Duncan Pritchard recently proposed a Wittgensteinian solution to closure-based skepticism. According to Wittgenstein, all epistemic systems assume certain truths. The notions that we are not disembodied brains, that the Earth has existed for a long time and that one’s name is such-and-such all function as “hinge commitments.” Pritchard views a hinge commitment as a positive propositional attitude that is not a belief. Because closure principles concern only knowledge-apt beliefs, they do not apply to hinge commitments. Thus, from the (...) fact that a subject knows that he is sitting in a room, and the fact that the subject’s sitting in a room entails his bodily existence, it does not follow that the subject also knows that he is not an envatted brain. This paper rejects Pritchard’s non-belief reading of hinge commitments. I start by showing that the non-belief reading fails to solve the skeptical paradox because the reasons that Pritchard uses to support the non-belief reading do not exempt hinge propositions from closure principles. I then proceed to argue that the non-belief reading is false as it claims that hinge commitments, unlike ordinary beliefs, are rationally unresponsive—with the help of a scenario in which a subject’s experience is internally chaotic, we can safely conclude that the hinge commitment that one is not systematically mistaken about the world is equally responsive to one’s evidential situations. (shrink)

Endocrinologists apply the idea of feedback loops to explain how hormones regulate certain bodily functions such as glucose metabolism. In particular, feedback loops focus on the maintenance of the plasma concentrations of glucose within a narrow range. Here, we put forward a different, organicist perspective on the endocrine regulation of glycaemia, by relying on the pivotal concept of closure of constraints. From this perspective, biological systems are understood as organized ones, which means that they are constituted of a set (...) of mutually dependent functional structures acting as constraints, whose maintenance depends on their reciprocal interactions. Closure refers specifically to the mutual dependence among functional constraints in an organism. We show that, when compared to feedback loops, organizational closure can generate much richer descriptions of the processes and constraints at play in the metabolism and regulation of glycaemia, by making explicit the different hierarchical orders involved. We expect that the proposed theoretical framework will open the way to the construction of original mathematical models, which would provide a better understanding of endocrine regulation from an organicist perspective. (shrink)

In this paper, we present a non-trivial and expressively complete paraconsistent naïve theory of truth, as a step in the route towards semantic closure. We achieve this goal by expressing self-reference with a weak procedure, that uses equivalences between expressions of the language, as opposed to a strong procedure, that uses identities. Finally, we make some remarks regarding the sense in which the theory of truth discussed has a property closely related to functional completeness, and we present a (...) sound and complete three-sided sequent calculus for this expressively rich theory. (shrink)

This article discusses the various dimensions of East Central Europe's closure with the communist past, and then assesses the impact of transitional justice measures in the closure with communism. Special attention is paid to the so called 'lustration', which in the view of the author performs important functions in transitions to democratic regimes, related to the reconstruction of a moral and rational community, and to the closure with the communist past. The article shows that the failures and (...) controversies surrounding 'lustration' were due to its radical potential of reconstruction of a moral–rational democratic community, and also to specific socio–political factors of the post–communist ECE. What specific features of ECE post–communist transitions and of lustration conducted to the recurrence of debates related to the communist past is a question that has not been addressed heretofore, despite a fairly well–developed literature on post–communist administrative justice. (shrink)

Trent Dougherty argues (contra Jonathan Matheson) that when taking into consideration the probabilities involving skeptical theism (ST) and gratuitous evils, an agent may reasonably affirm both ST and that gratuitous evils exist. In other words, Dougherty thinks that assigning a greater than .5 probability to ST is insufficient to defeat the commonsense problem of evil. I argue that Dougherty’s response assumes, incorrectly, that ST functions solely as an evidential defeater, and that, when understood as a closure of inquiry defeater, (...) ST may still defeat reasonable belief in gratuitous evils, even in the face of strong evidence that gratuitous evils exist. (shrink)

Traditionally, computational theory (CT) and dynamical systems theory (DST) have presented themselves as opposed and incompatible paradigms in cognitive science. There have been some efforts to reconcile these paradigms, mainly, by assimilating DST to CT at the expenses of its anti-representationalist commitments. In this paper, building on Piccinini’s mechanistic account of computation and the notion of functional closure, we explore an alternative conciliatory strategy. We try to assimilate CT to DST by dropping its representationalist commitments, and by inviting (...) CT to recognize the functionally closed nature of some computational systems. (shrink)

The Two Selves takes the position that the self is not a "thing" easily reduced to an object of scientific analysis. Rather, the self consists in a multiplicity of aspects, some of which have a neuro-cognitive basis (and thus are amenable to scientific inquiry) while other aspects are best construed as first-person subjectivity, lacking material instantiation. As a consequence of their potential immateriality, the subjective aspect of self cannot be taken as an object and therefore is not easily amenable to (...) treatment by current scientific methods. -/- Klein argues that to fully appreciate the self, its two aspects must be acknowledged, since it is only in virtue of their interaction that the self of everyday experience becomes a phenomenological reality. However, given their different metaphysical commitments (i.e., material and immaterial aspects of reality), a number of issues must be addressed. These include, but are not limited to, the possibility of interaction between metaphysically distinct aspects of reality, questions of causal closure under the physical, the principle of energy conservation. -/- After addressing these concerns, Klein presents evidence based on self-reports from case studies of individuals who suffer from a chronic or temporary loss of their sense of personal ownership of their mental states. Drawing on this evidence, he argues that personal ownership may be the factor that closes the metaphysical gap between the material and immaterial selves, linking these two disparate aspects of reality, thereby enabling us to experience a unified sense of self despite its underlying multiplicity. -/- . (shrink)

In this article I review the defining qualities of an ideology, and I introduce the notion of an ideological receptor, which I propose to be an innate cognitive structure in normal human beings. The existence of this ideological receptor would explain the common human tendency toward embracing ideologies essentially spontaneously and uncritically. This tendency is relevant to critical thinking due to the fact that some ideologies are comprised of beliefs that are false, or maladaptive, or beliefs that prescribe actions that (...) are immoral. The ideological receptor should be relatively uncontroversial since it simply groups together several widely acknowledged functions of normal human cognition into a larger, functional cognitive system. That being said, grouping these independent functions together under a unifying label can constitute an arguably useful model of yet another obstacle to critical thinking, and in theory, an awareness of this obstacle can potentially facilitate its neutralization. (shrink)

