Online research in philosophy

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)

Some argue for materialism claiming that a physical event cannot have a non-physical cause, or by claiming the 'Principle of Causal Closure' to be true. This I call a 'Sweeping Naturalistic Argument'. This article argues against this. It describes what it would be for a material event to have an immaterial cause.

It is widely thought that if knowledge requires sensitivity, knowledge is not closed because sensitivity is not closed. This paper argues that there is no valid argument from sensitivity failure to non-closure of knowledge. Sensitivity does not imply non-closure of knowledge. Closure considerations cannot be used to adjudicate between safety and sensitivity accounts of knowledge.

The standard contextualist solution to the skeptical paradox is intended to provide a way to retain epistemic closure while avoiding the excessive modesty of radical skepticism and the immodesty of Moorean dogmatism. However, contextualism’s opponents charge that its solution suffers from epistemic immodesty comparable to Moorean dogmatism. According to the standard contextualist solution, all contexts where an agent knows some ordinary proposition to be true are contexts where she also knows that the skeptical hypotheses are false. It has been (...) hoped that contrastivist theories of knowledge can mirror the contextualist solution while avoiding this epistemic immodesty. I review the main problems for contrastive closure and argue that none of the arguments currently in the literature pose an insurmountable problem for the contrastivist solution. However, I argue that contrastivist theories of knowledge, like their contextualist rivals, are indeed committed to epistemic immodesty. (shrink)

In this paper I present a more refined analysis of the principles of deductive closure and positive introspection. This analysis uses the expressive resources of logics for different types of group knowledge, and discriminates between aspects of closure and computation that are often conflated. The resulting model also yields a more fine-grained distinction between implicit and explicit knowledge, and places Hintikka’s original argument for positive introspection in a new perspective.

In this essay I present a new version of the Paradox of the Knower and show that this new paradox vitiates a certain argument against epistemic closure. I then prove a theorem that relates the new paradox to epistemological scepticism. I conclude by assessing the use of the Knower in arguments against syntactical treatments of knowledge.

Most solutions to the skeptical paradox about justified belief assume closure for justification, since the rejection of closure is widely regarded as a non-starter. I argue that the rejection of closure is not a non-starter, and that its problems are no greater than the problems associated with the more standard anti-skeptical strategies. I do this by sketching a simple version of the unpopular strategy and rebutting the three best objections to it. The general upshot for theories of (...) justification is that it is not a constraint on such theories that we must somehow have justification to believe that we are not massively deceived. (shrink)

Epistemic closure has been a central issue in epistemology over the last forty years. According to versions of the relevant alternatives and subjunctivist theories of knowledge, epistemic closure can fail: an agent who knows some propositions can fail to know a logical consequence of those propositions, even if the agent explicitly believes the consequence (having “competently deduced” it from the known propositions). In this sense, the claim that epistemic closure can fail must be distinguished from the fact (...) that agents do not always believe, let alone know, the consequences of what they know—a fact that raises the “problem of logical omniscience” that has been central in epistemic logic. -/- This paper, part I of II, is a study of epistemic closure from the perspective of epistemic logic. First, I introduce models for epistemic logic, based on Lewis’s models for counterfactuals, that correspond closely to the pictures of the relevant alternatives and subjunctivist theories of knowledge in epistemology. Second, I give an exact characterization of the closure properties of knowledge according to these theories, as formalized. Finally, I consider the relation between closure and higher-order knowledge. The philosophical repercussions of these results and results from part II, which prompt a reassessment of the issue of closure in epistemology, are discussed further in companion papers. -/- As a contribution to modal logic, this paper demonstrates an alternative approach to proving modal completeness theorems, without the standard canonical model construction. By “modal decomposition” I obtain completeness and other results for two non-normal modal logics with respect to new semantics. One of these logics, dubbed the logic of ranked relevant alternatives, appears not to have been previously identified in the modal logic literature. More broadly, the paper presents epistemology as a rich area for logical study. (shrink)

This essay corrects an error in the presentation of the Paradox of the Knowledge-Plus Knower, which is the variant of Kaplan and Montague’s Knower Paradox presented in C. Cross 2001: ‘The Paradox of the Knower without Epistemic Closure,’ MIND, 110, pp. 319–33. The correction adds a universally quantified transitivity principle for derivability as an additional assumption leading to paradox. This correction does not affect the status of the Knowledge-Plus paradox as a rebuttal to an argument against epistemic closure, (...) since the quantified transitivity principle is true in the standard model of arithmetic and therefore innocuous. (shrink)

In “The Paradox of the Knower without Epistemic Closure”, MIND 110:319-33, 2001, I develop a version of the Knower Paradox which does not assume epistemic closure, and I use it to argue that the original Knower Paradox does not support an argument against epistemic closure. In “The Paradox of the Knower without Epistemic Closure?”, MIND 113:95-107, 2004, Gabriel Uzquiano, using his own result, argues that my rebuttal to the anti-closure argument is not successful. I respond (...) here by arguing that in order to use Uzquiano’s result in an argument against closure, one must assume an implausible skepticism about arithmetic. (shrink)

Closure principles loom large in recent internalist critiques of epistemic externalism. Cohen (Philos Phenomenol Res 65:309–329, 2002, Philos Phenomenol Res 70:417–430, 2005), Vogel (J Philos 97:602–623, 2000), and Fumerton (Meta-Epistemology and skepticism. Rowman and Littlefield, Lanham, 1995) argue that, given closure, epistemic externalism is committed to the possibility of implausibly easy knowledge. By contrast, Zalabardo (Philos Rev 114:33–61, 2005) proposes that epistemic closure actually precludes the possibility of easy knowledge, and appeals to closure principles to solve (...) the problem of easy knowledge. In my view, disagreement over closure’s bearing on externalism and the problem of easy knowledge is rooted in a failure to bear in mind the familiar distinction between ex ante and ex post forms of epistemic justification and warrant. When this distinction is kept in focus, the result is clear: epistemic closure provides no relief from the problem of easy knowledge. (shrink)

Anthony Brueckner has argued that claims about underdetermination of evidence are suppressed in closure-based scepticism (“The Structure of the Skeptical Argument”, Philosophy and Phenomenological Research 54:4, 1994). He also argues that these claims about underdetermination themselves lead to a paradoxical sceptical argument—the underdetermination argument—which is more fundamental than the closure argument. If Brueckner is right, the status quo focus of some predominant anti-sceptical strategies may be misguided. In this paper I focus specifically on the relationship between these two (...) arguments. I provide support for Brueckner’s claim that the underdetermination argument is the more fundamental sceptical argument. I do so by responding to a challenge to this claim put forward by Stewart Cohen (“Two Kinds of Skeptical Argument”, Philosophy and Phenomenological Research 58:1, 1998). Cohen invokes an alternative epistemic principle which he thinks can be used to challenge Brueckner. Cohen’s principle raises interesting questions about the relationship between evidential considerations and explanatory considerations in the context of scepticism about our knowledge of the external world. I explore these questions in my defence of Brueckner. (shrink)

Peter Baumann and Nicholas Shackel defend me against a serious criticism by Christoph Jäger. They argue that my account of information is consistent with my denial of closure for knowledge. Information isn’t closed under known entailment either. I think that, technically speaking, they are right. But the way they are right doesn’t help me much in my effort to answer the skeptic. I describe a way in which information, like knowledge, fails to be closed in a way that makes (...) an information-based account of knowledge an effective tool in answering the skeptic. (shrink)

