It is a striking fact from reverse mathematics that almost all theorems of countable and countably representable mathematics are equivalent to just five subsystems of second order arithmetic. The standard view is that the significance of these equivalences lies in the set existence principles that are necessary and sufficient to prove those theorems. In this article I analyse the role of set existence principles in reverse mathematics, and argue that they are best understood as closure conditions on the (...) powerset of the natural numbers. (shrink)

Picking up the question of what FLaK might be, this editorial considers the relationship between openness and closure in feminist legal studies. How do we draw on feminist struggles for openness in common resources, from security to knowledge, as we inhabit a compromised space in commercial publishing? We think about this first in relation to the content of this issue: on image-based abuse continuums, asylum struggles, trials of protestors, customary justice, and not-so-timely reparations. Our thoughts take us through the (...) different ways that openness and closure work in struggles against violence, cruel welcomes, and re-arrangements of code and custom. Secondly, we share some reflections on methodological openness and closure as the roundtable conversation on asylum, and the interview with Riles, remind us of #FLaK2016 and its method of scattering sources as we think about how best to mix knowledges. Thirdly, prompted by the FLaK kitchen table conversations on openness, publishing and ‘getting the word out’, we respond to Kember’s call to ‘open up open access’. We explain the different current arrangements for opening up FLS content and how green open access, the sharedit initiative, author request and publisher discretion present alternatives to gold open access. Finally drawing on Franklin and Spade, we show how there are a range of ‘wench tactics’—adapting gifts, stalling and resting—which we deploy as academic editors who are trying to have an impact on the access, use and circulation of our journal, even though we do not own the journal we edit. These wench tactics are alternatives to the more obvious or reported tactic of resignation, or withdrawing academic labour from editing and reviewing altogether. They help us think about brewing editorial time, what ambivalence over our 25th birthday might mean, and how to inhabit painful places. In this, we respond in our own impure, compromised way to da Silva’s call not to forget the native and slave as we do FLaK, and repurpose shrapnel, in our common commitments. (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)

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.

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)

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)

This paper argues that if knowledge is defined in terms of probabilistic tracking then the benefits of epistemic closure follow without the addition of a closure clause. (This updates my definition of knowledge in Tracking Truth 2005.) An important condition on this result is found in "Closure Failure and Scientific Inquiry" (2017).

Closure principles say that if you know some proposition which entails a second and you meet further conditions then you know the second. In this paper I construct an argument against closure principles which turns on the idea that knowing a proposition requires that one’s belief-forming process be reliable. My argument parallels an influential argument offered by Timothy Williamson against KK principles–principles that say that if you know some proposition and you meet further conditions then you (...) know that you know the proposition. After offering my argument, I provisionally assess its damage to closure principles and also look at how responses to my argument against closure principles can be used to generate responses to Williamson’s argument against KK principles. (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)

In this paper, we prove that the lattice of all closure operators of a complete Almost Distributive Lattice L with fixed maximal element m is dual atomistic. We define the concept of a completely meet-irreducible element in a complete ADL and derive a necessary and sufficient condition for a dual atom of Φ (L) to be complemented.

It has been argued that an advantage of the safety account over the sensitivity account is that the safety account preserves epistemic closure, while the sensitivity account implies epistemic closure failure. However, the argument fails to take the method-relativity of the modal conditions on knowledge, viz., sensitivity and safety, into account. In this paper, I argue that the sensitivity account and the safety account are on a par with respect to epistemic closure once the method-relativity of (...) the modal conditions is taken into account. Therefore, epistemic closure is no longer an arbiter in the debate. (shrink)

