In lieu of an abstract, here is a brief excerpt of the content:Ethics & the Environment 7.1 (2002) 102-108 [Access article in PDF] Symposium IntroductionEric Katz's Nature As Subject Andrew Light Can and should we distinguish between nature and culture? The question has become a perennial one in environmental ethics, as well as in allied fields in environmental history, sociology, and politics. And just when we think it is settled—as many did after William Cronon's famous deconstruction of wilderness in (...) 1995—and it feels safe to go back into the metaphysical waters of environmental philosophy, such issues pop up again to send us running.What follows extends this debate in the context of Eric Katz's well-known work in environmental ethics. Brought together here are Katz and two critics sympathetic to his positions, Wayne Ouderkirk and Ned Hettinger. While the occasion for the exchange was the publication of Katz's book, Nature as Subject: Human Obligation and Natural Community (1997), which collects his papers from 1979 to 1996, the focus quickly became somewhat narrower, centering on possible problems with Katz's ontological dualism between natural and artifactual entities.Katz has exerted a strong influence on the development of environmental ethics, particularly through his persistent arguments for a distinction between individualist and holist approaches to obligations to the nonhuman natural world and his metaphysical and moral critique of attempts [End Page 102] by humans to create, repair, or restore nature. While these positions continue to be on display here, there is also something more. In this exchange Katz throws down the gauntlet on his nature-culture dualism, admitting at the start that the accusation is true. Like the Protestants in Monty Python's The Meaning of Life, he is a dualist—and fiercely proud of it! Despite whatever misgivings I or others may have about this position, I believe that in making this claim, Katz, similar to David Chalmers's open embrace and resuscitation of dualism in philosophy of mind, has done us a great service. For it is arguably true that many environmental philosophers, along with other academics, have settled into a complacent "labeling as argument" pattern. If we can label a position as "dualist" or "essentialist," then our work is done. But surely not every form of dualism is necessarily wrong, even if, as Ouderkirk reminds us in his reading of Val Plumwood's work, the logic of many forms of dualism has tended toward a pernicious pattern. Arguments are needed in both directions. If nature-culture dualism is mistaken or wrongheaded, or leads to unfortunate conclusions, then we must know why; if dualism is the only way to make sense of the hurdles presented to us in coming to a better understanding of the value of nature and culture then we must know why as well. And further, because many in the field have assumed a nature-culture dualism without admitting to it, we can admire that Katz chooses to "fess up" and finally put the issue squarely on the table.If this is the road we are heading down with this particular exchange, then we are off to a good start. Both critics of Katz do a thorough job of pressing home their arguments, rather than only pointing accusatory fingers. Ned Hettinger, in "The Problem of Finding a Positive Role for Humans in the Natural World," avoids using the term "dualism," but nonetheless argues that Katz has a tendency to "rigidly dichotomize the natural/artifactual distinction." Additionally, Hettinger argues that Katz tends to overlook his own admission that the differences between nature and artifacts admit to degrees rather than a strict dualism. Hettinger focuses the bulk of his critique on the unfortunate upshot of such a position: given such a distinction, there is little if any place for a positive role for human intervention in nature, or for a philosophical contribution to the formation of better environmental policies.Wayne Ouderkirk, in "Katz's Problematic Dualism and its 'Seismic' Effects on His Theory," is sympathetic with Katz's intuitions that some distinction is needed between artifacts and natural entities. The problem [End Page 103] he sees is that such... (shrink)

The ethical and social implications of data mining, algorithmic curation and automation in the context of social media have been of heightened concern for a range of researchers with interests in digital media in recent years, with particular concerns about privacy arising in the context of mobile and locative media. Despite their wide adoption and economic importance, mobile dating apps have received little scholarly attention from this perspective – but they are intense sites of data generation, algorithmic processing, and cross-platform (...) data-sharing; bound up with competing cultures of production, exploitation and use. In this paper, we describe the ways various forms of data are incorporated into, and emerge from, hook-up apps’ business logics, socio-technical arrangements, and cultures of use to produce multiple and intersecting data cultures. We propose a multi-layered research agenda for critical and empirical inquiry into this field, and suggest appropriate conceptual and methodological frameworks for exploring the social and political challenges of data cultures. (shrink)

