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A Logical Analysis of Some Value Concepts

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The purpose of this paper is to provide a partial logical analysis of a few concepts that may be classified as value concepts or as concepts that are closely related to value concepts. Among the concepts that will be considered are striving for, doing, believing, knowing, desiring, ability to do, obligation to do , and value for . Familiarity will be assumed with the concepts of logical necessity, logical possibility, and strict implication as formalized in standard systems of modal logic (such as S4), and with the concepts of obligation and permission as formalized in systems of deontic logic. It will also be assumed that quantifiers over propositions have been included in extensions of these systems.

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... Our contribution lies in this stream of research. Indeed, logical investigations have made formally explicit salient aspects for understanding the concept of a secret, such as the distinction between knowing a secret ( [25]) and keeping/intending to keep a secret 6 . Nevertheless, much more work needs to be done. ...
... The third conjunct expresses item 3. and represents an unveiling component formalizing the intention of agent a to sustain b's ignorance about ϕ. The specific condition of ignorance for agent b is called factive ignorance, which goes back to Fitch [6,11,1,2,23] and is formalized as T b ϕ := ϕ ∧ ¬K b ϕ (this motivates the reference to the concept of true secret ). ...
... This is coherent with our modeling framework, which is not dynamic and cannot represent situations in which evolving dynamics make ϕ not worth being protected anymore. Item (6) suggests that keeping something secret is equivalent to keeping secret its secrecy. At the same time, item (7) is a direct consequence of the notion of a true secret: no agent can know the secrecy of a given proposition ϕ without knowing that ϕ is true. ...
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Logical investigations of the notion of secrecy are typically concentrated on tools for deducing whether private information is well hidden from unauthorized, direct, or indirect access attempts. This paper proposes a multi-agent, normal multi-modal logic to capture salient features of secrecy. We focus on the intentions, beliefs and knowledge of secret keepers and, more generally, of all the actors involved in secret-keeping scenarios. In particular, we investigate the notion of a true secret, namely a secret concerning information known (and so true) by the secret keeper. The resulting characterization of intending to keep a true secret provides useful insights into conditions ensuring or undermining secrecy depending on agents' attitudes and links between secrets and their surrounding context. We present soundness, completeness, and decidability results of the proposed logical system. Furthermore, we outline some theorems with applications in several contexts from computer science and social sciences.
... 542-3). 30 See Fitch (1963). It is now known that the argument is originally due to Alonzo Church, appearing in a referee report of an earlier paper by Fitch. ...
... 36 What all of this suggests is that a characterization of valid inference in terms of the preservation of being in a position to know or is knowable or something similar simply does not work. 34 Moreover, this is where one can appeal to Fitch's (1963) paradox to argue that the claim that all truths are knowable entails the far more implausible claim that all truths are known. 35 There are other possible culprits. ...
... 13) for further details of his view. 47 It is also worth mentioning that even if A→OA is not immediately implausible for some operator O (e.g., "is in principle justifiable"), a version of Fitch's (1963) paradox may arise that leads to a much more implausible conclusion ("is in fact justified"). See San (2020) for discussion of how versions of Fitch's paradox can arise very generally, without the need for a factive operator. ...
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How should we understand validity? A standard way to characterize validity is in terms of the preservation of truth (or truth in a model). But there are several problems facing such characterizations. An alternative approach is to characterize validity epistemically, for instance in terms of the preservation of an epistemic status. In this paper, I raise a problem for such views. First, I argue that if the relevant epistemic status is factive, such as being in a position to know or having conclusive evidence for, then the account runs into trouble if we endorse certain familiar logical principles. Second, I argue that if the relevant epistemic status is non-factive, such as is rationally committed to or has justification for believing, then a similar problem arises if we endorse the logical principles as well as a sufficiently strong epistemic “level-bridging” principle. Finally, I argue that an analogous problem arises for the most natural characterization of validity in terms of rational credence.
... The title of this seminal book is Knowledge and beliefs: An introduction to the logic of the two notions (Hintikka 1962). In this work, Hintikka provides a propositional axiomatization of the modal operators K (for knowledge) and B (for belief), as well as insights on the meaning of various forms of knowledge and lack of knowledge, like, e.g., not knowing whether f (Hintikka 1962: 3), which is represented by Hintikka as ~Kf Ù ~K~f (Hintikka 1962: 12). 1 This form of lack of knowledge, which we refer as ignorance whether and will denote in the following with I(f), has indeed been widely investigated in several logical frameworks focused on ignorance, see, e.g., Fan et al. 2015, Steinsvold 2008, and van der Hoek and Lomuscio 2004 Such a representation of ignorance might be considered stronger than another classical definition, referred as unknown truth (Fitch 1963). In this alternative view, being ignorant of f simply stands for not knowing that f, which is formally expressed as fÙ~Kf. ...
... As mentioned above, Fan's ignorance Ñ(f) is the disjunction of I(f) and If(f), which are the two most prominent first-order definitions of Fig. 1. Factive ignorance triggers a loop of ignorance, as suggested by If(f) ® ~ K (If(f)), which is a result obtained in Fitch 1963. The main result given in Fine 2018 concerns the properties of second-order ignorance in the case of I(f), which induces both first-order ignorance and third-order ignorance, thus triggering the loop towards higher orders. ...
... In the following, we recall some results concerning the hierarchies of ignorance and we fill the gap concerning the conditions that trigger/block the passage from first-order ignorance to second-order ignorance in the case of the operator I. In Fitch 1963, it is demonstrated that ~K(If(f)) holds. Here we recast this result and its consequences in our formal system, by showing that If(f)® If (If(f)). ...
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We study different forms of ignorance and their correlations in a bi-modal logical language expressing the two modalities of knowledge and belief. In particular, we are mainly interested in clarifying which definitions of ignorance and which circumstances trigger higher-order forms of ignorance, inducing ignorance about ignorance and so on. To this aim, three ground conditions concerning knowledge and belief are presented, which may be seen as a cause of ignorance and can help us to identify the conditions enabling the emergence of higher-order forms of ignorance .
... Note that by elementary syntax they meant both 1 There are some paradoxes that go by names similar to the knower paradox with which the knower paradox should not be confused. One example is the knowledge or knowability paradox by Fitch [13]. This paradox of knowability is a logical result implying that, necessarily, if all truths are knowable in principle then all truths are in fact known. ...
... We derive the knower paradox as follows. Another way of formulating an apparently unacceptable conclusion from the assumptions and the definition of D is leaving out (13) and (14) and concluding ' ⊥' from (5) and (12). In both ways, the paradox is used to prove that a system in which assumptions E1, E2, and E3 are made is inconsistent. ...
