This paper is a criticism of Plantinga’s analysis of a version of the ontological argument. He thinks it is obvious that his version is valid and that the only question of interest is whether a key premise is true. The paper lays out two relevant semantical accounts of modal logic. It contends that Plantinga needs to show that one is preferable to the other.

The purpose of this paper is to demonstrate how the navya-Nyaya school of indian philosophy determines the truth or falsity of a sentence which contains an empty term, And to point out some similarities and differences between its method of analysis and truth-Value determinations of such sentences and that of bertrand russell.

In this article, I explore the encounter of the Mādhva philosopher Vyāsatīrtha with the works of the Navya-Naiyāyika Gaṅgeśa Upādhyāya. The article is based on original translations of passages from Vyāsatīrtha’s Nyāyāmr̥ta and Tarkatāṇḍava. Philosophically, the article focuses on the issue of empty-terms/nonexistent entities, particularly in the context of the theory of inference. I begin by outlining the origin of the Mādhva and Nyāya positions about these issues in their respective analyses of perceptual illusion. I then contrast the role (...) of Gaṅgeśa’s thought in the Nyāyāmr̥ta and Tarkatāṇḍava and show how Vyāsatīrtha responds to Gaṅgeśa’s ideas in different ways in those texts. I conclude with a discussion of how Vyāsatīrtha defends the Mādhva theory of “substrate-free” qualities against Gaṅgeśa in order to show that we can think and make valid inferences about nonexistent things. (shrink)

The problem of empty terms is one of the focal issues in analytic philosophy. Russell’s theory of descriptions, a proposal attempting to solve this problem, attracted much attention and is considered a hallmark of the analytic tradition. Scholars of Indian and Buddhist philosophy, e.g., McDermott, Matilal, Shaw and Perszyk, have studied discussions of empty terms in Indian and Buddhist philosophy. But most of these studies rely heavily on the Nyāya or Navya-Nyāya sources, in which Buddhists are portrayed as (...) opponents to be defeated, and thus do not truly reflect Buddhist views on this issue. The present paper will explore how Dignāga, the founder of Buddhist logic, deals with the issue of empty subject terms. His approach is subtle and complicated. On the one hand, he proposes a method of paraphrase that resembles Russell’s theory of descriptions. On the other, by relying on his philosophy of language—the apoha theory, he tends to fall into a panfictionalism. Through the efforts of his follower Dharmakīrti, the latter approach would become more acceptable among Indian and Tibetan Buddhists. Dignāga’s Chinese commentators, who were free from the influence of Dharmakīrti, dealt with the empty term issue in three ways: (1) by adhering to Dignāga’s method of paraphrase; (2) by allowing exceptions for non-implicative negation; and (3) by indicating the propositional attitude of a given proposition. Among these, the third proved most popular. (shrink)

The aim of this paper is to investigate the problem of existential import in Abelard's modal logic, and to ask whether the system of logical relationships that he proposes for modal propositions ma...

Mental files, in Recanati's framework, function as 'singular terms in the language of thought' ; they serve to think about objects in the world (and to store information about them). But they have a derived, metarepresentational function : they serve to represent how other subjects think about objects in the world. To account for the metarepresentational use of files, Recanati introduces the notion of an 'indexed file', i.e. a vicarious file that stands, in the subject's mind, for another subject's file (...) about an object. Using that notion, he argues, one can provide an analysis of attitude ascriptions and the conniving use of empty singular terms. (shrink)

This paper considers the problem of assigning meanings to empty natural kind terms. It does so in the context of the Twin-Earth externalist-internalist debate about whether the meanings of natural kind terms are individuated by the external physical environment of the speakers using these terms. The paper clarifies and outlines the different ways in which meanings could be assigned to empty natural kind terms. And it argues that externalists do not have the semantic resources to assign them meanings. (...) The paper ends on a sceptical note concerning the fruitfulness of using the Twin-Earth setting in debates about the semantics of empty natural kind terms. (shrink)

Several authors (Boghossian 1998; Segal 2000) allege that 'empty' would-be natural kind terms are a problem for anti-individualistic semantics. In this paper I rebut the charge by providing an anti-individualistic semantics for such terms.

Abstract: In this paper I offer an account of a particular variety of perception of absence, namely, visual perception of empty space. In so doing, I aim to make explicit the role that seeing empty space has, implicitly, in Mike Martin's account of the visual field. I suggest we should make sense of the claim that vision has a field—in Martin's sense—in terms of our being aware of its limitations or boundaries. I argue that the limits of the (...) visual field are our own sensory limitations, and that we are aware of them as such. Seeing empty space, I argue, involves a structural feature of experience that constitutes our awareness of our visual sensory limitations, and thus, in virtue of which vision has a field. (shrink)

Aristotle draws what are, by our lights, two unusual relationships between predication and existence. First, true universal affirmations carry existential import. If ‘All humans are mortal’ is true, for example, then at least one human exists. And secondly, although affirmations with empty terms in subject position are all false, empty negations are all true: if ‘Socrates’ lacks a referent, then both ‘Socrates is well’ and ‘Socrates is ill’ are false but both ‘Socrates is not well’ and ‘Socrates is (...) not ill’ are true. In this paper, I conjecture that for Aristotle predications have mereological truth conditions: for example, ‘Socrates is pale’ is true just in case Socrates is a part of the mereological sum of pale things. The existential import of universal affirmations and the semantic profile of empty negations follow from this mereological semantics. (shrink)