Attempts to ‘naturalize’ phenomenology challenge both traditional phenomenology and traditional approaches to cognitive science. They challenge Edmund Husserl’s rejection of naturalism and his attempt to establish phenomenology as a foundational transcendental discipline, and they challenge efforts to explain cognition through mainstream science. While appearing to be a retreat from the bold claims made for phenomenology, it is really its triumph. Naturalized phenomenology is spearheading a successful challenge to the heritage of Cartesian dualism. This converges with the reaction against Cartesian thought (...) within science itself. Descartes divided the universe between res cogitans, thinking substances, and res extensa, the mechanical world. The latter won with Newton and we have, in most of objective science since, literally lost our mind, hence our humanity. Despite Darwin, biologists remain children of Newton, and dream of a grand theory that is epistemologically complete and would allow lawful entailment of the evolution of the biosphere. This dream is no longer tenable. We now have to recognize that science and scientists are within and part of the world we are striving to comprehend, as proponents of endophysics have argued, and that physics, biology and mathematics have to be reconceived accordingly. Interpreting quantum mechanics from this perspective is shown to both illuminate conscious experience and reveal new paths for its further development. In biology we must now justify the use of the word “function”. As we shall see, we cannot prestate the ever new biological functions that arise and constitute the very phase space of evolution. Hence, we cannot mathematize the detailed becoming of the biosphere, nor write differential equations for functional variables we do not know ahead of time, nor integrate those equations, so no laws “entail” evolution. The dream of a grand theory fails. In place of entailing laws, a post-entailing law explanatory framework is proposed in which Actuals arise in evolution that constitute new boundary conditions that are enabling constraints that create new, typically unprestatable, Adjacent Possible opportunities for further evolution, in which new Actuals arise, in a persistent becoming. Evolution flows into a typically unprestatable succession of Adjacent Possibles. Given the concept of function, the concept of functional closure of an organism making a living in its world, becomes central. Implications for patterns in evolution include historical reconstruction, and statistical laws such as the distribution of extinction events, or species per genus, and the use of formal cause, not efficient cause, laws. (shrink)

Halbach has argued that Tarski biconditionals are not ontologically conservative over classical logic, but his argument is undermined by the fact that he cannot include a theory of arithmetic, which functions as a theory of syntax. This article is an improvement on Halbach's argument. By adding the Tarski biconditionals to inclusive negative free logic and the universal closure of minimal arithmetic, which is by itself an ontologically neutral combination, one can prove that at least one thing exists. The result (...) can then be strengthened to the conclusion that infinitely many things exist. Those things are not just all Gödel codes of sentences but rather all natural numbers. Against this background inclusive negative free logic collapses into noninclusive free logic, which collapses into classical logic. The consequences for ontological deflationism with respect to truth are discussed. (shrink)

When a group of processes achieves such closure that a set of states of affairs recurs continually, then the effect of that coherence on the world differs from what would occur in the absence of that closure. Such altered effectiveness is an attribute of the system as a whole, and would have consequences. This indicates that the network of processes, as a unit, has ontological significance. Whenever a network of processes generates continual return to a limited set of (...) states of affairs, the system may function as a “whole”— with respect to appropriate interaction partners. The balance achieved by the processes provides the form of definiteness of a unified agent. The causal powers of such coherent aggregates are indeed just the powers of the “constituents acting in concert”. However, the components act in concert in the specific way they do only because of their inclusion in the closed set of interactions that defines the coherence. This renders the causal powers of the coherence defined by that closure non-redundant, and hence the coherence, as a unit, is ontologically significant. The form of definiteness that provides internal coherence also grounds external efficacy of the societal aggregation. The closure is a structural feature of the coherence — possibly, but not necessarily, apparent in spatial structuring. This approach can provide a unified account that includes quantum microphysics, systems biology, and the philosophy of organism ─ without reducing any of these to another. (shrink)

The paper provides both a description of conspiracy beliefs and an insight into their cultural significance. On one side, it highlights their specific formal features, on the other, and this constitutes its peculiarity in the recent literature on the topic, it considers them within the broader genre of general conceptual beliefs, whose main characteristics are weak methodology and logical structure, strong affective and dispositional constraints, epistemic closure and mauvaise foi, and whose main function is practical and self-representative (not epistemic). (...) The paper also claims that political theory and social sciences had some influence in legitimizing certain ideas and stereotypes that are exacerbated in conspiracy beliefs, while other te(le)ological conceptions, such as finalism and providence, may be an important source of ideas that turn out to be widely spread in contemporary criticism towards those same beliefs. (shrink)

Agents are often assumed to have degrees of belief (“credences”) and also binary beliefs (“beliefs simpliciter”). How are these related to each other? A much-discussed answer asserts that it is rational to believe a proposition if and only if one has a high enough degree of belief in it. But this answer runs into the “lottery paradox”: the set of believed propositions may violate the key rationality conditions of consistency and deductive closure. In earlier work, we showed that this (...) problem generalizes: there exists no local function from degrees of belief to binary beliefs that satisfies some minimal conditions of rationality and non-triviality. “Locality” means that the binary belief in each proposition depends only on the degree of belief in that proposition, not on the degrees of belief in others. One might think that the impossibility can be avoided by dropping the assumption that binary beliefs are a function of degrees of belief. We prove that, even if we drop the “functionality” restriction, there still exists no local relation between degrees of belief and binary beliefs that satisfies some minimal conditions. Thus functionality is not the source of the impossibility; its source is the condition of locality. If there is any non-trivial relation between degrees of belief and binary beliefs at all, it must be a “holistic” one. We explore several concrete forms this “holistic” relation could take. (shrink)

It has recently been remarked that the argument for physicalism from the causal closure of the physical is incomplete. It is only effective against mental causation manifested in the action of putative mental forces that lead to acceleration of particles in the nervous system. Based on consideration of anomalous, physically unaccounted-for correlations of neural events, I argue that irreducible mental causation whose nature is at least prima facie probabilistic is conceivable. The manifestation of such causation should be accompanied by (...) a local violation of the Second Law of thermodynamics. I claim that mental causation can be viewed as the disposition of mental states to alter the state probability distribution within the nervous system, with no violation of the conservation laws. If confirmed by neurophysical research, it would indicate a kind of causal homogeneity of the world. Causation would manifest probabilistically in both quantum mechanical and psychophysical systems, and the dynamics of both would be determined by the temporal evolution of the corresponding system state function. Finally, I contend that a probabilistic account of mental causation can consistently explain the character of selectional states that ensure the uniformity of causal patterns, as well as the fact that different physical realizers of a mental property cause the same physical effects in different contexts. (shrink)