... As we stated above, T ω is consistent if Q is consistent. 13 Besides, theorems like D ↔ P rov('¬D') are not in this new theory which describes knowledge. This means that Skyrms's system satisfies Haack's first criterion as a solution to the knower paradox. ...
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The knower paradox states that the statement ‘We know that this statement is false’ leads to inconsistency. This article presents a fresh look at this paradox and some well-known solutions from the literature. Paul Égré discusses three possible solutions that modal provability logic provides for the paradox by surveying and comparing three different provability interpretations of modality, originally described by Skyrms, Anderson, and Solovay. In this article, some background is explained to clarify Égré’s solutions, all three of which hinge on intricacies of provability logic and its arithmetical interpretations. To check whether Égré’s solutions are satisfactory, we use the criteria for solutions to paradoxes defined by Susan Haack and we propose some refinements of them. This article aims to describe to what extent the knower paradox can be solved using provability logic and to what extent the solutions proposed in the literature satisfy Haack’s criteria. Finally, the article offers some reflections on the relation between knowledge, proof, and provability, as inspired by the knower paradox and its solutions.
... Contact bibliothèque : ddoc-theses-contact@univ-lorraine.fr (Cette adresse ne permet pas de contacter les auteurs) Fitch (1963) semble montrer que ce principe de connaissabilité se réduit à l'affirmation absurde qu'il n'existe aucune vérité qui ne soit jamais connue. ...
... Ce raisonnement, originalement dû à Frederic Fitch (1963), est le plus souvent présenté comme une réfutation du principe de connaissabilité et, par là même, des positions qui le défendent. Néanmoins, la littérature abondante que le paradoxe a suscitée porte essentiellement sur différentes stratégies pour la résoudre, sans entrer dans les détails des positions dont il est question. ...
... Il sera soutenu que la manière la plus fructueuse d'aborder le raisonnement est dans le contexte philosophique d'où le principe de connaissabilité découle et que son aspect paradoxal se trouve principalement dans le fait qu'il semble réfuter des positions philosophiques substantielles par un simple raisonnement formel sans offrir aucune indication de l'endroit où les positions réfutées font fausse route. Le raisonnement de Fitch fit son apparition dans un court article de Frederic B.Fitch (1963) dédié à l'analyse logique d'un nombre de concepts étroitement liés à la notion de valeur. En effet, ce résultat, pour lequel l'article est toujours connu, n'y apparaît que comme une question intermédiaire, voire secondaire, à une analyse logique de la notion de valeur et d'autre concepts connexes. ...
Thesis
Un certain nombre de philosophes ont soutenu que toute vérité peut être connue. Ils font valoir que la vérité ne s'applique qu'aux expressions linguistiques signifiantes et notre compréhension de ces expressions consiste en notre capacité à déterminer leur vérité dans les circonstances appropriées. Par conséquent, bien qu'il existe de nombreuses vérités que nous ne pouvons pas connaître en pratique, tout énoncé vrai est signifiant et donc connaissable en principe. Un argument logique simple dû à Frederic Fitch (1963) semble montrer que ce principe de connaissabilité se réduit à l'affirmation absurde qu'il n'existe aucune vérité qui ne soit jamais connue. Ce résultat, connu sous le nom de paradoxe de la connaissabilité, a fait l'objet d'un grand nombre de tentatives de résolution dans le cadre de la logique épistémique, et l'exposé et l'évaluation de ces solutions constituent une partie considérable de la présente étude. Cependant, son objectif premier est d'étudier les implications philosophiques du paradoxe. Je soutiens que l'intérêt principal du paradoxe ne réside pas dans la question de sa résolution potentielle, mais dans les perspectives que son étude offre sur les positions philosophiques qu'il est censé compromettre. La première partie de l'étude présente le vérificationnisme du cercle de Vienne avec un accent particulier sur la philosophie de Moritz Schlick et sa thèse de décidabilité universelle. La deuxième partie étudie le paradoxe de la connaissabilité et les stratégies qui ont été mises en avant pour le résoudre. La troisième partie est consacrée à l'anti-réalisme de Michael Dummett et tente de clarifier le rôle du principe de connaissabilité dans son défi bien connu au réalisme. La quatrième et dernière partie se concentre sur les approches du paradoxe qui sont directement motivées par les considérations vérificationnistes sous-jacentes au principe de connaissabilité. Plusieurs solutions vérificationnistes différentes au p aradoxe seront présentées, et leurs réponses diverses montrent que le raisonnement de Fitch touche à certaines des questions les plus fondamentales de cette philosophie.
... Such a representation of ignorance might be considered stronger than another classical definition, referred as unknown truth (Fitch, 1963). In this alternative view, being ignorant of φ simply stands for not knowing that φ , which is formally expressed as φ ∧ ¬K(φ ). ...
... This form of lack of knowledge, which we refer as ignorance whether and will denote in the following with I(φ ), has indeed been widely investigated in several logical frameworks focused on ignorance, see, e.g., (Fan et al., 2015;Steinsvold, 2008;van der Hoek and Lomuscio, 2004). which is a result by Fitch, 1963. The main result of (Fine, 2018) concerns the properties of second-order ignorance in the case of I(φ ), which induces both first-order ignorance and third-order ignorance, thus triggering the loop towards higher orders. ...
... In (Fitch, 1963), it is demonstrated that ¬K(I f (φ )) holds. Here we recast this result and its consequences in our formal system, by showing that I f (φ ) → I f (I f (φ )) 7 . ...
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We study different forms of ignorance and their correlations in a bi-modal logical language expressing the two modalities of knowledge and belief. In particular, we are mainly interested in clarifying which definitions of ignorance and which circumstances trigger higher-order forms of ignorance, inducing ignorance about ignorance and so on. To this aim, three ground conditions concerning own knowledge and belief are presented, which may be seen as a cause of ignorance and can help us to identify the conditions enabling the emergence of higher-order forms of ignorance.
... The knowability paradox is that if all truths are knowable, then all truths are actually known. The standard references for the knowability paradox are [8,13]. However, following Salerno's archival efforts the obligatory precursor to that Church's 'anonymous' referee report of what (much) later became [13]: [. . . ...
... The standard references for the knowability paradox are [8,13]. However, following Salerno's archival efforts the obligatory precursor to that Church's 'anonymous' referee report of what (much) later became [13]: [. . . ] there is always a true proposition which it is empirically impossible for a to know at time t. ...
... [13, p. 139] Formally, 'proposition ϕ is knowable' later became ♦Kϕ [8], where ♦ is some modal diamond, representing a process, or time, or some alethic modality of truth. This modal diamond does not yet occur in [13]. Let us sketch the paradox. ...