For thirty years at least, I have designated myself as a zoologist interested in the “philosophical aspects of biology”. But I have now to admit that not until within the last two or three years have I recognized that logic, particularly in its inductive aspect, is involved in such interest.For me as a zoologist with a predilection for natural history, observation has had a place of wide application and of implicit confidence. Until recently, I had rested in the supposition that (...) after all of the devices practiced by naturalists to insure certainty of their observations, nothing further was needed so far as the information and understanding they gained was concerned. I had gone ahead with my reasoning, my logic, with no more qualms about the trustworthiness of the “inductive” than of the “deductive” side of it. I had not taken seriously the assertion, made years ago in my hearing by an eminent philosopher, that no one can be a philosopher without being skeptical and critical of the use of his senses in getting knowledge. Latterly, however, I have come to see that traditional philosophers take themselves very seriously in such assertions. This realization has been due to the discovery that the recent advances in physics and mathematics have led most of the leaders in these fields to believe that observation and the inductive side of logic may be left where they were a hundred years ago. These scientists seem to be about where John Stuart Mill was in his effort to improve Lord Bacon's effort for a better understanding of both observation and induction. (shrink)

There seems to be something mysterious about applications of for- mal systems, including those of logic, to empirical reality 1 . If logic is to be applied to empirical situations, like those described in an ordinary lan- guage, then { it seems to some people { its statements cannot be necessary, or analytic, propositions. However, they are both applicable and necessary. This supposed puzzle constitutes a signicant part of the problem of philo- sophical foundations of logic 2 . To this (...) mind of the present writer, there is no mystery of applications, since any empirical, or even ostensive, predicate can be involved in cer- tain meaning postulates, e.g. \No red is green", which are both empirical and necessary propositions; empirical as they involve ostensibly dened predicated; necessary, as it is enough to know their meaning to state, their validity. The case of logical theorems can be considered at the same footing, logical constants being treated as dened ostensively. Nevertheless, not an individual state of mind but the actual state of scholarly discussions makes something a problem; in the present state of foundational discussions on logic, the \mystery of applications" preserves its vitality, hence any new way of dealing with it proves welcome. (shrink)

This book analyses and defends the deflationist claim that there is nothing deep about our notion of truth. According to this view, truth is a 'light' and innocent concept, devoid of any essence which could be revealed by scientific inquiry. Cezary Cieśliński considers this claim in light of recent formal results on axiomatic truth theories, which are crucial for understanding and evaluating the philosophical thesis of the innocence of truth. Providing an up-to-date discussion and original perspectives on this (...) central and controversial issue, his book will be important for those with a background in logic who are interested in formal truth theories and in current philosophical debates about the deflationary conception of truth. (shrink)

In this collection of essays written over a period of twenty years, Solomon Feferman explains advanced results in modern logic and employs them to cast light on significant problems in the foundations of mathematics. Most troubling among these is the revolutionary way in which Georg Cantor elaborated the nature of the infinite, and in doing so helped transform the face of twentieth-century mathematics. Feferman details the development of Cantorian concepts and the foundational difficulties they engendered. He argues that the (...) freedom provided by Cantorian set theory was purchased at a heavy philosophical price, namely adherence to a form of mathematical platonism that is difficult to support. -/- Beginning with a previously unpublished lecture for a general audience, Deciding the Undecidable, Feferman examines the famous list of twenty-three mathematical problems posed by David Hilbert, concentrating on three problems that have most to do with logic. Other chapters are devoted to the work and thought of Kurt Gödel, whose stunning results in the 1930s on the incompleteness of formal systems and the consistency of Cantors continuum hypothesis have been of utmost importance to all subsequent work in logic. Though Gödel has been identified as the leading defender of set-theoretical platonism, surprisingly even he at one point regarded it as unacceptable. -/- In his concluding chapters, Feferman uses tools from the special part of logic called proof theory to explain how the vast part--if not all--of scientifically applicable mathematics can be justified on the basis of purely arithmetical principles. At least to that extent, the question raised in two of the essays of the volume, Is Cantor Necessary?, is answered with a resounding no. -/- This volume of important and influential work by one of the leading figures in logic and the foundations of mathematics is essential reading for anyone interested in these subjects. (shrink)