Enactivism has the potential to provide a sense of teleology in purpose-directed action, but without violating the principles of efficient causation. Action can be distinguished from mere reaction by virtue of the fact that some systems are self-organizing. Self-organization in the brain is reflected in neural plasticity, and also in the primacy of motivational processes that initiate the release of neurotransmitters necessary for mental and conscious functions, and which guide selective attention processes. But in order to flesh out the enactivist (...) approach in a way that is plausible and not merely an epiphenomenon, it is necessary to confront the problem of causal closure in a serious way. Atoms and molecules in the brain do not violate the normal causal principles that govern them in other contexts. The theory of self-organizing dynamical systems must be developed in a way that is compatible with causal closure rather than contradicting it. (shrink)

Wolfgang Spohn's theory of ranking functions is an elegant and powerful theory of the structure and dynamics of doxastic states. In two recent papers, Spohn has applied it to the analysis of conditionals, claiming to have presented a unified account of indicative and subjunctive (counterfactual) conditionals. I argue that his analysis fails to account for counterfactuals that refer to indirect causes. The strategy of taking the transitive closure that Spohn employs in the theory of causation is not available for (...) counterfactuals. I have a close look at Spohn's treatment of the famous Oswald-Kennedy case in order to illustrate my points. I sketch an alternative view that seems to avoid the problems. (shrink)

We propose a new account of indicative conditionals, giving acceptability and logical closure conditions for them. We start from Adams’ Thesis: the claim that the acceptability of a simple indicative equals the corresponding conditional probability. The Thesis is widely endorsed, but arguably false and refuted by empirical research. To fix it, we submit, we need a relevance constraint: we accept a simple conditional 'If φ, then ψ' to the extent that (i) the conditional probability p(ψ|φ) is high, provided that (...) (ii) φ is relevant for ψ. How (i) should work is well-understood. It is (ii) that holds the key to improve our understanding of conditionals. Our account has (i) a probabilistic component, using Popper functions; (ii) a relevance component, given via an algebraic structure of topics or subject matters. We present a probabilistic logic for simple indicatives, and argue that its (in)validities are both theoretically desirable and in line with empirical results on how people reason with conditionals. (shrink)

In recent decades, several theories have claimed to explain the teleological causality of organisms as a function of self-organising and self-producing processes. The most widely cited theories of this sort are variations of autopoiesis, originally introduced by Maturana and Varela. More recent modifications of autopoietic theory have focused on system organisation, closure of constraints and autonomy to account for organism teleology. This article argues that the treatment of teleology in autopoiesis and other organisation theories is inconclusive for three reasons: (...) First, non-living self-organising processes like autocatalysis meet the defining features of autopoiesis without being teleological; second, organisational approaches, whether defined in terms of the closure of constraints, self-determination or autonomy, are unable to specify teleological normativity, that is, the individuation of an ultimate beneficiary; third, all self-organised systems produce local order by maximising the throughput of energy and/or material (obeying the maximum entropy production (MEP) principle) and thereby are specifically organised to undermine their own critical boundary conditions. Despite these inadequacies, an alternative approach called teleodynamics accounts for teleology. This theory shows how multiple self-organising processes can be collectively linked so that they counter each other’s MEP principle tendencies to become codependent. Teleodynamics embraces – not ignoring – the difficulties of self-organisation, but reinstates teleology as a radical phase transition distinguishing systems embodying an orientation towards their own beneficial ends from those that lack normative character. (shrink)

After a short preface, the first of the three sections of this paper is devoted to historical and philosophic aspects of categoricity. The second section is a self-contained exposition, including detailed definitions, of a proof that every mathematical system whose domain is the closure of its set of distinguished individuals under its distinguished functions is categorically characterized by its induction principle together with its true atoms (atomic sentences and negations of atomic sentences). The third section deals with applications especially (...) those involving the distinction between characterizing a system and axiomatizing the truths of a system. (shrink)

This paper argues that a necessary condition on inferential knowledge is that one knows all the propositions that knowledge depends on. That is, I will argue in support of a principle I call the Knowledge from Knowledge principle: (KFK) S knows that p via inference or reasoning only if S knows all the propositions on which p depends. KFK meshes well with the natural idea that (at least with respect to deductively valid or induc- tively strong arguments) the epistemic status (...) of one’s belief in the conclusion of an argument is a function of the epistemic status of the attitudes one has to the premises of that argument. One gets what one puts in: S’s belief in the conclusion C of an argument A has epistemic status E for S only if S’s belief in each of the premises C depends on also has epistemic status E.1 This is compatible, of course, with the epistemic status of S’s conclusion being greater than the epistemic status of the premises on which the conclusion of her reasoning depends. Furthermore, even though KFK applies more straightforwardly to non-suppositional inferences, we can also plausibly apply the principle to sup- positional reasoning if we treat S’s assuming for the sake of argument that p,2 as S assuming for the sake of argument that she knows that p.3 Although extremely plausible, philosophers have failed to present a concerted case in support of KFK. Most of their energy goes to articulating and/or defending some form of closure principle for knowledge.4 What is more, recently versions of this principle have been dismissed as problematic but I think without the reasons in its support been fully appreciated. This paper is an e fort to change the current state of affairs. Its focus is positive, however: my goal is to make the best case I can in support of KFK.5 To that end, the paper presents the reasons we have to accept KFK (§1), consid- ers an objection to the principle (§2), and situates the principle in the broader normative context of knowledge norms (§3). (shrink)

It is widely agreed that fiction is necessarily incomplete, but some recent work postulates the existence of universal fictions—stories according to which everything is true. Building such a story is supposedly straightforward: authors can either assert that everything is true in their story, define a complement function that does the assertoric work for them, or, most compellingly, write a story combining a contradiction with the principle of explosion. The case for universal fictions thus turns on the intuitive priority we assign (...) to the law of non-contradiction. My goal in this paper is to show that our critical and reflective literary practices set constraints on story-telling which preclude universal fictions. I will raise four stumbling blocks for universal fictionalists: the gap between saying and making true, our actual interpretive reactions to story-level contradictions, the criteria we accept for what counts as a story in our literary practices, and the undesirability of the universal fictionalist’s closure principles. (shrink)

