to the Leibniz and the Suszko operators, and obtain several characterizations of the main classes of logics in the Leibniz hierarchy by the form of their full generalized models, by old and new properties of the Leibniz operator, and by the behaviour of the Suszko operator. Some of these characterizations complete or extend known ones, for some classes in the hierarchy, thus offering an integrated approach to the Leibniz hierarchy that uncovers some new, nice symmetries. (shrink)

In this paper we study an alternative approach to the concept of abstract logic and to connectives in abstract logics. The notion of abstract logic was introduced by Brown and Suszko —nevertheless, similar concepts have been investigated by various authors. Considering abstract logics as intersection structures we extend several notions to their κ -versions, introduce a hierarchy of κ -prime theories, which is important for our treatment of infinite connectives, and study different concepts of κ -compactness. We are particularly (...) interested in non-topped intersection structures viewed as semi-lattices with a minimal meet-dense subset, i.e., with a minimal generator set. We study a chain condition which is sufficient for a minimal generator set, implies compactness of the logic, and in regular logics is equivalent to compactness of the consequence relation together with the existence of a inconsistent set, where κ is the cofinality of the cardinality of the logic. Some of these results are known in a similar form in the context of closure spaces, we give extensions to intersection structures and to big cardinals presenting new proofs based on set-theoretical tools. The existence of a minimal generator set is crucial for our way to define connectives. Although our method can be extended to further non-classical connectives we concentrate here on intuitionistic and infinite ones. Our approach leads us to the concept of the set of complete theories which is stable under all considered connectives and gives rise to the definition of the topological space of the logic. Topological representations of abstract logics by means of this space remain to be further investigated. (shrink)

dialectical frameworks constitute a recent and powerful generalisation of Dung's argumentation frameworks, where the relationship between the arguments can be specified via Boolean formulas. Recent results have shown that this enhancement comes with the price of higher complexity compared to AFs. In fact, acceptance problems in the world of ADFs can be hard even for the third level of the polynomial hierarchy. In order to implement reasoning problems on ADFs, systems for quantified Boolean formulas thus are suitable engines to (...) be employed. In this paper we give complexity sensitive QBF encodings of ADF problems generalising recent work on QBFs for AF labellings. Our encodings provide a uniform and modular way of translating reasoning in ADFs to QBFs, that can be used as the basis for novel systems for ADF reasoning. (shrink)

We explore the formal foundations of recent studies comparing aural pattern recognition capabilities of populations of human and non-human animals. To date, these experiments have focused on the boundary between the Regular and Context-Free stringsets. We argue that experiments directed at distinguishing capabilities with respect to the Subregular Hierarchy, which subdivides the class of Regular stringsets, are likely to provide better evidence about the distinctions between the cognitive mechanisms of humans and those of other species. Moreover, the classes of (...) the Subregular Hierarchy have the advantage of fully abstract descriptive (model-theoretic) characterizations in addition to characterizations in more familiar grammar- and automata-theoretic terms. Because the descriptive characterizations make no assumptions about implementation, they provide a sound basis for drawing conclusions about potential cognitive mechanisms from the experimental results. We review the Subregular Hierarchy and provide a concrete set of principles for the design and interpretation of these experiments. (shrink)

This paper addresses the methodological problem of analysing what it is to explain observed behaviours of engineered computing systems, focusing on the crucial role that abstraction and idealization play in explanations of both correct and incorrect BECS. First, it is argued that an understanding of explanatory requests about observed miscomputations crucially involves reference to the rich background afforded by hierarchies of functional specifications. Second, many explanations concerning incorrect BECS are found to abstract away from descriptions of physical components and processes (...) of computing systems that one finds below the logic circuit and gate layer of functional specification hierarchies. Third, model-based explanations of both correct and incorrect BECS that are provided in the framework of formal verification methods often involve idealizations. Moreover, a distinction between restrictive and permissive idealizations is introduced and their roles in BECS explanations are analysed. (shrink)

We show how to encode context-free string grammars, linear context-free tree grammars, and linear context-free rewriting systems as Abstract Categorial Grammars. These three encodings share the same constructs, the only difference being the interpretation of the composition of the production rules. It is interpreted as a first-order operation in the case of context-free string grammars, as a second-order operation in the case of linear context-free tree grammars, and as a third-order operation in the case of linear context-free rewriting systems. This (...) suggest the possibility of defining an Abstract Categorial Hierarchy. (shrink)