In his earlier writings, Fred Dretske proposed an anti-skeptical strategy that is based on a rejection of the view that knowledge is closed under known entailment. This strategy is seemingly congenial with a sensitivity condition for knowledge, which is often associated with Dretske’s epistemology. However, it is not obvious how Dretske’s early account meshes with the information-theoretic view developed in Knowledge and the Flow of Information. One aim of this paper is to elucidate the connections between these accounts. First I (...) argue that, contrary to an objection raised by Christoph Jäger, the information-theoretic account is compatible with Dretske’s anti-skeptical strategy based on the rejection of closure. This strategy invokes the notion of channel conditions, which are roughly speaking those conditions that are necessary and jointly sufficient for a signal to carry information. I propose an interpretation of the account that is based on the idea that a signal’s carrying information requires that the channel conditions are stable. It is shown that the resulting account incorporates both a sensitivity condition and a safety condition for knowledge. Finally, I demonstrate how this proposal allows for knowledge of modally robust propositions without making its acquisition too easy, as simple safety accounts do. I end with a suggestion concerning the direction that future research should take, based on the fact that in its present form the information-theoretic account does not capture inferential knowledge. (shrink)

The topic of this article is the closure of a priori knowability under a priori knowable material implication: if a material conditional is a priori knowable and if the antecedent is a priori knowable, then the consequent is a priori knowable as well. This principle is arguably correct under certain conditions, but there is at least one counterexample when completely unrestricted. To deal with this, Anderson proposes to restrict the closure principle to necessary truths and Horsten suggests (...) to restrict it to formulas that belong to less expressive languages. In this article it is argued that Horsten’s restriction strategy fails, because one can deduce that knowable ignorance entails necessary ignorance from the closure principle and some modest background assumptions, even if the expressive resources do not go beyond those needed to formulate the closure principle itself. It is also argued that it is hard to find a justification for Anderson’s restricted closure principle, because one cannot deduce it even if one assumes very strong modal and epistemic background principles. In addition, there is an independently plausible alternative closure principle that avoids all the problems without the need for restriction. (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)

Radical scepticism contends that our knowledge of the external world is impossible. Particularly, radical scepticism can be motivated by the closure principle. Several commentators have noted that a straightforward way to respond to such arguments is via externalist strategies, e.g., Goldman, Greco, Bergmann. However, these externalist strategies are not effective against a slightly weaker form of the argument, a closure principle for rationally grounded knowledge, closureRK.\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${closure}_{RK.}$$\end{document} The sceptical argument, (...) framed around the closureRK\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\hbox {closure}_{\mathrm{RK}}$$\end{document} principle, targets rationally grounded knowledge. Although externalist strategies are ineffective against this form of argument, its conclusion can nonetheless be resisted by combining, in a novel way, the resources of Wittgenstein and Davidson. In particular, I argue that the sceptic is assuming an unrestricted way of using the closureRK\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\hbox {closure}_{\mathrm{RK}}$$\end{document} principle, which is incorrect. Alternatively, I argue for The Conditionality of Rational Support Thesis, i.e., the thesis that rational support via competent deduction is conditional. In particular, rational support must be provided within an evaluative system in which hinge propositions are presupposed and contentful beliefs are being evaluated. In the end, we can resist the closureRK\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\hbox {closure}_{\mathrm{RK }}$$\end{document}-based sceptical argument while retaining the closureRK\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\hbox {closure}_{\mathrm{RK}}$$\end{document} principle. (shrink)

Ernest Sosa and others have proposed a safety condition on knowledge: If S knows p, then in the nearest (non-actual) worlds in which S believes p, p is true.1 Colloquially, this is the idea that knowing requires not being easily mistaken. Here, I will argue that like another condition requiring a counterfactual relation between a subject’s belief and the world, viz. Robert Nozick’s sensitivity condition, safety leads, in certain cases, to the unacceptable result that knowledge is not closed under known (...) implication. (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)