Previous discussions of Robb’s work on space and time have offered a philosophical focus on causal interpretations of relativity theory or a historical focus on his use of non-Euclidean geometry, or else ignored altogether in discussions of relativity at Cambridge. In this paper I focus on how Robb’s work made contact with those same foundational developments in mathematics and with their applications. This contact with applications of new mathematical logic at Göttingen and Cambridge explains the transition from his electron research (...) to his treatment of relativity in 1911 and finally to the axiomatic presentation in 1914 in terms of postulates. At the heart of Robb’s physical optics was the model of the light cone. The model underwent a transition from a working mechanical model in the Maxwellian Cambridge sense of a pedagogical and research tool to the semantic model, in the logical, model-theoretic sense. Robb tracked this transition from the 19th- to the 20th-century conception with the earliest use of the term ‘model’ in the new sense. I place his cone models in a genealogy of similar models and use their evolution to track how Robb’s physical researches were informed by his interest in geometry, logic and the foundations of mathematics. (shrink)

Logical theory codifies rules of correct inferences. On the other hand, logical reasoning is typically considered as one of the most fundamental cognitive activities. Thus, cognitive science is a natural meeting-point for investigations about the place of logic in human cognition. Investigations in this perspective strongly depend on a possible understanding of logic. This paper focuses on logic in the strict sense; that is, the theory of deductive inferences. Two problems are taken into account, namely: do humans apply logical rules (...) in ordinary reasoning?; the genesis of logic. The second issue is analyzed from the naturalistic point of view. (shrink)

We investigate a subsystem of minimal logic related to D. Vakarelov’s logic \, using the framework of subminimal logics by A. Colacito, D. de Jongh and A. L. Vargas. In the course of it, the relationship between the two semantics in the respective frameworks is clarified. In addition, we introduce a sequent calculus for the investigated subsystem, and some proof-theoretic properties are established. Lastly, we formulate a new infinite class of subsystems of minimal logics.

Uses erotetic logic to model the courtroom objection "Asked and Answered!".

논리적 귀추주의란 논리학의 형이상학적, 인식론적 예외성을 인정하지 않고, 논리학 또한 과학과 유사하게 귀추적으로 평가되어야 한다는 주장이다. 필자는 이 글에서, 고전논리학과 양진주의의 사례를 통해, 귀추주의가 성립하지 않음을 보일 것이다. 이러한 필자의 주장은 ‘이론적 폐쇄성’에 기초한다. 이론적 폐쇄성이란, 논리학 이론은 형식적인 연산체계뿐 아니라 논리철학적 주장을 포함하는데, 이러한 논리철학적 주장은 논리학을 평가하는 귀추적 기준을 함축한다는 것이다. 간단히 말해, 논리이론은 귀추적 평가기준을 포함하는 폐쇄적 구조를 갖기 때문에, 경쟁하는 논리이론을 평가하는 독립적 기준을 제시할 수 없다는 것이다. 물론, 이러한 귀추주의의 문제점을 필자가 처음 제시한 것은 아니다. (...) 귀추주의에 대한 기존의 비판과의 차이점은, 필자의 논의는 귀추주의의 문제를 ‘이론적 폐쇄성’이라는 논리학의 특성에 기초해서 제시한다는 것 그리고 그것에 기초해서 귀추주의자들의 가능한 반박에 대응한다는 것이다. (shrink)

I urge return by the lights of logic and commonsense to a dialectical tabula rasa – according to which: (1) consciousness, in the ordinary pre-analytic sense of the term, is identified with P-consciousness, and “A-consciousness” is supplanted by suitably configured terms from its Blockian definition; (2) the supposedly fallacious Searlean argument for the view that a function of P-consciousness is to allow flexible and creative cognition is enthymematic and, when charitably specified, quite formidable.

Light Linear Logic (LLL) and Intuitionistic Light Affine Logic (ILAL) are logics that capture polynomial time computation. It is known that every polynomial time function can be represented by a proof of these logics via the proofs-as-programs correspondence. Furthermore, there is a reduction strategy which normalizes a given proof in polynomial time. Given the latter polynomial time “weak” normalization theorem, it is natural to ask whether a “strong” form of polynomial time normalization theorem holds or not. (...) In this paper, we introduce an untyped term calculus, called Light Affine Lambda Calculus (λLA), which corresponds to ILAL. λLA is a bi-modal λ-calculus with certain constraints, endowed with very simple reduction rules. The main property of LALC is the polynomial time strong normalization: any reduction strategy normalizes a given λLA term in a polynomial number of reduction steps, and indeed in polynomial time. Since proofs of ILAL are structurally representable by terms of λLA, we conclude that the same holds for ILAL. (shrink)

When my son Maxwell was a toddler, he did not believe he was ever an infant. This skepticism became manifest when he started identifying himself in photographs. Maxwell was accurate with photographs that were taken after age six months. But he dismissed earlier pictures as photographs of "BABIES".

