It’s sometimes suggested that we can (in a sense) settle the truth-value of some statements in the language of number theory by stipulation, adopting either φ or ¬φ as an additional axiom. For example, in Clarke-Doane (2020b) and a series of recent APA presentations, Clarke-Doane suggests that any Σ01 sound expansion of our current arithmetical practice would express a truth. In this paper, I’ll argue that (given a certain popular assumption about the model-theoretic representability of languages like ours) (...) we can’t know ourselves to have any such freedom. (shrink)

Fools rush in. ALEXANDER POPE, AN ESSAY ON CRITICISM (London, 1711). The full quotation is, “For Fools rush in where Angels fear to tread.” Id. at 66. She who hesitates is lost. Adaptation of the line, “The woman that deliberates is lost.” JOSEPH ADDISON, CATO: A TRAGEDY, AND SELECTED ESSAYS 30 (2004). See also OLIVER WENDALL HOLMES, SR., THE AUTOCRAT AT THE BREAKFAST TABLE 29 (1858) (“The woman who ‘calculates’ is lost.”). American legal realism numbers among the most important theoretical (...) contributions of legal academia to date. Given the movement’s influence, as well as the common centrality of certain key figures, it is surprising that privacy scholarship in the United States has paid next to no attention to the movement. This inattention is unfortunate for several reasons, including that privacy law furnishes rich examples of the indeterminacy thesis — a key concept of American legal realism — and because the interdisciplinary efforts of privacy scholars to explore extra-legal influences on privacy law arguably further the plot of legal realism itself. (shrink)

Modern social theory, by and large, has aimed at reducing the complexity of action situations to a set of manageable abstractions. But these abstractions, whether functionalist or linguistic, fail to grasp the indeterminacy of action situations.Action proceeds by discovery and combination. The logic of action is serendipitous and combinative. From these characteristics, a number of consequences flow: The whole field of our intentions is engaged in each action situation, and cannot really be understood apart from the situation itself. (...) In action situations we remain aware of the problems of categorization, including the dangers of infinite regress and the difficulties of specifying borders and ranges of categories. In action situations, attention is in permanent danger of being overwhelmed. We must deal with many features of action situations outside of attention; in doing so, we must entertain simultaneously numerous possibilities of action. Emotional expression is a way of talking about the kinds of possibilities we entertain. Expression and action have a rebound effect on attention. "Effort" is required to find appropriate expressions and actions, and rebound effects play a role in such effort, making it either easier or more difficult.Recent theoretical trends have failed to capture these irreducible characteristics of action situations, and have slipped into a number of errors. Language is not rich in meanings or multivocal, except as put to use in action situations. The role of "convention" in action situations is problematic, and therefore one ought not to talk of "culture." Contrary to the assertions of certain theorists, actors do not follow strategies, except when they decide to do so. Actors do not "communicate," in the sense of exchanging information, except in specially arranged situations. More frequently, they intervene in the effortful management of attention of their interlocutors. Dialogue, that is, very commonly becomes a form of cooperative emotional effort.From these considerations, it follows that the proper method for gaining social knowledge is to examine the history of action and of emotional effort, and to report findings in the form of narrative. (shrink)

In line with recent interest in weak and often not fully determinate effects of communication permeating relevance-theoretic research, I contribute a discussion on two possible sources of speaker-intended indeterminacy within explicit import of an utterance: one residing in an intentionally underspecified location of an ad hoc concept between literal or non-literal (e.g. metaphorical or hyperbolic) interpretation, and the other lying in the higher-level explicature of an utterance, and being related to propositional attitude (e.g. pretence, reporting, dissociation) or speech-act (...) description (e.g. complementing, reproaching). In both cases, the speaker leaves a certain amount of indeterminacy concerning the degree of her commitment to the content communicated, by not indicating where precisely the concept or the attitude is to be located on the literalness-metaphor continuum or on the continuum of various propositional attitudes/speech-act descriptions, respectively. Using a number of illustrative examples drawn from television discourse, I try to show that this kind of intended indeterminacy gives rise to communicative effects of a non-propositional kind. Additionally, the analyses presented here support the deflationary view of figurative uses of language, on which they form continua with non-figurative utterances. (shrink)

Since 1976 Hilary Putnam has on many occasions proposed an argument, founded on some model-theoretic results, to the effect that any philosophical programme whose purpose is to naturalize semantics would fail to account for an important feature of every natural language, the determinacy of reference. Here, after having presented the argument, I will suggest that it does not work, because it simply assumes what it should prove, that is that we cannot extend the metatheory: Putnam appears to think that (...) all we may determinately say about the relations between words and entities in the world is what the model theory tells us, but he has never offered justifications for that. At the end of the article, I will discuss the apparently reliable intuition that seems to me to be at the root of the argument, that is that, given a formal theory, there is an infinite number of ways of connecting it to, or of projecting it onto, the world. I will suggest that we should resist this intuition, because it rests on a very doubtful notion of world, which assumes that for any class of objects there is a corresponding property. (shrink)

