After spinal cord injury (SCI), motoneuron death occurs at and around the level of injury which induces changes in function and organization throughout the nervous system, including cortical changes. Muscle affected by SCI may consist of both innervated (accessible to voluntary drive) and denervated (inaccessible to voluntary drive) muscle fibers. Voluntary activation measured with transcranial magnetic stimulation (VATMS) can quantify voluntary cortical/subcortical drive to muscle but is limited by technical challenges including suboptimal stimulation of target muscle relative to its antagonist. (...) The motor evoked potential (MEP) in the biceps compared to the triceps (i.e., MEP ratio) may be a key parameter in the measurement of biceps VATMS after SCI. We used paired pulse TMS, which can inhibit or facilitate MEPs, to determine whether the MEP ratio affects VATMS in individuals with tetraplegia. Ten individuals with tetraplegia following cervical SCI and ten non-impaired individuals completed single pulse and paired pulse VATMS protocols. Paired pulse stimulation was delivered at 1.5, 10, and 30 ms inter-stimulus intervals (ISI). In both the SCI and non-impaired groups, the main effect of the stimulation pulse (paired pulse compared to single pulse) on VATMS was not significant in the linear mixed-effects models. In both groups for the stimulation parameters we tested, the MEP ratio was not modulated across all effort levels and did not affect VATMS. Linearity of the voluntary moment and superimposed twitch moment relation was lower in SCI participants compared to non-impaired. Poor linearity in the SCI group limits interpretation of VATMS. Future work is needed to address methodological issues that limit clinical application of VATMS. (shrink)

A new technique is presented for proving that a consequence system enjoys Craig interpolation or Maehara interpolation based on the fact that these properties hold in another consequence system. This technique is based on the existence of a back and forth translation satisfying some properties between the consequence systems. Some examples of translations satisfying those properties are described. Namely a translation between the global/local consequence systems induced by fragments of linear logic, a Kolmogorov-Gentzen-Gödel style translation, and a new (...) translation between the global consequence systems induced by full Lambek calculus and linear logic, mixing features of a Kiriyama-Ono style translation with features of a Kolmogorov-Gentzen-Gödel style translation. These translations establish a strong relationship between the logics involved and are used to obtain new results about whether Craig interpolation and Maehara interpolation hold in that logics . (shrink)

We prove that certain natural sequent systems for bi-intuitionistic logic have the analytic cut property. In the process we show that the (global) subformula property implies the (local) analytic cut property, thereby demonstrating their equivalence. Applying a version of Maehara technique modified in several ways, we prove that bi-intuitionistic logic enjoys the classical Craig interpolation property and Maximova variable separation property; its Halldén completeness follows.

In this paper we study the interpolation property in fragments of intuitionistic and propositional logic, using both proof theoretic and semantic techniques. We will also sketch some computational methods, based on the semantical techniques introduced, to obtain counterexamples in fragment where interpolation does not hold.

Acquiring expertise in a task is often thought of as an automatic process that follows inevitably with practice according to the log‐log law (aka: power law) of learning. However, as Ericsson, Chase, and Faloon (1980) showed, this is not true for digit‐span experts and, as we show, it is certainly not true for Tetris players at any level of expertise. Although some people may simply “twitch” faster than others, the limit to Tetris expertise is not raw keypress time but (...) the techniques acquired by players that allow them to use the tools provided by the hardware and software to compensate for the game's relentlessly increasing drop speed. Unfortunately, these increases in drop speed between Tetris levels make performance plateaus very short and quickly followed by game death. Hence, a player's success at discovering, exploring, and practicing new techniques for the tasks of board preparation, board maintenance, optimal placement discovery, zoid rotation, lateral movement of zoids, and other tasks important to expertise in Tetris is limited. In this paper, we analyze data collected from 492 Tetris players to reveal the challenges they confronted while constructing expertise via the discovery of new techniques for gameplay at increasingly difficult levels of Tetris. (shrink)

Acquiring expertise in a task is often thought of as an automatic process that follows inevitably with practice according to the log‐log law (aka: power law) of learning. However, as Ericsson, Chase, and Faloon (1980) showed, this is not true for digit‐span experts and, as we show, it is certainly not true for Tetris players at any level of expertise. Although some people may simply “twitch” faster than others, the limit to Tetris expertise is not raw keypress time but (...) the techniques acquired by players that allow them to use the tools provided by the hardware and software to compensate for the game's relentlessly increasing drop speed. Unfortunately, these increases in drop speed between Tetris levels make performance plateaus very short and quickly followed by game death. Hence, a player's success at discovering, exploring, and practicing new techniques for the tasks of board preparation, board maintenance, optimal placement discovery, zoid rotation, lateral movement of zoids, and other tasks important to expertise in Tetris is limited. In this paper, we analyze data collected from 492 Tetris players to reveal the challenges they confronted while constructing expertise via the discovery of new techniques for gameplay at increasingly difficult levels of Tetris. (shrink)