Is there a connection between power struggles and urban context? How the urban space used for the symbolic manifestation of power and social control? How urban space becomes the site of conflict and resistance? How urban nodes like squares became political apparatus in social demonstrations and revolutions? How do specific squares become symbols of revolutions? This thesis investigated these questions by viewing the city as a place formed by politics, which built upon the central concept of Meydan (Public Square), as (...) the "political order in the ideogram of the city". Focusing on public squares, it traces their sociopolitical transformations as well as their role in instigating social transformations through examples that span from the pre-modern times to the present. As the embodiment of the square in the image of the city, the historical, social and theological concept of Meydan- a term, which has mostly applied, for Iranian public squares has been studied. The Empty Locus of Power rereads squares as peculiar urban forms and representations of political ideas, when the squares of the city has become the stage for the process of politicizing, de-politicizing or neutralizing space. This thesis tries to analyze the square beyond an architectural element in the city, but weaves this blank slate, with its contemporary socio political atmosphere as a new paradigm. This interpretation, suggested the idea of Meydan as the core of the projects on the city, which historically exposed in formalization of theological ideologies. Regarding this issue, urban space of traditional Iranian cities introduced as the medium through which theological ideologies and political sovereignty took place. In pursuing such analyses, this research engages with issues ranging from details of political histories of the case studies in public squares to the master plan of the city of VIII Tehran. During the recent century, various political events and social demonstrations have been staged in Tehran as Middle Eastern Capital, which emphasize the further discussions for analyzing the relationship between socio-political dimensions of city and its urban projects that ultimately led to occupy the city and reclaim the public spaces in Iranian Islamic Revolution of 1979. In this sense, analyzing the major political events in modern Tehran as a city loaded with politics will lead to trace the processes of its spatial transformation. In this regard, the thesis examines the relationship between affordances of public spaces, their histories, and the emergence of social events and movements. Building on this theoretical framework, this thesis stresses on the transformative dynamism of autocratic modernization, which motivate or shape a creative tension in the form of the city. The emergence of representative pseudo political public space for demonstration of power and national identity during the First Pahlavi Era (1925-41), reoccupation of public spaces by social movements and political parties during the nationalization of Iranian oil movement till 1953 Iranian coup d'etat will be the second part and socio- political arrangement of capital as ‘Metropolis of Tomorrow' and its urban transformation during the second Pahlavi Era (1941-79) till Islamic Revolution have been examined. Analysis suggests that spatial transformations and modernization politics have led to or facilitated (directly or indirectly and, or inadvertently) political changes. Building on the foundation of knowledge established in this research, the final part of research focuses on the centrality of squares in recent social protests. Using Middle East sociologist, Asef Bayat's theory of ‘Spatialities of discontent', the final chapter explores the spatial dimension of political spaces of the city and aims to theorize the necessity of urban social movements to approach democratic space in a global context. (shrink)

This is an entry on Empty Names for the Routledge Companion to the Philosophy of Language, edited by Delia Graff Fara and Gillian Russell.

Human perception is often believed to function holistically, especially in the tradition of Gestalt psychology, involving a focused item and its surrounding. This holistic approach can allow us to explain something that is not directly experienced in our perception, meaning that the absence as well as the presence of something can have a significant impact on how we perceive the world. The way we perceive the presence is more or less the same cross-culturally, but the prominence of the absence, or (...) what is termed emptiness in this volume, varies considerably from one culture to another. The aim of this volume is to identify what emptiness is like and how different cultures incorporate this concept from various perspectives. It turns out that emptiness plays a key role in identifying socio-cultural diversity in a broader sense, including arts and languages. This volume consists of contributions from different fields covering a wide range of topics such as history, literary studies, mythology, film studies, architecture, linguistics, social-anthropology, ethnology and cognitive science. Due to the range covered in this volume, studies presented here are highly interdisciplinary, but all chapters deal with the sense of emptiness, which suggest that the underlying idea of the significance of emptiness is pervasive. Yet, this topic has not previously been systematically compared across different disciplines. It is hoped that this volume will offer a first overview of the pervasiveness and integration of disciplines concerning the sense of emptiness. (shrink)

A pivotal focus of exegesis of Nāgārjuna's Mūlamadhyamakakārïkā (MMK) for the past half century has been the attempt to decipher the text's philosophy of language, and determine how this best aids us in characterizing Madhyamaka thought as a whole. In this vein, MMK 24:18 has been judged of particular weight insofar as it purportedly insists that the concepts pratītyasamutpāda (conditioned co-arising) and śūnyatā (emptiness), both indispensable to Buddhist praxis, are themselves only "nominal" or "conventional," that is, they are merely labels (...) that do not referentially signify anything that can be taken to be an ontologically ultimate reality. In various guises, as a result of this explication, Nāgārjuna's thought has been seen to embrace an overarching linguistic nominalism or conventionalism in which words, whether they are used for the purposes of theory or practice, though they serve as commonly accepted currency in the transactions of worldly business (vyavahāra), are in the end only ideas (prajñapti) or metaphysical fabrications (prapanca). This interpretation is largely due to tenaciously inaccurate translations and expositions of MMK 24:18 and their dependence on Candrakīrti's peculiar analysis of this verse in his Prasannapadā. This essay will attempt to correct both the diction of the major translations of MMK 24:18 and the fictions of nominalism and conventionalism that the consequent interpretations of this stanza have perpetuated. The argument that will be developed in the course of this essay is that Candrakīrti's reading of this verse proffers a strong form of linguistic nominalism that Nāgārjuna himself does not embrace. It will be shown, based on everything else found in the MMK, that Nāgārjuna, rather than advocating the mere nominal or conventional status of terms such as pratītyasamutpāda and sunyata, demands they be accepted as both pedagogically useful and even referentially accurate descriptions of the world as it is. (shrink)

The variations in how subjects respond to positively or negatively framed descriptions of the same issue have received attention from social science research, where, nevertheless, a naïve understanding of speech interpretation has undermined the different explanations offered. The present paper explores the semantic-pragmatic side of framing effects and provides a unifying explanation of this phenomenon in terms of a combined effect of pragmatic presuppositions and default implicatures. The paper contributes to a more comprehensive understanding of representations and cognitive processes involved (...) in the framing bias by showing how well-entrenched linguistic practices associated to frame choice, and conducive to an implicit focus, result in default implicatures on the addressee’s side. (shrink)

Sengzhao (374?−414 CE), a leading Chinese Mādhyamika philosopher, holds that the myriad things are empty, and that they are, at bottom, the same as emptiness qua the way things truly are. In this paper, I distinguish the level of the myriad things from that of the way things truly are and call them, respectively, the ontic and the ontological levels. For Sengzhao, the myriad things at the ontic level are indeterminate and empty, and he equates the way things (...) truly are at the ontological level with emptiness. This paper explicates the ontic status of the myriad things and the ontological notion of emptiness and elucidates the relationship between them. I read Sengzhao as dismissing the idea that reality has a mind-independent structure that comprises discrete entities waiting to be captured by concepts. I show that his notion of emptiness points to a subject-object unity wherein both the subject and the myriad objects are conceptually undifferentiated. It will be seen that Sengzhao’s overall theory of emptiness is philosophically interesting in terms of its relevance for ontology and the relationship between language and reality. It provides a refreshing and challenging perspective on how we may perceive the world and understand our relationship to it. (shrink)

We study the intersection type assignment system as defined by Barendregt, Coppo and Dezani. For the four essential variants of the system (with and without a universal type and with and without subtyping) we show that the emptiness (inhabitation) problem is recursively unsolvable. That is, there is no effective algorithm to decide if there is a closed term of a given type. It follows that provability in the logic of "strong conjunction" of Mints and Lopez-Escobar is also undecidable.