A formal theory of oppositions and opposites is proposed on the basis of a non- Fregean semantics, where opposites are negation-forming operators that shed some new light on the connection between opposition and negation. The paper proceeds as follows. After recalling the historical background, oppositions and opposites are compared from a mathematical perspective: the first occurs as a relation, the second as a function. Then the main point of the paper appears with a calculus of oppositions, by means of a (...) non-Fregean semantics that redefines the logical values of various sorts of sentences. A num- ber of topics are then addressed in the light of this algebraic semantics, namely: how to construct value-functional operators for any logical opposition, beyond the classical case of contradiction; Blanché's "closure problem", i.e., how to find a complete structure connecting the sixteen binary sentences with one another. All of this is meant to devise an abstract theory of opposition: it encompasses the relation of consequence as subalternation, while relying upon the use of a primary "proto- negation" that turns any relatum into an opposite. This results in sentential negations that proceed as intensional operators, while negation is broadly viewed as a difference-forming operator without special constraints on it. (shrink)

Choices confront us with questions. How we act depends on our answers to those questions. So the way our beliefs guide our choices is not just a function of their informational content, but also depends systematically on the questions those beliefs address. This paper gives a precise account of the interplay between choices, questions and beliefs, and harnesses this account to obtain a principled approach to the problem of deduction. The result is a novel theory of belief-guided action that explains (...) and predicts the decisions of agents who, like ourselves, fail to be logically omniscient: that is, of agents whose beliefs may not be deductively closed, or even consistent. (Winner of the 2021 Isaac Levi Prize.). (shrink)

Reasoning from a naturalistic perspective, viewing the mind as an evolved biological organ with a particular structure and function, a number of influential philosophers and cognitive scientists claim that science is constrained by human nature. How exactly our genetic constitution constrains scientific representations of the world remains unclear. This is problematic for two reasons. Firstly, it often leads to the unwarranted conclusion that we are cognitively closed to certain aspects or properties of the world. Secondly, it stands in the way (...) of a nuanced account of the relationship between our cognitive and perceptual wiring and scientific theory. In response, I propose a typology or classification of the different kinds of biological constraints and their sources on science. Using Boden’s notion of a conceptual space, I distinguish between constraints relating to the ease with which we can reach representations within our conceptual space and constraints causing possible representations to fall outside of our conceptual space. This last kind of constraints does not entail that some aspects or properties of the world cannot be represented by us – as argued by advocates of ‘cognitive closure’ – merely that some ways of representing the world are inaccessible to us. It relates to what Clark and Rescher have framed as ‘the alien scientist hypothesis’. The purpose of this typology is to provide some much needed clarity and structure to the debate about biological constraints on science. (shrink)

It is well known that the following features hold of AR + T under the strong Kleene scheme, regardless of the way the language is Gödel numbered: 1. There exist sentences that are neither paradoxical nor grounded. 2. There are 2ℵ0 fixed points. 3. In the minimal fixed point the weakly definable sets (i.e., sets definable as {n∣ A(n) is true in the minimal fixed point where A(x) is a formula of AR + T) are precisely the Π1 1 sets. (...) 4. In the minimal fixed point the totally defined sets (sets weakly defined by formulae all of whose instances are true or false) are precisely the ▵1 1 sets. 5. The closure ordinal for Kripke's construction of the minimal fixed point is ωCK 1. In contrast, we show that under the weak Kleene scheme, depending on the way the Gödel numbering is chosen: 1. There may or may not exist nonparadoxical, ungrounded sentences. 2. The number of fixed points may be any positive finite number, ℵ0, or 2ℵ0 . 3. In the minimal fixed point, the sets that are weakly definable may range from a subclass of the sets 1-1 reducible to the truth set of AR to the Π1 1 sets, including intermediate cases. 4. Similarly, the totally definable sets in the minimal fixed point range from precisely the arithmetical sets up to precisely the ▵1 1 sets. 5. The closure ordinal for the construction of the minimal fixed point may be ω, ωCK 1, or any successor limit ordinal in between. In addition we suggest how one may supplement AR + T with a function symbol interpreted by a certain primitive recursive function so that, irrespective of the choice of the Godel numbering, the resulting language based on the weak Kleene scheme has the five features noted above for the strong Kleene language. (shrink)

This paper applies homotopy type theory to formal semantics of natural languages and proposes a new model for the linguistic phenomenon of copredication. Copredication refers to sentences where two predicates which assume different requirements for their arguments are asserted for one single entity, e.g., "the lunch was delicious but took forever". This paper is particularly concerned with copredication sentences with quantification, i.e., cases where the two predicates impose distinct criteria of quantification and individuation, e.g., "Fred picked up and mastered three (...) books." In our solution developed in homotopy type theory and using the rule of existential closure following Heim analysis of indefinites, common nouns are modeled as identifications of their aspects using HoTT identity types, e.g., the common noun book is modeled as identifications of its physical and informational aspects. The previous treatments of copredication in systems of semantics which are based on simple type theory and dependent type theories make the correct predictions but at the expense of ad hoc extensions (e.g., partial functions, dot types and coercive subtyping). The model proposed here, also predicts the correct results but using a conceptually simpler foundation and no ad hoc extensions. (shrink)

The principle of the causal closure of the physical plays a central role in the philosophy of mind, especially as a premise in arguments for physicalism. In this paper I explore what role the principle can play in the argument for physicalism and whether the evidence we have for the principle allows for such a role or function.

Several recent results on the aggregation of judgments over logically connected propositions show that, under certain conditions, dictatorships are the only propositionwise aggregation functions generating fully rational (i.e., complete and consistent) collective judgments. A frequently mentioned route to avoid dictatorships is to allow incomplete collective judgments. We show that this route does not lead very far: we obtain oligarchies rather than dictatorships if instead of full rationality we merely require that collective judgments be deductively closed, arguably a minimal condition of (...) rationality, compatible even with empty judgment sets. We derive several characterizations of oligarchies and provide illustrative applications to Arrowian preference aggregation and Kasher and Rubinsteinís group identification problem. (shrink)