In this article, “Narrative Closure,” a theory of the nature of narrative closure is developed. Narrative closure is identified as the phenomenological feeling of finality that is generated when all the questions saliently posed by the narrative are answered. The article also includes a discussion of the intelligibility of attributing questions to narratives as well as a discussion of the mechanisms that achieve this. The article concludes by addressing certain recent criticisms of the view of narrative expounded (...) by this article. (shrink)

Closure for justification is the claim that thinkers are justified in believing the logical consequences of their justified beliefs, at least when those consequences are competently deduced. Many have found this principle to be very plausible. Even more attractive is the special case of Closure known as Single-Premise Closure. In this paper, I present a challenge to Single-Premise Closure. The challenge is based on the phenomenon of rational self-doubt – it can be rational to be less (...) than fully confident in one's beliefs and patterns of reasoning. In rough outline, the argument is as follows: Consider a thinker who deduces a conclusion from a justified initial premise via an incredibly long sequence of small competent deductions. Surely, such a thinker should suspect that he has made a mistake somewhere. And surely, given this, he should not believe the conclusion of the deduction even though he has a justified belief in the initial premise. (shrink)

In early essays and in more recent work, Fred Dretske argues against the closure of perception, perceptual knowledge, and knowledge itself. In this essay I review his case and suggest that, in a useful sense, perception is closed, and that, while perceptual knowledge is not closed under entailment, perceptually based knowledge is closed, and so is knowledge itself. On my approach, which emphasizes the safe indication account of knowledge, we can both perceive, and know, that sceptical scenarios (such as (...) being a brain in a vat) do not hold. (shrink)

Anthony Brueckner argues for a strong connection between the closure and the underdetermination argument for scepticism. Moreover, he claims that both arguments rest on infallibilism: In order to motivate the premises of the arguments, the sceptic has to refer to an infallibility principle. If this were true, fallibilists would be right in not taking the problems posed by these sceptical arguments seriously. As many epistemologists are sympathetic to fallibilism, this would be a very interesting result. However, in this paper (...) I will argue that Brueckner’s claims are wrong: The closure and the underdetermination argument are not as closely related as he assumes and neither rests on infallibilism. Thus even a fallibilist should take these arguments to raise serious problems that must be dealt with somehow. (shrink)

Knowledge closure is, roughly, the following claim: For every agent S and propositions P and Q, if S knows P, knows that P implies Q, and believes Q because it is so implied, then S knows Q. Almost every epistemologist believes that closure is true. Indeed, they often believe that it so obviously true that any theory implying its denial is thereby refuted. Some prominent epistemologists have nevertheless denied it, most famously Fred Dretske and Robert Nozick. There are (...) closure advocates who see other virtues in those accounts, however, and so who introduce revisions of one sort or another in order to preserve closure while maintaining their spirit. One popular approach is to replace the “sensitivity” constraint at the heart of both of those accounts with a “safety” constraint, as advocated by Timothy Williamson, Duncan Pritchard, Ernest Sosa, Stephen Luper, and others. The purpose of this essay is to show that this approach does not succeed: safety does not save closure. And neither does a popular variation on the safety theme, the safe-basis or safe-indicator account. (shrink)

Contextualists claim two important virtues for their view. First, contextualism is a non-skeptical epistemology, given the plausible idea that not all contexts invoke the high standards for knowledge needed to generate the skeptical conclusion that we know little or nothing. Second, contextualism is able to preserve closure concerning knowledge – the idea that knowledge is extendable on the basis of competent deduction from known premises. As long as one keeps the context fixed, it is plausible to think that some (...) closure principle can be articulated that will survive scrutiny. Opponents of contextualism often try to gain an advantage over it by claiming that their view mimics these virtues of contextualism as well as having other virtues. A recent example of the same is termed ‘contrastivism," as presented by Jonathan Schaffer. I will argue that the representation made is chimerical, that in fact contrastivism has no hope of mirroring these twin virtues of contextualism. (shrink)

Epistemic closure, the idea that knowledge is closed under known implication, plays a central role in current discussions of skepticism and the semantics of knowledge reports. Contextualists in particular rely heavily on the truth of epistemic closure in staking out their distinctive response to the so-called "skeptical paradox." I argue that contextualists should re-think their commitment to closure. Closure principles strong enough to force the skeptical paradox on us are too strong, and closure principles weak (...) enough to express unobjectionable epistemic principles are too weak to generate the skeptical paradox. I briefly consider how the contextualist might live without (strong) closure. (shrink)

In this paper I articulate a fictionalist solution to the closure problem that affects constructive empiricism. Relying on Stephen Yablo’s recent study of closure puzzles, I show how we can partition the content of a theory in terms of its truthmakers and claim that a constructive empiricist can believe that all the observable conditions that are necessary to make a part of her theory true obtain and remain agnostic about whether or not the other truthmakers for the other (...) parts of her theory obtain. This can be done even though she asserts her theory as if it was wholly true. (shrink)

How should the contrastivist formulate closure? That is, given that knowledge is a ternary contrastive state Kspq (s knows that p rather than q), how does this state extend under entailment? In what follows, I will identify adequacy conditions for closure, criticize the extant invariantist and contextualist closure schemas, and provide a contrastive schema based on the idea of extending answers. I will conclude that only the contrastivist can adequately formulate closure.

The most prominent arguments for scepticism in modern epistemology employ closure principles of some kind. To begin my discussion of such arguments, consider Simple Knowledge Closure (SKC): (SKC) (Kxt[p] ∧ (p → q)) → Kxt[q].1 Assuming its truth for the time being, the sceptic can use (SKC) to reason from the two assumptions that, firstly, we don’t know ¬sh and that, secondly, op entails ¬sh to the conclusion that we don’t know op, where ‘op’ and ‘sh’ are shorthand (...) for ‘ordinary proposition’ and ‘sceptical hypothesis’ respectively. (SKC), however, fails for familiar reasons: since knowledge entails belief (KB), we can derive the falsity (F) from (SKC) by hypothetical syllogism, and thus reduce (SKC) to absurdity: (KB) Kxt[p] → Bxt[p]. (F) (Kxt[p] ∧ (p → q)) → Bxt[q]. (shrink)

Christoph Jäger (2004) argues that Dretske’s information theory of knowledge raises a serious problem for his denial of closure of knowledge under known entailment: Information is closed under known entailment (even under entailment simpliciter); given that Dretske explains the concept of knowledge in terms of “information”, it is hard to stick with his denial of closure for knowledge. Thus, one of the two basic claims of Dretske would have to go. Since giving up the denial of closure (...) would commit Dretske to skepticism, it would most probably be better to rather give up the information-theoretic account of knowledge. But that means that one of the best externalist views of knowledge has to be given up. I argue here that Jäger is mistaken and that there is no problem for Dretske. There is a rather easy way out of Jäger’s problem. (shrink)

JUSTIFICATION and knowledge are thought to be closed under known implication..1 This widely shared assumption is embodied in the following principles of epistemic closure.

This paper argues for a solution to a problem that contrastivism faces. The problem is that contrastivism cannot preserve closure, in spite of claims to the contrary by its defenders. The problem is explained and a response developed.

Kripke’s puzzle has puts pressure on the intuitive idea that one can believe that Superman can fly without believing that Clark Kent can fly. If this idea is wrong then many theories of belief and belief ascription are built from faulty data. I argue that part of the proper analysis of Kripke’s puzzle refutes the closure principles that show up in many important arguments in epistemology, e.g., if S is rational and knows that P and that P entails Q, (...) then if she considers these two beliefs and Q, then she is in a position to know that.. (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)

This article is a response to an important objection that Sherrilyn Roush has made to the standard closure-based argument for skepticism, an argument that has been studied over the past couple of decades. If Roush's objection is on the mark, then this would be a quite significant finding. We argue that her objection fails.