Intuitionistic logic is presented here as part of familiar classical logic which allows mechanical extraction of programs from proofs. to make the material more accessible, basic techniques are presented first for propositional logic; Part II contains extensions to predicate logic. This material provides an introduction and a safe background for reading research literature in logic and computer science as well as advanced monographs. Readers are assumed to be familiar with basic notions of first order logic. One device for making this (...) book short was inventing new proofs of several theorems. The presentation is based on natural deduction. The topics include programming interpretation of intuitionistic logic by simply typed lambda-calculus (Curry-Howard isomorphism), negative translation of classical into intuitionistic logic, normalization of natural deductions, applications to category theory, Kripke models, algebraic and topological semantics, proof-search methods, interpolation theorem. The text developed from materal for several courses taught at Stanford University in 1992-1999. (shrink)

Alasdair Urquhart’s work on models for relevant logics is distinctive in a number of different ways. One key theme, present in both his undecidability proof for the relevant logic R and his proof of the failure of interpolation in R, is the use of techniques from geometry. In this paper, inspired by Urquhart’s work, I explore ways to generate natural models of R+\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$^+$$\end{document} from geometries, and different constraints that an accessibility relation (...) in such a model might satisfy. I end by showing that a set of natural conditions on an accessibility relation, motivated by geometric considerations, is jointly unsatisfiable. (shrink)

Some time in the late 1590s, the Welsh amateur mathematician John Bulkeley wrote to Thomas Harriot asking his opinion about the properties of a truly gargantuan (but totally imaginary) plano-spherical convex lens, 48 feet in diameter. While Bulkeley’s original letter is lost, Harriot devoted several pages to the optical properties of “Mr Bulkeley his Glasse” in his optical papers (now in British Library MS Add. 6789), paying particular attention to the place of its burning point. Harriot’s calculational methods in these (...) papers are almost unique in Harriot’s optical remains, in that he uses both the sine law of refraction and interpolation from Witelo’s refraction tables in order to analyze the passage of light through the glass. For this and other reasons, it is very likely that Harriot wrote his papers on Bulkeley’s glass very shortly after his discovery of the law and while still working closely with Witelo’s great Optics; the papers represent, perhaps, his very first application of the law. His and Bulkeley’s interest in this giant glass conform to a long English tradition of curiosity about the optical and burning properties of large glasses, which grew more intense in late sixteenth-century England. In particular, Thomas Digges’s bold and widely known assertions about his father’s glasses that could see things several miles distant and could burn objects a half-mile or further away may have attracted Harriot and Bulkeley’s skeptical attention; for Harriot’s analysis of the burning distance and the intensity of Bulkeley’s fantastic lens, it shows that Digges’s claims could never have been true about any real lens (and this, I propose, was what Bulkeley had asked about in his original letter to Harriot). There was also a deeper, mathematical relevance to the problem that may have caught Harriot’s attention. His most recent source on refraction—Giambattista della Porta’s De refractione of 1593—identified a mathematical flaw in Witelo’s cursory suggestion about the optics of a lens (the only place that lenses appear, however fleetingly, in the writings of the thirteenth-century Perspectivist authors). In his early notes on optics, in a copy of Witelo’s optics, Harriot highlighted Witelo’s remarks on the lens and della Porta’s criticism (which he found unsatisfactory). The most significant problem with Witelo’s theorem would disappear as the radius of curvature of the lens approached infinity. Bulkeley’s gigantic glass, then, may have provided Harriot an opportunity to test out Witelo’s claims about a plano-spherical glass, at a time when he was still intensely concerned with the problems and methods of the Perspectivist school. (shrink)

In this paper we consider the theory of predicate logics in which the principle of Bivalence or the principle of Non-Contradiction or both fail. Such logics are partial or paraconsistent or both. We consider sequent calculi for these logics and prove Model Existence. For L4, the most general logic under consideration, we also prove a version of the Craig-Lyndon Interpolation Theorem. The paper shows that many techniques used for classical predicate logic generalise to partial and paraconsistent logics once the right (...) set-up is chosen. Our logic L4 has a semantics that also underlies Belnap’s [4] and is related to the logic of bilattices. L4 is in focus most of the time, but it is also shown how results obtained for L4 can be transferred to several variants. (shrink)

This article is about ‘genogeographic’ maps produced by late-Soviet geneticists and published during post-Soviet time. It focuses on the visual and numerical techniques scientists used to project genetic data onto geographic space. Rather than discussing their representational character, I follow these visuals as ‘folded objects’, describing the layering and realigning of measurements and temporalities as well as the shifts in the practices and meanings of genetics. In the 1970s Soviet biological anthropologists transformed scattered data points by means of spatial statistics (...) into visually coherent maps for a ‘genogeographical atlas’, by interpolating data for the entire USSR territory. Computer-aided modelling rendered ‘populations’ as systemic entities and enacted specific cybernetic versions of population, evolution and difference. Tracing the history of their making helps one in understanding what these folded objects hold in store, in terms of data ranging from Russian imperial and colonial anthropology, through early Soviet traditions, to cold war technologies. Folded into those maps in intricate ways, they have co-shaped post-Soviet human genetics as an ever-active site for possible reinscriptions of difference. (shrink)