This essay argues two main points by analyzing Sengzhao’s contentions regarding several basic Buddhist concepts such as emptiness, being, and nonbeing. First, Sengzhao synthesizes Daoist methods of argumentation into his description of the middle path and other Buddhist concepts. Second, he revives Daoist concepts, giving them Buddhist meaning and expressing them in Buddhist terms. In the process, he consciously differentiates Madhyamika Buddhism from earlier Buddhism as understood from a Daoist perspective, such as the teachings of the School of Original Non-Being (...) and the School of No-Mind. However, because his understanding of the middle path is not sufficiently deep, he wavers between Buddhist and Daoist understandings of terms such as prajñā and nirvāṇa, rest and movement, and name and reality. Despite these inconsistencies, his discourse, as the first attempt to differentiate Madhyamika from Daoism, is still a milestone in the development of Chinese Buddhism and has exerted a far-reaching influence on later generations. (shrink)

Mental files have a referential role—they serve to think about objects in the world—but they also have a meta-representational role: when ‘indexed’, they serve to represent how other subjects think about objects in the world. This additional, meta-representational function of files is invoked to shed light on the uses of empty singular terms in negative existentials and pseudo-singular attitude ascriptions. -/- For a longer version see "Empty Singular Terms in the Mental-File Framework" In Manuel Garcia-Carpintero & Genoveva Marti (...) (eds.), Empty Representations. Oxford 162-185. (shrink)

The article analyzes the ongoing Russian–Ukrainian war in terms of a colonial seizure undertaken by a fading but aggressive Russian empire. This highly political adventure is translated into more abstract terms, that is, an irresolvable conflict between existence, which is always the experience of coexistence devoid of any essence whatsoever, and imperial expansion, which is an infinite conquest of space indifferent to all forms of life. The dualism in question is backed up by the writings of two important scholars, namely, (...) the French philosopher Jean-Luc Nancy and the Russian scientist and revolutionary Peter Kropotkin. Nancy’s philosophy of being-in-common is the development of Heidegger’s concept of Mitsein, where the meaning of Mit- (co-, with) is existential and not categorical. Nancy’s philosophy challenges all forms of communal essence, of which empire is one blatant example. For Kropotkin, existence is even more all-embracing, implying a continuity of different life forms. His doctrine of mutual aid is a law that governs both society and nature, as well as evolution itself. Even morality springs from “solidarity” and the joys of social life, while sympathy or love are too narrow a basis for human ethics. Viewed from this perspective, a colonial war would not only be the greatest waste of human lives and natural resources—it would contradict the very order of things. (shrink)

Frank Kermode uses the term aevum to question the links between origin, order, and time, associating experience with spatial form. Without end or beginning, aevum identifies an intersubjective order of time where we participate in the “relation between the fictions by which we order our world and the increasing complexity of what we take to be the ‘real’ history of that world”; being “in-between” time is a primary quality of the aevum. Regarding cinema, aevum identifies this third duration as (...) emotional experience, occuring as traumatic time. It facilitates thinking beyond lived temporal experience of everyday life to a philosophy of experience that accounts for alternative sensoria of time, similar to the traumatic encounter. The cinematic aevum is equally not of the material, corporeal world; concurrently associating human reality with the myths of the human condition. To say that a cinematic aevum exists following traumatic scenes, is to specify a visual “time-fiction” in film, to recognise a spatial form that belongs neither to the finite time of the film's narrative, or of the “eternal” time outside the film's diegesis, but participates in the order (and linking) of both. Wilfred Bion's psychoanalytic works are used to discuss the traumatic symptom of “empty time”: the inability to recollect, to make links between memory and experience, demonstrating a version of empty time that works as an external violence to spectator perception. Bion's theories offer fresh psychoanalytic perspective on trauma and its relationship to time by challenging classical ways of thinking about inner and outer perception. (shrink)

The Chinese word “bukong” 不空 appearing in the Āgama texts is a rendering of Pāli words such as aritta (not discarded), asuñña (not empty), amogha (not vain). Whereas the Madhyamika texts never affirm the term non‐empty as a counterpart of the concept emtpy, the Yogācāra texts overlay it with a slightly negative connotation as a false imagination. However, Tathāgatagarbha thought affirms that term positively, and Chinese strands of Buddhism further adopt it as an absolute affirmation by (...) identifying emptiness with non‐emptiness, thereby turning the concept of mutual dependence or counterpart into that of mutual identification. In the Huayan jing, the concepts such as aśūnya, aśūnyatā, amogha, amoghatā are equally rendered as “bukong”, and this term thus comes to have a wide range of meanings including non‐empty, without any errors, beneficial, not deluded, or fully endowed with wisdom and compassion, and so on, which are attributes of tathāgatagarbha. Therefore, we see that the Huayan jing has such diverse meaning in the word “bukong” in according to the context. In the case of the typical Chinese Buddhist text Dasheng qixin lun, “bukong” indicates the state in which the pure dharmas are fulfilled. It seems that this text regards the essence as empty while considering the function to be non‐empty. When we contemplate on the meanings of “bukong” appearing in the Huayan jing, we may say that this scripture played a role of bridging Indian Buddhism and Chinese Buddhism. In particular, the comparisons of the meanings of “bukong” in the scriptures and treatises will reveal the affirmative spirit of the Chinese Buddhists, which is implicit in Indian Mahāhyāna Buddhism. 『아함경』에서 ‘不空’이라 한역된 원어는 Pā. aritta(버려지지 않은), asuñña(비어 있지 않은), amogha(헛되지 않은) 등이다. 중관에서는 공의 상대(相待)인 불공을 절대 부정하는 반면, 유식에서는 허망 분별로 인한 부정적인 이미지가 살짝 보이다가 여래장 사상이 도입되면서 긍정적인 이미지로 전환하는데, 그것이 중국불교에서는 절대 긍정으로 바뀌어서 ‘空即不空’인 상의상즉(相依相卽)의 의미로 공과 분리가 불가능해진다. 『화엄경』에서는 ‘不空’이란 역어가 Skt. aśūnya, aśūnyatā, amogha, amoghatā로, 그 의미가 어리석지 않은 지혜와 자비가 충만한 여래장의 뜻으로 무한히 펼쳐져서 불공이란 꽃이 활짝 핀다. 이렇게 불공은 문맥에 따라 그 의미가 다르나, 중국불교의 전형적인 논인 『기신론』에서는 정법(淨法)이 만족스러우면 ‘불공’이라 하면서, 체(體)가 공이고 용(用)이 불공이라고 말하는 것으로 보인다. ‘不空’이라는 역어만을 놓고 대조해 본다면, 초기불교 경전인 『아함경』에서의 불공의 의미가 『화엄경』에서 되살아나는 듯하여 『화엄경』이 인도불교와 중국불교의 가교역할을 한다고 감히 추정할 수 있는데, 특히 경론에서 불공의 의미만 가지고 비교한다면, 『화엄경』에는 인도 대승불교 속에 자연스레 녹아 있는 긍정적인 중국 불자들의 마음이 엿보인다. (shrink)