It is commonly held that the natural numbers sequence 0, 1, 2,... possesses a unique structure. Yet by a well known model theoretic argument, there exist non-standard models of the formal theory which is generally taken to axiomatize all of our practices and intentions pertaining to use of the term “natural number.” Despite the structural similarity of this argument to the influential set theoretic indeterminacy argument based on the downward L ̈owenheim-Skolem theorem, most theorists agree that (...) the number theoretic version does not have skeptical consequences about the reference of “natural number” analogous to the ‘relativity’ Skolem claimed pertains to notions such as “uncountable” and “cardinal.” In this paper I argue that recent proposals by Shapiro, Lavine, McGee and Field which aim to distinguish the number and set theoretic indeterminacy arguments by locating extra-mathematical constraints on the interpretation of our number theoretic vocabulary are inadequate. I then suggest that if we consider the manner in which the natural numbers figure in our computational practices – e.g. in characterizing how we compute sums and products or determine prime factorizations – it follows that our application of computational terms such as “computable function” and “tractable computational problem” indirectly uniquely determines their structure. (shrink)

This paper introduces a novel theory of vagueness. Its main aim is to show how naïve judgments about tolerance and indeterminacy can be preserved while departing from classical logic only in ways which are independently motivated. -/- The theory makes use of a bilateral approach to acceptance and rejection. Combined with a standard account of validity, this approach gives rise to an entailment relation which is non-transitive. I argue that this is desirable: it is both pre-theoretically plausible and provides (...) a compelling solution to a number of longstanding puzzles. The theory aims to preserve the naïve picture of vagueness. It combines tools from expressivist and dynamic treatments of information in conversation to show how principles which are generally assumed to be in tension are compatible. The resulting logic departs from classical logic in precisely those places we should expect it to. -/- . (shrink)

A proposal by Baron, Colyvan, and Ripley to extend the counterfactual theory of explanation to include counterfactual reasoning about mathematical explanations of physical facts is discussed. Their suggestion is that the explanatory role of mathematics can best be captured counterfactually. This paper focuses on their example with a number-theoretic antecedent. Incorporating discussions on the structure and de re knowledge of numbers, it is argued that the approach leads to a change in the structure of numbers. As a result, (...) the counterfactual is not about the natural numbers anymore. Linking the antecedent and consequent of the counterfactual also becomes problematic. (shrink)

Summary One of the central questions concerning theories of reference has been the problem of how the reference of scientific terms gets fixed. Descriptive causal theories of reference, as discussed in this paper, have re-introduced the role of theoretical beliefs and conceptualisations in term introductions and reference-fixing. The present paper argues that the idea of reference-fixing as a dot-like event (baptism) is wrong: a number of episodes from the history of science are discussed to support the claim that reference-fixing (...) is a historical, drawn-out process. This, however, does not stand in the way of successful reference. The two processes are simply separated. A criterion is suggested to determine successful reference. From this approach two further ideas follow: not all scientific terms actually have the power of referring and even those that do will always retain a residual indeterminacy. (shrink)

We propose a group-theoretical interpretation of the fact that the transition from classical to quantum mechanics entails a reduction in the number of observables needed to define a physical state and \ to \ or \ in the simplest case). We argue that, in analogy to gauge theories, such a reduction results from the action of a symmetry group. To do so, we propose a conceptual analysis of formal tools coming from symplectic geometry and group representation theory, notably Souriau’s (...) moment map, the Mardsen–Weinstein symplectic reduction, the symplectic “category” introduced by Weinstein, and the conjecture according to which “quantization commutes with reduction”. By using the generalization of this conjecture to the non-zero coadjoint orbits of an abelian Hamiltonian action, we argue that phase invariance in quantum mechanics and gauge invariance have a common geometric underpinning, namely the symplectic reduction formalism. This stance points towards a gauge-theoretical interpretation of Heisenberg indeterminacy principle. We revisit this principle in the light of the difference between the set-theoretic points of a phase space and its category-theoretic symplectic points. (shrink)

Indeterminate Bodies organizes a number of theoretical and empirical studies around the concept and actuality of indeterminacy, as it relates to body and society. Located within the struggle to apprehend different categories of ‘body’ in the volatile flows of late-capital, indeterminacy is considered through such multiple incarnations as economy, contingency, inheritance, question, force, uncertainty, materiality and affective resistance to determination. While indeterminacy is often positioned as the ‘trouble’ or friction in subject/object knowledge-formation (framed as ontological or (...) empirical challenge), it also engenders affects such that some subjects are both in and out of recognition. Questions of indeterminacy overlap with work on imperceptibility, giving rise to interlocked questions about the modes of representation, categorization, inclusion, exclusion and sensibility in the production of bodies. We address the hesitancies, difficulties and necessities of working with and through indeterminacy in order open up new descriptions, visions and modes of political work. (shrink)