Most of us think we can always enlarge our knowledge base by accepting things that are entailed by (or logically implied by) things we know. The set of things we know is closed under entailment (or under deduction or logical implication), which means that we know that a given claim is true upon recognizing, and accepting thereby, that it follows from what we know. However, some theorists deny that knowledge is closed under entailment, and the issue remains controversial. The arguments (...) against closure include the following. (shrink)

Is the physical world causally closed? Can something immaterial have any causal role within physics? This article seeks to answer these questions by explaining the theory of Causal Closure. Causal Closure says that nothing immaterial can have any causal efficacy upon the material world. Physicalists have long held this position and have used it as an argument against Dualism, but does it hold? The hope of this article is that we may better understand the arguments for and against (...) Causal Closure in order to discover a cogent philosophy of science. (shrink)

I argue against unqualified acceptance of the principle of deductive closure (DC): that, if p follows deductively from premises that are already known, we are in a position to know p. DC, I claim, is a sorites premise; it seems intuitively irresistible, but indiscriminate application of it leads to absurd conclusions. Furthermore, a theory on which the application of DC is restricted explains our practice of deriving new knowledge from old knowledge better than a theory on which our application (...) of DC is unrestricted. This restriction on the application of DC allows contextualists to meet an argument of Hawthorne’s that contextualism must lead either to absurd knowledge attributions or to constant shifting of the standards for knowledge. Even if the standard of knowledge remains constant, the absurd knowledge attribution is the conclusion of a sorites argument and should be rejected. (shrink)

According to Fred Dretskes externalist theory of knowledge a subject knows that p if and only if she believes that p and this belief is caused or causally sustained by the information that p. Another famous feature of Dretskes epistemology is his denial that knowledge is closed under known logical entailment. I argue that, given Dretskes construal of information, he is in fact committed to the view that both information and knowledge are closed under known entailment. This has far-reaching consequences. (...) For if it is true that, as Dretske also believes, accepting closure leads to skepticism, he must either embrace skepticism or abandon his information theory of knowledge. The latter alternative would seem to be preferable. But taking this route would deprive one of the most powerfully developed externalist epistemologies of its foundation. (shrink)

From the mid-1980‘s to the early 2000‘s the wide-ranging resources of the concept we now call sensitivity , which Robert Nozick used to give an analysis of the concepts of knowledge and evidence , went largely unappreciated in epistemology. This was in part because these resources were upstaged by a glamorous implication the condition has for skepticism, and in part because of loss of faith in the project of giving a theory of knowledge at all, due to the failure time (...) and again to construct a theory without counterexamples. The sensitivity condition, or as Nozick called it the variation condition, which requires that were p to be false you wouldn‘t believe it, had its own apparent counterexamples. And while the implication of this condition for skepticism was elegant and principled – it is possible to know that there is a table in front of you without knowing you are not a brain in a vat – it had the price of denying closure of knowledge under known implication, that is, denying that knowing q and knowing that q implies p are together sufficient to make the belief in p that you have on that basis knowledge. (shrink)

This paper examines various claims by Noël Carroll about narrative closure and its relationship to narrative connections, which are, roughly, causal connections generously conceived to include necessary conditions for sufficient conditions for an effect. I propose supplementing the expanded notion of a cause with Michael Bratman’s notion of a psychological connection to account for the particular role that human agents play in narratives. A novel and a film are used as examples to illustrate how the concept of a psychological (...) connection eliminates the need for Carroll's condition that narratives must be globally forward-looking. (shrink)

This paper contributes to the current debate about radical scepticism and the structure of warrant. After a presentation of the standard version of the radical sceptic’s challenge, both in its barest and its more refined form, three anti-sceptical responses, and their respective commitments, are being identified: the Dogmatist response, the Conservativist response and the Dretskean response. It is then argued that both the Dretskean and the Conservativist are right that the anti-sceptical hypothesis cannot inherit any perceptual warrants from ordinary propositions (...) about the environment—and so the Dogmatist response founders. However, if this is so Epistemic Closure lacks any clear rationale. There is therefore good reason to agree with both the Dretskean and the Dogmatist that perceptual warrants for ordinary propositions about the environment are enough in order for those propositions to enjoy a positive epistemic status—and so the Conservativist response founders. However, the Conservativist is nonetheless right that a warrant for the anti-sceptical hypothesis is needed. For contrary to what much of the recent literature suggests, the radical sceptic need not appeal to Epistemic Closure in order to cast doubt on the legitimacy of our beliefs in ordinary propositions about the environment: there is a Pyrrhonian version of scepticism that, though equally radical, is consistent with failure of Epistemic Closure. For this reason, the Dretskean response is insufficient to answer scepticism. (shrink)

The question whether epistemological concepts are closed under deduction is an important one since many skeptical arguments depend on closure. Such skepticism can be avoided if closure is not true of knowledge (or justification). This response to skepticism is rejected by Peter Klein and others. Klein argues that closure is true, and that far from providing the skeptic with a powerful weapon for undermining our knowledge, it provides a tool for attacking the skeptic directly. This paper examines (...) various arguments in favor of closure and Klein's attempted use of closure to refute skepticism. Such a refutation of skepticism is mistaken. But the closure principle is in any case false, so the skepticism that depends on it is undermined. The appeal of the closure principle derives from a failure to recognize an important feature of our epistemological concepts, namely, their context relativity. (shrink)

Is there a plausible argument for external world skepticism? Robert Nozick’s well–known discussion focuses upon arguments which utilize the Sensitivity Requirement and the Closure Principle. Nozick claims, correctly, that no such argument succeeds. But he gets almost all the details wrong. The Sensitivity Requirement and the Closure Principle are compatible; the Sensitivity Requirement is incorrect; and even if true, the Closure Principle is structurally incapable of generating a plausible and valid global skeptical argument. It is therefore (...) a mistake to take the Closure Principle as central in discussions of skepticism. The paper concludes by examining the prospects for a plausible skeptical argument. (shrink)

It is received wisdom that the skeptic has a devastating line of argument in the following. You probably think, he says, that you know that you have hands. But if you knew that you had hands, then you would also know that you were not a brain in a vat, a brain suspended in fluid with electrodes feeding you perfectly coordinated impressions that are generated by a supercomputer, of a world that looks and moves just like this one. You would (...) know you weren’t in this state if you knew you had hands, since having hands implies you are no brain in a vat. You obviously don’t know you’re not a brain in a vat, though—you have no evidence that would distinguish that state from the normal one you think you’re in. Therefore, by modus tollens, you don’t know you have hands. At least, the skeptic has a devastating argument, it is thought, if we grant him closure of knowledge under known implication, which many of us are inclined to do: roughly, if you know p, and you know that p implies q, then you know q.i To say that this is an intuitively compelling argument is an understatement; the project of finding a reply that isn’t table-thumping, or obfuscating, or special-pleading has exercised philosophers for a very long time. The steps of the argument have been scoured in detail to try to find cracks that will yield under pressure. Some of these efforts have been intriguing, and illuminating, and some even appear to provide dialectical victories that shift the burden of proof back to the skeptic. However, as refutations they all come up short. I will argue that we have missed a very simple point: though the skeptical argument above is valid, it has a false premise, namely, the claim that the thing we obviously know implies the thing we seem obviously not to know. This premise, I will argue, cannot be repaired, so we have a refutation; if the skeptic wants to convince us to worry about our ordinary knowledge, he will have to come up with a completely different argument. Closure of knowledge under known implication (hereafter “closure”), is obviously necessary for the skeptical argument presented above.. (shrink)