In lieu of an abstract, here is a brief excerpt of the content:Divine Emptiness and Historical Fullness: A Buddhist-Jewish-Christian Conversation with Masao AbeEdward L. ShirleyDivine Emptiness and Historical Fullness: A Buddhist-Jewish-Christian Conversation with Masao Abe. Edited by Christopher Ives. Valley Forge, PA: Trinity Press International, 1995. 272 pp.This book is a continuation of a discussion begun by Masao Abe in 1984, previous incarnations of which have been published elsewhere. In the present volume, Abe’s expanded essay serves as the first part (...) of a four-part book. The second part includes responses by six Jewish, Christian, and Buddhist thinkers. The third part is Abe’s responses to questions and criticisms raised in Part Two. The fourth part consists of critiques by Hans Küng and Wolfhart Pannenberg and Abe’s separate responses to each.Part One consists of Abe’s well-known essay, “Kenotic God and Dynamic Sunyata.” There are many issues that Abe raises in this essay, of which only two—his understanding of a self-emptying God and his interpretation of the Holocaust—raise any serious concerns among his responders. Abe begins his theological reflection on the self-emptying of God by focusing on the kenosis hymn of Philippians 2:5–11. Abe raises the issue of the kenosis of Christ from a moral to ontological level, asking whether the self-emptying of Christ in either the Incarnation or the Crucifixion can be understood apart from the self-emptying “nature” of the Logos itself. And then, upping the ante, he asks whether one can speak of the kenosis of Christ without also speaking of the kenosis of the God whom Christ reveals.Abe then turns his attention to the Buddhist idea of Sunyata and proposes that Sunyata, far from being either a static or merely negative term, is dynamic and positive. This, he believes, will help to distinguish the Buddhist notion of Sunyata from Nietzschean nihilism. Abe also writes of the place of human reason, free will, the concept of evil, and time and history in Buddhism. It is in this context that he addresses the question of the Holocaust, speaking of his own responsibility not in the sense of having been directly involved, but in light of “collective karma.” At the same time, he does not want to suggest joint responsibility of the victims in the ethical dimension, but rather that such solidarity is realized in what he terms “the most fundamental religious dimension,” this dimension being identical with dynamic Sunyata. [End Page 207]The second part of the book consists of responses by six religious scholars. The first two respondents write from a Jewish perspective. Richard Rubenstein begins by asserting that the Philippians hymn can be understood only in light of Paul’s Jewish-Christian experience of the Covenant. Next, Rubenstein turns to his primary concern, which is Abe’s treatment of the Holocaust. He begins by outlining various Jewish responses to the Holocaust and rejecting each of them. Like Abe, Rubenstein denies the continuity of the ethical and ultimate dimensions, and accepts ultimate reality as ultimate nothingness.Sandra B. Lubarsky also appreciates Abe’s concept of Sunyata, but for different reasons than Rubenstein. Lubarsky sees it as a healthy antidote to Jewish tendencies to absolutize the Jewish community, which has led to an idolization of the community. On the other hand, she questions the Zen assumptions that lie behind Abe’s treatment of history and justice. While there is a higher vantage point than the level of human understanding from which to understand historical events, this level is reserved for God alone and to pretend to that level is arrogant. Likewise, from a Jewish perspective, it is impossible to remove God from ethical considerations.Christian responses begin with those of Heinrich Ott, who argues that the proper locus of interreligious dialogue lies not in the discussion of doctrinal or philosophical similarities, but rather in similar psychological experiences. Ott finds something in Christian experience similar to Sunyata. From this starting point, he proposes a Christian understanding of Nothingness that encompasses both human interconnectedness as well as the interconnectedness of humans with nonhuman reality.Marjorie Hewitt Suchocki maintains that speaking of the self-emptying God revealed in the self-emptying Christ has... (shrink)

During the past few decades a growing interest in what is often called the ‘Kyoto School’ of philosophy has evidenced itself here and there in the West, especially in discussions of comparative religious thought and in the pages of journals which are sensitive, in the post-colonial world, to the value of giving attention to contemporary thought that originates outside the Anglo-American and continental contexts. What has made the so-called Kyoto School especially interesting is the fact that those thinkers identified with (...) it obviously possess a wide acquaintance with Western thought but also have a programme of clarifying points at which they, as Japanese philosophers, find Western philosophy either in sum or in its parts inadequate or objectionable. Moreover, inasmuch as the philosophers of the Kyoto School have deliberately reached back into the Mahayana Buddhist component in Japanese civilization in order to find terms, perspectives, and even foundations for their own analyses and constructions, Western students of comparative religion and comparative thought have in the study of this school a unique aperture for observing how a group of thinkers, while sharing modernity and its problems with us, reates both of these to a religious tradition which is in many ways strikingly different from that of the West. (shrink)

Abstract. On the Direct Reference thesis, proper names are what I call ‘genuine terms’, terms whose sole semantic contributions to the propositions expressed by their use are the terms’ semantic referents. But unless qualified, this thesis implies the false consequence that sentences containing names that fail to refer can never express true or false propositions. (Consider ‘The ancient Greeks worshipped Zeus’, for instance.) I suggest that while names are typically and fundamentally used as genuine terms, there is a small class (...) of names, which I call ‘descriptive names’, whose reference is fixed by commonly associated definite descriptions, and I also suggest that there is an idiom of natural language on which such names can be used as abbreviated definite descriptions in a limited set of sentential contexts, including (1) positive and negative existentials, (2) cognitive ascriptions, and (3) uses of names to talk about myth. Uses of empty descriptive names in such contexts can then be either true or false. Relying on Gregory Currie’s theory of truth in fiction, I also propose an idiom on which fictional names can be used as short for a certain type of description in talk about fiction. Along the way, I provide arguments that names are used as short for descriptions in substantive existential statements as well as in both metamythic and metafictive contexts. I also discuss and criticize alternative views of these matters, including the views of David Braun, Saul Kripke, Peter van Inwagen, and others. (shrink)