The aim of this paper is to strengthen the point made by Horty about the relationship between reason holism and moral particularism. In the literature prima facie obligations have been considered as the only source of reason holism. I strengthen Horty’s point in two ways. First, I show that contrary-to-duties provide another independent support for reason holism. Next I outline a formal theory that is able to capture these two sources of holism. While in simple settings the proposed account coincides (...) with Horty’s one, this is not true in more complicated or “realistic” settings in which more than two norms collide. My chosen formalism is so-called input/output logic. A bottom-line example is introduced. It raises the issue of whether the conventional wisdom is right in assuming that normative reasons run parallel to epistemic ones. (shrink)

In this paper I discuss the consistency and accuracy of Husserl’s sketch of a theory about non-declarative sentences in the last chapter of Logical Investigations. Whereas the consistency is acknowledged, the accuracy is denied, because Husserl’s treatment of non-declarative phrases such as questions or orders implies that those phrases contain, in some way, a declarative sentence and an objectifying act. To construct a question like »is A B?« as being equivalent to a declarative sentence such as »I ask whether A (...) is B« is, indeed, a false phenomenological analysis, because to ask or to order or to beg is not to assert. I turn, then, to John Austin’s theory of performative utterances and illocutionary acts in order to find a more accurate approach to the logical-semantic content of non-declarative sentences. Eventually, I show how this Husserlian theory of non-declarative sentences has a negative impact on the phenomenological theory of social acts and communication. (shrink)

Light version of DEMO for composing epistemic models, based on the code for the ESSLLI 2008 course on Dynamic Epistemic Logic (see http:// homepages.cwi.nl/~jve/courses/esslli08/) extended with vocabulary information [EWS10]. Factual change is also treated. The piece ends with some examples: the muddy children, and hat puzzles, dealing with the interaction of perception and change [Eijar].

Excerpt from Logic, Vol. 1: Deduction The present work aims at embracing a full course of Logic, both Formal and Inductive. In an introductory chapter, are set forth such doctrines of psychology as have a bearing on Logic, the nature of knowledge in general, and the classification of the sciences the intention being to avoid doctrinal digressions in the course of the work. Although preparatory to the under standing of what follows, this chapter may be passed over lightly on a (...) first perusal of the work. The part on Deduction contains the usual doctrines of the Syllogism, with the additions of Hamilton, and a full abstract of the novel and elaborate schemes of De Morgan and Boole. The Inductive portion comprises the methods of induc tive research, and all those collateral topics brought for ward by Mr. Mill, as part of the problem of Induction; various modifications being made in the manner of state ment, the order of topics, and the proportion of the hand ling. The greatest innovation is the rendering of Cause by the new doctrine called the Conservation, Persistence, or Correlation of Force. Mr. Mill's view of the relation of Deduction and Induc tion is fully adopted, as being the solution of the otherwise inextricable puzzle of the syllogism, and the means of giving unity and comprehensiveness to Logic. About the Publisher Forgotten Books publishes hundreds of thousands of rare and classic books. Find more at www.forgottenbooks.com This book is a reproduction of an important historical work. Forgotten Books uses state-of-the-art technology to digitally reconstruct the work, preserving the original format whilst repairing imperfections present in the aged copy. In rare cases, an imperfection in the original, such as a blemish or missing page, may be replicated in our edition. We do, however, repair the vast majority of imperfections successfully; any imperfections that remain are intentionally left to preserve the state of such historical works. (shrink)

'From Logic to Politics' is a study of a promise: the promise of philosophy as universal science that could comprehend, connect and guide, and of politics elucidated and made transparent for thinking. Hegel's Philosophy of Right is built on that promise, and set on the goal of grasping its time and providing the ground of modern politics in the idea of right. The fulfilment of this task is made possible in the light of the first, fundamental science, which for (...) Hegel, has to replace metaphysics - the Science of Logic. This is, therefore, a study of the inner transformation of a logical doctrine into an ethico-political one. It attempts to answer the following questions: How does an onto-logical doctrine extend its central vision to the reality of human passion, action and self-interest? Can this be done and is this central vision strong enough to sustain itself in such an extension? What is the status of philosophical truth when exposed to the light, or darkness, of politics? (shrink)