_Strange Wonder_ confronts Western philosophy's ambivalent relationship to the Platonic "wonder" that reveals the strangeness of the everyday. On the one hand, this wonder is said to be the origin of all philosophy. On the other hand, it is associated with a kind of ignorance that ought to be extinguished as swiftly as possible. By endeavoring to resolve wonder's indeterminacy into certainty and calculability, philosophy paradoxically secures itself at the expense of its own condition of possibility. _Strange Wonder_ locates a (...) reopening of wonder's primordial uncertainty in the work of Martin Heidegger, for whom wonder is first experienced as the shock at the groundlessness of things and then as an astonishment that things nevertheless _are_. Mary-Jane Rubenstein traces this double movement through the thought of Emmanuel Levinas, Jean-Luc Nancy, and Jacques Derrida, ultimately thematizing wonder as the awesome, awful opening that exposes thinking to devastation as well as transformation. Rubenstein's study shows that wonder reveals the extraordinary in and through the ordinary, and is therefore crucial to the task of reimagining political, religious, and ethical terrain. (shrink)

_Strange Wonder_ confronts Western philosophy's ambivalent relationship to the Platonic "wonder" that reveals the strangeness of the everyday. On the one hand, this wonder is said to be the origin of all philosophy. On the other hand, it is associated with a kind of ignorance that ought to be extinguished as swiftly as possible. By endeavoring to resolve wonder's indeterminacy into certainty and calculability, philosophy paradoxically secures itself at the expense of its own condition of possibility. _Strange Wonder_ locates a (...) reopening of wonder's primordial uncertainty in the work of Martin Heidegger, for whom wonder is first experienced as the shock at the groundlessness of things and then as an astonishment that things nevertheless _are_. Mary-Jane Rubenstein traces this double movement through the thought of Emmanuel Levinas, Jean-Luc Nancy, and Jacques Derrida, ultimately thematizing wonder as the awesome, awful opening that exposes thinking to devastation as well as transformation. Rubenstein's study shows that wonder reveals the extraordinary in and through the ordinary, and is therefore crucial to the task of reimagining political, religious, and ethical terrain. (shrink)

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)

Carter and Pritchard (2016) and Pritchard (2010, 2012, 2016) have tried to reconcile the intuition that perceptual knowledge requires only limited discriminatory abilities with the closure principle. To this end, they have introduced two theoretical innovations: a contrast between two ways of introducing error-possibilities and a distinction between discriminating and favoring evidence. I argue that their solution faces the “sufficiency problem”: it is unclear whether the evidence that is normally available to adult humans is sufficient to retain knowledge of (...) the entailing proposition and come to know the entailed proposition. I submit that, on either infallibilist or fallibilist views of evidence, Carter and Pritchard have set the bar for deductive knowledge too low. At the end, I offer an alternative solution. I suggest that the knowledge-retention condition of the closure principle is not satisfied in zebra-like scenarios. (shrink)

All standard epistemic logics legitimate something akin to the principle of closure, according to which knowledge is closed under competent deductive inference. And yet the principle of closure, particularly in its multiple premise guise, has a somewhat ambivalent status within epistemology. One might think that serious concerns about closure point us away from epistemic logic altogether—away from the very idea that the knowledge relation could be fruitfully treated as a kind of modal operator. This, however, need not (...) be so. The abandonment of closure may yet leave in place plenty of formal structure amenable to systematic logical treatment. In this paper we describe a family of weak epistemic logics in which closure fails, and describe two alternative semantic frameworks in which these logics can be modelled. One of these—which we term plurality semantics—is relatively unfamiliar. We explore under what conditions plurality frames validate certain much-discussed principles of epistemic logic. It turns out that plurality frames can be interpreted in a very natural way in light of one motivation for rejecting closure, adding to the significance of our technical work. The second framework that we employ—neighbourhood semantics—is much better known. But we show that it too can be interpreted in a way that comports with a certain motivation for rejecting closure. (shrink)

Transmission arguments against closure of knowledge base the case against closure on the premise that a necessary condition for knowledge is not closed. Warfield argues that this kind of argument is fallacious whereas Brueckner, Murphy and Yan try to rescue it. According to them, the transmission argument is no longer fallacious once an implicit assumption is made explicit. I defend Warfield’s objection by arguing that the various proposals for the unstated assumption either do not avoid the fallacy or (...) turn the central premise of the transmission argument, namely that a necessary condition is not closed, into a redundant and superfluous premise. I conclude that Warfield’s advice is still to be heeded: Arguments against closure must not rely essentially on the premise that a necessary condition for knowledge is not closed. (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.