I argue that, pace Chomsky (2000, 2003), standard theories of linguistic competence are committed to taking talk of representations seriously, in particular, to recognizing that the “of x” clause that invariably follows “representation” is a way of specifying that representation’s intentional content. One reason to insist upon intentional content in such cases is that the “x” in “of x” may not exist (as in "of Zeus"). This issue is especially relevant to linguistics since, recapitulating considerations raised by many linguists, I (...) go on to argue that most of the SLEs themselves seldom, if ever exist: it is doubtful there are many, if any, tokens of them in space and time; indeed, their existence is by and large inessential to the needs of either communication or serious linguistic theory. All that linguistic theory requires to be real in this regard are the representations, presumably entokened in people’s brains, understood, however, in terms of their intentional contents. (shrink)

In a short but important article, the Polish philosopher Izydora Dambska criticized the thesis-endorsed by Tadeusz Kotarbinski the effect that there are "empty" terms which denote no objects at all, besides the usual general and singular terms. Dambska remarked that "we usually find cited as examples of empty names such self-contradictory names as or, or names of mythical deities-fictitious figures that exist only in legends, poems, novels, etc." She also pointed out, however, that the basic semantic function of (...) names consists in denoting. This means that, if we admit the concept of a name which denotes nothing, either we arbitrarily change the meaning of the term "name" or run into contradiction. In fact-she observed- "for the everyday interpretation of the term, a name which does not denote anything is practically a non-name.". (shrink)

Abstract:Unlike previous comparisons between Nāgārjuna and the later Wittgenstein, this article denies that either philosophy intends to replace ontological pluralism with holism (see, e.g., Streng 1967, Hudson 1973, Gudmunsen 1974 and 1977, Waldo 1975 and 1978, and Katz 1981). Instead, employing a "therapeutic" reading of the Philosophical Investigations associated with the later Gordon Baker, it is proposed that Nāgārjuna and the later Wittgenstein claim that an ontology consisting of discrete particulars is pictorially indeterminate, visible under either aspect of parts or (...) a whole. What Nāgārjuna and Wittgenstein share is an understanding of contrary positions like pluralism and holism as interchangeable Gestalten rather than matters of fact. As a result, I argue that Nāgārjuna's philosophy may be understood in terms of a broader non-dualism employed as a skill-in-means (upāyakauśalya). Akin to Wittgenstein's notion of an "object of comparison" (PI §131), Nāgārjuna's chief insight amounts to a useful perspective rather than a philosophical thesis. (shrink)

Leah Kalmanson's Cross-Cultural Existentialism offers an original and provocative interpretation of existentialist themes and threads running through classical and modern East Asian Buddhist and Ruist philosophical sources. The book takes its point of departure in existential questions concerning meaningfulness and meaning-formative practices, as articulated in European existentialism and postexistentialism, and traces how these questions are and can be addressed in their own terms in dharmic and Song dynasty Ruist discourses of karma, vital force, and ritual propriety. The book elucidates existential (...) techniques and practices encompassing case studies of Buddhist... (shrink)

The paper deals with the topic of empty terms as considered in chapter six of Devitt’s book Designation. Devitt’s proposal is that a statement about fiction is (usually) implicitly preceded by a fiction operator roughly paraphrasable by “it is pretended that” or “in fiction”. The causal chain that forms the network for a fictitious name are not d(esignational)-chains, for they are not grounded in an object. Nevertheless, although the fictitious name does not designate, we could say that it stands (...) in some other referential relationship to the world: it ‘F(ictionally)-designates’. It seem that we could then state truth conditions of a F-sentence using F-designation: the F-operator direct us to look not for the designatum of name but for its F-designatum. The name (for example, “Napoleon” in War and Peace), though nonempty, is just like a fictitious name. In proposing that nonempty names in fiction refer to real items, I argue against the view that “Napoleon” in the War and Peace designates the famous general, Napoleon, and F-designates parts of War and Peace. The referent of F-designation is Napoleon, famous general, too. At the end of the paper I claim that Devitt, if he wants to remain a modal fictionalist, has to renounce the view of one-world metaphysics. (shrink)

One strategy Mādhyamikas use to support their claim that nothing has intrinsic nature (svabhāva) is to argue that things with intrinsic nature could not enter into causal relations. But it is not clear that there is a good Madhyamaka argument against ultimate causation that understands causation in ‘Humean’ terms and understands dharmas as tropes. After exploring the rationale behind the intrinsic-nature criterion of dharma-hood, I survey the arguments Mādhyamikas actually give for their claim that anything dependently originated must be devoid (...) of intrinsic nature, and suggest that none actually succeeds in ruling out this hypothesis about how ultimate causation might work. (shrink)

Fred Adams and collaborators advocate a view on which empty-name sentences semantically encode incomplete propositions, but which can be used to conversationally implicate descriptive propositions. This account has come under criticism recently from Marga Reimer and Anthony Everett. Reimer correctly observes that their account does not pass a natural test for conversational implicatures, namely, that an explanation of our intuitions in terms of implicature should be such that we upon hearing it recognize it to be roughly correct. Everett argues (...) that the implicature view provides an explanation of only some of our intuitions, and is in fact incompatible with others, especially those concerning the modal profile of sentences containing empty names. I offer a pragmatist treatment of empty names based upon the recognition that the Gricean distinction between what is said and what is implicated is not exhaustive, and argue that such a solution avoids both Everett's and Reimer's criticisms. (shrink)