This work examines how the conceptualization of knowledge as both problem and solution reconfigured intelligence and law enforcement after 9/11. The idea was that more information should be collected, and better analyzed. If the intelligence that resulted was shared, then terrorists could be identified, their acts predicted, and ultimately prevented. Law enforcement entered into this scenario in the United States, and internationally. "Policing terrorism" refers to the engagement of state and local law enforcement in intelligence, as well as approaching terrorism (...) as a legal crime, in addition to or as opposed to an act of war. Two venues are explored: fusion centers in the United States and the international organization of police, Interpol. The configuration can be thought of schematically as operating through the set of law, discipline and security. Intelligence is predominantly a security approach. It modulates that within its purview, wielding the techniques and technologies that are here discussed.The dissertation is divided into two sections: Intelligence and Policing Terrorism. In the first, intelligence is taken up as a term, and its changes in referent and concept are examined. The Preface and Chapter One present a general introduction to the contemporary situation and intelligence, via Sherman Kent, as knowledge, organization and activities. Chapter Two traces the development of intelligence in the United States as a craft and profession. Chapter Three discusses some of the issues involving the intersection of intelligence and policy, and how those manifested in the aftermath of 9/11 and the lead up to the 2003 invasion of Iraq. The second section examines the turn to policing terrorism, beginning, in Chapter Four, with how Interpol has dealt with bioterrorism, and an examination of the shifting conceptualization of biological threats in international law. Moving from threats to their consequences, Chapter Five takes up the concept of an event in order to analyze the common comparison of Pearl Harbor and 9/11. Chapters Six and Seven turn to fieldwork done in the United States, with an examination of the suspicious activity reporting system and law enforcement's inclusion in the Information Sharing Environment, focusing on fusion centers and data mining. (shrink)

Given that Lucretius offers a systematic and cohesive explanation of the workings of nature, we should not expect inconsistencies in his poem. The explanation presented by Lucretius emphatically rejects any interventionist divine machinery of the cosmos, offering in its place the eminently regular dynamics of atomic configuration and dissolution, which can explain everything that pertains to natural philosophy without necessitating the activity of any divinity. The reader who understands the basics of Lucretius’ philosophy, therefore, should be surprised that theDRNbegins with (...) an anthropomorphic description of the goddess Venus, whom the poet petitions to intervene actively in human affairs. Lucretius gradually refines this initial presentation of the goddess’ essence throughout the poem, suggesting that the Venus who inaugurates the poem must be nuanced once one has learned the basic principles of Epicurean philosophy. At the end ofDRN4, Venus is presented not as a divine being but as nothing more than the sexual drive shared by all creatures in nature (haec Venus est nobis, 4.1058). Historically scholars have been keen to ‘correct’ perceived inconsistencies such as this, either by arguing that such imperfections derive from later interpolators or by appealing to the incomplete nature of the text as we possess it. Of course, no one has suggested that Lucretius would have changed or removed the proem, so the Venus problem remains an issue. One major aim of this article is to show that the process we witness with Lucretius’ Venus is not inconsistent but programmatic, a point grasped by many with regard to Venus herself but not in respect to theDRNas a whole. In other words, this article extends this insight about Lucretius’ Venus, whose providential attributes are only provisionally attached to her at the outset of the poem and, in due course, are completely removed. In fact, I will argue that the technique of gradually redefining initial propositions—a process I will refer to as provisional argumentation—fundamentally informs Lucretius’ famous metaphor of the honeyed cup, and I will suggest that we ought to recognize this technique as a major aspect of Lucretius’ discursive method throughout theDRN. (shrink)

We review, and in the process unify two techniques , for proving results concerning amalgamation in several classes studied in algebraic logic. The logical counterpart of these results adress interpolation and definability properties in modal and algebraic logic. Presenting them in a functorial context as adjoint situations, we show that both techniques can indeed be seen as instances of the use of the Keisler-Shelah ultrapower Theorem in proving Robinson's Joint Consistency Theorem. Some new results are surveyed. The results of this (...) paper are presented in such a way that further establishes the hitherto existing well developed duality theory between boolean algebras with operators and modal logic. (shrink)

Introduced by Leon Henkin back in the fifties, the notion of neat reducts is an old venerable notion in algebraic logic. But it is often the case that an unexpected viewpoint yields new insights. Indeed, the repercussions of the fact that the class of neat reducts is not closed under forming subalgebras turn out to be enormous. In this paper we review and, in the process, discuss, some of these repercussions in connection with the algebraic notion of amalgamation. Some new (...) unpublished results concerning neat reducts and amalgamation are given. . Several counterexamples which convey the gist of techniques used in this area are presented two of which are new It is known that the algebraic notion of amalgamation in a class of algebras corresponds to the metalogical notion of interpolation in the corresponding logic. Answers to open question in the recent paper [31] concerning both amalgamation and interpolation are summarized in tabular form at the end of this paper. This paper appears in two parts. The first part contains results on neat reducts. The present second part contains results relating the notion of neat embeddings to various amalgamation properties.1. (shrink)