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In this paper, we propose three knowability logics LK, LK−, and LK=. In the single-agent case, LK is equally expressive as arbitrary public announcement logic APAL and public announcement logic PAL, whereas in the multi-agent case, LK is more expressive than PAL. In contrast, both LK− and LK= are equally expressive as classical propositional logic PL. We present the axiomatizations of the three knowability logics and show their soundness and completeness. We show that all three knowability logics possess the properties of Church-Rosser and McKinsey. Although LK is undecidable when at least three agents are involved, LK− and LK= are both decidable.
... A considerable amount of attention by logicians and philosophers has been attracted by a few brief paragraphs that formed part of a short paper by Frederic Brenton Fitch (1963). The paper's objective was to provide a logical analysis of a small group of value concepts. ...
... A few sample informative publications that will lead the reader further includeBeall (2000),Edgington (2010),Fara (2010),Fitch (1963),Mackie (1980),Rescher (2005Rescher ( , 2009,Routley (2010Routley ( /1981,Salerno (2009), Williamson (2000. ...
... For Fitch's own formulation of his result, seeFitch (1963), and for alternative representations of Fitch's result, commentary, and criticism, see the examples of sources listed in the previous note. ...
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NOTE, September, 2021: This is the corrected second eBook edition of this work. Readers are asked to use this edition. The _Critique of Impure Reason_ has now also been published in a printed edition. To reduce the otherwise high price of this scholarly/technical book of nearly 900 pages and make it more widely available beyond university libraries to individual readers, the non-profit publisher and the author have agreed to issue the printed edition at cost. The printed edition was released on September 1, 2021 and is now available through all booksellers, including Barnes & Noble, Amazon, and brick-and-mortar bookstores under the following ISBN: 978-0-578-88646-6 Commendations of this work, from the back cover of the published edition: “I admire its range of philosophical vision.” – Nicholas Rescher, Distinguished University Professor of Philosophy, University of Pittsburgh, author of more than 100 books. “Bartlett’s _Critique of Impure Reason_ is an impressive, bold, and ambitious work. Careful scholarship is balanced by original analyses that lead the reader to recognize the limits of meaning, knowledge, and conceptual possibility. The work addresses a host of traditional philosophical problems, among them the nature of space, time, causality, consciousness, the self, other minds, ontology, free will and determinism, and others. The book culminates in a fascinating and profound new understanding of relativity physics and quantum theory.” – Gerhard Preyer, Professor of Philosophy, Goethe-University, Frankfurt am Main, Germany, author of many books including _Concepts of Meaning_, _Beyond Semantics and Pragmatics_, _Intention and Practical Thought_, and _Contextualism in Philosophy_. “[This work’s] goal is of a unique and difficult species: Dr. Bartlett seeks to develop a formal logical calculus on the basis of transcendental philosophical arguments; in fact, he hopes that this calculus will be the formal expression of the transcendental foundation of knowledge.... I consider Dr. Bartlett’s work soundly conceived and executed with great skill.” – C. F. von Weizsäcker, philosopher and physicist, former Director, Max-Planck-Institute, Starnberg, Germany. “Bartlett has written an American “Prolegomena to All Future Metaphysics.” He aims rigorously to eliminate meaningless assertions, reach bedrock, and place philosophy on a firm foundation that will enable it, like science and mathematics, to produce lasting results that generations to come can build on. This is a great book, the fruit of a lifetime of research and reflection, and it deserves serious attention.” — Martin X. Moleski, former Professor, Canisius College, Buffalo, NY, studies of scientific method, the presuppositions of thought, and the self-referential nature of epistemology. “Bartlett has written a book on what might be called the underpinnings of philosophy. It has fascinating depth and breadth, and is all the more striking due to its unifying perspective based on the concepts of reference and self-reference.” – Don Perlis, Professor of Computer Science, University of Maryland, author of numerous publications on self-adjusting autonomous systems and philosophical issues concerning self-reference, mind, and consciousness. +++++++++++++++++++++++++++++++++++++++++ The _Critique of Impure Reason: Horizons of Possibility and Meaning_ comprises a major and important contribution to the philosophy of science. Thanks to the generosity of its publisher, this massive volume of 885 pages has been published as a free open access eBook. It inaugurates a revolutionary paradigm shift in philosophical thought by providing compelling and long-sought-for solutions to a wide range of problems that have concerned philosophers of science as well as epistemologists. The book includes a Foreword by the celebrated German physicist and philosopher of science Carl Friedrich von Weizsäcker. The principal objective of the study is to identify the unavoidable limitations of the conceptual frameworks in terms of which knowledge, reference, and meaning are possible. The book establishes a bridge between, on the one hand, a model of philosophy as a science — i.e., rigorous proof-based scientific philosophy — and, on the other, the philosophy of science. The study develops a logically compelling method that enables us both to recognize the boundaries of what is referentially forbidden — the limits beyond which reference becomes meaningless — and second, to avoid falling victims to a certain broad class of conceptual confusions which lie at the heart of major problems of philosophy of science and epistemology. With these ends in view, individual chapters are devoted to a critique of a wide range of fundamental concepts studied by philosophy of science, among them, space, time, space-time, causality, the problem of discovery or invention in general problem-solving, mathematics, and physics, the role of the observer, the perturbation theory of measurement, indeterminacy and uncertainty, complementarity, the ontological status of physical reality, and others. The study culminates in a group of chapters devoted to special and general relativity and quantum theory. In these concluding chapters the purpose is to show the extent to which both relativity physics and quantum theory bear out results that have been reached in a logically compelling manner wholly by means of the approach to conceptual analysis developed in the book. Based on original research and rigorous analysis combined with extensive scholarship, the _Critique of Impure Reason_ develops a logically self-validating method that at last provides provable and constructive solutions to a significant number of major philosophical problems in philosophy of science and epistemology. Bartlett is the author or editor of more than 20 books and numerous papers.
... Our result is also relevant to broader issues in epistemic logic: from Γ Const , one can derive the knowability paradox due to Fitch (2009), according to which if all truths are knowable, then all truths are known. Taking ϕ to range over all L-formulas and α to range over all agents in A, Fitch's paradox is: ...
... Given that actual knowledge seems to be contingent in a way that truth is not, this result seems wrong (especially since if knowledge is factive, knowledge and truth would then become definitionally equivalent). Traditional epistemology has acknowledged the difficulties of conceding Fitch's paradox (see Williamson (1982); Edgington (1985); Williamson (1987); Percival (1990); Kvanvig (1995); Tennant (1997); Hand and Kvanvig (1999);DiVidi and Solomon (2001);Fitch (2009);Dummett (2009);Restall (2009);Edgington (2010)). From our result, we find that if DIST and NCK are true of quantum epistemology, then quantum epistemology does not admit Fitch's paradox (at least not via its usual deduction). ...