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)

It is well known that, in a topological space, the open sets can be characterized using ?lter convergence. In ZF , we cannot replace filters by ultrafilters. It is proven that the ultra?lter convergence determines the open sets for every topological space if and only if the Ultrafilter Theorem holds. More, we can also prove that the Ultra?lter Theorem is equivalent to the fact that uX = kX for every topological space X, where k is the usual Kuratowski closure (...) operator and u is the Ultra?lter Closure with uX := {x ∈ X: [U converges to x and A ∈ U ]}. However, it is possible to built a topological space X for which uX ≠ kX, but the open sets are characterized by the ultra?lter convergence. To do so, it is proved that if every set has a free ultra?lter, then the Axiom of Countable Choice holds for families of non-empty finite sets. It is also investigated under which set theoretic conditions the equality u = k is true in some subclasses of topological spaces, such as metric spaces, second countable T0-spaces or {ℝ}. (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)

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)

The goal of synthetic biology is to create artificial organisms. To achieve this it is essential to understand what life is. Metabolism-replacement systems, or (M, R)-systems, constitute a theory of life developed by Robert Rosen, characterized in the statement that organisms are closed to efficient causation, which means that they must themselves produce all the catalysts they need. This theory overlaps in part with other current theories, including autopoiesis, the chemoton, and autocatalytic sets, all of them invoking some idea of (...) closure. A simple model of an (M, R)-system has been implemented in the computer, and behaves in ways that may shed light on the requirements for a prebiotic self-organizing system. In addition to a trivial steady state in which nothing happens, it can establish a non-trivial steady state in which all intermediates have finite concentrations, with their rates of degradation balanced by their rates of synthesis. The system can be regenerated from the set of food components plus a single intermediate, and maintain itself in that state indefinitely, despite continuous degradation. At the very low compartment volumes that may have existed in prebiotic conditions, for example in cavities in minerals, or in micelles formed by simple amphiphiles, statistical fluctuations in the numbers of molecules need to be taken into account. With the stochastic approach there is no non-trivial steady state in strict mathematical terms, because the system will always collapse to the trivial state after sufficient time. However, the average time before collapse is so long for volumes greater than $10^{-19}\;{\textsc{l}}$ (much smaller than the volume of the order of $10^{-15}\;{\textsc{l}}$ for a typical bacterial cell) that for practical purposes the self-maintaining state of non-null concentrations becomes significant, recalling the situation of bistability that is observed in deterministic analysis. In turn, there exists a minimum size below which the self-organizing system cannot maintain itself on chemically relevant time scales. The value of the critical volume depends on the particular concentrations and rate constants assumed, but the principle could apply generally. (shrink)

Skeptical puzzles and arguments often employ knowledge-closure principles . Epistemologists widely believe that an adequate reply to the skeptic should explain why her reasoning is appealing albeit misleading; but it’s unclear what would explain the appeal of the skeptic’s closure principle, if not for its truth. In this paper, I aim to challenge the widespread commitment to knowledge-closure. But I proceed by first examining a new puzzle about failing to know—what I call the New Ignorance Puzzle . (...) This puzzle resembles to the Old Ignorance Puzzle , although it does not involve a closure principle. It instead centers on the standard view of ignorance, a highly intuitive principle stating that ignorance is merely a failure to know. In Sects. 2 and 3, I argue that the best way to solve the New Ignorance Puzzle is to reject the standard view of ignorance and to explain away its appeal via conversational implicature. I then use this solution to the New Ignorance Puzzle as a way of explaining why knowledge-closure principles would seem true, and why abominable conjunctions would seem abominable, even if such principles were false . The upshot is a new way of explaining why the skeptic’s reasoning is appealing albeit misleading. (shrink)

Let A be a subset of the constructive real line. What are the necessary and sufficient conditions for the set A such that A is continuously separated from other reals, i.e., there exists a continuous function f with f−1(0) = A? In this paper, we study the notions of closed sets and closure maps in constructive reverse mathematics.