This response to Robert Knox’s very kind and constructive review1 of my 2008 book The Degradation of the International Legal Order: The Rehabilitation of Law and the Possibility of Politics gives me the opportunity not only to answer some of his criticisms, but also, on the basis of my own reflections since 2008, to fill in some gaps. Indeed, to revise a number of my arguments. First, I restate my attempt at a materialist account of human rights. Next I (...) explain why, for me, the right of peoples to self-determination is absolutely central to a materialist understanding of human rights; and also fill a serious gap in my own account in the book. This leads me not only to a reply to Robert Knox on the question of ‘indeterminacy’ in international law, but also to a disagreement with him on the use or misuse of the language of self-determination. My fourth section returns to our very different evaluations of the significance and meaning of the work of Yevgeny Pashukanis, and what, for me, is Pashukanis’s misunderstanding, for reasons consistent with his general theoretical trajectory, of Marx and Lenin on the Irish question. Finally, I present an outline of a re-evaluation of Marx’s principled position on self-determination. (shrink)

It is commonly believed that Quine's principal argument for the Indeterminacy of Translation requires an untenably strong account of the underdetermination of theories by evidence, namely that that two theories may be compatible with all possible evidence for them and yet incompatible with each other. In this article, I argue that Quine's conclusion that translation is indeterminate can be based upon the weaker, uncontroversial conception of theoretical underdetermination, in conjunction with a weak reading of the ‘Gavagai’ argument which establishes (...) the underdetermination of the sense and reference of subsentential terms. If underdetermination is considered to be a widespread phenomenon in science, or in inductive reasoning more generally, then the Indeterminacy of Translation will be widespread too. Finally, I briefly consider two issues concerning the scope of this conclusion about the Indeterminacy of Translation: first, whether the argument presupposes behaviourism; and second, whether indeterminacy is restricted to the case of radical translation. I argue that the answer to both these questions is negative, and thus that the thesis of semantic indeterminacy remains relevant to those who disagree with Quine about some issues concerning the nature of mind and language. (shrink)

There is currently great anxiety among literary critics and theorists about literary criticism’s loss of identity, both as an identifiable, coherent discipline with a recognizable set of problems and as a body of authoritative and well-founded convictions about literature and its history. Part of the reason for this anxiety is that everything that was until recently considered relatively unproblematical has now been rendered problematical. The hermeneutic of suspicion emerges as an interpretative strategy, pitting itself against the hermeneutic of belief; radical (...) indeterminacy of meanings challenges the certainty of traditional epistemology which authorized the objectivity of meaning, text, and context; and the whole of the Western metaphysical tradition is “inscribed” in any piece of writing which undermines the traditional notions of canon-formation, evaluation, and responsible criticism. Traditional literary theory assumes literary works to be structures of determinate meaning accessible by objective critical procedures. It is this assumption, and along with it a number of critical practices, which are challenged by the contemporary interest in the indeterminacy of meaning. In deconstruction, the thesis of indeterminacy takes its most radical and thoroughgoing form, so that all modernist critical stances from new Criticism to phenomenological, psychoanalytical, structuralist, marxist, and reader-response criticisms are put into question. This renders the scene of contemporary criticism and its theory highly volatile, since every theoretical claim or critical practice has become vulnerable to the deconstructive critique, even if this critique itself requires some form of certainty to question any critical theory or practice. (shrink)

Ellsberg (The Quarterly Journal of Economics 75, 643–669 (1961); Risk, Ambiguity and Decision, Garland Publishing (2001)) argued that uncertainty is not reducible to risk. At the center of Ellsberg’s argument lies a thought experiment that has come to be known as the three-color example. It has been observed that a significant number of sophisticated decision makers violate the requirements of subjective expected utility theory when they are confronted with Ellsberg’s three-color example. More generally, such decision makers are in conflict (...) with either the ordering assumption or the independence assumption of subjective expected utility theory. While a clear majority of the theoretical responses to these violations have advocated maintaining ordering while relaxing independence, a persistent minority has advocated abandoning the ordering assumption. The purpose of this paper is to consider a similar dilemma that exists within the context of multiattribute models, where it arises by considering indeterminacy in the weighting of attributes rather than indeterminacy in the determination of probabilities as in Ellsberg’s example. (shrink)

In this article we show how to extract with the use of the Buchholz -Cichon-Weiermann approach to subrecursive hierarchies from Rathjen's 1991 ordinal analysis of KPM a characterization of the provably total number-theoretic functions of KPM and some of its subsystems in a uniform and direct way.

C.F Gauss’s computational work in number theory attracted renewed interest in the twentieth century due to, on the one hand, the edition of Gauss’s Werke, and, on the other hand, the birth of the digital electronic computer. The involvement of the U.S. American mathematicians Derrick Henry Lehmer and Daniel Shanks with Gauss’s work is analysed, especially their continuation of work on topics as arccotangents, factors of n2 + a2, composition of binary quadratic forms. In general, this strand in Gauss’s (...) reception is part of a more general phenomenon, i.e. the influence of the computer on mathematics and one of its effects, the reappraisal of mathematical exploration. (shrink)