The third volume of “Advancing Uncertain Combinatorics through Graphization, Hyperization, and Uncertainization: Fuzzy, Neutrosophic, Soft, Rough, and Beyond” presents an in-depth exploration of the cutting-edge developments in uncertain combinatorics and set theory. This comprehensive collection highlights innovative methodologies such as graphization, hyperization, and uncertainization, which enhance combinatorics by incorporating foundational concepts from fuzzy, neutrosophic, soft, and rough set theories. These advancements open new mathematical horizons, offering novel approaches to managing uncertainty within complex systems. Combinatorics, a discipline focused on counting, arrangement, (...) and structure, often faces challenges when uncertainty is present. Set theory, which underpins combinatorial problems, has evolved to tackle these challenges. The introduction of fuzzy and neutrosophic sets has expanded the toolkit for modeling uncertainty by incorporating elements of truth, indeterminacy, and falsehood into decision-making processes. These innovations seamlessly intersect with graph theory, providing new ways to represent uncertain structures through "graphized" forms such as hypergraphs and superhypergraphs. This volume also introduces advanced concepts like Neutrosophic Oversets, Undersets, and Offsets, which push the boundaries of classical graph theory and offer deeper insights into the mathematical and practical challenges posed by real-world systems. By blending combinatorics, set theory, and graph theory, the authors have created a robust framework for addressing uncertainty in both mathematical systems and their real-world applications. This foundation sets the stage for future breakthroughs in combinatorics, set theory, and related fields. Each chapter in this volume contributes both theoretical foundations and practical applications, demonstrating the power of integrating graph theory, set theory, and uncertainty models. The new ideas, algorithms, and mathematical tools presented here will drive the future of combinatorial research and its applications in uncertain environments. In the first chapter, “Introduction to Upside-Down Logic: Its Deep Relation to Neutrosophic Logic and Applications”, the authors present Upside-Down Logic, a novel logical framework that systematically transforms truths into falsehoods and vice versa, based on contextual shifts. Introduced by F. Smarandache, this paper provides a mathematical definition of Upside-Down Logic, including applications related to the Japanese language. The chapter also introduces Contextual Upside-Down Logic, an extension that adjusts logical connectives alongside flipped truth values, as well as Indeterm-Upside-Down Logic and Certain Upside-Down Logic to address indeterminacy. A simple algorithm is also proposed to demonstrate the computational aspects of this logic. In the second chapter, “Local-Neutrosophic Logic and Local-Neutrosophic Sets: Incorporating Locality with Applications”, the authors introduce Local-Neutrosophic Logic and Local-Neutrosophic Sets, which integrate the concept of locality into Neutrosophic Logic. By defining locality as the influence of immediate surroundings on an object or system, this chapter explores how it affects indeterminacy in real-world problems. The paper also examines potential applications and provides mathematical definitions for these new concepts. The third chapter, “A Review of Fuzzy and Neutrosophic Offsets: Connections to Some Set Concepts and Normalization Function”, extends the concept of offsets in uncertain set-theoretic frameworks, such as Fuzzy Sets, Neutrosophic Sets, and Plithogenic Sets. This chapter introduces several advanced types of offsets, including Nonstationary Fuzzy Offset, Multi-valued Plithogenic Offset, and Subset-valued Neutrosophic Offset, offering deeper insights into handling uncertainty in mathematical models. In the fourth chapter, “Review of Plithogenic Directed, Mixed, Bidirected, and Pangene OffGraph”, the authors build upon Plithogenic Graphs to propose extensions such as Plithogenic Directed OffGraph, Plithogenic BiDirected OffGraph, and Plithogenic Mixed OffGraph. These new concepts, including the Plithogenic Pangene OffGraph, are explored in detail, with a focus on their mathematical properties and potential applications in uncertain graph theory. The fifth chapter, “Short Note on Neutrosophic Closure Matroids”, explores the extension of matroid concepts into Neutrosophic and Turiyam Neutrosophic set theories, introducing Neutrosophic closure matroids. This concept integrates uncertainty, indeterminacy, and liberal states into matroid theory, enhancing its applicability in optimization and combinatorial problems. In the sixth chapter, “Some Graph Parameters for Superhypertree-width and Neutrosophic Tree-width”, the authors discuss graph parameters such as Superhypertree-width and Neutrosophic tree-width. These parameters play a crucial role in the study of graph characteristics, particularly in algorithms and real-world applications. The chapter explores the generalization of hypergraphs to SuperHyperGraphs and examines how these concepts extend tree-width parameters within the context of Neutrosophic logic. (shrink)

The scientific study of consciousness, also referred to as consciousness science, is a young scientific field devoted to understanding how conscious experiences and the brain relate. It comprises a host of theories, experiments, and analyses that aim to investigate the problem of consciousness empirically, theoretically, and conceptually. This thesis addresses some of the questions that arise in these investigations from a formal and mathematical perspective. These questions concern theories of consciousness, experimental paradigms, methodology, and artificial consciousness. -/- Regarding theories of (...) consciousness, the thesis contributes to the understanding of the mathematical structure that some of the formal theories in the field propose. The work presented here targets the theory of consciousness known as Integrated Information Theory (IIT) and the neuroscientific theory known as Predictive Processing Theory or Free Energy Principle in its Active Inference form (AI-PP). The thesis provides axiomatic definitions of the mathematical structures that constitute these theories, and uses these definitions to address some of the open questions surrounding the theories. For AI-PP, this includes a rigorous derivation of the formula for Active Inference via Free Energy minimisation and a proof of compositionality of Free Energy. For IIT, this includes resolutions of some of the criticisms of IIT's formal scope and applications, but also the identification of new issues that concern the formalism and its derivation. When possible, the definitions are provided in the mathematical framework of category theory. -/- Regarding experiments, the thesis addresses the main paradigm for testing and falsifying theories of consciousness currently applied in the field. This paradigm consists of comparing the conscious experience that a theory predicts with the conscious experience that is inferred from behavioural data or report by use of measures of consciousness. The thesis provides a formal model of this paradigm and shows that under a certain condition—if inference and prediction are independent—, any minimally informative theory of consciousness can always be falsified. This is deeply problematic since the field’s reliance on report or behaviour to infer conscious experiences, in conjunction with the general structure of most contemporary theories of consciousness, implies such independence. This observation provides the exact formal underpinning of the well-known unfolding argument. The thesis analyses the origin of the problem and identifies precisely which changes are required to avoid this problem in future research. The thesis furthermore shows that the problem of falsifying theories of consciousness, and of empirical comparisons of theories of consciousness more generally, follows from a pervasive closure paradigm in consciousness science, that consists of taking a neuroscientific account of the brain as input to a theory of consciousness, so as to explain what consciousness is, without allowing for modifications or adaptations of the neuroscientific account that would accommodate consciousness as part of the brain's functioning. As is shown in the thesis, this paradigm has implications that point to a fundamental need of revision. -/- Regarding methodological and conceptual questions, the thesis contributes to the foundations of structural research in consciousness science. Structural research aims to use mathematical structures or mathematical spaces, instead of verbal descriptions or simple categorisations, to represent conscious experiences scientifically, for example when building theories of consciousness, or when exploring new empirical avenues to measure consciousness. Despite considerable advances in this realm, there was, prior to this thesis, no explicit definition of what a mathematical structure of conscious experience should be; that is, how the attribution of mathematical structure to conscious experiences should be systematically understood. Perhaps the most important contribution of this thesis to the field is to propose such a definition. The definition, a structural concept, extends existing approaches wherever available, and provides a basis for developing a common formal language to study consciousness, bridging developments as far apart as psychophysics and phenomenology. In addition, and independently of this proposal, the thesis offers a critical analysis of which metaphysical premises need to be presumed in structural research, whether the use of particular formal tools (such as structure-preserving mappings or homomorphisms) is justified, and how structural theories of consciousness could otherwise be built in the first place. An attempt to expand the results from consciousness to more general problems in philosophy of science is made in the context of the well-known Newman problem. -/- Regarding the question of artificial consciousness—can AI feel?—, the thesis contributes two results that take the form of no-go theorems, as known from physics. The first no-go theorem shows that if consciousness is relevant for the temporal evolution of a system's states—if it is dynamically relevant—, then contemporary AI systems cannot be conscious. That is because AI systems run on CPUs, GPUs, TPUs or other processors which have been designed and verified to adhere to computational dynamics that systematically preclude or suppress deviations. The second no-go theorem is situated in the context of computational functionalism, a view which posits that consciousness is a computation. The theorem shows that if computational functionalism holds true, consciousness cannot be a Turing computation. Rather, it must be a novel type of computation that has recently been proposed by Geoffrey Hinton, called mortal computation. -/- This thesis is part of a global effort to pioneer a mathematical perspective in consciousness science, now called Mathematical Consciousness Science. The hope behind the research carried out in this PhD is to illustrate the power and usefulness of mathematical approaches in different areas of consciousness science, and in doing so, to lay the foundations for future mathematical work that complements and supports empirical and theoretical work in the further development of this exciting field. (shrink)