In lieu of an abstract, here is a brief excerpt of the content:The Enchantment of Art: Abstraction and Empathy from German Romanticism to ExpressionismDavid MorganA familiar tradition since the eighteenth century has invested art with the power to heal a decadent human condition. Inheriting this ability from religion—the romantic enthusiast Wilhelm Wackenroder considered artistic inspiration to originate in “divine inspiration” in the case of his hero, Raphael 1 —art eventually replaced institutionalized belief in an evolutionary schedule of cultural development determined (...) by German idealism. Hegel, for instance, theorized the evolution of spirit from religion to art as a qualitative advancement over time. Rejecting the metaphysics of German idealism, Friedrich Nietzsche nevertheless contended that “Art raises its head where religions decline.” By assuming the “feelings and moods” of religion which had been dismantled by enlightenment, Nietzsche claimed that art gave new form to the life of feeling and imbued human endeavors with a “loftier, darker coloration.” 2 In the early twentieth century Wassily Kandinsky believed that the “spiritual in art” was the historical manifestation of an “inner necessity” that was in itself timeless and universal. Kandinsky portrayed the “life of the spirit” as “a large acute-angled triangle divided horizontally into unequal parts.” In a kind of “trickle down” aesthetic, the compartments of the triangle move slowly upward toward the apex, which is inhabited by the solitary genius of an age whose vision gradually elevates all those who occupy the lower stages. 3 [End Page 317]The point for proponents of the spiritual in art from romanticism to expressionism (and beyond) was to find in works of art what can best be described as an enchantment. 4 As the negotiation of the gaps or disjunctures separating human beings from complete power over their lives, enchantment conjoins the plane of immediate human experience to a desirable state of affairs through the mediation of such symbolic devices as rituals, incantations, or charms. Thus, evil can be warded off by the use of an apotropaic amulet, and fertility can be enhanced by the recitation of a suitable prayer or the performance of a certain ritual. In a modern analogy a sense of impotence is often alleviated by a trip to the mall where the bored therapeutically lose themselves in symbolic acts of self-renewing consumption. In the traditions of philosophy and science, systems of representation such as astrology, the microcosm, the harmony of the spheres, the great chain of being, the celestial hierarchies, or the mechanisms of theurgy and divination situate humanity within a cosmos that follows a particular order accessible to and in some degree manipulable by human beings. Enchantment consists of an apparatus of belief whose form of representation or enactment is thought to exert influence over the realities it configures.But enchantment is not limited to the spells and magic of premodern metaphysics (or modern consumerism). By means of religion, morality, and aesthetics, Nietzsche once averred, “man likes to believe that... he is touching the heart of the world.” 5 This study will focus on the manner in which art as constructed within the idealist tradition of German art theory functioned as an enchantment. Specifically, it will explore what the history of ideas about abstraction and empathy has to tell us about the spiritualization or enchantment of art. After identifying the philosophical and aesthetic wherewithal of enchantment in nineteenth-century Germany, attention focusses on Franz Marc’s early twentieth-century theorization of abstraction. It is certainly more customary to study Kandinsky, whose writings and early efforts at abstraction are much better known, but Marc’s ideas offer more immediate connections to German intellectual history and clarify the aims of his abstract art as a strategy of enchantment. Moreover, Marc was unusually articulate and wrote often, exploring ideas and artistic efforts with great insight. Focusing on Marc therefore offers a corrective to the historiography that has favored Kandinsky’s writings. Finally, Marc differed from his Russian friend over the cause of the war. Manifesting a difference that is portentous for this study, Kandinsky left Germany for Russia in 1914, while Marc enthusiastically took up arms for the fatherland. Given these circumstances and the unfortunate tendency of art historians to valorize expressionist art rather than situate it in the ideological context of... (shrink)

This paper presents a new algorithm to find an appropriate similarityunder which we apply legal rules analogically. Since there may exist a lotof similarities between the premises of rule and a case in inquiry, we haveto select an appropriate similarity that is relevant to both thelegal rule and a top goal of our legal reasoning. For this purpose, a newcriterion to distinguish the appropriate similarities from the others isproposed and tested. The criterion is based on Goal-DependentAbstraction (GDA) to select a (...) similarity such that an abstraction basedon the similarity never loses the necessary information to prove the ground (purpose of legislation) of the legal rule. In order to cope withour huge space of similarities, our GDA algorithm uses some constraintsto prune useless similarities. (shrink)

This paper presents a methodology to design and implement programs intended to decide cases, described as sets of factors, according to a theory of a particular domain based on a set of precedent cases relating to that domain. We useDialectical Frameworks, a recent development in AI knowledge representation, as the central feature of our design method. ADFs will play a role akin to that played by Entity–Relationship models in the design of database systems. First, we explain how the factor (...) class='Hi'>hierarchy of the well-known legal reasoning system CATO can be used to instantiate an ADF for the domain of US Trade Secrets. This is intended to demonstrate the suitability of ADFs for expressing the design of legal cased based systems. The method is then applied to two other legal domains often used in the literature of AI and Law. In each domain, the design is provided by the domain analyst expressing the cases in terms of factors organised into an ADF from which an executable program can be implemented in a straightforward way by taking advantage of the closeness of the acceptance conditions of the ADF to components of an executable program. We evaluate the ease of implementation, the performance and efficacy of the resulting program, ease of refinement of the program and the transparency of the reasoning. This evaluation suggests ways in which factor based systems, which are limited by taking as their starting point the representation of cases as sets of factors and so abstracting away the particular facts, can be extended to address open issues in AI and Law by incorporating the case facts to improve the decision, and by considering justification and reasoning using portion of precedents. (shrink)

This paper continues the investigation, started in Lávička and Noguera : 521–551, 2017), of infinitary propositional logics from the perspective of their algebraic completeness and filter extension properties in abstract algebraic logic. If follows from the Lindenbaum Lemma used in standard proofs of algebraic completeness that, in every finitary logic, intersection-prime theories form a basis of the closure system of all theories. In this article we consider the open problem of whether these properties can be transferred to lattices of filters (...) over arbitrary algebras of the logic. We show that in general the answer is negative, obtaining a richer hierarchy of pairwise different classes of infinitary logics that we separate with natural examples. As by-products we obtain a characterization of subdirect representation for arbitrary logics, develop a fruitful new notion of natural expansion, and contribute to the understanding of semilinear logics. (shrink)