The existing literature on indeterminacy in the law focuses mostly on the use of vague terms in legislation – terms the use of which makes the content of the relevant utterance to some extent indeterminate. As I aim to show, however, not only is the content of a legislative utterance often indeterminate, it is often indeterminate what the content of such an utterance is. In the first two sections of the paper, I discuss in some detail the conditions for (...) successful non-literal speech and address the question whether these conditions are satisfied in the legal context. I argue that due to the fact that legislative contexts generally contain little unequivocal information about legislative intent, interpreters are typically not warranted in taking the legislature to have intended to communicate something non-literal. In the third section, I consider what I take to be the strongest case against my argument: the wealth of actual cases in which the courts have taken the content of the law to be something other than its literal content, seemingly based on relatively straightforward inferences about the legislature’s communicative intentions. As I hope to show, however, very few of these cases are as straightforward as they appear to be. In the fourth, and final section, I argue that the argument from the first two sections has important consequences for the extent to which we should take the content of the law to be determinate. This has significant implications for the analysis of a number of important but controversial legal cases, which I discuss in some detail. (shrink)

This paper provides an exposition of the structuralist approach to underdetermination, which aims to resolve the underdetermination of theories by identifying their common theoretical structure. Applications of the structuralist approach can be found in many areas of philosophy. I present a schema of the structuralist approach, which conceptually unifies such applications in different subject matters. It is argued that two classic arguments in the literature, Paul Benacerraf’s argument on natural numbers and W. V. O. Quine’s argument for the indeterminacy (...) of translation, can be analyzed as instances of the structuralist schema. These two applications illustrate different kinds of conclusions that can be drawn through the structuralist approach; Benacerraf’s argument shows that we can derive an ontological conclusion about the given subject matter, while Quine’s structuralist approach leads to a semantic conclusion about how to determine linguistic meanings given radical translation. Then, as a case study, I review a recent debate in metaphysics between Shamik Dasgupta, Jason Turner, and Catharine Diehl to consider the extent to which different instances of the structuralist schema are conceptually unified. Both sides of the debate can be interpreted as utilizing the structuralist approach; one side uses the structuralist approach for an ontological conclusion, while the other side relies on a semantic conclusion. I argue that this has a strong dialectical consequence, which sheds light on the conceptual unity of the structuralist approach. (shrink)