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Constructivist epistemology posits that all truths are knowable. One might ask to what extent constructivism is compatible with naturalized epistemology and knowledge obtained from inference-making using successful scientific theories. If quantum theory correctly describes the structure of the physical world, and if quantum theoretic inferences about which measurement outcomes will be observed with unit probability count as knowledge, we demonstrate that constructivism cannot be upheld. Our derivation is compatible with both intuitionistic and quantum propositional logic. This result is implied by the Frauchiger-Renner theorem, though it is of independent importance as well.
... According to Jenkins, an actually true proposition p is knowable if and only if there is a possible world w where someone recognizes that the very same state of affairs that renders p true at the actual world holds in w. Symbolizing 'someone recognizes that the state of affairs that renders p true at the actual world holds' as R[p], Jenkins represents KP as follows: 10 On p. 534 the principle is written as 'p ⊃ R[p]' but the absence of the diamond in the consequent is obviously a misprint as is shown by the application of the principle on p. 538. ...
... The paradox was first presented, in a more general form, inFitch (1963), and many authors refer to it as 'Fitch's paradox' . But Fitch attributes the paradox argument to an anonymous referee of an earlier paper by him that was not published. ...
Article
The most straightforward interpretation of the principle of knowability is that every true proposition may be known. This, taken together with some intuitively appealing ideas, raises a problem known as the Church–Fitch paradox. There is a wide variety of alternative interpretations of the principle of knowability that have been offered to avoid the paradox. Some of them are based on rigidification of certain aspects of what is knowable. I examine three proposals representing this strategy, those by Edgington, Rückert and Jenkins. Edgington defines what is knowable as a proposition prefixed by the actuality operator. Rückert and Jenkins maintain that what makes a proposition knowable is the possibility of knowing de re (Rückert) or recognizing (Jenkins) the state of affairs that renders the proposition actually true. In both cases, the link to the actual world (or situation) rigidifies what is knowable in some aspect or other. I argue that all three theories have strongly counterintuitive consequences, and I offer an interpretation of the principle of knowability that is both free from rigidity and immune to the Church–Fitch argument.
... 2 This paradox was first published by Frederic Fitch (1963), but originates with Alonzo Church, who relayed it to Fitch as an anonymous referee. Church's reports have since been published in Church (2009); see Salerno (2009) for more on the history of the paradox. ...
... If all sensible objects can be distinctly conceived, then all sensible objects are conceived. This collapse is strongly reminiscent of Alonzo Church's and Frederic Fitch's paradox of knowability (Church 2009, Fitch 1963, which proves (at least to most people's satisfaction) that if all truths are knowable, then all truths are known. The rough idea is this: if all truths are knowable and yet not all truths are known, then some p is both true and unknown. ...
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We can find in the passages that set out the Master Argument a precursor to the paradox of knowability. That paradox shows that if all truths are knowable, all truths are known. Similarly, Berkeley might be read as proposing that if all sensible objects are (distinctly) conceivable, then all sensible objects are conceived.
... To represent agentive obligations, a modal logic of agency is needed. While there are many ideas for formulating such a logic [1,11,17,23,37,38], the most prominent theory of agency is a branch called Seeing-To-It-That (STIT) logic that originated in the works of Belnap [5] and Belnap and Perloff [6]. STIT theory introduces a STIT operator of the form which expresses that an agent a sees to it that F holds. ...
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The paper presents a comprehensive analysis of the European AI Act in terms of its logical modalities, with the aim of preparing its formal representation, for example, within the logic-pluralistic Knowledge Engineering Framework and Methodology (LogiKEy). LogiKEy develops computational tools for normative reasoning based on formal methods, employing Higher-Order Logic (HOL) as a unifying meta-logic to integrate diverse logics through shallow semantic embeddings. This integration is facilitated by Isabelle/HOL, a proof assistant tool equipped with several automated theorem provers. The modalities within the AI Act and the logics suitable for their representation are discussed. For a selection of these logics, embeddings in HOL are created, which are then used to encode sample paragraphs. Initial experiments evaluate the suitability of these embeddings for automated reasoning, and highlight key challenges on the way to more robust reasoning capabilities.
... In an epistemic setting, accident is read 'unknown truths', which is an important notion in philosophy and formal epistemology. For example, it is a source of Fitch's 'paradox of knowability' [10]. As another example, it is an important kind of Moore sentences, which is in turn essential to Moore's paradox [14,20]. ...
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Contingency and accident are two important notions in philosophy and philosophical logic. Their meanings are so close that they are mixed sometimes, in both everyday discourse and academic research. This indicates that it is necessary to study them in a unified framework. However, there has been no logical research on them together. In this paper, we propose a language of a bimodal logic with these two concepts, investigate its model-theoretical properties such as expressivity and frame definability. We axiomatize this logic over various classes of frames, whose completeness proofs are shown with the help of a crucial schema. The interactions between contingency and accident can sharpen our understanding of both notions. Then we extend the logic to a dynamic case: public announcements. By finding the required reduction axioms, we obtain a complete axiomatization, which gives us a good application to Moore sentences.
... 30 After writing this paper, I learned from Harvey Lederman that Elliot 2022 raises similar doubts about the axioms. 31 E ⊓ ¬K(E) is the classic example of an unknowable event from what is known as Fitch's paradox (Fitch 1963); for example, can Ann know the event expressed by "Bob played left but Ann doesn't know it"? We could use this simpler event and the axiom K(E ⊓ ¬K(E)) = 0, but we will instead derive K(E ⊓ ¬U (E) ⊓ ¬K(E)) = 0 from other axioms. ...
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We propose a model of unawareness that remains close to the paradigm of Aumann's model for knowledge [R. J. Aumann, International Journal of Game Theory 28 (1999) 263-300]: just as Aumann uses a correspondence on a state space to define an agent's knowledge operator on events, we use a correspondence on a state space to define an agent's awareness operator on events. This is made possible by three ideas. First, like the model of [A. Heifetz, M. Meier, and B. Schipper, Journal of Economic Theory 130 (2006) 78-94], ours is based on a space of partial specifications of the world, partially ordered by a relation of further specification or refinement, and the idea that agents may be aware of some coarser-grained specifications while unaware of some finer-grained specifications; however, our model is based on a different implementation of this idea, related to forcing in set theory. Second, we depart from a tradition in the literature, initiated by [S. Modica and A. Rustichini, Theory and Decision 37 (1994) 107-124] and adopted by Heifetz et al. and [J. Li, Journal of Economic Theory 144 (2009) 977-993], of taking awareness to be definable in terms of knowledge. Third, we show that the negative conclusion of a well-known impossibility theorem concerning unawareness in [Dekel, Lipman, and Rustichini, Econometrica 66 (1998) 159-173] can be escaped by a slight weakening of a key axiom. Together these points demonstrate that a correspondence on a partial-state space is sufficient to model unawareness of events. Indeed, we prove a representation theorem showing that any abstract Boolean algebra equipped with awareness, knowledge, and belief operators satisfying some plausible axioms is representable as the algebra of events arising from a partial-state space with awareness, knowledge, and belief correspondences.