In [14], we studied the computational behaviour of various first-order and modal languages interpreted in metric or weaker distance spaces. [13] gave an axiomatisation of an expressive and decidable metric logic. The main result of this paper is in showing that the technique of representing metric spaces by means of Kripke frames can be extended to cover the modal (hybrid) language that is expressively complete over metric spaces for the (undecidable) two-variable fragment of first-order logic with binary pred-icates interpreting (...) the metric. The frame conditions needed correspond rather directly with a Boolean modal logic that is, again, of the same expressivity as the two-variable fragment. We use this representation to derive an axiomatisation of the modal hybrid variant of the two-variable fragment, discuss the compactness property in distance logics, and derive some results on (the failure of) interpolation in distance logics of various expressive power. (shrink)

If well-designed, the results of a Randomised Clinical Trial can justify a causal claim between treatment and effect in the study population; however, additional information might be needed to carry over this result to another population. RCTs have been criticized exactly on grounds of failing to provide this sort of information Evidence, inference and enquiry. Oxford University Press, New York, 2011), as well as to black-box important details regarding the mechanisms underpinning the causal law instantiated by the RCT result. On (...) the other side, so-called In Silico Clinical Trials face the same criticisms addressed against standard modelling and simulation techniques, and cannot be equated to experiments Philosophy of molecular medicine: foundational issues in research and practice, Routledge, New York, 2017; Parker in Synthese 169:483–496, 2009; Parke in Philos Sci 81:516–536, 2014; Diez Roux in Am J Epidemiol 181:100–102, 2015 and related discussions in Frigg and Reiss in Synthese 169:593–613, 2009; Winsberg in Synthese 169:575–592, 2009; Beisbart and Norton in Int Stud Philos Sci 26:403–422, 2012). We undertake a formal analysis of both methods in order to identify their distinct contribution to causal inference in the clinical setting. Britton et al.’s study :E2098–E2105, 2013) on the impact of ion current variability on cardiac electrophysiology is used for illustrative purposes. We deduce that, by predicting variability through interpolation, ISCTs aid with problems regarding extrapolation of RCTs results, and therefore in assessing their external validity. Furthermore, ISCTs can be said to encode “thick” causal knowledge —as opposed to “thin” difference-making information inferred from RCTs. Hence, ISCTs and RCTs cannot replace one another but rather, they are complementary in that the former provide information about the determinants of variability of causal effects, while the latter can, under certain conditions, establish causality in the first place. (shrink)

Imitations of the works of Aśvaghoṣa, especially the Buddhacarita, are widely attested, both in the form of extra verses interpolated into the texts themselves and of entire texts in Sanskrit and Tocharian which are restructured versions of Aśvaghoṣa’s work. Such imitations and restructurings are here evaluated from the point of view of Sanskrit literary theorists, who describe similar techniques of refashioning pre-existing poems, showing that such works should not be considered as plagiarism but rather as tributes to the original (...) author. (shrink)

Theiones is a work in medieval logic from the second half of the 13th century. Clearly a product of the British university culture and much cited, quoted and imitated, it is attributed in two manuscripts to 'Master Richard the Sophist'. This Richard is referred to by other philosophers and logicians as 'The Master of Abstractions' - an honorific title which indicates that his work was a standard textbook. The Abstractiones is a collection of sophismata, or logical puzzles of increasing complexity (...) and difficulty which have been gathered under logical operators like 'all'. Each sophisma is introduced by a proposition that appears to be both provably true and provably false, like 'God knows whatever he knew'. The Master determines the truth or falsity of the proposition and analyses the defects of the arguments that have been offered by detecting logical fallacies, equivocal expressions and the like. The work as we have it is clearly the result of a process of development, modification, and interpolation, probably extending over at least a generation. Although there came to be works that imitated the Abstractions and followed some of its plan and style, these are 'descendants,' rather than variations.The Abstractions gives us a better sense than does an independent and original work of medieval logic like William of Ockham's Summa Totius Logicae of how instruction in techniques of argumentation and reasoning, often of a fairly sophisticated sort, was carried on in British universities in the latter part of the 13th century and well into the 14th century. (shrink)