For nonmonotonic logics, Cumulativity is an important logical rule. We show here that Cumulativity fans out into an infinity of different conditions, if the domain is not closed under finite unions.

Let $\Omega:= \aleph_1$ . For any $\alpha \Omega:\xi = \omega^\xi\}$ let EΩ (α) be the finite set of ε-numbers below Ω which are needed for the unique representation of α in Cantor-normal form using 0, Ω, +, and ω. Let $\alpha^\ast:= \max (E_\Omega(\alpha) \cup \{0\})$ . A function f: εΩ + 1 → Ω is called essentially increasing, if for any $\alpha < \varepsilon_{\Omega + 1}; f(\alpha) \geq \alpha^\ast: f$ is called essentially monotonic, if for any $\alpha,\beta < \varepsilon_{\Omega + (...) 1}$; $\alpha \leq \beta \wedge \alpha^\ast \leq \beta^\ast \Rightarrow f(\alpha) \leq f(\beta).$ Let Clf(0) be the least set of ordinals which contains 0 as an element and which satisfies the following two conditions: (a) $\alpha,\beta \epsilon \mathrm{Cl}_f(0) \Rightarrow \omega^\alpha + \beta \epsilon \mathrm{Cl}_f(0)$ , (b) $E_\Omega\alpha \subseteq \mathrm{Cl}_f(0) \Rightarrow f(\alpha) \epsilon \mathrm{Cl}_f(0)$ . Let ϑεΩ + 1 be the Howard-Bachmann ordinal, which is, for example, defined in [3]. The following theorem is shown: If f:εΩ + 1 → Ω is essentially monotonic and essentially increasing, then the order type of Clf(0) is less than or equal to ϑεΩ + 1. (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)

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)

Schwichtenberg showed that the System T definable functionals are closed under a rule-like version Spector’s bar recursion of lowest type levels 0 and 1. More precisely, if the functional Y which controls the stopping condition of Spector’s bar recursor is T-definable, then the corresponding bar recursion of type levels 0 and 1 is already T-definable. Schwichtenberg’s original proof, however, relies on a detour through Tait’s infinitary terms and the correspondence between ordinal recursion for α < ε₀ and primitive recursion over (...) finite types. This detour makes it hard to calculate on given concrete system T input, what the corresponding system T output would look like. In this paper we present an alternative (more direct) proof based on an explicit construction which we prove correct via a suitably defined logical relation. We show through an example how this gives a straightforward mechanism for converting bar recursive definitions into T-definitions under the conditions of Schwichtenberg’s theorem. Finally, with the explicit construction we can also easily state a sharper result: if Y is in the fragment Tᵢ then terms built from BR^ℕ,σ for this particular Y are definable in the fragment T_i+max{1,level(σ)}+2. (shrink)

The notion of the rational closure of a positive knowledge base K of conditional assertions θ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$i$$ \end{document} |∼ φ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$i$$ \end{document} (standing for if θ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$i$$ \end{document} then normally φ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$i$$ \end{document}) 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, θ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$i$$ \end{document} |∼ φ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$i$$ \end{document}, but also negative conditional assertions, θi φ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$i$$ \end{document} (standing for not if θ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$i$$ \end{document} then normally φ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$i$$ \end{document}. 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)