... 30 After writing this paper, I learned from Harvey Lederman that Elliot 2020 raises similar doubts about the axioms. 31 E ⊓ ¬K(E) is the classic example of an unknowable event from what is known as Fitch's paradox (Fitch 1963); for example, can Ann know the event expressed by "Bob played left but Ann doesn't know it"? We could use this simpler event and the axiom K(E ⊓ ¬K(E)) = 0, but we will instead derive K(E ⊓ ¬U (E) ⊓ ¬K(E)) = 0 from other axioms. ...
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Full-text available
We propose a model of unawareness that remains close to the paradigm of Aumann's model for knowledge [R. J. Aumann, International Journal of Game Theory 28 (1999) 263-300]: just as Aumann uses a correspondence on a state space to define an agent's knowledge operator on events, we use a correspondence on a state space to define an agent's awareness operator on events. This is made possible by three ideas. First, like the model of [A. Heifetz, M. Meier, and B. Schipper, Journal of Economic Theory 130 (2006) 78-94], ours is based on a space of partial specifications of the world, partially ordered by a relation of further specification or refinement, and the idea that agents may be aware of some coarser-grained specifications while unaware of some finer-grained specifications; however, our model is based on a different implementation of this idea, related to forcing in set theory. Second, we depart from a tradition in the literature, initiated by [S. Modica and A. Rustichini, Theory and Decision 37 (1994) 107-124] and adopted by Heifetz et al. and [J. Li, Journal of Economic Theory 144 (2009) 977-993], of taking awareness to be definable in terms of knowledge. Third, we show that the negative conclusion of a well-known impossibility theorem concerning unawareness in [Dekel, Lipman, and Rustichini, Econometrica 66 (1998) 159-173] can be escaped by a slight weakening of a key axiom. Together these points demonstrate that a correspondence on a partial-state space is sufficient to model unawareness of events. Indeed, we prove a representation theorem showing that any abstract Boolean algebra equipped with awareness, knowledge, and belief operators satisfying some plausible axioms is representable as the algebra of events arising from a partial-state space with awareness, knowledge, and belief correspondences.
... What is known as the Paradox of Knowability goes back to Fitch (1963), who, in beautiful irony, credits it to an anonymous 1945 referee report for a paper that was never published. In 2005 this referee was identified as Alonzo Church (Salerno, 2009, pp. ...
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Husserl’s theory of fulfilment conceives of empty acts, such as symbolic thought, and fulfilling acts, such as sensory perceptions, in a strict parallel. This parallelism is the basis for Husserl’s semantics, epistemology, and conception of truth. It also entails that any true proposition can be known in principle, which Church and Fitch have shown to explode into the claim that every proposition is actually known. I assess this logical challenge and discuss a recent response by James Kinkaid. While Kinkaid’s proposal saves one direction of the parallel for semantics, it gives up the parallelism for truth. I spell out a different response which meshes naturally with Husserl’s account of meaning. If the parallelism is restricted to a class of basic propositions, the truth of non-basic propositions can be defined inductively, without leading to the paradox. I then discuss objections that have been raised against a similar proposal by Dummett. The result is that exegetically plausible and popular interpretations of Husserl’s correlationism are indeed challenged by Church and Fitch. But when taking into account the ‘logical adumbration’ of propositional blindspots, truth and possible fulfilment can be connected without paradox.
... An issue that came up fairly early (cf. Church, 2009;Fitch, 1963) with the Naive conception of knowability is that, in the context of theories committed to the thesis that all truths are knowable and under some fairly standard assumptions about the behaviour of the ◇ and K operators involved, it gives rise to what is now-called Fitch's paradox. The paradox states that, if all truths are possibly known, then every truth is in effect known. ...
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Many philosophical discussions hinge on the concept of knowability. For example, there is a blooming literature on the so-called paradox of knowability. How to understand this notion, however? In this paper, we examine several approaches to the notion: the naive approach to take knowability as the possibility to know, the counterfactual approach endorsed by Edgington (1985) and Schlöder (2019) , approaches based on the notion of a capacity or ability to know (Fara 2010, Humphreys 2011), and finally, approaches that make use of the resources of dynamic epistemic logic (van Benthem 2004, Holliday 2017).
... You close the very last window thereby bringing it about that all the windows in the house are closed (which 9 That is, ∀p∃a(p → ◊ a p) implies ∃a∀p(p → a p) where a ranges over some finite set of agents. This is essentially Theorem 3 of Fitch (1963). 10 Here we understand collective agency in the weak sense in which each true proposition in a certain class is brought about by at least one agent. ...
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Humberstone has shown that if some set of agents is collectively omniscient (every true proposition is known by at least one agent) then one of them alone must be omniscient. The result is paradoxical as it seems possible for a set of agents to partition resources whereby at the level of the whole community they enjoy eventual omniscience. The Humberstone paradox only requires the assumption that knowledge distributes over conjunction and as such can be viewed as a reductio against the universal validity of that principle. A new route to this paradox is presented which does not require the distribution principle, building on earlier work of Jago and Williamson on Fitch’s paradox. The result relies on an axiom strictly weaker than one necessary for the Jago-Fitch variant. It is shown that the same reasoning behind the variant form of Humberstone’s paradox can recover Bigelow’s results in action theory in a way that is immune to an objection brought against it by Guigon and Humberstone.
... (1) φ → ◊Kφ Данная формализация естественна, поскольку трактует познаваемость как возможность быть известным, что вполне соответствует интуитивному смыслу слова «познаваемость». Проблема состоит в том, что, как показал Ф. Фитч в [Fitch, 1963], (1) вместе с двумя интуитивно привлекательными принципами имеет следствием крайне контринтуитивный тезис, что все факты известны. Интуитивно привлекательные принципы, о которых идет речь, таковы: ...