ASSERTIONS ARE A SYMBOLIC FORM that exists only within species--specifically human language. Language, of course, allows for many other conventions of symbolic expression: greetings, exclamations, commands, exhortations, imprecations, interrogatives, and so forth. But assertions are unique in possessing, of themselves, a truth-value--that is to say, in being adjudicable as true or false. All other varieties of discourse are adjudicable as true or false by reason of assertions they presuppose, contain, or imply. But the assertion as such is what is directly (...) so adjudicable. (shrink)

Logical form has always been a prime concern for philosophers belonging to the analytic tradition. For at least one century, the study of logical form has been widely adopted as a method of investigation, relying on its capacity to reveal the structure of thoughts or the constitution of facts. This book focuses on the very idea of logical form, which is directly relevant to any principled reflection on that method. Its central thesis is that there is no such thing as (...) a correct answer to the question of what is logical form: two significantly different notions of logical form are needed to fulfil two major theoretical roles that pertain respectively to logic and to semantics. This thesis has a negative and a positive side. The negative side is that a deeply rooted presumption about logical form turns out to be overly optimistic: there is no unique notion of logical form that can play both roles. The positive side is that the distinction between two notions of logical form, once properly spelled out, sheds light on some fundamental issues concerning the relation between logic and language. (shrink)

The rapid innovations in digital technology deeply influence views on language and information processing. These exciting developments raise many questions for researchers, and shed new light on old approaches. Researchers are drawn to closely investigate the relation between form and content, the ways that linguistic utterances change information content, and the dynamics of information change. Logic, as an established method of valid argumentation, is a tool that researchers can use to gain insight in these questions of language and computation. (...) In answer to these linguistic and computational questions, this book presents a broad range of logical investigations into language and information processing. The topics surveyed include the notion of 'reasonable belief' in commonsense reasoning, perceptual reports in natural languages, and the logic of creation and modification of objects. The book also presents work on the verification of temporal aspects of reactive systems, analysis of scope by combining model theory and situation semantics and semantic analysis of the information articulation of linguistic statements. This volume exemplifies the abundant development of logical methods and tools in an interdisciplinary context. (shrink)

Ponieważ każda księga heglowskiej "Logiki" przedstawia tylko jej właściwy sposób przejawiania się bytu, więc zakłada ona także - określoną przez ten sposób przejawiania się bytu - charakterystyczną wykładnię prawdy. Niedowolna kolejność poszczególnych ksiąg określa wtedy także porządek, w jaki układają się założone w nich wykładnie prawdy. Skorelowanie klasycznej i innych wykładni prawdy z odpowiednimi księgami "Logiki" pozwala więc wykorzystać porządek "Logiki” do odtworzenia struktury całości prawdy, tzn. także do objaśnienia miejsca, jakie klasyczna wykładnia zajmuje zarówno w obrębie całości prawdy, jak (...) i wśród wykładni pozostałych. (shrink)

The unique relation between logic and truth is crucial for understanding Fregean conception of logic. Frege has an insight that the nature of logic resides in the “truth“, which he finally locates in the assertoric-force of a sentence. Though Frege admits that assertoric-force is ineffable in ordinary language, he coins in his conceptual notation for such a force a much-disputed sign, i.e., judgment-stroke. In this paper, I will try to demonstrate that judgment-stroke is not adequate for the task its inventor (...) has assigned to it. Accordingly, it is misconceived and inconducive to clarify Frege’s vague insight into the protorelation. The mistake of judgment-stroke for the sign of assertoric-force has its root in Frege’s ignorance of the significant difference between “judgment” and assertion”, which will be elucidated at length in the light of Husserl’s theory of “doxic-modification“. In the end, based on a further elucidation of the activity of assertion, I will advance a tentative interpretation of the vague insight Frege has concerning the protorelation. (shrink)

This volume offers a wide range of both reconstructions of Nikolai Vasiliev’s original logical ideas and their implementations in the modern logic and philosophy. A collection of works put together through the international workshop "Nikolai Vasiliev’s Logical Legacy and the Modern Logic," this book also covers foundations of logic in the light of Vasiliev’s contradictory ontology. Chapters range from a look at the Heuristic and Conceptual Background of Vasiliev's Imaginary Logic to Generalized Vasiliev-style Propositions. It includes works which cover (...) Imaginary and Non-Aristotelian Logics, Inconsistent Set Theory and the Expansion of Mathematical Thinking, Plurivalent Logic, and the Impact of Vasiliev's Imaginary Logic on Epistemic Logic. The Russian logician, Vasiliev, was widely recognized as one of the forerunners of modern non-classical logic. His "imaginary logic" developed in some of his work at the beginning of 20th century is often considered to be one of the first systems of paraconsistent and multi-valued logic. The novelty of his logical project has opened up prospects for modern logic as well as for non-classical science in general. This volume contains a selection of papers written by modern specialists in the field and deals with various aspects of Vasiliev's logical ideas. The logical legacy of Nikolai Vasiliev can serve as a promising source for developing an impressive range of philosophical interpretations, as it marries promising technical innovations with challenging philosophical insights. (shrink)