In this paper, I consider some issues involving a certain closure principle for Structural Justification, a relation between a cognitive subject and a proposition that’s expressed by locutions like ‘S has a source of justification for p’ and ‘p is justifiable for S’. I begin by summarizing recent work by Peter Klein that advances the thesis that the indicated closure principle is plausible but lacks Skeptical utility. I then assess objections to Klein’s thesis based on work by Robert (...) Audi and Anthony Brueckner. One finding is that the typical statement of the relevant closure principle can express a number of different closure principles, and that recognizing this helps to resolve certain disputes. (shrink)

The Knower Paradox has had a brief but eventful history, and principles of epistemic closure (which say that a subject automatically knows any proposition she knows to be materially implied, or logically entailed, by a proposition she already knows) have been the subject of tremendous debate in epistemic logic and epistemology more generally, especially because the fate of standard arguments for and against skepticism seems to turn on the fate of closure. As far as I can tell, however, (...) no one working in either area has emphasized the result I emphasize in this paper: the Knower Paradox just falsifies even the most widely accepted general principles of epistemic closure. After establishing that result, I discuss five of its more important consequences. (shrink)

The general principle of epistemic closure stipulates that epistemic properties are transmissible through logical means. According to this principle, an epistemic operator, say ε, should satisfy any valid scheme of inference, such as: if ε(p entails q), then ε(p) entails ε(q). The principle of epistemic closure under known entailment (ECKE), a particular instance of epistemic closure, has received a good deal of attention since the last thirty years or so. ECKE states that: if one knows that p (...) entails q, and she knows that p, then she knows that q. It is widely accepted that ECKE constitutes an important piece of the skeptical argument, but the acceptance of an unrestricted version of ECKE is still a matter of debate. On the side of the defenders of ECKE, one finds Stine (1976), Brueckner (1985), Vogel (1990), and Feldman (1995). Others proposed a refutation or a limitation of the principle, like Dretske (1970), Nozick (1981), Hales (1995), Williams (1996), and Sosa (1999). As it turns out, the relevant alternatives view (RAV) elaborated by Dretske, which restricts the scope of ECKE, has been discussed extensively and acknowledged as one of the most important contributions. There is nonetheless a major unsolved difficulty pertaining to Dretske-RAV: the notion of relevant alternatives is defined in such a way that it is bounded by counterfactual possibilities. This ontological import leaves open the door to the skeptic. Some have tried to give more precision to this notion, like Stine (1976), who appealed to a Gricean approach to define relevant alternatives in conversational contexts. My proposal is in accordance with the gist of Dretske’s strategy, i.e. to restrict the validity of ECKE, and I claim that in order to escape the difficulties inherent to RAV one has to introduce a more robust notion, the notion of epistemic context. Epistemic contexts are a subclass of propositional contexts. In that perspective, the closure property is expressed in terms of a property of a relation between epistemic contexts. ECKE holds when and only when either the epistemic context of the premisses is the same as the epistemic context of the conclusion, or the epistemic context change between the premisses and the conclusion is permissible. Permissibility of epistemic context change is a function of consistency. By means of this epistemic context approach, I will show that: (1) epistemic contexts are defined by basic propositions (unchallenged justified beliefs), (2) ECKE holds only under very specific constraints, and (3) the skeptical argument involves a non-permissible change of epistemic context and, by the same token, cannot rely upon ECKE. (shrink)

Closure is the principle that a person, who knows a proposition p and knows that p entails q, also knows q. Closure is usually regarded as expressing the commonplace assumption that persons can increase their knowledge through inference from propositions they already know. In this paper, I will not discuss whether closure as a general principle is true. The aim of this paper is to explore the various relations between closure and knowledge through inference. I will (...) show that closure can hold for two propositions p and q for numerous different reasons. The standard reason that S knows q through inference from p, if S knows p and knows that p entails q, is only one of them. Therefore, the relations between closure and inferential knowledge are more complex than one might suspect. (shrink)

E. J. Coffman defends Peter Klein’s work on epistemic closure against various objections that I raised in an earlier paper. In this paper, I respond to Coffman.

The focus of this paper is the prima facie plausible view, expressed by the principle of Counter-Closure, that knowledge-yielding competent deductive inference must issue from known premises. I construct a case that arguably falsifies this principle and consider five available lines of response that might help retain Counter-Closure. I argue that three are problematic. Of the two remaining lines of response, the first relies on non-universal intuitions and forces one to view the case I construct as exhibiting a (...) justified, true belief to which none of the usual diagnoses of knowledge failure in Gettier cases apply. The second line involves claiming that Fake Barns and its ilk are misdiagnosed by epistemological orthodoxy as Gettier cases. We are thus confronted by a trilemma: either the case I discuss undermines the first-blush plausible principle of Counter-Closure; or the case I discuss instantiates a novel kind of Gettier case; or a popular conception of a key range of alleged Gettier cases must be rejected. No matter which horn we choose, the case points to a philosophically curious conclusion. (shrink)

This paper evaluates a number of closure principles (for both knowledge and justification) that have appeared in the literature. Counterexamples are presented to all but one of these principles, which is conceded to be true but trivially so. It is argued that a consequence of the failure of these closure principles is that certain projects of doxastic logic are doomed, and that doxastic logic is of dubious merit for epistemologists interested in actual knowers in the actual world.

This paper looks at an argument strategy for assessing the epistemic closure principle. This is the principle that says knowledge is closed under known entailment; or (roughly) if S knows p and S knows that p entails q, then S knows that q. The strategy in question looks to the individual conditions on knowledge to see if they are closed. According to one conjecture, if all the individual conditions are closed, then so too is knowledge. I give a deductive (...) argument for this conjecture. According to a second conjecture, if one (or more) condition is not closed, then neither is knowledge. I give an inductive argument for this conjecture. In sum, I defend the strategy by defending the claim that knowledge is closed if, and only if, all the conditions on knowledge are closed. After making my case, I look at what this means for the debate over whether knowledge is closed. (shrink)

Graham and Maitzen think my CORNEA principle is in trouble because it entails “intolerable violations of closure under known entailment.” I argue that the trouble arises from current befuddlement about closure itself, and that a distinction drawn by Rudolph Carnap, suitably extended, shows how closure, when properly understood, works in tandem with CORNEA. CORNEA does not obey Closure because it shouldn’t: it applies to “dynamic” epistemic operators, whereas closure principles hold only for “static” ones. What (...) the authors see as an intolerable vice of CORNEA is actually a virtue, helping us see what closure principles should—and shouldn’t—themselves be about. (shrink)

This paper argues for tlie claims that a) a natural language such as English is semanticaly closed b) semantic closure implies inconsistency. A corollary of these is that the semantics of English must be paraconsistent. The first part of the paper formulates a definition of semantic closure which applies to natural languages and shows that this implies inconsistency. The second section argues that English is semeantically closed. The preceding discussion is predicated on the assumption that there are no (...) truth value gaps. The next section of the paper considers whether the possibility of these makes any difference to the substantive conclusions of the previous sections, and argues that it does not. The crux of the preceding arguments is that none of the consistent semantical accounts that have been offered for solving the semantical paradoxes is a semantic ofEnglish. The final section of the paper produces a general argument as to why this must always be the case. (shrink)

This paper complicates, extends, and modifies Pinch and Bijker's original social construction of technology, specifically their concepts of relevant social groups, closure, and stabilization, in order to gain insight into long-term processes of how we use and understand technology. First, this paper identifies four broad categories of relevant social groups in the social construction of technology based on stake holdings and compares them according to their activities, resources, and directionality. Second, the paper discusses the distinctions between closure and (...) stabilization of technological artifacts, introducing temporary closure and structural flexibility as a means of understanding how different technologies can relate to each other. Third, using Rosch's cognitive approach to categorization, the paper suggests structural flexibility as a means of operationalizing stabilization. These reconceptualizations offer researchers a broader scale with which to understand the social construction of technology. (shrink)

Summary The article argues thatceteris paribus clauses have to be separated from another type of clauses called closure clauses. The former are associated with laws and theories, the latter with test situations of a particular kind. It is also argued that closure clauses, but notceteris paribus clauses, make Popper's falsifiability principle untenable. In that way, it also resolves the quarrel between Popper and Lakatos aboutceteris paribus clauses and falsifiability by saying that both are partly wrong and partly right.