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The formalization of the principle of knowability suggested by Dorothy Edgington is examined. This formalization has been suggested as a solution to the Fitch problem. It is interesting in that it blocks the Fitch argument and, in informal reading, makes a clear and intuitively appealing sense. On the other hand, as is shown in the paper, the semantic theory behind this formalization has two significant gaps: 1) it does not define the interpretation of actuality operator, and 2) it does not define the semantic way of representing the agent’s knowledge. The main outcome of the papers is critical. It is to the effect that unless those gaps are filled, Edgington’s theory cannot count as a solution to the Fitch problem.
... Remark 5.3.41. First, we note a connection to "Fitch's paradox" of knowability [64,34]: under weak assumptions, the "verifiability principle" that every truth could in principle be known (at some time) by some agent or other, p → ∃v2 v p (where is some kind of possibility modal), entails the stricter verificationist principle that every truth is known (at some time) by some agent or other, p → ∃v2 v p. Now we add that world frames commit the strict verificationist to the implausible principle that there is (at some time) a single omniscient agent. ...
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In traditional semantics for classical logic and its extensions, such as modal logic, propositions are interpreted as subsets of a set, as in discrete duality, or as clopen sets of a Stone space, as in topological duality. A point in such a set can be viewed as a "possible world," with the key property of a world being primeness—a world makes a disjunction true only if it makes one of the disjuncts true—which classically implies totality—for each proposition, a world either makes the proposition true or makes its negation true. This chapter surveys a more general approach to logical semantics, known as possibility semantics, which replaces possible worlds with possibly partial "possibilities." In classical possibility semantics, propositions are interpreted as regular open sets of a poset, as in set-theoretic forcing, or as compact regular open sets of an upper Vietoris space, as in the recent theory of "choice-free Stone duality." The elements of these sets, viewed as possibilities, may be partial in the sense of making a disjunction true without settling which disjunct is true. We explain how possibilities may be used in semantics for classical logic and modal logics and generalized to semantics for intuitionistic logics. The goals are to overcome or deepen incompleteness results for traditional semantics, to avoid the nonconstructivity of traditional semantics, and to provide richer structures for the interpretation of new languages.
... 253-254). Again, since we are in a position to figure out that it is impossible to know propositions of that form (Fitch, 1963), (3*) explains this phenomenon equally well (Rosenkranz, 2021, pp. 88-91, 243-251). ...
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The knowledge account of assertion construes assertion as subject to constitutive norms. In its standard version, it combines a wide scope obligation not to assert p without knowing p, with narrow scope principles specifying conditions under which it is permissible to assert p, where the notions of obligation and permission are duals and behave uniformly for variable p. It is argued that, given natural assumptions about the logic of ‘ought’, the account proves incoherent. The argument generalizes to accounts that substitute other factive notions for knowledge. A recent non-standard version of the knowledge account employs proposition-relative norms and circumvents the problem. However, it still leads to intolerable combinations of verdicts. Again, the problem arises because knowledge is factive, and it generalizes to other factive notions. It is shown that non-factive accounts face none of the diagnosed difficulties and can do much of the explanatory work that the knowledge account is alleged to do.
... In other words, the notion gives rise to a well-known counterexample to Verificationism. This is the so-called Fitch's 'paradox of knowability' [15]. 1 To take another example: it gives rise to an important type of Moore sentences, which is essential to Moore's paradox, which says that one cannot claim the paradoxical sentence "p but I do not know it" [20,26]. It is known that such a Moore sentence is unsuccessful and self-refuting (see, e.g. ...
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In this article, we study logics of unknown truths and false beliefs under neighborhood semantics. We compare the relative expressivity of the two logics. It turns out that they are incomparable over various classes of neighborhood models, and the combination of the two logics are equally expressive as standard modal logic over any class of neighborhood models. We propose morphisms for each logic, which can help us explore the frame definability problem, show a general soundness and completeness result, and generalize some results in the literature. We axiomatize the two logics over various classes of neighborhood frames. Most importantly, by adopting the intersection semantics and the subset semantics in the literature, we extend the results to the case of public announcements, which gives us the desired reduction axioms and has good applications related to Moore sentences, successful formulas and self-refuting formulas. Also, we can say something about the comparative merits of the intersection semantics and the subset semantics.
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I present two arguments that aim to establish logical limits on what we can know. More specifically, I argue for two results concerning what we can know about questions that we cannot answer. I also discuss a line of thought, found in the writings of Pierce and of Rescher, in support of the idea that we cannot identify specific scientific questions that will never be answered.
Chapter
In the last twenty years, knowledge-centered approaches have become increasingly popular in analytic epistemology. Rather than trying to account for knowledge in other terms, these approaches take knowledge as the starting point for the elucidation of other epistemic notions (such as belief, justification, and rationality). Knowledge-centered approaches have been so influential that it now looks as if epistemology is undergoing a factive turn. However, relatively little has been done to explore how knowledge-centered views fare in new fields inside and beyond epistemology strictly understood. This volume aims to remedy this situation by putting together contributions that investigate the significance of knowledge in debates where its roles have been less explored. The goal is to see how far knowledge-centered views can go by exploring new prospects and identifying new trends of research for the knowledge-first program. Extending knowledge-centered approaches in this way promises not only to deliver novel insights into these neglected fields but also to revisit more traditional debates from a fresh perspective. As a whole, the volume develops and evaluates the knowledge-first program in original and fertile ways.
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A course of dialogical reasoning involving the atheist and the theist reveals a connection between the Curry phenomenon and the step‐wise construction of a sound version of the modal ontological argument. The exercise is both adversarial and cooperative as the participants are committed to the norms of shared truth‐seeking, respect for one's opponents and a desire to continue the dialectic for as long as possible. The theist relies on the interaction between the properties of a Curry‐style sentence and the structure of implication in order to show that the atheist's own commitments imply Anselm's principle (God necessarily exists if He actually exists at all). As Anselm's principle and the possibility premise are the only assumptions required for the modal ontological argument it follows that the theist has, given the norms of the dialogue, a winning strategy against the atheist. This follows since the possibility premise is granted by the atheist as part of their commitment to the norms governing the dialectic though the theist in virtue of those same norms must accept that God is at best maximally perfect in the light of the argument from evil and the Stone paradox.
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This article surveys the philosophy of logic of John Corcoran by focusing on some of its characteristic themes: his understanding of logic as formal epistemology articulating the ontic-epistemic distinction of classical metaphysics, the Socratic belief-knowledge distinction, and the Aristotelian truth-knowledge distinction; his conception of mathematical logic as instrumental when considering mathematical logics as models of underlying reasoning found in the practice of proof; his tireless search for a careful and successful communication in a community of thinkers eliminating ambiguity of key terms and embracing ethical values; his discussion of argumentations and logic as a philosophical realization of the previous dichotomies, allowing precise definitions of key concepts, such as argument, argumentation, proof, deduction, fallacy, and paradox; finally, his recovering and articulation of the nineteenth century information-theoretic conception of validity, exploring its heuristic power in the study of omega arguments and suggesting the existence of different paradigms of logical consequence equally entrenched in the theory and practice of logic.