The second volume of “Advancing Uncertain Combinatorics through Graphization, Hyperization, and Uncertainization: Fuzzy, Neutrosophic, Soft, Rough, and Beyond” presents a deep exploration of the progress in uncertain combinatorics through innovative methodologies like graphization, hyperization, and uncertainization. This volume integrates foundational concepts from fuzzy, neutrosophic, soft, and rough set theory, among others, to further advance the field. Combinatorics and set theory, two central pillars of mathematics, focus on counting, arrangement, and the study of collections under defined rules. Combinatorics excels in handling (...) uncertainty, while set theory has evolved with concepts such as fuzzy and neutrosophic sets, which enable the modeling of complex real-world uncertainties by addressing truth, indeterminacy, and falsehood. These advancements, when combined with graph theory, give rise to novel forms of uncertain sets in "graphized" structures, including hypergraphs and superhypergraphs. Innovations such as Neutrosophic Oversets, Undersets, and Offsets, as well as the Nonstandard Real Set, build upon traditional graph concepts, pushing both theoretical and practical boundaries. The synthesis of combinatorics, set theory, and graph theory in this volume provides a robust framework for addressing the complexities and uncertainties inherent in both mathematical and real-world systems, paving the way for future research and application. In the first chapter, “A Review of the Hierarchy of Plithogenic, Neutrosophic, and Fuzzy Graphs: Survey and Applications”, the authors investigate the interrelationships among various graph classes, including Plithogenic graphs, and explore other related structures. Graph theory, a fundamental branch of mathematics, focuses on networks of nodes and edges, studying their paths, structures, and properties. A Fuzzy Graph extends this concept by assigning a membership degree between 0 and 1 to each edge and vertex, representing the level of uncertainty. The Turiyam Neutrosophic Graph is introduced as an extension of both Neutrosophic and Fuzzy Graphs, while Plithogenic graphs offer a potent method for managing uncertainty. The second chapter, “Review of Some Superhypergraph Classes: Directed, Bidirected, Soft, and Rough”, examines advanced graph structures such as directed superhypergraphs, bidirected hypergraphs, soft superhypergraphs, and rough superhypergraphs. Classical graph classes include undirected graphs, where edges lack orientation, and directed graphs, where edges have specific directions. Recent innovations, including bidirected graphs, have sparked ongoing research and significant advancements in the field. Soft Sets and their extension to Soft Graphs provide a flexible framework for managing uncertainty, while Rough Sets and Rough Graphs address uncertainty by using lower and upper approximations to handle imprecise data. Hypergraphs generalize traditional graphs by allowing edges, or hyperedges, to connect more than two vertices. Superhypergraphs further extend this by allowing both vertices and edges to represent subsets, facilitating the modeling of hierarchical and group-based relationships. The third chapter, “Survey of Intersection Graphs, Fuzzy Graphs, and Neutrosophic Graphs”, explores the intersection graph models within the realms of Fuzzy Graphs, Intuitionistic Fuzzy Graphs, Neutrosophic Graphs, Turiyam Neutrosophic Graphs, and Plithogenic Graphs. The chapter highlights their mathematical properties and interrelationships, reflecting the growing number of graph classes being developed in these areas. Intersection graphs, such as Unit Square Graphs, Circle Graphs, and Ray Intersection Graphs, are crucial for understanding complex graph structures in uncertain environments. The fourth chapter, “Fundamental Computational Problems and Algorithms for SuperHyperGraphs”, addresses optimization problems within the SuperHypergraph framework, such as the SuperHypergraph Partition Problem, Reachability, and Minimum Spanning SuperHypertree. The chapter also adapts classical problems like the Traveling Salesman Problem and the Chinese Postman Problem to the SuperHypergraph context, exploring how hypergraphs, which allow hyperedges to connect more than two vertices, can be used to solve complex hierarchical and relational problems. The fifth chapter, “A Short Note on the Basic Graph Construction Algorithm for Plithogenic Graphs”, delves into algorithms designed for Plithogenic Graphs and Intuitionistic Plithogenic Graphs, analyzing their complexity and validity. Plithogenic Graphs model multi-valued attributes by incorporating membership and contradiction functions, offering a nuanced representation of complex relationships. The sixth chapter, “Short Note of Bunch Graph in Fuzzy, Neutrosophic, and Plithogenic Graphs”, generalizes traditional graph theory by representing nodes as groups (bunches) rather than individual entities. This approach enables the modeling of both competition and collaboration within a network. The chapter explores various uncertain models of bunch graphs, including Fuzzy Graphs, Neutrosophic Graphs, Turiyam Neutrosophic Graphs, and Plithogenic Graphs. In the seventh chapter, “A Reconsideration of Advanced Concepts in Neutrosophic Graphs: Smart, Zero Divisor, Layered, Weak, Semi, and Chemical Graphs”, the authors extend several fuzzy graph classes to Neutrosophic graphs and analyze their properties. Neutrosophic Graphs, a generalization of fuzzy graphs, incorporate degrees of truth, indeterminacy, and falsity to model uncertainty more effectively. The eighth chapter, “Short Note of Even-Hole-Graph for Uncertain Graph”, focuses on Even-Hole-Free and Meyniel Graphs analyzed within the frameworks of Fuzzy, Neutrosophic, Turiyam Neutrosophic, and Plithogenic Graphs. The study investigates the structure of these graphs, with an emphasis on their implications for uncertainty modeling. The ninth chapter, “Survey of Planar and Outerplanar Graphs in Fuzzy and Neutrosophic Graphs”, explores planar and outerplanar graphs, as well as apex graphs, within the contexts of fuzzy, neutrosophic, Turiyam Neutrosophic, and plithogenic graphs. The chapter examines how these types of graphs are used to model uncertain parameters and relationships in mathematical and real-world systems. The tenth chapter, “General Plithogenic Soft Rough Graphs and Some Related Graph Classes”, introduces and explores new concepts such as Turiyam Neutrosophic Soft Graphs and General Plithogenic Soft Graphs. The chapter also examines models of uncertain graphs, including Fuzzy, Intuitionistic Fuzzy, Neutrosophic, and Plithogenic Graphs, all designed to handle uncertainty in diverse contexts. The eleventh chapter, “Survey of Trees, Forests, and Paths in Fuzzy and Neutrosophic Graphs”, provides a comprehensive study of Trees, Forests, and Paths within the framework of Fuzzy and Neutrosophic Graphs. This chapter focuses on classifying and analyzing graph structures like trees and paths in uncertain environments, contributing to the ongoing development of graph theory in the context of uncertainty. (shrink)

QUINE'S DOCTRINE of the indeterminacy of translation is made possible by the principle of substitution characteristic of extensional logic. The same characteristic makes it impossible, in philosophy of science, to choose among theoretical models no one of which is obviously best suited to explain the facts. Hilary Putnam achieved a sort of closure to the problem of reference in philosophy of science, when he pointed out the implications of the Skolem-Löwenheim theorem. He said that besides the facts a theory (...) is designed to explain, there are any number of unintended interpretations of the same theory that assign the correct truth values to all sentences of the theory. There can be no sound basis for deciding upon the sole interpretation for a theory. Following Putnam, some philosophers have abandoned the problem of reference in the context of axiomatic systems. Ian Hacking, for instance, prefers a notion of reference "that is not tied by any specific, binding theory about what is referred to." He has turned to the study of approximations and practical models, spurred on by the difficulty of reconciling axiomatic systems to scientific practice. (shrink)