This paper looks at an argument strategy for assessing the epistemic closure principle. This is the principle that says knowledge is closed under known entailment; or (roughly) if S knows p and S knows that p entails q, then S knows that q. The strategy in question looks to the individual conditions on knowledge to see if they are closed. According to one conjecture, if all the individual conditions are closed, then so too is knowledge. I give a deductive (...) argument for this conjecture. According to a second conjecture, if one (or more) condition is not closed, then neither is knowledge. I give an inductive argument for this conjecture. In sum, I defend the strategy by defending the claim that knowledge is closed if, and only if, all the conditions on knowledge are closed. After making my case, I look at what this means for the debate over whether knowledge is closed. (shrink)

Ted A. Warfield reviews the history of epistemology and argues that epistemologists mistakenly take for granted the inference that the failure of closure of some necessary condition on knowledge is sufficient for the failure of epistemic closure. So he concludes that epistemologists should avoid using this inference to explain the failure of epistemic closure. However, I will defend the inference that epistemologists often employ in their discussions. My thesis is that although this inference is invalid, one can (...) still legitimately conclude the failure of epistemic closure from the failure of closure of some necessary condition on knowledge. (shrink)

Is there a plausible argument for external world skepticism? Robert Nozick’s well-known discussion focuses upon arguments which utilize the Sensitivity Requirement and the Closure Principle. Nozick claims, correctly, that no such argument succeeds. But he gets almost all the details wrong. The Sensitivity Requirement and the Closure Principle are compatible; the Sensitivity Requirement is incorrect; and even if true, the Closure Principle is structurally incapable of generating a plausible and valid global skeptical argument. It is therefore a (...) mistake to take the Closure Principle as central in discussions of skepticism. The paper concludes by examining the prospects for a plausible skeptical argument. (shrink)

This paper argues that Epistemic Contextualism, Knowledge Closure, and the Knowledge Account of Assertion are inconsistent. The argument is developed by considering an objection to Contextualism that is unsuccessful. Some Contextualist responses are canvassed and rejected. Finally, it is argued that an analogue of the inconsistency arises for those who accept that justification is closed under known entailment.

In this note some epistemological problems in general theories about living systems are considered; in particular, the question of hidden connections between different areas of experience, such as folk biology and scientific biology, and hidden connections between central concepts of theoretical biology, such as function, semiosis, closure and life.

In this note some epistemological problems in general theories about living systems are considered; in particular, the question of hidden connections between different areas of experience, such as folk biology and scientific biology, and hidden connections between central concepts of theoretical biology, such as function, semiosis, closure and life.

Acceptance of the quantization of the elementary electrical charge (e) was preceded by a bitter dispute between Robert Millikan (1868–1953) and Felix Ehrenhaft (1879–1952), which lasted for many years (1910–25). Both Millikan and Ehrenhaft obtained very similar experimental results and yet Millikan was led to formulate the elementary electrical charge (electron) and Ehrenhaft to fractional charges (subelectron). There have been four major attempts to reconstruct the historical events that led to the controversy: Holton ([1978]); Franklin ([1981]); Barnes et al. ([1996]); (...) Goodstein ([2001]). So we have the controversy not only among the original protagonists but also among those who have interpreted the experiment. The objective of this study is a critical appraisal of the four interpretations and an attempt to provide closure to the controversy. It is plausible to suggest that Ehrenhaft's methodology approximated the traditional scientific method, which did not allow him to discard anomalous data. Millikan, on the other hand, in his publications espoused the scientific method but in private (handwritten notebooks) was fully aware of the dilemma faced and was forced to select data to uphold his presuppositions. A closure to the controversy is possible if we recognize that Millikan's data selection procedure depended primarily on his commitment to his presuppositions (existence of e). Franklin's ([1981]) finding that the selection of the drops did not change the value of e but only its statistical error carries little weight as Millikan did not perform Franklin-style analyses that could have justified the exclusion of drops. It is plausible to suggest that had Millikan performed such analyses, he would have included them in his publication in order to provide support for his data selection procedures. In the absence of his presuppositions, Millikan could not tell which was the ‘expected correct’ value of e and the degree of statistical error. Finally, if we try to understand Millikan's handling of data with no reference to his presuppositions, then some degree of ‘misconduct’ can be perceived. Introduction An appraisal of Holton's interpretation An appraisal of Franklin's interpretation An appraisal of Barnes, Bloor and Henry's interpretation An appraisal of Goodstein's interpretation A crucial test: the second drop (reading) of 15 March 1912 Conclusion: Is closure possible? (shrink)

It is received wisdom that the skeptic has a devastating line of argument in the following. You probably think, he says, that you know that you have hands. But if you knew that you had hands, then you would also know that you were not a brain in a vat, a brain suspended in fluid with electrodes feeding you perfectly coordinated impressions that are generated by a supercomputer, of a world that looks and moves just like this one. You would (...) know you weren’t in this state if you knew you had hands, since having hands implies you are no brain in a vat. You obviously don’t know you’re not a brain in a vat, though—you have no evidence that would distinguish that state from the normal one you think you’re in. Therefore, by modus tollens, you don’t know you have hands. At least, the skeptic has a devastating argument, it is thought, if we grant him closure of knowledge under known implication, which many of us are inclined to do: roughly, if you know p, and you know that p implies q, then you know q. (shrink)

The notion of the rational closure of a positive knowledge base K of conditional assertions | (standing for if then normally ) was first introduced by Lehmann (1989) and developed by Lehmann and Magidor (1992). Following those authors we would also argue that the rational closure is, in a strong sense, the minimal information, or simplest, rational consequence relation satisfying K. In practice, however, one might expect a knowledge base to consist not just of positive conditional assertions, | (...) , but also negative conditional assertions, i (standing for not if then normally . Restricting ourselves to a finite language we show that the rational closure still exists for satisfiable knowledge bases containing both positive and negative conditional assertions and has similar properties to those exhibited in Lehmann and Magidor (1992). In particular an algorithm in Lehmann and Magidor (1992) which constructs the rational closure can be adapted to this case and yields, in turn, completeness theorems for the conditional assertions entailed by such a mixed knowledge base. (shrink)