This paper examines to what extent Davidsonian truth-theoretic semantics can give an adequate account for empty names in natural languages. It argues that the prospect is dim because of a tension between metaphysical austerity, non-vacuousness of theorems and empirical adequacy. Sainsbury (2005) proposed a Davidsonian account of empty names called ‘Reference Without Referents’ (RWR), which explicates reference in terms of reference-condition rather than referent, thus avoiding the issue of existence. This is an inspiring account. However, it meets several (...) difficulties. First, there is no non-vacuous, interpretive truth-condition available for any T-sentence containing an empty name, because by stipulation there is no way to compose a true atomic sentence using an empty name in RWR. The absence of an interpretive truth-condition implies the absence of an interpretive reference-condition. It also entails empirical inadequacy to support an interpretation because the related propositional attitude can never be truly satisfied. These considerations cast doubt on whether truth is the proper vehicle to articulate meaning. This paper suggests that we can alleviating the problem by employing truth-conditions varying across contexts of utterance, rather than just contexts of evaluation. The result is a two-dimensional model-theoretic approach to empty names. It involves a new function coined counterfactual reference and the account is compatible with both descriptivism and Millianism. (shrink)

This paper examines the evolution of Husserl’s philosophy of nonintuitive intentions. The analysis has two stages. First, I expose a mistake in Husserl’s account of non-intuitive acts from his 1901 Logical Investigations. I demonstrate that Husserl employs the term “signitive” too broadly, as he concludes that all non-intuitive acts are signitive. He states that not only meaning acts, but also the contiguity intentions of perception are signitive acts. Second, I show how Husserl, in his 1913/14 Revisions to the Sixth (...) Logical Investigation, amends his 1901 theory of non-intuitive acts, which he now calls “empty” intentions. He there accurately distinguishes empty meaning acts from the empty intentions of perception. In the conclusion, I reveal how Husserl’s alterations to his theory of non-intuitive intentions can inform our understanding of a larger shift in his philosophy. (shrink)

This is one of the best studies to date on the philosophy of emptiness, established by the Buddhist scholar Nägärjuna. It not only presents an exposition of emptiness, the lack of self-existent entities, but also gives the background in India at the time of the formulation of the Mädhyamika and analyzes the structures of religious apprehension in Indian thought. Streng finds three types of religious realization: mythic, intuitive, and dialectical. He clearly sees and demonstrates that the doctrine of emptiness is (...) not a teaching of an unqualified base of phenomena, and thus classifies this system as a dialectical structure. The second part is devoted to a study of the system itself; the third, to placing that system in the context of Indian religious thought; the fourth, to relating the doctrine of emptiness to the general problem of religious knowledge as a means for ultimate transformation. Thus, the book is by no means limited to Buddhologists or Indologists. The almost constant translation of Buddhist Sanskrit terms into English makes this work available to all interested in philosophy and religious thought. The appendix contains translations of the whole of the Mülamadhyamakakärikäs and of the Vigrahavyävartanï both by Nägärjuna. Because the texts are root or fundamental texts, and thus brief, the translations are not easily comprehensible; however, those parts can be skipped over as the principle being exemplified is the same in every instance. Because this book places the concept of emptiness in its proper perspective, distinguishing emptiness from an all-pervasive base out of which phenomena are produced, and yet appreciates the spiritual value of the doctrine, it is a must for all who wish to know more of the more profound aspects of Buddhist philosophy.--P. J. H. (shrink)

In lieu of an abstract, here is a brief excerpt of the content:Desideri's Understanding of EmptinessEnzo Gualtiero BargiacchiThe works of Ippolito Desideri (1684–1733)1 lay forgotten in the archives for a very long time;2 had they been studied, European studies of Tibet and Buddhism would have begun a century earlier. The partial publication of his Relazione in 1904 was not enough to make scholars of Buddhism interested in the subject and resulted only a modest enthusiasm in the geographical and anthropological fields.3 (...) Later the Swedish geographer and explorer Sven Hedin (1865–1952) stated that Desideri "had accomplished a journey which ought to make his name forever famous,"4 and also that "by far the greatest part of Desideri's narrative is a description of the cult and religion of the Tibetans, which is not less admirable than the geographical part."5 But it was yet another geographer and explorer, Filippo De Filippi (1869–1938), who made available the Jesuit missionary's work by publishing an English version of the Relazione6 in 1932. Charles Oldham recognized that "there will no longer be any excuse for the ignorance and misconceptions regarding Desideri, to which Cornelis Wessels and De Filippi have drawn such pointed attention."7In reality, however, Desideri's work would still be awarded inadequate consideration for some time, apart from the generous praise of Giuseppe Tucci, which appeared exclusively in Italian articles and books, and of a few other scholars of Buddhism. This continued even after the critical edition of the Relazione and letters, expertly edited by Luciano Petech,8 despite the prestigious reputation of this academic and the ISMEO, the institute directed by Tucci that sponsored the series; very little attention was also paid to the translations by Saverian missionary Giuseppe Toscano9 (1911–2003) of Desideri's works in the Tibetan language.10 Around the turn of the millennium academics such as Donald Lopez, Rudolf Kaschewsky, and Martin Brauen began to pay closer attention to Desideri's work.In recent years there has been a renewed interest in the Jesuit missionary's work, as shown by the panel "Buddhism in the Writings of Ippolito Desideri" at the Fifteenth Congress of the International Association of Buddhist Studies (Atlanta, 23–28 June 2008), as well as two fundamental events concerning it; now Michael Sweet is completing the new English edition and translation of the Relazione.11Even today Desideri's writings can still provide academics in various disciplines with useful material for reflection, and not only in historical terms. As far as Tibetan religion is concerned, Desideri extensively discussed many important topics, in the Relazione and in his works in Tibetan, with considerable interpretative skill. [End Page 101] Unfortunately, none of this information reached the scholarly community, with the exception of a few rough ideas.12Roy Andrew Miller13 has pointed out his correct interpretation of the mantra oṁmaṇi padme hūṁ; more recently Rudolf Kaschewsky has maintained that Desideri's excellent linguistic analysis of this mantra "may mark the memorable beginning of Tibetology in the West."14 But the most surprising result is without doubt his perfect understanding of the Buddhist concept of emptiness. Tucci states thatSome aspects of Tibetan theology are very difficult to understand: for example, the concept of Ton pa gni—insubstantiality of all phenomenal things—in which, following the ancient mystical tendencies of the Mahayana, Buddhists dissolve all appearance and illusion, men and things, inexistent and unreal like a dream or mirage. Even today very few Westerners are able to attain a perfect understanding of the depth of this doctrine. It is therefore amazing how Desideri, with no knowledge of Indian philosophy, was able to grasp the real meaning of these difficult ideas.15Let us define here the reasons for this discovery and consider its profound meaning.Desideri certainly possessed remarkable linguistic and logical-philosophical aptitude, and he had honed and developed these skills during the challenging studies typical of Jesuit schools. He was furthermore assisted by a tenacious and heroic determination to fulfill his mission, but one of the decisive factors in explaining the reasons for his cognitive success lies in the complete honesty of his approach, as a man and a priest convinced of... (shrink)