Can we understand (German) idealism as emancipatory today, after the new realist critique? In this paper, I argue that we can do so by identifying a political theology of revolution and utopia at the theoretical heart of German Idealism. First, idealism implies a certain revolutionary event at its foundation. Kant’s Copernicanism is ingrained, methodologically and ontologically, into the idealist system itself. Secondly, this revolutionary origin remains a “non-place” for the idealist system, which thereby receives a utopian character. I define the (...) utopian as the ideal gap, produced by and from within the real, between the non-place of the real as origin and its reduplication as the non-place of knowledge’s closure, as well as the impulse, inherent in idealism, to attempt to close that gap and fully replace the old with the new. Based on this definition, I outline how the utopian functions in Kant, Fichte and Hegel. Furthermore, I suggest that idealism may be seen as a political-theological offshoot of realism, via the objective creation of a revolutionary condition. The origin of the ideal remains in the real, maintaining the utopian gap and the essentially critical character of idealism, both at the level of theory and as social critique. (shrink)

Empirical knowledge exists in the form of antiskeptical conditionals, which are propositions like [if I am not undetectably deceived, then I am holding a pen]. Such conditionals, despite their trivial appearance, have the same essential content as the categorical propositions that we usually discuss, and can serve the same functions in science and practical reasoning. This paper sketches out two versions of a general response to skepticism that employs these conditionals. The first says that our ordinary knowledge attributions can safely (...) be replaced by statements using antiskeptical conditionals, which provides a way around the standard sort of skeptical argument while accepting its soundness with respect to the usual targets. The second analyzes the objects of our ordinary knowledge attributions as antiskeptical conditionals, which allows us to refute, not just evade, the skeptic's argument. Both versions compare favorably to the best-knowncurrent approaches to skepticism, including semantic contextualism. (shrink)

Causal closure arguments against interactionist dualism are currently popular amongst physicalists. Such an argument appeals to some principles of the causal closure of the physical, together with certain other premises, to conclude that at least some mental events are identical with physical events. However, it is crucial to the success of any such argument that the physical causal closure principle to which it appeals is neither too strong nor too weak by certain standards. In this paper, it (...) is argued that various forms of naturalistic dualism, of an emergentist character, are consistent with the strongest physical causal closure principles that can plausibly be advocated. (shrink)

We report the results of four empirical studies designed to investigate the extent to which an epistemic closure principle for knowledge is reflected in folk epistemology. Previous work by Turri (2015a) suggested that our shared epistemic practices may only include a source-relative closure principle—one that applies to perceptual beliefs but not to inferential beliefs. We argue that the results of our studies provide reason for thinking that individuals are making a performance error when their knowledge attributions and denials (...) conflict with the closure principle. When we used research materials that overcome what we think are difficulties with Turri’s original materials, we found that participants did not reject closure. Furthermore, when we presented Turri’s original materials to non-philosophers with expertise in deductive reasoning (viz., professional mathematicians), they endorsed closure for both perceptual and inferential beliefs. Our results suggest that an unrestricted closure principle—one that applies to all beliefs, regardless of their source—provides a better model of folk patterns of knowledge attribution than a source-relative closure principle. (shrink)

Certain puzzling cases have been discussed in the literature recently which appear to support the thought that knowledge can be obtained by way of deduction from a falsehood; moreover, these cases put pressure, prima facie, on the thesis of counter closure for knowledge. We argue that the cases do not involve knowledge from falsehood; despite appearances, the false beliefs in the cases in question are causally, and therefore epistemologically, incidental, and knowledge is achieved despite falsehood. We also show that (...) the principle of counter closure, and the concomitant denial of knowledge from falsehood, is well motivated by considerations in epistemological theory--in particular, by the view that knowledge is first in the epistemological order of things. (shrink)

Recent research has identified a tension between the Safety principle that knowledge is belief without risk of error, and the Closure principle that knowledge is preserved by competent deduction. Timothy Williamson reconciles Safety and Closure by proposing that when an agent deduces a conclusion from some premises, the agent’s method for believing the conclusion includes their method for believing each premise. We argue that this theory is untenable because it implies problematically easy epistemic access to one’s methods. Several (...) possible solutions are explored and rejected. (shrink)

This article gives an overview over different principles of epistemic closure, their attractions and their problems.