Many people, such as Adam Smith, Milton Friedman, Irving Fisher, and William Sharpe, assume that free markets full of rational people automatically lead to ethical actions and outcomes. After all, at its equilibrium point, a perfectly competitive free market maximizes utility, respects autonomy, and fulfills justice’s dictates. Unfortunately, in some technology markets, there are a significant number of people who have undergone epistemic closure. Epistemic closure entails that all reliable evidence that would challenge deeply held beliefs is dismissed (...) as corrupted, whereas all supporting evidence, no matter how unreliable, is accepted as incontrovertible. Those who have the condition act irrationally within that domain. As a result, business decisions become much more difficult than they would be in a rational market. In this article, epistemic closure’s ethical issues are developed. First, although they are acting irrationally within the closure’s domain, those with epistemic closure can still be held accountable for their actions. Second, to deal ethically with epistemic closure and its consequences, then it is vital to know what it is and its root causes, as well as to have a practical principle that can assist in making pragmatic decisions. Because some new technologies face epistemic closure, then focusing on a particular representative case of it will help to illustrate the issue’s ethical dimensions. (shrink)

In The Advancement of Science (1993) Philip Kitcher develops what he calls the 'Compromise Model' of the closure of scientific debates. The model is designed to acknowledge significant elements from 'Rationalist' and 'Antirationalist' accounts of science, without succumbing to the one-sidedness of either. As part of an ambitious naturalistic account of scientific progress, Kitcher's model succeeds to the extent that transitions in the history of science satisfy its several conditions. I critically evaluate the Compromise Model by identifying its crucial (...) assumptions and by attempting to apply the model to a major transition in the history of biology: the rejection of 'naive group selectionism' in the 1960s. I argue that the weaknesses and limitations of Kitcher's model exemplify general problems facing philosophical models of scientific change, and that recognition of these problems supports a more modest vision of the relationship between historical and philosophical accounts of science. (shrink)

Let R be a binary relation on some domain. Use R∗ for the reflexive transitive closure of R, i.e., the smallest binary relation S with R ⊆ S that is reflexive and transitive. Use R+ for the transitive closure of R, i.e., the smallest binary relation S with R ⊆ S that is transitive. Use I for the identity relation on the domain. Let n range over natural numbers. Define Rn as follows, by induction: R0 := I Rn+1 (...) := R ◦ R.. (shrink)

Cabe argumentar en favor del fisicismo a partir de consideraciones metodológicas o epistémicas, o desde un punto de vista ontológico. En los últimos años se ha venido presentando un potente argumento ontológico que hace un uso esencial de lo que se ha dado en llamar el "principio del cierre causal del mundo físico". En este artículo examino si es posible que sea la propia física quien fundamente este principio. Propongo que, con la ayuda de las contemporáneas teorías reductivas de la (...) causalidad a intercambio o transferencia de cantidades conservadas, las leyes de conservación pueden proporcionar tal fundamento. También evalúo qué fuerza modal puede tener este principio del cierre. /// It is possible to argue for physicalism from methodological or epistemic considerations or from an ontological position. In the last years one can find a powerful ontological argument for physicalism which makes essential use of what has been labeled "the principle of the causal closure of the physical world". In this paper I examine whether this principle can be grounded in physics itself. I propose that, with the aid of contemporary reductive transference or exchange theories of causation, conservation laws can provide such a basis to the principle of the causal closure. I also consider what modal force the principle may have. (shrink)

Could our observations of apparently pointless evil ever justify the conclusion that God does not exist? Not according to Stephen Wykstra, who several years ago announced the “Condition of Reasonable Epistemic Access,” or “CORNEA,” a principle that has sustained critiques of atheistic arguments from evil ever since. Despite numerous criticisms aimed at CORNEA in recent years, the principle continues to be invoked and defended. We raise a new objection: CORNEA is false because it entails intolerable violations of closure.

An analysis is presented of the relationships between consumers ethical beliefs, ethical ideology, Machiavellianism, political preference and the individual difference variable "need for closure". It is based on a representative survey of 286 Belgian respondents. Standard measurement tools of proven reliability and robustness are used to measure ethical beliefs (consumer ethics scale), ethical ideology (ethical positioning), Machiavellianism (Mach IV scale) and need for closure. The analysis finds the following. First, individuals with a high need for closure tend (...) to have beliefs that are more ethical as regards possible consumer actions, and score higher on idealism and lower on Machiavellianism, than those with a low need for closure. Second, a correlation exists between political preference and ethical beliefs. Third, a significant relationship exists between ethical ideology and political preference for the two largest political parties. Fourth, individuals with a high and low need for closure have different political preferences for right-wing and left-wing parties. (shrink)

In his logical papers, Leo Esakia studied corresponding ordered topological spaces and order-preserving mappings. Similar spaces and mappings appear in many other application areas such the analysis of causality in space-time. It is known that under reasonable conditions, both the topology and the original order relation $${\preccurlyeq}$$ can be uniquely reconstructed if we know the “interior” $${\prec}$$ of the order relation. It is also known that in some cases, we can uniquely reconstruct $${\prec}$$ (and hence, topology) from $${\preccurlyeq}$$. In this (...) paper, we show that, in general, under reasonable conditions, the open order $${\prec}$$ (and hence, the corresponding topology) can be uniquely determined from its closure $${\preccurlyeq}$$. (shrink)

This paper has two main purposes. First, it will provide an introductory discussion of hyperset theory, and show that it is useful for modeling complex systems. Second, it will use hyperset theory to analyze Robert Rosen’s metabolismrepair systems and his claim that living things are closed to efficient cause. It will also briefly compare closure to efficient cause to two other understandings of autonomy, operational closure and catalytic closure.

Temporal logic can be used to describe processes: their behaviour ischaracterized by a set of temporal models axiomatized by a temporaltheory. Two types of models are most often used for this purpose: linearand branching time models. In this paper a third approach, based onsocalled joint closure models, is studied using models which incorporateall possible behaviour in one model. Relations between this approach andthe other two are studied. In order to define constructions needed torelate branching time models, appropriate algebraic notions (...) are defined(in a category theoretical manner) and exploited. In particular, thenotion of joint closure is used to construct one model subsuming a setof models. Using this universal algebraic construction we show that aset of linear models can be merged to a unique branching time model.Logical properties of the described algebraic constructions are studied.The proposed approach has been successfully aplied to obtain anappropriate semantics for non-monotonic reasoning processes based ondefault logic. References are discussed that show the details of theseapplications. (shrink)

In this paper we consider the structure of the class FGModS of full generalized models of a deductive system S from a universal-algebraic point of view, and the structure of the set of all the full generalized models of S on a fixed algebra A from the lattice-theoretical point of view; this set is represented by the lattice FACSs A of all algebraic closed-set systems C on A such that (A, C) ε FGModS. We relate some properties of these structures (...) with tipically logical properties of the sentential logic S. The main algebraic properties we consider are the closure of FGModS under substructures and under reduced products, and the property that for any A the lattice FACSs A is a complete sublattice of the lattice of all algebraic closed-set systems over A. The logical properties are the existence of a fully adequate Gentzen system for S, the Local Deduction Theorem and the Deduction Theorem for S. Some of the results are established for arbitrary deductive systems, while some are found to hold only for deductive systems in more restricted classes like the protoalgebraic or the weakly algebraizable ones. The paper ends with a section on examples and counterexamples. (shrink)

This paper presents an enhanced ontology formalization, combining previous work in Conceptual Structure Theory and Order-Sorted Logic. Most existing ontology formalisms place greater importance on concept types, but in this paper we focus on relation types, which are in essence predicates on concept types. We formalize the notion of ‘predicate of predicates’ as meta-relation type and introduce the new hierarchy of meta-relation types as part of the ontology definition. The new notion of closure of a relation or meta-relation type (...) is presented as a means to complete that relation or meta-relation type by transferring extra arguments and properties from other related types. The end result is an expanded ontology, called the closure of the original ontology, on which automated inference could be more easily performed. Our proposal could be viewed as a novel and improved ontology formalization within Conceptual Structure Theory and a contribution to knowledge representation and formal reasoning (e.g., to build a query-answering system for legal knowledge). (shrink)