This book is the fifth volume in the series of Collected Papers on Advancing Uncertain Combinatorics through Graphization, Hyperization, and Uncertainization: Fuzzy, Neutrosophic, Soft, Rough, and Beyond. This volume specifically delves into the concept of Various SuperHyperConcepts, building on the foundational advancements introduced in previous volumes. The series aims to explore the ongoing evolution of uncertain combinatorics through innovative methodologies such as graphization, hyperization, and uncertainization. These approaches integrate and extend core concepts from fuzzy, neutrosophic, soft, and rough set theories, (...) providing robust frameworks to model and analyze the inherent complexity of real-world uncertainties. At the heart of this series lies combinatorics and set theory—cornerstones of mathematics that address the study of counting, arrangements, and the relationships between collections under defined rules. Traditionally, combinatorics has excelled in solving problems involving uncertainty, while advancements in set theory have expanded its scope to include powerful constructs like fuzzy and neutrosophic sets. These advanced sets bring new dimensions to uncertainty modeling by capturing not just binary truth but also indeterminacy and falsity. In this fifth volume, the exploration of Various SuperHyperConcepts provides an innovative lens to address uncertainty, complexity, and hierarchical relationships. It synthesizes key methodologies introduced in earlier volumes, such as hyperization and neutrosophic extensions, while advancing new theories and applications. From pioneering hyperstructures to applications in advanced decision-making, language modeling, and neural networks, this book represents a significant leap forward in uncertain combinatorics and its practical implications across disciplines. -/- The book is structured into 17 chapters, each contributing unique perspectives and advancements in the realm of Various SuperHyperConcepts and their related frameworks: Chapter 1 introduces the concept of Body-Mind-Soul-Spirit Fluidity within psychology and phenomenology, while examining established social science frameworks like PDCA and DMAIC. It extends these frameworks using Neutrosophic Sets, a flexible extension of Fuzzy Sets, to improve their adaptability for mathematical and programming applications. The chapter emphasizes the potential of Neutrosophic theory to address multi-dimensional challenges in social sciences. Chapter 2 delves into the theoretical foundation of Hyperfunctions and their generalizations, such as Hyperrandomness and Hyperdecision-Making. It explores higher-order frameworks like Weak Hyperstructures, Hypergraphs, and Cognitive Hypermaps, aiming to establish their versatility in addressing multi-layered problems and setting a foundation for further studies. Chapter 3 extends traditional decision-making methodologies into HyperDecision-Making and n-SuperHyperDecision-Making. By building on approaches like MCDM and TOPSIS, this chapter develops frameworks capable of addressing complex decision-making scenarios, emphasizing their applicability in dynamic, multi-objective contexts. Chapter 4 explores integrating uncertainty frameworks, including Fuzzy, Neutrosophic, and Plithogenic Sets, into Large Language Models (LLMs). It proposes innovative models like Large Uncertain Language Models and Natural Uncertain Language Processing, integrating hierarchical and generalized structures to advance the handling of uncertainty in linguistic representation and processing. Chapter 5 introduces the Natural n-Superhyper Plithogenic Language by synthesizing natural language, plithogenic frameworks, and superhyperstructures. This innovative construct seeks to address challenges in advanced linguistic and structural modeling, blending attributes of uncertainty, complexity, and hierarchical abstraction. Chapter 6 defines mathematical extensions such as NeutroHyperstructures and AntiHyperstructures using the Neutrosophic Triplet framework. It formalizes structures like neutro-superhyperstructures, advancing classical frameworks into higher-dimensional realms. Chapter 7 explores the extension of Binary Code, Gray Code, and Floorplans through hyperstructures and superhyperstructures. It highlights their iterative and hierarchical applications, demonstrating their adaptability for complex data encoding and geometric arrangement challenges. Chapter 8 investigates the Neutrosophic TwoFold SuperhyperAlgebra, combining classical algebraic operations with neutrosophic components. This chapter expands upon existing algebraic structures like Hyperalgebra and AntiAlgebra, exploring hybrid frameworks for advanced mathematical modeling. Chapter 9 introduces Hyper Z-Numbers and SuperHyper Z-Numbers by extending the traditional Z-Number framework with hyperstructures. These extensions aim to represent uncertain information in more complex and multidimensional contexts. Chapter 10 revisits category theory through the lens of hypercategories and superhypercategories. By incorporating hierarchical and iterative abstractions, this chapter extends the foundational principles of category theory to more complex and layered structures. Chapter 11 formalizes the concept of n-SuperHyperBranch-width and its theoretical properties. By extending hypergraphs into superhypergraphs, the chapter explores recursive structures and their potential for representing intricate hierarchical relationships. Chapter 12 examines superhyperstructures of partitions, integrals, and spaces, proposing a framework for advancing mathematical abstraction. It highlights the potential applications of these generalizations in addressing hierarchical and multi-layered problems. Chapter 13 revisits Rough, HyperRough, and SuperHyperRough Sets, introducing new concepts like Tree-HyperRough Sets. The chapter connects these frameworks to advanced approaches for modeling uncertainty and complex relationships. Chapter 14 explores Plithogenic SuperHyperStructures and their applications in decision-making, control, and neuro systems. By integrating these advanced frameworks, the chapter proposes innovative directions for extending existing systems to handle multi-attribute and contradictory properties. Chapter 15 focuses on superhypergraphs, expanding hypergraph concepts to model complex structural types like arboreal and molecular superhypergraphs. It introduces Generalized n-th Powersets as a unifying framework for broader mathematical applications, while also touching on hyperlanguage processing. Chapter 16 defines NeutroHypergeometry and AntiHypergeometry as extensions of classical geometric structures. Using the Geometric Neutrosophic Triplet, the chapter demonstrates the flexibility of these frameworks in representing multi-dimensional and uncertain relationships. Chapter 17 establishes the theoretical groundwork for SuperHyperGraph Neural Networks and Plithogenic Graph Neural Networks. By integrating advanced graph structures, this chapter opens pathways for applying neural networks to more intricate and uncertain data representations. (shrink)