Epistemologists understand radical skepticism as arising from two principles: Closure and Underdetermination. Both possess intuitive prima facie support for their endorsement. Understanding how they engender skepticism is crucial for any reasonable anti-skeptical attempt. The contemporary discussion has focused on elucidating the relationship between them to ascertain whether they establish distinct skeptical questions and which of the two constitutes the ultimately fundamental threat. Major contributions to this debate are due to Brueckner, Cohen, and Pritchard. This contribution aims at defending Brueckner’s (...) contention that underdetermination expresses the fundamental skeptical threat and that the closure-based argument can ultimately be reduced to it, at least concerning skeptical contexts. This will be achieved by undermining Cohen’s objections to Brueckner on both counts. Cohen’s argument endorses a picture of evidential underdetermination, which, while apt for non-skeptical contexts, cannot be applied to radical skepticism. A comparison with a case of scientific underdetermination is developed to argue for this idea. Our argument is then applied to the principle equivalence issue and to objections against it. It is then shown how this analysis possesses a distinctive effect on our understanding of the skeptical threat and the assessment of successful anti-skeptical strategies. (shrink)

Duncan Pritchard’s Epistemic Angst promises a novel solution to the closure-based sceptical problem that, unlike more traditional solutions, does not entail revising our fundamental epistemological commitments. In order to do this, it appeals to a Wittgensteinian account of rational evaluation, the overarching theme of which is that it neither makes sense to doubt nor to believe in our anti-sceptical hinge commitments. The purpose of this paper is to show that the argument for the claim that there can be no (...) rational basis to believe our anti-sceptical hinge commitments relies upon an implicit assumption about rational support that I label The Pritchensteinian Rational Grounds Principle. I argue that, insofar as this principle is intended to apply to closure-style inferences, it leads to irrational doxastic attitudes. I consider a seemingly plausible modification of the principle that would avoid this result but show that this modified principle faces serious problems of its own. -/- . (shrink)

In this article, I define and then defend the principle of information closure (pic) against a sceptical objection similar to the one discussed by Dretske in relation to the principle of epistemic closure. If I am successful, given that pic is equivalent to the axiom of distribution and that the latter is one of the conditions that discriminate between normal and non-normal modal logics, a main result of such a defence is that one potentially good reason to look (...) for a formalization of the logic of “ $S$ is informed that $p$ ” among the non-normal modal logics, which reject the axiom, is also removed. This is not to argue that the logic of “ $S$ is informed that $p$ ” should be a normal modal logic, but that it could still be insofar as the objection that it could not be, based on the sceptical objection against pic, has been removed. In other word, I shall argue that the sceptical objection against pic fails, so such an objection provides no ground to abandon the normal modal logic B (also known as KTB) as a formalization of “ $S$ is informed that $p$ ”, which remains plausible insofar as this specific obstacle is concerned. (shrink)

The idea that knowledge can be extended by inference from what is known seems highly plausible. Yet, as shown by familiar preface paradox and lottery-type cases, the possibility of aggregating uncertainty casts doubt on its tenability. We show that these considerations go much further than previously recognized and significantly restrict the kinds of closure ordinary theories of knowledge can endorse. Meeting the challenge of uncertainty aggregation requires either the restriction of knowledge-extending inferences to single premises, or eliminating epistemic uncertainty (...) in known premises. The first strategy, while effective, retains little of the original idea—conclusions even of modus ponens inferences from known premises are not always known. We then look at the second strategy, inspecting the most elaborate and promising attempt to secure the epistemic role of basic inferences, namely Timothy Williamson’s safety theory of knowledge. We argue that while it indeed has the merit of allowing basic inferences such as modus ponens to extend knowledge, Williamson’s theory faces formidable difficulties. These difficulties, moreover, arise from the very feature responsible for its virtue- the infallibilism of knowledge. (shrink)

The principle of epistemic closure is the claim that what is known to follow from knowledge is known to be true. This intuitively plausible idea is endorsed by a vast majority of knowledge theorists. There are significant problems, however, that have to be addressed if epistemic closure – closed knowledge – is endorsed. The present essay locates the problem for closed knowledge in the separation it imposes between knowledge and evidence. Although it might appear that all that stands (...) between knowing the truth of the premises of a valid inference and knowledge of its conclusion is inferring it from the premises, the evidence for each of the premises may jointly count against the conclusion. The intuitive view regarding inferred knowledge says one thing, the evidence says another. One epistemological framework that seems to have the resources to resolve this tension endorses the view that knowledge always requires conclusive evidence. A second framework resolves the tension by limiting the scope of the closure principle. Only inferences drawn directly from propositions contained in the scope of a single knowledge operator are considered closed. The aim of the present essay is to revive the unpopular third option, the idea that knowledge is open. The essay proceeds by arguing that in different ways the two former frameworks only succeed in relocating the problem, not in resolving it. The first framework, the infallibilist view, relocates the problem to a sharp separation between knowledge of the occurrences of events from knowledge of their chance of occurring, a separation leading to several significant additional problems. The fallibilist view, the second framework, in endorsing closure neglects to take into full account the ways in which evidence fails to be transitive. For instance, evidence can count in favor of a conjunction while counting against each of its conjuncts. This fact, which is argued for in the essay on probabilistic as well as non-probabilistic grounds, is used as the foundation of an argument against closed knowledge that can be used as a way to understand several of the most fundamental challenges of epistemology. Not only can an open knowledge view that is based on open evidence resolve all these problems in a simple and natural way, it can also respond to formidable challenges that significantly hinder other open knowledge views. There are good reasons, then, to view both knowledge and evidence as open. (shrink)

According to the principle of epistemic closure, knowledge is closed under known implication. The principle is intuitive but it is problematic in some cases. Suppose you know you have hands and you know that ‘I have hands’ implies ‘I am not a brain-in-a-vat’. Does it follow that you know you are not a brain-in-a-vat? It seems not; it should not be so easy to refute skepticism. In this and similar cases, we are confronted with a puzzle: epistemic closure (...) is an intuitive principle, but at times, it does not seem that we know by implication. In response to this puzzle, the literature has been mostly polarized between those who are willing to do away with epistemic closure and those who think we cannot live without it. But there is a third way. Here I formulate a restricted version of the principle of epistemic closure. In the standard version, the principle can range over any proposition; in the restricted version, it can only range over those propositions that are within the limits of a given epistemic inquiry and that do not constitute the underlying assumptions of the inquiry. If we adopt the restricted version, I argue, we can preserve the advantages associated with closure, while at the same time avoiding the puzzle I’ve described. My discussion also yields an insight into the nature of knowledge. I argue that knowledge is best understood as a topic-restricted notion, and that such a conception is a natural one given our limited cognitive resources. (shrink)