In lieu of an abstract, here is a brief excerpt of the content:88 Reviews EPISTEMOLOGY OFMODERNISM WILLIAM R. EVERDELL History/ St. Ann'sSchool Brooklyn, NY 11201, USA [email protected] Ann Banfield. The Phantom Table:Woolf,Fry,Russelland the Epistemology of Modernism. Cambridge and New York: Cambridge U.P., 2000. £35.00; US$49.95. In Virginia Woolf's difficult masterpiece, The Waves(1931),each of several separate interior monologues-"streams of consciousness" in the American critical idiom-is separated from the next by an interpolated "Interlude". The interior monologues are assigned co different characters, (...) bur the Interludes are rendered from no perspectiveat all, verymuch like the "Time Passes"chapter of Woolf's earlier and better known novel, Tothe Lighthouse (1927).The passive voice and the demonstrative and indefinite pronotms are essential, removing from the scene not only the characters' points of view, but the author's as well. Numbered among the consequences of the new twentieth-century technique of interior monologue, the Interludes can be accepted today as one more trick of literary "Modernism", but for Woolf it was new, and the task of preventing any of her Interludes from becoming just another report from the rypical "omniscient narrator" of Victorian fiction was one of the most demanding she ever set herself as a writer. Like her friend the Post-Impressionist critic Roger Fry, she wanted to analyze the world outside the self into its smallest sensible elements, then show human selvesasthey formed by accumulating the experience of these myriad, random, discontinuous components, and finally to recombine these components in their two kinds, selvesand sense-able things, into a meaningful whole-a work of art. Rejecting both the idealism of Bradley and the British philosophical generation of the 1890sand the naive positivist materialism coming out of the 1850s, Bertrand Russell eventually settled on an ontology most philosophers call "realist " in that it rejected the view that the world accessible to the senses was no more than a mental construct and considered it naive to assign the origin of subjectivity to matter. The material world was, Russell thought, actually external -"out there". He did think, however,that all we knew of the external world was what wassensed by the sensesand pictured by the mind. He alsoconcluded (in essayslike "The Relation of Sense-data to Physics"written in 1910-17)d1at both that world and what we knew of it was presented in very small parts, and Reviews 89 analyzable down to them, or, in other words, that both ontology and epistemology were atomized. Each mind, with its senses, occupied, he thought, one of an infinite number of occupied and unoccupied "perspectives", each of which Lmiquely focused che "sensibilia"-whatever was perceived in che external world-like a Leibnizian monad. Machemacicallogic, on which Russell set che highest store, would be able, he thought, to bring together perspectives by providing a minimal secof cruchswhich, though abscracc,were objective in che sense chaccheywere noc private and could be agreed to by all. Something similar though rougher mighc be done for chesensibilia in cheworld "ouc chere" by che science of physics. Did Virginia Woolflearn from Bertrand Russell?To a latcer-dayacademic ic sounds too urrerly improbable, for che simple reason (obvious to academics) chat anyone who has made icto the upper levelsof English literature must have chosen cheseparticular heights to scale because the trail was so blessedlyfreeof chosenasry mathematical burrs and logical reazles,and the air so filledwith che pure chatter of French phenomenologisrs, that no hiker would ever have to struggle with analytical philosophy again. Lessmetaphorically, the separation of departments of literature from those of philosophy has come so close to divorce chat few in U.S. English Departments can make head or cail of che graduate thesis that one of Russell's philosophy students wrote on F.H. Bradley, and fewer still can trace the term "objective correlative" used by thac student (who was in fact T. S. Eliot) back to an early work of Husserl. Meanwhile, ic is taken for granted by those in Philosophy that there are no ideas worth discussing in Woolf, or, indeed, in any other novelise,greacthough she may be, Lmlessone is to descend from ontology and epistemology to the foothills of ethics and politics, or worse, aesthetics. (Among... (shrink)

This paper continues the investigation into the relationship between good approximations and jump inversion initiated by Griffith. Firstly it is shown that there is a \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\Pi^{0}_{2}}$$\end{document} set A whose enumeration degree a is bad—i.e. such that no set \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${X \in a}$$\end{document} is good approximable—and whose complement \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\overline{A}}$$\end{document} has lowest possible jump, in other words (...) is low2. This also ensures that the degrees y ≤ a only contain \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\Delta^{0}_{3}}$$\end{document} sets and thus yields a tight lower bound for the complexity of both a set of bad enumeration degree, and of its complement, in terms of the high/low jump hierarchy. Extending the author’s previous characterisation of the double jump of good approximable sets, the triple jump of a \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\Sigma^{0}_{2}}$$\end{document} set A is characterised in terms of the index set of coinfinite sets enumeration reducible to A. The paper concludes by using Griffith’s jump interpolation technique to show that there exists a high quasiminimal \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\Delta^{0}_{2}}$$\end{document} enumeration degree. (shrink)