We prove some results about the limitations of the expressive power of quantifiers on finite structures. We define the concept of a bounded quantifier and prove that every relativizing quantifier which is bounded is already first-order definable (Theorem 3.8). We weaken the concept of congruence closed (see [6]) to weakly congruence closed by restricting to congruence relations where all classes have the same size. Adapting the concept of a thin quantifier (Caicedo [1]) to the framework of finite structures, we define (...) the concept of a meager quantifier. We show that no proper extension of first-order logic by means of meager quantifiers is weakly congruence closed (Theorem 4.9). We prove the failure of the full congruence closure property for logics which extend first-order logic by means of meager quantifiers, arbitrary monadic quantifiers, and the Härtig quantifier (Theorem 6.1). (shrink)

In our previous paper Algebraic Logic for Classical Conjunction and Disjunction we studied some relations between the fragmentL of classical logic having just conjunction and disjunction and the varietyD of distributive lattices, within the context of Algebraic Logic. The central tool in that study was a class of closure operators which we calleddistributive, and one of its main results was that for any algebraA of type (2,2) there is an isomorphism between the lattices of allD-congruences ofA and of all (...) distributive closure operators overA. In the present paper we study the lattice structure of this last set, give a description of its finite and infinite operations, and obtain a topological representation. We also apply the mentioned isomorphism and other results to obtain proofs with a logical flavour for several new or well-known lattice-theoretical properties, like Hashimoto's characterization of distributive lattices, and Priestley's topological representation of the congruence lattice of a bounded distributive lattice. (shrink)

Nozick’s contribution to the epistemology of the last half of the twentieth century includes addressing the question of whether knowledge is closed under known implication. I argue that the question of closure provides a serious obstacle to Nozickian approaches to epistemology.

Economic stories with a rational choice structure usually entail closure or equilibrium. This paper argues that Knightian uncertainty and Kirznerian alertness allow economists to construct plausible accounts of open-ended processes such as virtuous cycles and vicious circles without abandoning the centrality of instrumental rationality. The basic form of such stories is explored and two example cases are put forward.

In the categorical approach to the foundations of quantum theory, one begins with a symmetric monoidal category, the objects of which represent physical systems, and the morphisms of which represent physical processes. Usually, this category is taken to be at least compact closed, and more often, dagger compact, enforcing a certain self-duality, whereby preparation processes (roughly, states) are interconvertible with processes of registration (roughly, measurement outcomes). This is in contrast to the more concrete “operational” approach, in which the states and (...) measurement outcomes associated with a physical system are represented in terms of what we here call a convex operational model: a certain dual pair of ordered linear spaces–generally, not isomorphic to one another. On the other hand, state spaces for which there is such an isomorphism, which we term weakly self-dual, play an important role in reconstructions of various quantum-information theoretic protocols, including teleportation and ensemble steering. In this paper, we characterize compact closure of symmetric monoidal categories of convex operational models in two ways: as a statement about the existence of teleportation protocols, and as the principle that every process allowed by that theory can be realized as an instance of a remote evaluation protocol—hence, as a form of classical probabilistic conditioning. In a large class of cases, which includes both the classical and quantum cases, the relevant compact closed categories are degenerate, in the weak sense that every object is its own dual. We characterize the dagger-compactness of such a category (with respect to the natural adjoint) in terms of the existence, for each system, of a symmetric bipartite state, the associated conditioning map of which is an isomorphism. (shrink)

This paper is closely related to investigations of abstract properties of basic logical notions expressible in terms of closure spaces as they were begun by A. Tarski (see [6]). We shall prove many properties of -conjunctive closure spaces (X is -conjunctive provided that for every two elements of X their conjunction in X exists). For example we prove the following theorems:1. For every closed and proper subset of an -conjunctive closure space its interior is empty (i.e. it (...) is a boundary set). 2. If X is an -conjunctive closure space which satisfies the -compactness theorem and [X] is a meet-distributive semilattice (see [3]), then the lattice of all closed subsets in X is a Heyting lattice. 3. A closure space is linear iff it is an -conjunctive and topological space. 4. Every continuous function preserves all conjunctions. (shrink)

The main result of this paper is the following theorem: a closure space X has an , , Q-regular base of the power iff X is Q-embeddable in It is a generalization of the following theorems:(i) Stone representation theorem for distributive lattices ( = 0, = , Q = ), (ii) universality of the Alexandroff's cube for T 0-topological spaces ( = , = , Q = 0), (iii) universality of the closure space of filters in the lattice (...) of all subsets for , -closure spaces (Q = 0). By this theorem we obtain some characterizations of the closure space given by the consequence operator for the classical propositional calculus over a formalized language of the zero order with the set of propositional variables of the power . In particular we prove that a countable closure space X is embeddable with finite disjunctions preserved into F iff X is a consistent closure space satisfying the compactness theorem and X contains a 0, -base consisting of -prime sets. (shrink)

Gyula Klima maintains that Anselm's ontological argument is best understood in terms of a theory of reference that was made fully explicit only by later medievals. I accept the interpretative claim but offer here two objections to the argument so interpreted. The first points up a certain ambiguity in Klima's formulation of the argument, the correction of which requires a substantive revision of the argument's conclusion. The second exploits the notion of semantic closure introduced by Tarski. Klima offers the (...) atheist an ?out? by drawing a distinction between constitutive and parasitic reference. I argue that using Klima's preferred description (?the thought object than which no thought object can be thought to be greater?) to refer constitutively to God results in conceptual closure, a condition analogous to semantic closure that renders the instant conceptual scheme inconsistent and subject to paradox. Although the proof ultimately fails, Klima's development of the notions of constitutive and parasitic reference has important and far-reaching implications. (shrink)

Attempts to define life should focus on the transition from molecules to cells and the “closure” aspects of this event. Rather than classifying existing objects into living and non-living entities I believe the challenge is to understand how the transition from non-life to life can take place, that is, the how the closure in Jagers op Akkerhuis’s hierarchical classification of operators, comes about.

The process is traced whereby crucially important, multiple denotations of classical sociology's key notion referring to social position-the Weberian German concept of Stand-have been stripped to create a simplified and inaccurate representation of social inequalities. Some historical material from central Europe is surveyed, with a brief look at Japan, to demonstrate validity problems created by blanket application of the culturally specific, streamlined notions of status/class. As an alternative, a notion of contingent social closure argues that relaxing the modernizationist assumptions (...) of a single transition from estate to status/class increases the comparative-historical sensitivity of research on social structure, inequality, and stratification. A dynamic reading of Polanyi suggests a reconceptualization of institutions as the "raw material" of social change. This might help to avoid the outdated contrast of the "West" vs. its "Others.". (shrink)

This paper describes a tentative model for how discrete memories transform into an interconnected conceptual network, or worldview, wherein relationships between memories are forged by way of abstractions. The model draws on Kauffman’s theory of how an information-evolving system could emerge through the formation and closure of an autocatalytic network. Here, the information units are not catalytic molecules, but memories and abstractions, and the process that connects them is not catalysis but reminding events (i.e. one memory evokes another). The (...) result is a worldview that both structures, and is structured by, self-triggered streams of thought. (shrink)

We give an idea of uniform approach to the problem of characterization of absolute extensors for categories of topological spaces [21], closure spaces [15], Boolean algebras [22], and distributive lattices [4]. In this characterization we use the notion of retract of the closure space of filters in the lattice of all subsets.