According to the principle of Conjunction Closure, if one has justification for believing each of a set of propositions, one has justification for believing their conjunction. The lottery and preface paradoxes can both be seen as posing challenges for Closure, but leave open familiar strategies for preserving the principle. While this is all relatively well-trodden ground, a new Closure-challenging paradox has recently emerged, in two somewhat different forms, due to Backes :3773–3787, 2019a) and Praolini :715–726, 2019). This (...) paradox synthesises elements of the lottery and the preface and is designed to close off the familiar Closure-preserving strategies. By appealing to a normic theory of justification, I will defend Closure in the face of this new paradox. Along the way I will draw more general conclusions about justification, normalcy and defeat, which bear upon what Backes :2877–2895, 2019b) has dubbed the ‘easy defeat’ problem for the normic theory. (shrink)

In response to the claim that certain epistemically defective inferences such as Moore’s argument lead us to the conclusion that we ought to abandon closure, Crispin Wright suggests that we can avoid doing so by distinguishing it from a stronger principle, namely transmission. Where closure says that knowledge of a proposition is a necessary condition on knowledge of anything one knows to entail it, transmission makes a stronger claim, saying that by reasoning deductively from known premises one can (...) thereby acquire knowledge of or justification for the conclusion. Wright’s thought is that in cases such as Moore what is really going on is a failure of transmission, not a failure of closure. The problem with this claim is that it relies on an outdated formulation of closure which few nowadays would find plausible. Once we stipulate that it is the intuitive closure principle that deserves our attention, it becomes far less obvious that the project of diagnosing what is wrong with arguments such as Moore in terms of transmission failure but not closure failure can be vindicated. In order to demonstrate this claim, I consider intuitive closure of propositional justification vs. transmission of propositional justification and concede that perhaps these can come apart in the way that Wright suggests. I then argue that despite this concession, we also need an answer to the question of whether intuitive closure for doxastic justification and transmission for doxastic justification can come apart. We need an answer to this question because our ultimate interest in these issues stems from our interest in whether knowledge obeys closure, not merely whether propositional justification does. I argue that, even on the assumption that there is transmission failure of propositional justification, there is no transmission failure of doxastic justification. And if this is true, then there is no transmission failure of knowledge either. The transmission failure diagnosis of Moore’s argument and its ilk is thus in trouble. (shrink)

Todd and Rabern (2021) have argued that if we assume that future contingents are untrue and if we accept the Retro-closure principle (???? → PF(????)), then the existence of a temporal omniscient entity becomes metaphysically impossible. Since the truth of a metaphysical and theological theory should not be dependent on questions of temporal semantics, Todd and Rabern conclude that, if one wishes to maintain that future contingents are untrue, one must abandon the Retro-closure principle. The aim of this (...) paper is to propose a temporal semantics system in which future contingents are untrue, the Retro-closure principle is valid, and the possibility of the existence of an omniscient and temporal being is guaranteed. (shrink)

Closure is the epistemological thesis that if S knows that P and knows that P implies Q, then if S infers that Q, S knows that Q. Fred Dretske acknowledges that closure is plausible but contends that it should be rejected because it conflicts with the plausible thesis: Conclusive reasons (CR): S knows that P only if S believes P on the basis of conclusive reasons, i.e., reasons S wouldn‘t have if it weren‘t the case that P. Dretske (...) develops an analysis of knowing that centers on CR, and argues that the requirement undermines skepticism by implying the falsity of closure. We develop a Dretske-style analysis of knowing that incorporates CR, and we argue that this analysis not only accords with closure, but also implies it. In addition, we argue that the analysis accounts for the prima facie plausibility of closure-invoking skeptical arguments, and nonetheless implies that they are fallacious. If our arguments turn out to be sound, the acceptability of Dretske‘s analysis of knowing will be significantly enhanced by the fact that, despite implying closure, it undermines closure-based skepticism. (shrink)

In spite of the intuitiveness of epistemic closure, there has been a stubborn stalemate regarding whether it is true, largely because some of the “Moorean” things we seem to know easily seem clearly to entail “heavyweight” philosophical things that we apparently cannot know easily—or perhaps even at all. In this paper, I will show that two widely accepted facts about what we do and don’t know—facts with which any minimally acceptable understanding of knowledge must comport—are jointly inconsistent with the (...) truth of CLR. The proof works by supposing the truth of “Categorialism,” a thesis about the relation between basic categories and common nouns and predicates, which is itself a heavyweight claim that cannot be easily known to be either true or false. (shrink)

Deduction is important to scientific inquiry because it can extend knowledge efficiently, bypassing the need to investigate everything directly. The existence of closure failure—where one knows the premises and that the premises imply the conclusion but nevertheless does not know the conclusion—is a problem because it threatens this usage. It means that we cannot trust deduction for gaining new knowledge unless we can identify such cases ahead of time so as to avoid them. For philosophically engineered examples we have (...) “inner alarm bells” to detect closure failure, but in scientific investigation we would want to use deduction for extension of our knowledge to matters we don’t already know that we couldn’t know. Through a quantitative treatment of how fast probabilistic sensitivity is lost over steps of deduction, I identify a condition that guarantees that the growth of potential error will be gradual; thus, dramatic closure failure is avoided. Whether the condition is fulfilled is often obvious, but sometimes it requires substantive investigation. I illustrate that not only safe deduction but the discovery of dramatic closure failures can lead to scientific advances. (shrink)

The acceptance of causal closure has had a profoundly limiting effect on the philosophical treatment of sui generis mental causation in recent decades. Philosophical treatments of special divine action have been likewise hampered by a widespread commitment to closure. If fundamental reality is as closure tells us it is, then nonphysical minds—human and divine— are either causally impotent or redundant. In this paper, I reject this limitation as baseless. Specifically, I will show how Hempel’s dilemma poses a (...) far greater challenge when applied to a causal closure principle than it did to the task of defining physicalism. Reponses to the dilemma can be used to disambiguate closure, but not without significant cost. In light of Hempel’s dilemma, there is no conception of closure that can do all of the following: rule out sui generis mental causation, have a reasonable chance of being true, and avoid problematic circularity. At the end of the day, I maintain that closure, properly understood, is, or ought to be, just one methodological stance among many. If this is correct, it is a stance that a theist can freely reject. (shrink)