He who has seen everything empty itself is close to knowing what everything is filled with. Emptiness is probably the most important philosophical and religious concept of Mahayana Buddhism. Its precise meaning has been explained differently by different schools and in different Buddhist cultures, but almost all Mahāyāna Buddhists would agree with the following characterization: Philosophically , emptiness is the term that describes the ultimate mode of existence of all phenomena, namely, as naturally ‘empty’ of enduring substance, (...) or self-existence : rather than being independently self-originated, phenomena are dependently originated from causes and conditions. Emptiness, thus, explains how it is that phenomena change and interact as they do, how it is that the world goes on as it does. Religiously , emptiness is the single principle whose direct comprehension is the basis of liberation from samsāra, and ignorance of which, embodied in self-gasping is the basis of continued rebirth – hence suffering – in samsāra. (shrink)

Lorsque nous parlons généralement d’un espace vide, ce vide n’est pas réellement vide physiquement. L’espace vide est rempli d’air, de matière invisible, mais il ne peut pas être vu ou capturé. Pourtant, nous savons bien que l’air, élément indispensable pour tous les êtres vivants, est également présent autour de nous. Devrions-nous alors considérer différemment l’espace vide et l’air dans l’installation? Pour Yasuaki Onishi, sculpteur et installateur in situ, l’espace vide est un lieu essentiel pour l’installation et l’inspiration. Il s’intéresse à (...) l’invisible, à l’espace, à l’air, à l’espace négatif, etc. Il révèle ce qui est invisible, en utilisant un matériau léger tel qu’un adhésif liquide noir et le support de matériaux immatériels tels que la lumière, l’air ou d’autres matériaux intangibles. Parmi ses travaux, en particulier, en analysant les œuvres de la série Vertical Volume, Daily Distance, Gawa, Shaping Air – A Breath of Mobility et Vertical Space, nous cherchons dans cette étude la signification de l’air invisible, du ki, du svi, du wonsang et le sens du vide du Tao et du bouddhisme. Quel est le sens du vide dans une installation de Yasuaki Onishi? Et quel est l’enjeu entre l’espace vide et l’espace plein, l’espace visible et l’espace invisible, la matérialité et l’immatérialité? Dans son travail, qui ne suit pas le processus général de la sculpture traditionnelle, nous trouverons le lien avec la signification de l’espace négatif et avec le terme coréen gan, l’espace-temps, avec le mot japonais ma qui constitue un concept esthétique au Japon. When we generally speak of an empty space, this empty is not really physically empty. The empty space is filled with air, invisible matter, but it cannot be seen or captured. However, we know that air, an essential element for all living things, is also present around us. So should we consider differently the empty space and air in the installation? For Yasuaki Onishi, sculptor and installator in situ, the empty space is an essential place for installation work and inspiration. The empty space that we think is empty is filled with substances that are invisible. The empty space is filled with air, but it can’t be seen or captured. He is interested in the invisible, space, air, negative space, etc. He reveals what is invisible, using a light material such as a black liquid adhesive, with the support of non-material materials such as light, air or other intangible materials. Among his works, in particular, by analyzing the works of the series Vertical Volume, Daily Distance, Gawa, Shaping Air – A Breath of Mobility and Vertical Space, we look for the meaning of invisible air, breath, ki, svi, the etymology of sunyata, wonsang 원상, and the meaning of emptiness of Tao and Buddhism. Through the work of Yasuaki Onishi, what is the meaning of the emptiness in his installation? And what is at stake between empty space and full space, visible space and invisible space, materialism and immateriality. In his work, which does not follow the general process of typical sculpture, we will find the connection with the meaning of “negative space” and the term Korean gan “간 ”, the space-time, as the Japanese word “ma ” that use as a concept of aesthetics in Japan. (shrink)

In lieu of an abstract, here is a brief excerpt of the content:Toward a Dialectics of Emptiness: Overcoming Nihilism and Combatting Mechanization in Nishitani Keiji’s Postwar ThoughtGriffin WernerIn his postwar writings on nihilism in modernity, Nishitani Keiji (1900–90) does not explicitly articulate the structure of the relationship between the mechanization of the world and nihilism. Instead, he discusses mechanization with respect to his critique of modern worldviews such as atheism, scientism, and liberalism and how they have contributed to the advent (...) of nihilism and the loss of human autonomy in modern Europe and Japan.1 According to Nishitani, before the advent of scientism and the subsequent progression of unfettered technological development, pre-Western-Enlightenment and pre-Meiji-period worldviews consisted of a fundamental embodied connection between a people or culture and their spiritual traditions. In Europe’s case, due to the death of God and the subsequent shift from a Christian worldview to an atheistic and scientistic one, a historical period of nihilism began. Modern mechanistic worldviews like Descartes’s, Nishitani argues, present the world as consisting of dead matter that is fundamentally separated from the indubitable ego. No longer is the world considered to be “alive” in the sense of there being a fundamental connection between human life and the living, breathing world around it. Rather, from the perspective of the ego, the modern world is simply a cold and lifeless world of dead machines.As discussed in Religion and Nothingness, the development of structurally Cartesian worldviews led to or caused the advent of nihilism in Europe. Consequently, European culture lost its cultural ground in Christianity and individuals lost their foundational meaning which supported their understanding of themselves in relation [End Page 129] to others, the world, and a transcendent God. As Japan began to modernize, or, arguably, “Westernize,” it inherited a Cartesian-like worldview and, over time, subsequently lost touch with its traditional philosophical and religious ground as well. As a result, the modern period of nihilism began in both Europe and Japan as both peoples lost touch with the traditional meaning that supported them.While this description between mechanization and nihilism implies a causal relationship, I argue that the relationship between nihilism and the mechanization of the world can be more accurately articulated in terms of a dialectical relationship. In the modern context, not only does the mechanization of the world lead to or exacerbate a culture’s plunge into nihilism or an individual’s confrontation with the abyss of nihility, but nihilism also reinforces and exacerbates the mechanization of the world. As partners in this dialectical oscillation, each reinforces the other as one (culture or person) continuously spirals deeper and deeper into meaninglessness, ultimately to the culminating point that Nishitani calls the “mechanization of man.”2Just as Nishitani argues that the problem of nihilism can only be solved by way of radicalizing nihilism to the “field of emptiness,” the structural oscillation between nihilism and mechanization will continue at the levels of the individual and the culture unless nihilism and mechanization are overcome. However, rather than only overcoming nihilism to reach emptiness in the individual, and that leading to social and political change, I argue that overcoming the dialectical tension between nihilism and mechanization will require a “dialectics of emptiness.” Such a dialectic will combat nihilism and mechanization with a mutually reinforcing tension between personal transformation, which overcomes nihilism by way of stepping back to the field of emptiness, and socio-political transformation by way of a politics of emptiness. Solving the problem of nihilism in modernity—insofar as it is always already in relation to mechanization—will require more than individual existential confrontation and transformation. Through one’s individual overcoming of nihilism, one must also aim to “empty” those institutions which contribute to the mechanization of the world. While Nishitani highlights the individual’s religious quest as the impetus for the social and political change necessary to overcome the problem of nihilism at the cultural level, an analysis of how that might be done and what [End Page 130] may be politically necessary in order to do so is missing from his thought.As such, my aim in this article is twofold: I attempt to shed light on the structure of nihilism and... (shrink)