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En este artículo discutiré algunos problemas lógicos de la omnipotencia que van más allá de las clásicas paradojas ligadas a esta noción. Presentaré una versión refinada de la paradoja de Fitch sobre la omnipotencia que tiene en cuenta la distinción entre acciones básicas y derivadas, así como la distinción entre la capacidad de hacer algo y la mera posibilidad metafísica de hacerlo. También explico cómo esta paradoja puede reformularse para obtener una versión afín a la paradoja del mentiroso que afecta a la consistencia de ciertas nociones de omnipotencia. Por último, evalúo algunas posibles respuestas disponibles para el teísta y un intento de usar la paradoja de Fitch como argumento a favor de la existencia de Dios.
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Fitch's knowability paradox shows that for each unknown truth there is also an unknowable truth, a result which has been thought both odd in itself and at odds with views which impose epistemic constraints on truth and/or meaningfulness. Here a solution is considered which has received little attention in the debate but which carries prima facie plausibility. The decidability solution is to accept that Fitch sentences are unknowably true but deny the significance of this on the grounds that Fitch sentences are nevertheless decidable. The decidability solution is particularly attractive for those whose primary concern is an epistemic constraint on meaningfulness (‘verificationists’). For those whose main concern is truth (‘anti‐realists’), the situation is more complex: Melia takes the solution to exonerate anti‐realism completely; Williamson sees it as completely irrelevant. The truth lies between these two extremes: there is one broad anti‐realist commitment to which the solution does not apply, but there is also one, the “fundamental tenet” of anti‐realism according to Dummett, to which it does.
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A minimal constraint on normative reasons seems to be that if some fact is a reason for an agent to ϕ (act, believe, or feel), the agent could come to know that fact. This constraint is threatened by a well known type of counterexamples. Self-effacing reasons are facts that intuitively constitute reasons for an agent to ϕ, but that if they were to become known, they would cease to be reasons for that agent. The challenge posed by self-effacing reasons bears important structural similarities with a range of epistemic paradoxes, most notably the Knowability Paradox. In this article, we investigate the similarities and differences between the two arguments. Moreover, we assess whether some of the approaches to the Knowability Paradox could help solve the challenge posed by self-effacing reasons. We argue that at least two popular approaches to the paradox can be turned into promising strategies for addressing the self-effacing reasons problem.
Chapter
Notions of unknown truths and unknowable truths are important in formal epistemology, which are related to each other in e.g. Fitch’s paradox of knowability. Although there have been some logical research on the notion of unknown truths and some philosophical discussion on the two notions, there seems to be no logical research on unknowable truths. In this paper, we propose a logic of unknowable truths, investigate the logical properties of unknown truths and unknowable truths, which includes the similarities of the two notions and the relationship between the two notions, and axiomatize this logic.
Chapter
Historically, most of the best-known modal logics had axiomatic characterizations long before either tableau systems or semantical approaches were available. While early modal axiom systems were somewhat circuitous by today’s standards, a natural and elegant system for S4\mathbf {S4} was given in Gödel (1933), and this has become the paradigm for axiomatizing modal logics ever since. It is how we do things here.
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Stephenson (2022) has argued that Kant’s thesis that all transcendental truths are transcendentally a priori knowable leads to omniscience of all transcendental truths. His arguments depend on luminosity principles and closure principles for transcendental knowability. We will argue that one pair of a luminosity and a closure principle should not be used, because the closure principle is too strong, while the other pair of a luminosity and a closure principle should not be used, because the luminosity principle is too strong. Stephenson’s argument also depends on a factivity principle for transcendental knowability, which we will argue to be false.
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En este artículo una paradoja es un tipo de argumentación con respecto a un sujeto X (sea un individuo o una comunidad) en un determinado momento T. Muchas argumentaciones paradójicas tienen lugar en el desarrollo histórico y práctico de las ciencias. Algunas suponen grandes sorpresas acompañadas de profundas crisis, como ocurre con las llamadas antinomias. Solventar, y eventualmente resolver, una paradoja en este sentido supone avances revolucionarios que se obtienen al precio de rechazar creencias previamente asumidas o tenidas por verdaderas por la comunidad científica. El concepto de paradoja que se propone es relativo a sujetos y dinámico en el tiempo. Una argumentación que resulta paradójica para X pudiera no ser paradójica para Y, simplemente porque X e Y no tienen por qué compartir las mismas creencias. Asimismo, X puede descubrir que tiene una paradoja en T, y dejar de tenerla ulteriormente en virtud de un cambio de sus creencias.
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kk states that knowing entails knowing that one knows, and K¬K states that not knowing entails knowing that one does not know. In light of the arguments against kk and K¬K, one might consider modally qualified variants of those principles. According to weak kk, knowing entails the possibility of knowing that one knows. And according to weak K¬K, not knowing entails the possibility of knowing that one does not know. This paper shows that weak kk and weak K¬K are much stronger than they initially appear. Jointly, they entail kk and K¬K. And they are susceptible to variants of the standard arguments against kk and K¬K. This has interesting implications for the debate on positive introspection and for deeper issues concerning the structure and limits of knowability.
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Intuitionistic epistemic logic by Artemov and Protopopescu (Rev Symb Log 9:266–298, 2016) accepts the axiom “if A, then A is known” (written AKAA \supset K A) in terms of the Brouwer–Heyting–Kolmogorov interpretation. There are two variants of intuitionistic epistemic logic: one with the axiom “KA¬¬AKA \supset \lnot \lnot A” and one without it. The former is called IEL\textbf{IEL}, and the latter is called IEL\textbf{IEL}^{-}. The aim of this paper is to study first-order expansions (with equality and function symbols) of these two intuitionistic epistemic logics. We define Hilbert systems with additional axioms called geometric axioms and sequent calculi with the corresponding rules to geometric axioms and prove that they are sound and complete for the intended semantics. We also prove the cut-elimination theorems for both sequent calculi. As a consequence, the disjunction property and existence property are established for the sequent calculi without geometric implications. Finally, we establish that our sequent calculi can also be formulated with admissible structural rules (i.e., in terms of a G3-style sequent calculus).