My purpose in this paper is to shed some light on two questions: In what sense is logic philosophical? And what is philosophical logic? I take these two questions as co-extensive: an answer to one of them is also (or can easily be converted into) an answer to the other. I approach the problem from three perspectives: a conceptual, a descriptive and a prescriptive perspective. In other words, I try to answer the following questions: (i) In what sense can (...) logic be taken as philosophical? (ii) In what sense has logic been taken as philosophical? (iii) In what sense should logic be taken as philosophical? To this end, I analyze excerpts from five works in which meanings are attributed to the expression “philosophical logic”. The meanings are then object of critical analysis: I try to assess which of these semantical alternatives do not hold up as satisfactory answers to (iii). The result of this analysis is then used as a starting point for my own answer to question (iii). (shrink)

In “ A Light Theory of Color”, Walter Sinnott-Armstrong and David Sparrow argue that color is neither a primary quality of objects, nor a disposition that objects have, nor a property of our visual fields. Rather, according to the view they present, color is a property of light. The present paper aims to show, first, that the light theory is vulnerable to many of the very same objections that Sinnott-Armstrong and Sparrow raise against rival views. Second, the (...) paper argues that the strategies that Sinnott-Armstrong and Sparrow use to avoid certain objections are also available to proponents of rival accounts. This might only seem to show that the light theory is in the same shaky boat as other theories: suffering from the same problems but having the same tools for solving them. The paper concludes with a suggestion as to why this is not the case, but why the existence of the light theory is nevertheless likely to bring increased clarity to the debate about color realism. (shrink)

Broadly speaking, two views of modernity are prevalent in contemporary debates. According to the first view, i.e. “modernization theory,” there is one single form of modernity, which is tantamount to liberal, capitalist modernity. The West has already and fully achieved modernity; non-Western societies have lagged behind and must simply catch up with the West. In contrast, according to the second view, “post-colonial theory,” there is no such thing as modernity. What the West erroneously calls “modernity” is nothing but a highly (...) parochial way of thought, and of economic and political organization, that developed by accident first in the West and then by the exercise of military violence or economic power was subsequently “universalized.” In this chapter, I argue that Hegel, read properly, provides a third conception of modernity, one which, contrary to the modernization theory, respects differences between societies, and yet, contrary to post-colonial theory, does not engage in dividing the human species into fundamentally different or incommensurate groups. I argue that Hegel’s theory of modernity must be sought at the conjunction of the Science of Logic (specifically the logic of the Concept) and the Philosophy of Right. I then elaborate on Hegel’s theory of modernity, thus construed. (shrink)

Informal logic offers a radical new perspective on the evaluation of arguments. But little work has been done on how deep concepts in the logical foundations of argument need to be modified in light of such efforts. This paper offers an indication of what might be done by sketching a new approach to the theory of entailment, truth and relevance.

This paper propounds a systematic examination of the link between the Knower Paradox and provability interpretations of modal logic. The aim of the paper is threefold: to give a streamlined presentation of the Knower Paradox and related results; to clarify the notion of a syntactical treatment of modalities; finally, to discuss the kind of solution that modal provability logic provides to the Paradox. I discuss the respective strength of different versions of the Knower Paradox, both in the framework of first-order (...) arithmetic and in that of modal logic with fixed point operators. It is shown that the notion of a syntactical treatment of modalities is ambiguous between a self-referential treatment and a metalinguistic treatment of modalities, and that these two notions are independent. I survey and compare the provability interpretations of modality respectively given by Skyrms, B. (1978, The Journal of Philosophy 75: 368–387) Anderson, C.A. (1983, The Journal of Philosophy 80: 338–355) and Solovay, R. (1976, Israel Journal of Mathematics 25: 287–304). I examine how these interpretations enable us to bypass the limitations imposed by the Knower Paradox while preserving the laws of classical logic, each time by appeal to a distinct form of hierarchy. (shrink)