Christopher Hom has recently argued that the best-overall account of the meaning of pejorative terms is a semantic account according to which pejoratives make a distinctive truth-conditional contribution, and in particular express complex, negative socially constructed properties. In addition, Hom supplements the semantic account with a pragmatic strategy to deal with the derogatory content of occurrences of pejorative terms in negations, conditionals, attitude reports, and so on, according to which those occurrences give rise to conversational implicatures to the effect that (...) the pejorative terms are non-empty, which explains the offensiveness of those occurrences. In this paper, I aim to defend this semantic strategy from several recent objections, and I will also present a novel objection, which in my view shows that we should understand the semantic account as a version of inferentialism, rather than radical externalism. (shrink)

This article seeks to introduce a greater degree of precision into our understanding of Madhyamaka Buddhist ontological non-foundationalism, focussing specifically on the Madhyamaka founder Nāgārjuna (c. 150–250 CE). It distinguishes four senses of what the ‘foundation’ whose existence Mādhyamikas deny means; that is, (1) as ‘something that stands under or grounds things’ (a position known as generic substantialism); (2) as ‘a particular kind of basic entity’ (specific substantialism); (3) as ‘an individual essence (a haecceity or thisness of that object) by (...) means of which it is identical to that very object, to itself’ (modal essentialism); and (4) as ‘an essence in the absence of which an object could be of a radically different kind or sort of object than it in fact is’ (sortal essentialism). It then proceeds to delineate the Madhyamaka refutation of the specific substantialist position in terms of its argued denial of dharma as basic entity; of generic substantialism and modal essentialism in terms of its argued denial of svabhāva as both foundation for and essence of putative entities; and of sortal essentialism in terms of its argued denial of essentialist conceptions of conceptual thought (vikalpa), mental construction (prapañca), and in short the entire domain of ratiocination (kalpanā), by means of its notion of conceptual imputation (prajñaptir upādāya)—a denial strictly speaking ontological, but of what are putative epistemic entities. The final portion of the article explains the relationship of ontological to other forms of non-foundationalism according to Madhyamaka. (shrink)

The present study has attempted to artriculate a central issue of Mahäyäna soteriology through an examination of the writings of two Mädhyamika masters, Bhävaviveka and Candrakïrti. The purpose here has been to demonstrate a further criterion for the retrospective designation of their respective philosophies with the terms “Svātantrika” and “Prasangika” an exhaustive study of the nature of the Hinayäna wisdom according to the Mädhyamika school would entail an analysis of the writings of many other masters, especially those who produced what (...) has been called the Yogäcära-Mädhyamika synthesis. To attempt to determine the position of Maitreyanätha, for example, on this issue would entail an analysis of the famous “Five Treatises” (the Dharmadharmatävibhańa, the Madhyäntavibhańga, the Mahayanasutrlamkara, the Uttratantra, and the Abhisamayälamkära) as well as the myriad commentaries on these works. It is possible to speculate briefly here on what the position of Nägärjuna may have been on this issue and then go on to discuss the implications and possible motivations of the views of Bhävaviveka and Candrakïrti. (shrink)

Is there a vacuum in nature? This is a question which preoccupied natural philosophers for millennia. Great thinkers including Democritus and Newton maintained the existence of a vacuum, while Aristotle, Descartes and Leibniz argued strongly that there was not, and perhaps could not be, any such thing. A casual glance at the literature of contemporary physics may leave the impression that scientific progress has produced a definitive positive answer, so that the philosophers' debates are now of only historical interest. Not (...) only is the attainment of high vacua a multimillion dollar industry, but almost every text and research paper in theoretical high energy or condensed matter physics or cosmology includes multiple references to the vacuum and its often surprising properties. And yet we have it on the authority of no less a scientist than Einstein himself that his general theory of relativity vindicates Descartes' conclusion that there could be no vacuum since the idea of space without matter is unintelligible! Does this mean that there is after all no scientific consensus on this issue, and that the ancient philosophical debate should simply be resumed with renewed vigor? I shall argue that it does not. The progress of physics has given rise to such a proliferation of different uses of the word 'vacuum' as to irretrievably alter the terms of that debate. Rather than asking whether there is a vacuum in nature, one should ask how well what counts as the vacuum in some physical theory represents reality. This in turn splits up into a semantic question and an empirical question I shall focus mainly on the prior, semantic question here. (shrink)

This paper is concerned with techniques for identifying simple and quantified lattice points in 2SAT polytopes. 2SAT polytopes generalize the polyhedra corresponding to Boolean 2SAT formulas, Vertex-Packing and Network flow problems; they find wide application in the domains of Program verification and State-Space search . Our techniques are based on the symbolic elimination strategy called the Fourier-Motzkin elimination procedure and thus have the advantages of being extremely simple and incremental. We also provide a characterization of a 2SAT polytope in terms (...) of its extreme points and derive some interesting hardness results for associated optimization problems. Finally, we provide a brief discussion on the maximum size of a subdeterminant of the linear system representing a 2SAT polytope; this parameter plays a vital role in deriving analytical bounds on the size of the search space for checking whether the polyhedron includes a lattice point. (shrink)