In lieu of an abstract, here is a brief excerpt of the content:Erasmus and the Problem of the Johannine CommaJoseph M. LevineWhen Edward Gibbon decided to banish primary causes from the Decline and Fall and integrate secular and ecclesiastical history, he was completing a revolution that had begun unwittingly two centuries before. 1 To bring into his narrative of empire a consideration of the “Johannine comma” (the interpolation in 1 John 5:7–8) was not perhaps either digressive or inevitable; but it (...) would certainly have surprised and dismayed some of Gibbon’s immediate predecessors, such as the Catholic Frenchmen, Bossuet and Tillemont, each of whom had written about his subject and exercised an important effect on him at different times in his life; and it would have seemed even more incomprehensible to earlier generations. 2 Something had happened in the two centuries between Erasmus and Gibbon to make this new vantage point possible and congenial.It was, I think, nothing less than a victory for the idea of history. At the beginning of the sixteenth century theology still ruled as queen of the sciences and the claims of religion were largely justified by an appeal to reason and authority, that is, to Aristotelian logic and the dictates of the Church. By the end of the eighteenth century theology had become dependent on history, and religion was now justified by an appeal to “matter of fact.” 3 Even such mysteries as the doctrine of the Trinity, which had eluded the reason of St. Thomas and the schoolmen and had remained dependent on the dictates of Church Councils from Chalcedon to Trent, had come now to depend in some fashion on the [End Page 573] evidence of Scripture considered as history. At the same time the removal of final causes from the narrative of human events threatened to leave it—with every other Christian doctrine and event—to the arbitrament of ordinary scholarship. Gibbon brings us to the edge of religious modernity, and the history of the comma exactly spans the period of that development. It was Erasmus, however, as Gibbon himself suspected, who had first set some of the key problems, which he and his enlightened friends believed they had solved.For Erasmus the decision to edit the Greek New Testament was the result of an early commitment to Renaissance humanism and a response in particular to several lively currents in the contemporary intellectual scene. On the one hand Erasmus, encouraged perhaps by his new friend John Colet, began to think of the Bible principally as a record of history rather than as an arsenal of theological texts, above all as the story of Christ on earth—Christ as the supreme exemplar to be followed and imitated. Apparently, he had been well prepared for this by his acquaintance with the Brothers of the Common Life, and by a characteristic humanist distaste for logic and metaphysics—and a preference for rhetoric and history—which he displayed in all his early writing. 4 It seems to have been Colet’s Lectures on the New Testament that encouraged his view that Paul and the apostles, if not Christ himself, were best understood in their historical setting and that their message was primarily ethical. 5 In effect the churchman, Colet, and the wandering monk, Erasmus, wanted to place Christian responsibility back in the hands of the laity, in the conscience and behavior of the Christian believer, rather than in what they took to be the mechanical observances of Church ritual. This required that the Bible be placed at the very center of worship, in the hands of every man (and woman), accessible and intelligible as the Vulgate was not, and read in the first place for its story. There were others on the Continent who shared the same impulse—which, of course, the Protestant Reformation was also to endorse. 6Secondly, Erasmus was encouraged to his work by his accidental discovery of Lorenzo Valla’s Notes on the New Testament. 7 He had already pledged [End Page 574] himself to the culture of Italian humanism, to its determination to recover the Greek and Latin texts of classical antiquity and put them to use by imitation. Valla was the master of humanist technique, the greatest... (shrink)

Weakly Aggregative Modal Logic ) is a collection of disguised polyadic modal logics with n-ary modalities whose arguments are all the same. \ has interesting applications on epistemic logic, deontic logic, and the logic of belief. In this paper, we study some basic model theoretical aspects of \. Specifically, we first give a van Benthem–Rosen characterization theorem of \ based on an intuitive notion of bisimulation. Then, in contrast to many well known normal or non-normal modal logics, we show that (...) each basic \ system \ lacks Craig interpolation. Finally, by model theoretical techniques, we show that an extension of \ does have Craig interpolation, as an example of amending the interpolation problem of \. (shrink)

Introduced by Leon Henkin back in the fifties, the notion of neat reducts is an old venerable notion in algebraic logic. But it is often the case that an unexpected viewpoint yields new insights. Indeed, the repercussions of the fact that the class of neat reducts is not closed under forming subalgebras turn out to be enormous. In this paper we review and, in the process, discuss, some of these repercussions in connection with the algebraic notion of amalgamation. Some new (...) unpublished results concerning neat reducts and amalgamation are given. . Several counterexamples which convey the gist of techniques used in this area are presented two of which are new . It is known that the algebraic notion of amalgamation in a class of algebras corresponds to the metalogical notion of interpolation in the corresponding logic. Answers to open question in the recent paper [54] concerning both amalgamation and interpolation are summarized in tabular form at the end of this paper. This paper appears in two parts. The present first part contains results on neat reducts. The second part contains results relating the notion of neat embeddings to various amalgamation properties.1. (shrink)

This chapter examines global policing as it takes shape through the work of Interpol, the International Criminal Police Organization. Global policing emerges in the legal, political and technological amalgam through which transnational police cooperation is carried out, and includes the police practices inflected and made possible by this phenomenon. Interpol’s role is predominantly in the circulation of information, through which it enters into relationships and provides services that affect aspects of governance, from the local to national, regional and global. The (...) chapter describes this assemblage as a noteworthy experiment in developing what McKeon called a frame for common action. Drawing on Interpol publications, news stories, interviews with staff, and fieldwork at the General Secretariat in Lyon, France, the history, institutional structure, and daily practices are described. Three cases are analyzed, concerning Red Notices, national sovereignty, and terrorism, in order to explore some of the problems arising in Interpol’s political and technical operating arrangements. In conclusion, international and global policing are compared schematically, together with Interpol’s attempts to give institutional and procedural direction to the still-evolving form of global policing. (shrink)