The article considers pros and cons for a theoretic-measurement analogy, proposed by some philosophers as an illustration of semantic indeterminacy. Within this analogy ascribing of meanings to a certain linguistic expressions is compared with attribution of numbers according to a certain theory of measurement. Donald Davidson used this analogy in order to extend W.V.O. Quine's thesis of indeterminacy of translation to the interpretation of all human behavior. In other words, not only linguistic meanings, but all mental states (...) are considered as indeterminate. The article explains the failure of some counter-arguments put forward against Davidson's use of this analogy and against the thesis of indeterminacy on the whole. Particularly, instrumentalist version of the analogy is rejected for there is no direct relation between indeterminacy and underdetermination of theories by empirical evidences. The article concludes that the semantic indeterminacy is largely based on indeterminacy of rationality. (shrink)

A quantum measurement-like event can produce any of a number of macroscopically distinct results, with corresponding macroscopically distinct gravitational fields, from the same initial state. Hence the probabilistically evolving large-scale structure of space-time is not precisely or even always approximately described by the deterministic Einstein equations.Since the standard treatment of gravitational wave propagation assumes the validity of the Einstein equations, it is questionable whether we should expect all its predictions to be empirically verified. In particular, one might expect the (...) stochasticity of amplified quantum indeterminacy to cause coherent gravitational wave signals to decay faster than standard predictions suggest. This need not imply that the radiated energy flux from gravitational wave sources differs from standard theoretical predictions. An underappreciated bonus of gravitational wave astronomy is that either detecting or failing to detect predicted gravitational wave signals would constrain the form of the semi-classical theory of gravity that we presently lack. (shrink)

In many of his writings, Quine has argued that language is indeterminate in various ways. He has pursued, at length and often, an ingenious conclusion about one such way, which he sometimes calls the inscrutability of reference and, sometimes, the inscrutability of terms. It is the conclusion that one dimension of indeterminacy leaves the references of general terms unfixed among a number of alternatives; further, that no sort of scrutiny of the terms or of the occasions of their (...) utterance could, in principle, provide a means for settling objectively which referent to assign to a term. This single doctrine assumes various guises: there is a firm claim about incompatible but equally acceptable translations of certain Japanese classifiers; there is a somewhat less clear commitment to the inscrutability of a choice between expressions and their Godel numbers as referents for quoted expressions; further, there is a yet more tentative endorsement of Harman's example of the various referents of numerical expressions given by competing set theoretic reductions of number. (shrink)

This comment focuses on Chapter 9 of The Boundary Stones of Thought and the argument, due to William Tait, that Ian Rumfitt there sustains for the indeterminacy of set. I argue that Michael Dummett’s argument, based on the notion of indefinite extensibility and set aside by Rumfitt, provides a more powerful basis for the same conclusion. In addition, I outline two difficulties for the way Rumfitt attempts to save classical logic from acknowledged failures of the principle of bivalence, one (...) specifically for his treatment of the set-theoretic case, the other of more general bearing but especially germane to the case of vagueness. (shrink)

A number of analytical philosophers have recently endorsed the view that the world itself is indeterminate in some respect. Intriguingly, ideas similar to the view are expressed by thinkers from Chinese Madhyamaka Buddhism, which may shed light on the current discussion of worldly indeterminacy. Using as a basis Chinese Madhyamaka thought, together with Jessica Wilson’s account of indeterminacy, I develop an ontological conception of indeterminacy, termed ontic indeterminacy, which centres on two complementary ideas—conclusive indeterminability and (...) provisional determinability. I show that OI is well-equipped to tackle several issues of worldly indeterminacy. My overarching aim is to present a viable and sustainable perspective on the subject of indeterminacy to enrich analytical philosophers’ insights into the intricate nature of reality. (shrink)

Developing earlier studies of the system of numbers in Mundurucu, this paper argues that the Mundurucu numeral system is far more complex than usually assumed. The Mundurucu numeral system provides indirect but insightful arguments for a modular approach to numbers and numerals. It is argued that distinct components must be distinguished, such as a system of representation of numbers in the format of internal magnitudes, a system of representation for individuals and sets, and one-to-one correspondences between the numerosity expressed by (...) the number and its metrics. It is shown that while many-number systems involve a compositionality of units, sets and sets composed of units, few-number languages, such as Mundurucu, do not have access to sets composed of units in the usual way. The nonconfigurational character of the Mundurucu language, which is related to a property for which we coin the term 'low compositionality power', accounts for this and explains the curious fact that Mundurucus make use of marked one-to-one correspondence strategies in order to overcome the limitations of the core system at the perceptual/motor interface of the language faculty. We develop an analysis of a particular construction, parallel numbers, which has not been studied before, elucidating the whole system. This analysis, we argue, sheds new light on classical philosophical, psychological and linguistic debates about numbers and numerals and their relation to language, and more particularly, sheds light on few-number languages. (shrink)