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We develop intuitionistic epistemic logics with distributed knowledge, which is more general than a logic proposed by (Jäger & Marti 2016) in that a distributed knowledge operator is parameterized by a group of agents. Specifically, we present Hilbert systems of intuitionistic K, KT, KD, K4, K4D, and S4 with distributed knowledge. The semantic completeness of the logics with regard to suitable Kripke frames is shown by modifying the standard argument of the semantic completeness of classical distributed knowledge logics via the concept of pseudo-model. We also present cut-free sequent calculi for the logics, based on which we establish Craig interpolation theorem and decidability.
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Recently, there have been several attempts to use the kind of reasoning found in Fitch’s knowability paradox to argue for rather sweeping metaphysical claims: Jago (2020) uses such reasoning to argue that every truth has a truthmaker, and Loss (2021) does so to argue that every fact is grounded. This strategy has been criticized by Trueman (2021), who points out that the same kind of reasoning could be used to establish entirely opposite conclusions. In response, Jago (2021) has offered a revised argument that is meant to avoid Trueman’s objection. I argue that this revised argument is in fact undermined by an objection quite similar to Trueman’s.
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In article is undertaken the criticism of one of dogmas «naive» epistemology, according to which procedure of generalization conduct to occurrence and growth «new» knowledge, but, nevertheless, which acceptance as shows the logic analysis, conducts to occurrence of a series of paradoxes.
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How can we overcome the rapidly ageing postmodernist paradigm, which has become sterile orthodoxy in marketing? This book answers this crucial question using fresh philosophical tools developed by New Realism. It indicates the opportunities missed by marketing due to the pervasiveness of postmodernist attitudes and proposes a new and fruitful approach pivoting on the signifcance of reality to marketing analyses and models. Intensifying reference to reality will boost marketing research and practice, rather than impair them; conversely, neglecting such a reference will prevent marketing from realising its full potential, in several contexts. The aim of the book is foundational: its purpose is not a return to traditional realism but to break new ground and overcome theoretical obstacles in marketing and management by revising some of their assumptions and enriching their categories, thereby paving the way to fresh approaches and methodological innovations. In that sense, the book encourages theoretical innovation and experimentation and introduces new concepts, like invitation and attrition, which can fnd fruitful applications in marketing theory and practice. That is meant to be conductive to the solution of important difculties and to the uncovering of new phenomena. The last chapter of the book applies the new approach to eight case studies from business contexts. This book will be of interest to philosophers interested in New Realism and to researchers, scholars, and marketing professionals sensitive to the importance and fruitfulness of reference to reality, for their own purposes.
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This essay examines the philosophical significance of Ω\Omega-logic in Zermelo-Fraenkel set theory with choice (ZFC). The duality between coalgebra and algebra permits Boolean-valued algebraic models of ZFC to be interpreted as coalgebras. The modal profile of Ω\Omega-logical validity can then be countenanced within a coalgebraic logic, and Ω\Omega-logical validity can be defined via deterministic automata. I argue that the philosophical significance of the foregoing is two-fold. First, because the epistemic and modal profiles of Ω\Omega-logical validity correspond to those of second-order logical consequence, Ω\Omega-logical validity is genuinely logical, and thus vindicates a neo-logicist conception of mathematical truth in the set-theoretic multiverse. Second, the foregoing provides a modal-computational account of the interpretation of mathematical vocabulary, adducing in favor of a realist conception of the cumulative hierarchy of sets.
Book
Die Autorinnen und Autoren präsentieren in diesem Buch Argumente, die die Unmöglichkeit des Reduktionismus aus philosophischer, naturwissenschaftlicher bzw. mathematisch-logischer Perspektive zu begründen suchen. Der Reduktionismus behauptet, dass Eigenschaften auch von komplexen Systemen (bis hin zu Lebensvorgängen und menschlichem Bewusstsein) vollständig auf ihre Bestandteile zurückgeführt werden können. Diese Position ist einflussreich, aber umstritten. Im Jahr 2019 hat der Kurt Gödel Freundeskreis einen Essaywettbewerb veranstaltet, um schlagende Argumente gegen den Reduktionismus zu finden. Unter den internationalen Teilnehmern waren neben weltweit führenden Forschern auch Wissenschaftlerinnen und Wissenschaftler, die noch am Beginn ihrer Kariere stehen. Dieser Band versammelt die Beiträge der Preisträger und weitere ausgewählte Aufsätze. Aus dem Inhalt: · Kausalität als antireduktionistisches Hausmittel – Martin Breul · Reduktionismus im Diskurs – Hanna Hueske · Monads, Types, and Branching Time – Kurt Gödel’s approach towards a theory of the soul – Tim Lethen · The limits of reductionism: thought, life, and reality – Jesse M. Mulder · True or Rational? A Problem for a Mind-Body Reductionist – Michał Pawłowski · Why reductionism does not work – George F. R. Ellis · Physik ohne Reduktion – Rico Gutschmidt · Is there an Axiom for everything? – Jean-Yves Béziau · Unerklärliche Wahrheiten – Marco Hausmann · Gödel, mathematischer Realismus und Antireduktionismus – Reinhard Kahle Die Herausgeber Oliver Passon ist Privatdozent an der Bergischen Universität Wuppertal und lehrt Physik und ihre Didaktik. Zu seinen Hauptarbeits- und Interessensgebieten gehört die Didaktik, Geschichte und Philosophie der modernen Physik. Christoph Benzmüller ist Professor für KI/Informatik, Logik und Mathematik an der Freien Universität Berlin. Er war der erste UNA Europa Gastprofessor und er kooperiert derzeit mit einem Berliner Startup Unternehmen.
Chapter
Manche Menschen haben hohe Erwartungen an die Wissenschaft. Manche Menschen erwarten von der Wissenschaft, dass sie die Welt vollständig beschreiben wird. Die Wissenschaft wird diese Erwartung aber sicher nie erfüllen – zumindest dann nicht, wenn sie versucht, alle Wahrheiten über die Welt einzeln aufzuzählen. Aber auch der Versuch aus einer endlichen Anzahl von Annahmen eine vollständige Beschreibung zu reduzieren ist zum Scheitern verurteilt.
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Two proofs are given which show that if some set of truths fall under finitely many concepts (so‐called ), then they all fall under at least one of them even if we do not know which one. Examples are given in which the result seems paradoxical. The first proof crucially involves Moorean propositions while the second is a reconstruction and generalization of a proof due to Humberstone free from any reference to such propositions. We survey a few solution routes including Tennant‐style restriction strategies. It is concluded that accepting for some set of truths while also denying that any of the involved concepts in isolation capture all of them requires that one of these concepts cannot be closed under conjunction elimination. This is surprising since the paper surveys several applications in which and the latter closure condition seemed jointly satisfiable for concepts of actual philosophical interest.
Modalities and quantification
  • Carnap