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

Abstract:This article explores the representation of human‒animal interaction in D. H. Lawrence's poems "Bat" and "Man and Bat." Many influential critics interpret the poems as emphasizing the lack of connection, hospitality, and empathy between the poet and the bats, focusing on the relentless objectification of the animals and the poet's negative attitude towards them. We argue, however, that these poems can also invite different types of readings, by investigating the ways in which Lawrence employs perceptual and kinetic imagery to create (...) a certain degree of embodied, kinesthetic empathy with the bats. Using theoretical and methodological frameworks from cognitive-literary approaches to kinesthesia and human‒animal studies, we analyze Lawrence's multilayered poetic rendition of human‒animal interaction, to understand how the poet stages the tension between the symbolic/cultural connotations associated with bats and humans' perception of their embodied, affective, and kinetic being. (shrink)

Some patients awaken from their coma but only show reflex motor activity. This condition of wakeful (eyes open) unawareness is called the vegetative state. In 2002, a new clinical entity coined ‘‘minimally conscious state’’ defined patients who show more than reflex responsiveness but remain unable to communicate their thoughts and feelings. Emergence from the minimally conscious state is defined by functional recovery of verbal or nonverbal communication.1 Our empirical medical definitions aim to propose clearcut borders separating disorders of consciousness such (...) as coma, vegetative state and minimally conscious state but clinical reality shows that these boundaries can often be fuzzy (fig 1). Recent clinical, electrophysiological and neuroimaging studies are shedding light on these challenging limits of consciousness encountered following severe acute brain damage. At the patient’s bedside, it is very challenging to differentiate reflex or automatic motor behaviour from movements indicating signs of consciousness, and hence some minimally conscious patients might be misdiagnosed as being vegetative. For some motor responses (eg, blinking to visual threat, brief fixation, normal flexion response to pain, etc) it remains unclear whether they truly are voluntary or willed because we lack convincing scientific evidence. We also lack consensus on how to practically assess some of these behavioural responses. For example, there is no agreement on what stimulus to employ in the assessment of visual pursuit movements— often one of the first clinical signs heralding the transition from the vegetative to the minimally conscious state. Vanhaudenhuyse and colleagues2 recently studied visual pursuit in 51 post-comatose patients comparing eye tracking of a moving object, person or mirror. It was shown that more.. (shrink)

In this article we show that bi-intuitionistic predicate logic lacks the Craig Interpolation Property. We proceed by adapting the counterexample given by Mints, Olkhovikov and Urquhart for intuitionistic predicate logic with constant domains [13]. More precisely, we show that there is a valid implication $\phi \rightarrow \psi $ with no interpolant. Importantly, this result does not contradict the unfortunately named ‘Craig interpolation’ theorem established by Rauszer in [24] since that article is about the property more correctly named ‘deductive interpolation’ (see (...) Galatos, Jipsen, Kowalski and Ono’s use of this term in [5]) for global consequence. Given that the deduction theorem fails for bi-intuitionistic logic with global consequence, the two formulations of the property are not equivalent. (shrink)

Interpolation has become one of the standard properties that logicians investigate when designing a logic. In this paper, we provide strong evidence that the presence of interpolants is not only cogent for scientific reasoning but has also important practical implications in computer science. We illustrate that interpolation in general, and uniform splitting interpolants, in particular, play an important role in applications where formality and modularity are invoked. In recognition of the fact that common logical formalisms often lack uniform interpolants, we (...) advocate the need for developing general methods to engineer a specification logic so that some critical uniform interpolants become available. (shrink)

Preservation and interpolation results are obtained for L ∞ω and sublogics $\mathscr{L} \subseteq L_{\infty\omega}$ such that equivalence in L can be characterized by suitable back-and-forth conditions on sets of partial isomorphisms.

Preservation and interpolation results are obtained for L ∞ω and sublogics $\mathscr{L} \subseteq L_{\infty\omega}$ such that equivalence in L can be characterized by suitable back-and-forth conditions on sets of partial isomorphisms.

We analyse the structure of propositional proofs in the sequent calculus focusing on the well-known procedures of Interpolation and Cut Elimination. We are motivated in part by the desire to understand why a tautology might be ‘hard to prove’. Given a proof we associate to it a logical graph tracing the flow of formulas in it . We show some general facts about logical graphs such as acyclicity of cut-free proofs and acyclicity of contraction-free proofs , and we give a (...) proof of a strengthened version of the Craig Interpolation Theorem based on flows of formulas. We show that tautologies having minimal interpolants of non-linear size must have proofs with certain precise structural properties. We then show that given a proof Π and a cut-free form Π′ associated to it , certain subgraphs of Π′ which are logical graphs correspond to subgraphs of Π which are logical graphs for the same sequent. This locality property of cut elimination leads to new results on the complexity of interpolants, which cannot follow from the known constructions proving the Craig Interpolation Theorem. (shrink)

We investigate interpolation properties of many-valued propositional logics related to continuous t-norms. In case of failure of interpolation, we characterize the minimal interpolating extensions of the languages. For finite-valued logics, we count the number of interpolating extensions by Fibonacci sequences.

We investigate uniform interpolants in propositional modal logics from the proof-theoretical point of view. Our approach is adopted from Pitts’ proof of uniform interpolationin intuitionistic propositional logic [15]. The method is based on a simulation of certain quantifiers ranging over propositional variables and uses a terminating sequent calculus for which structural rules are admissible.