One might suppose that Everettian quantum mechanics (EQM) is inhospitable to metaphysial indeterminacy (MI), given that, as A. Wilson (2020) puts it, "the central idea of EQM is to replace indeterminacy with multiplicity" (77). But as Wilson goes on to suggest, the popular decoherence-based understanding of EQM (henceforth: DEQM) appears to admit of indeterminacy in both world number and world nature, where the latter indeterminacy---our focus here---is plausibly metaphysical. After a brief presentation of DEQM (S1), (...) we bolster the case for there being MI in world nature in DEQM (S2). The remainder of the paper is devoted to a comparative assessment of the two main approaches to MI for purposes of accommodating this MI---namely, a metaphysical supervaluationist approach (as per Barnes and Williams 2011) and a determinable-based approach (as per Wilson 2013 and Calosi and Wilson 2018 and 2021). We briefly describe each approach (S3), then offer arguments in favour of a determinable-based approach to world nature MI in DEQM (S4). (shrink)

In semiotics, final causation can be related to the process of determination (PAPE, 1993). From Peirce’s point of view, determination is not a causal determinism, but a delimitation of a range of possibilities. One starts from objects towards interpretants, in a process mediated by the sign, in which the dynamic object works as a force that constrains interpretants to correspond to their objects. The correspondence between object and interpretant is important because it is through a generated interpretant that the object (...) of a sign can be known. Even though this process of determination coincides with the idea of final causation, there is a certain indeterminacy in it. For Peirce (EP 2:353, 1905), vagueness and generality are two types of indeterminacy. In the terms of the phenomenological categories, vagueness is an indeterminacy of the order of firstness, generality an indeterminacy of the order of thirdness, and both, to some extent, are opposed to that which is defined, which belongs to secondness. Each aspect of the sign may vary according to the three phenomenological categories. Consequently, degrees of imprecision are added to the semiotic process, which is a determination process. Peirce asserts that the perfect precision of thought is theoretically unattainable (SS 11, 1903). Every sign is vague or general at least to some degree. In this paper, we seek to perceive degrees of indetermination and causality from an analysis of the kinds of objects and interpretants proposed by Peirce in the system of 28 sign classes. (shrink)

In simple action theory, when people choose between courses of action, they know what the outcome will be. When an individual is making a choice "against nature," such as switching on a light, that assumption may hold true. But in strategic interaction outcomes, indeterminacy is pervasive and often intractable. Whether one is choosing for oneself or making a choice about a policy matter, it is usually possible only to make a guess about the outcome, one based on anticipating what (...) other actors will do. In this book Russell Hardin asserts, in his characteristically clear and uncompromising prose, "Indeterminacy in contexts of strategic interaction... Is an issue that is constantly swept under the rug because it is often disruptive to pristine social theory. But the theory is fake: the indeterminacy is real."In the course of the book, Hardin thus outlines the various ways in which theorists from Hobbes to Rawls have gone wrong in denying or ignoring indeterminacy, and suggests how social theories would be enhanced--and how certain problems could be resolved effectively or successfully--if they assumed from the beginning that indeterminacy was the normal state of affairs, not the exception. Representing a bold challenge to widely held theoretical assumptions and habits of thought, Indeterminacy and Society will be debated across a range of fields including politics, law, philosophy, economics, and business management. (shrink)

The question of whether Everettian quantum mechanics (EQM) justifies the existence of metaphysical indeterminacy has recently come to the fore. Metaphysical indeterminacy has been argued to emerge from three sources: coherent superpositions, the indefinite number of branches in the quantum multiverse and the nature of these branches. This paper reviews the evidence and concludes that those arguments don’t rely on EQM alone and rest on metaphysical auxiliary assumptions that transcend the physics of EQM. We show how EQM (...) can be ontologically interpreted without positing metaphysical indeterminacy by adopting a deflationary attitude towards branches. Two ways of developing the deflationary view are then proposed: one where branches are eliminated, and another where they are reduced to the universal quantum state. (shrink)

In ‘Fair Infinite Lotteries’ (FIL), Wenmackers and Horsten use non-standard analysis to construct a family of nicely-behaved hyperrational-valued probability measures on sets of natural numbers. Each probability measure in FIL is determined by a free ultrafilter on the natural numbers: distinct free ultrafilters determine distinct probability measures. The authors reply to a worry about a consequent ‘arbitrariness’ by remarking, “A different choice of free ultrafilter produces a different ... probability function with the same standard part but infinitesimal differences.” They illustrate (...) this remark with the example of the sets of odd and even numbers. Depending on the ultrafilter, either each of these sets has probability 1/2, or the set of odd numbers has a probability infinitesimally higher than 1/2 and the set of even numbers infinitesimally lower. The point of the current paper is simply that the amount of indeterminacy is much greater than acknowledged in FIL: there are sets of natural numbers whose probability is far more indeterminate than that of the set of odd or the set of even numbers. (shrink)