Could our observations of apparently pointless evil ever justify the conclusion that God does not exist? Not according to Stephen Wykstra, who several years ago announced the “Condition of Reasonable Epistemic Access,” or “CORNEA,” a principle that has sustained critiques of atheistic arguments from evil ever since. Despite numerous criticisms aimed at CORNEA in recent years, the principle continues to be invoked and defended. We raise a new objection: CORNEA is false because it entails intolerable violations of closure.

We prove that if ℵα is uncountable and regular, then the Beth-closure of Lωω(Qα) is not a sublogic of L∞ω(Qn), where Qn is the class of all n-ary generalized quantifiers. In particular, B(Lωω(Qα)) is not a sublogic of any finitely generated logic; i.e., there does not exist a finite set Q of Lindstrom quantifiers such that B(Lωω(Qα)) ≤ Lωω(Q).

In this paper we show that some standard topological constructions may be fruitfully used in the theory of closure spaces (see [5], [4]). These possibilities are exemplified by the classical theorem on the universality of the Alexandroff's cube for T 0-closure spaces. It turns out that the closure space of all filters in the lattice of all subsets forms a generalized Alexandroff's cube that is universal for T 0-closure spaces. By this theorem we obtain the following (...) characterization of the consequence operator of the classical logic: If is a countable set and C: P() P() is a closure operator on X, then C satisfies the compactness theorem iff the closure space ,C is homeomorphically embeddable in the closure space of the consequence operator of the classical logic.We also prove that for every closure space X with a countable base such that the cardinality of X is not greater than 2 there exists a subset X of irrationals and a subset X of the Cantor's set such that X is both a continuous image of X and a continuous image of X. (shrink)

The sociotechnical concept of closure requires researchers to identify the relevant social groups and technological frames associated with a technology, and also to map the social, political, economic, and other forces which, over time, reduce an artifacts's interpretative flexibility to a more singular and homogeneous sociotechnical formation. The closure concept has proven very useful, but I argue that its success has led it to acquire a quasi-objective status that can unnecessarily restrict the power of sociotechnical analyses. Rather than (...) being used as a meta-theoretical key that can unlock the genealogy of an artifact, closure can become a theoretical thing whose history has to be explained. Rather than using closure to explain the artifact, therefore, we wind up looking for closure itself. I argue instead for a sociotechnical theory in which artifacts, relevant social groups, and technological frames are never closed but are always open to new interpretations. (shrink)

We prove that any positive elementary (least fixed point) induction expressing the negation of transitive closure on finite nondirected graphs requires at least two recursion variables.

(2013). Motivational determinants of reasoning about social relations: The role of need for cognitive closure. Thinking & Reasoning. ???aop.label???. doi: 10.1080/13546783.2012.752407.

In recent years an increasing amount of information leaves no doubt that the costs to the victims of plant closures are more than economic. The stress occasioned by job loss often results in ill health. These findings aside, little systematic research has been done of the consequences of unemployment for the spouses of the unemployed. In this article, a comparison is made between the effects of a closure on unemployed male employees and their wives. It is found that both (...) groups suffer a high degree of anxiety over future job prospects and both experience a high level of stress as a result of the closure. However, for wives, anxiety, but not general stress, leads to ill health. For men, neither appears to have health implications: post-closure illness is related to illness prior to the shutdown. In one sense, two months after the closure, it can be argued that the impact of the shutdown was greater on wives than unemployed former employees. (shrink)

The principle of the anomalousness of the mental (PAM) is one of the most controversial principles in Donald Davidson’s argument for anomalous monism (AM). It states that there cannot be any laws (psychophysical or psychological) on the basis of which mental events can be predicted and explained. The argument against such psychological laws rests on the claim that psychology is not a comprehensive closed system (though physics is). Here I sketch the argument for AM, focusing on the role of PAM (...) and the concept of closure. I present characterizations of the notion of closure offered by William Stanton and Brian McLaughlin. McLaughlin argues that Stanton’s characterization makes the argument for AM circular. McLaughlin offers a different characterization, but I argue that given Davidson’s criterion of event identity and individuation, the two are equivalent and thus both are subject to McLaughlin’s objection. If I’m right about this, there are still a couple of options open to Davidson and the defenders of Anomalous Monism. However, I conclude by indicating why neither seems promising to me. (shrink)

Call a set A n-correctable if every set Turing reducible to A via a Turing machine that on any input makes at most n queries is Turing reducible to A via a Turing machine that on any input makes at most n-queries and on any input halts no matter what answers are given to its queries. We show that if a c.e. set A is n-correctable for some n ≥ 2, then it is n-correctable for all n. We show that (...) this is the optimal such result by constructing a c.e. set that is 1-correctable but not 2-correctable. The former result is obtained by examining the logical closure properties of c.e. sets that are 2-correctable. (shrink)

While attempting to avoid closure, it can be argued that two of the analytical techniques employed by Lawson (1997) strongly imply closure. First, while ostensibly directed at liberating analysis from all forms of closure, the demi?reg is shown to effectively rely on implied closure. Second, when the use of control groups is compared to Mäki's method of isolation, it can be shown that Lawson implies substantially similar closure to that which is proposed by Mäki. Such (...) implied forms of closure generally indicate that the complete exclusion of closure remains somewhat problematic. Accordingly, it is suggested that Lawson's effective prohibition on closure might be moderated by adopting the judicious and careful use of closure, within the context of an open system. (shrink)

Este artigo examina a objeção ao fechamento [dedutivo] que surge no contexto de certos paradoxos epistêmicos, paradoxos cuja conclusão é que a crença justificada pode ser inconsistente. É universalmente aceito que, se essa conclusão é correta, o fechamento deve ser rejeitado, para que se evite a crença justificada em enunciados contraditórios (P, ~P). Mas, mesmo que os argumentos desses paradoxos – o paradoxo da falibilidade (do prefácio) e o paradoxo da loteria – sejam mal-sucedidos, eles, ainda assim, sugerem a existência (...) de evidência independente para uma objeção mais direta contra o fechamento. O exame do argumento da falibilidade revela uma exigência de modéstia epistêmica que viola o fechamento a partir de múltiplas premissas. A reflexão sobre o paradoxo da loteria nos confronta com um dilema em que cada alternativa fornece um contra-exemplo ao fechamento a partir de uma única premissa. Seja ou não possível a inconsistência racional, há uma objeção contra o fechamento. PALAVRAS-CHAVE – Fechamento dedutivo. Falibilidade. Paradoxo da Loteria. Paradoxo do Prefácio. Justificação. Inconsistência. ABSTRACT This paper examines the case against closure that arises in the context of certain epistemic paradoxes, paradoxes whose conclusion is that it is possible for justified belief to be inconsistent. It is generally agreed that if this conclusion is correct, closure must be rejected in order to avoid justified belief in contradictory statements (P, ~P). But even if the arguments of these paradoxes – the fallibility (preface) paradox and the lottery paradox – are unsuccessful, they nonetheless suggest independent grounds for a more direct case against closure. Examination of the fallibility argument reveals a requirement of epistemic modesty that violates multiple premise closure. Reflection on the lottery paradox presents us with a dilemma in which each alternative provides a counterexample to single premise closure. Whether or not rational inconsistency is possible, there is a case against closure. KEY WORDS – Closure. Fallibility. Lottery paradox. Preface paradox. Justification. Inconsistency. (shrink)