Several extensions of the basic modal language are characterized in terms of interpolation. Our main results are of the following form: Language ℒ' is the least expressive extension of ℒ with interpolation. For instance, let ℳ be the extension of the basic modal language with a difference operator [7]. First-order logic is the least expressive extension of ℳ with interpolation. These characterizations are subsequently used to derive new results about hybrid logic, relation algebra and the guarded fragment.

We provide a direct method for proving Craig interpolation for a range of modal and intuitionistic logics, including those containing a "converse" modality. We demonstrate this method for classical tense logic, its extensions with path axioms, and for bi-intuitionistic logic. These logics do not have straightforward formalisations in the traditional Gentzen-style sequent calculus, but have all been shown to have cut-free nested sequent calculi. The proof of the interpolation theorem uses these calculi and is purely syntactic, without resorting to embeddings, (...) semantic arguments, or interpreted connectives external to the underlying logical language. A novel feature of our proof includes an orthogonality condition for defining duality between interpolants. (shrink)

Craig's interpolation lemma (if φ → ψ is valid, then φ → θ and θ → ψ are valid, for θ a formula constructed using only primitive symbols which occur both in φ and ψ) fails for many propositional and first order modal logics. The interpolation property is often regarded as a sign of well-matched syntax and semantics. Hybrid logicians claim that modal logic is missing important syntactic machinery, namely tools for referring to worlds, and that adding such machinery solves (...) many technical problems. The paper presents strong evidence for this claim by defining interpolation algorithms for both propositional and first order hybrid logic. These algorithms produce interpolants for the hybrid logic of every elementary class of frames satisfying the property that a frame is in the class if and only if all its point-generated subframes are in the class. In addition, on the class of all frames, the basic algorithm is conservative: on purely modal input it computes interpolants in which the hybrid syntactic machinery does not occur. (shrink)

Arrow and turnstile interpolations are investigated in UCL [introduced by Sernadas et al. ], a logic that is a complete extension of classical propositional logic for reasoning about connectives that only behave as expected with a given probability. Arrow interpolation is shown to hold in general and turnstile interpolation is established under some provisos.

An interpolation theorem holds for many standard modal logics, but first order $S5$ is a prominent example of a logic for which it fails. In this paper it is shown that a first order $S5$ interpolation theorem can be proved provided the logic is extended to contain propositional quantifiers. A proper statement of the result involves some subtleties, but this is the essence of it.

An interpolation Theorem is proved for first order logic with infinitary predicates. Our proof is algebraic via cylindric algebras.1.

The Interpolation Theorem, first formulated and proved by W. Craig fifty years ago for predicate logic, has been extended to many other logical frameworks and is being applied in several areas of computer science. We give a short overview, and focus on the theory of software systems and modules. An algebra of theories TA is presented, with a nonstandard interpretation of the existential quantifier . In TA, the interpolation property of the underlying logic corresponds with the quantifier combination property . (...) It is shown how the Modularization Theorem, the Factorization Lemma and the Normal Form Theorem for module expressions can be proved in TA. (shrink)

Decision theory requires agents to assign probabilities to states of the world and utilities to the possible outcomes of different actions. When agents commit to having the probabilities and/or utilities in a decision problem defined by objective features of the world, they may find themselves unable to decide which actions maximize expected utility. Decision theory has long recognized that work-around strategies are available in special cases; this is where dominance reasoning, minimax, and maximin play a role. Here we describe a (...) different work around, wherein a rational decision about one decision problem can be reached by ‘interpolating’ information from another problem that the agent believes has already been rationally solved. (shrink)

We show in this paper that all fragments of intuitionistic propostional logic based on a subset of the connectives $\wedge, \vee, \rightarrow, \neg$ satisfy interpolation. Fragments containing $\leftrightarrow$ or $\neg\neg$ are briefly considered.

A method is presented that connects the existence of uniform interpolants to the existence of certain sequent calculi. This method is applied to several modal logics and is shown to cover known results from the literature, such as the existence of uniform interpolants for the modal logic \. New is the result that \ has uniform interpolation. The results imply that for modal logics \ and \, which are known not to have uniform interpolation, certain sequent calculi cannot exist.

We present tableau proof systems for the annotated version of propositional justification logics, that is, justification logics which are formulated using annotated application operators. We show that the tableau systems are sound and complete with respect to Mkrtychev models, and some tableau systems are analytic and provide a decision procedure for the annotated justification logics. We further show Craig’s interpolation property and Beth’s definability theorem for some annotated justification logics.

We study interpolation for elementary fragments of classical linear logic. Unlike in intuitionistic logic (see [Renardel de Lavalette, 1989]) there are fragments in linear logic for which interpolation does not hold. We prove interpolation for a lot of fragments and refute it for the multiplicative fragment (→, +), using proof nets and quantum graphs. We give a separate proof for the fragment with implication and product, but without the structural rule of permutation. This is nearly the Lambek calculus. There is (...) an appendix explaining what quantum graphs are and how they relate to proof nets. (shrink)