The canonical version of external world scepticism has the following structure: Premise1: If P knows that she is not a brain in a vat, then P does not have knowledge of the external world. Premise2: P does not know that she is not a brain in a vat. Conclusion: Therefore, P does not have knowledge of the external world. Some philosophers attack premise1 by denying the underlying closure principle. I will investigate possible argumentations for and against premise2. I will (...) show that premise2 is only true if additional internalistic conditions are fulfilled. I will conclude that the canonical version of the argument is inadequate and that premise1 must be replaced by a stronger internalistic premise. After presenting the adequate argument, I will argue that this internalistic premise contradicts the closure principle underlying premise1. I will conclude that external world scepticism can only hold if the closure principle is rejected. (shrink)

In a finitary closure space, irreducible sets behave like two-valued models, with membership playing the role of satisfaction. If f is a function on such a space and the membership of in an irreducible set is determined by the presence or absence of the inputs in that set, then f is a kind of truth function. The existence of some of these truth functions is enough to guarantee that every irreducible set is maximally consistent. The closure space is (...) then said to be expressive. This paper identifies the two-valued truth functional conditions that guarantee expressiveness. (shrink)

In biscuit conditionals (BCs) such as If you’re hungry, there’s pizza in the fridge, the if clause appears to apply to the illocutionary act performed in uttering the main clause, rather than to its propositional content. Accordingly, previous analyses of BCs have focused on illocutionary acts, and, this, I argue, leads them to yield incorrect paraphrases. I propose, instead, that BCs involve existential quantification over potential literal acts such as assertions, questions, commands, and exclamations, the semantic objects associated with declarative, (...) interrogative, imperative, and exclamative sentences, respectively. Such an existential interpretation of BCs requires only that we add potential literal acts to our inventory of individuals, and it produces reasonable paraphrases in which if has its normal meaning: If you’re hungry,[there’s a (relevant/salient) assertion that] there’s pizza in the fridge. These potential literal act variables are introduced into semantic interpretations and then undergo Existential Closure. Hence, we would expect to see similar interpretations in contexts other than BCs, that is, with other if constructions, with connectives other than if, with potential literal acts other than assertion, and in root sentences. This prediction is borne out, along with the parallel prediction that we cannot quantify over purely illocutionary acts like offers, but only over potential literal acts, those conventionally associated with a particular morphosyntactic shape. (shrink)

An ethics consultant’s scope of practice is best understood as equipping stakeholders to achieve closure over time following after the ethics consultation. This is in contrast to Autumn Fiester’s position in the article, “Neglected Ends: Clinical Ethics Consultation and the Prospects for Closure,” where she claims that moral closure is a necessary condition for the proper completion of an ethics consultation case.

Closure space has been proven to be a useful tool to restructure lattices and various order structures. This paper aims to provide an approach to characterizing domains by means of closure spaces. The notion of an interpolative generalized closure space is presented and shown to generate exactly domains, and the notion of an approximable mapping between interpolative generalized closure spaces is identified to represent Scott continuous functions between domains. These produce a category equivalent to that of (...) domains with Scott continuous functions. Meanwhile, some important subclasses of domains are discussed, such as algebraic domains, _L_-domains, bounded-complete domains, and continuous lattices. Conditions are presented which, when fulfilled by an interpolative generalized closure space, make the generated domain fulfill some restrictive conditions. (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)

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)

An assertion of high conditional probability or, more briefly, an HCP assertion is a statement of the type: The conditional probability of B given A is close to one. The goal of this paper is to construct logics of HCP assertions whose conclusions are highly likely to be correct rather than certain to be correct. Such logics would allow useful conclusions to be drawn when the premises are not strong enough to allow conclusions to be reached with certainty. This goal (...) is achieved by taking Adams' (1966) logic, changing its intended application from conditionals to HCP assertions, and then weakening its criterion for entailment. According to the weakened entailment criterion, called the Criterion of Near Surety and which may be loosely interpreted as a Bayesian criterion, a conclusion is entailed if and only if nearly every model of the premises is a model of the conclusion. The resulting logic, called NSL, is nonmonotonic. Entailment in this logic, although not as strict as entailment in Adams' logic, is more strict than entailment in the propositional logic of material conditionals. Next, NSL was modified by requiring that each HCP assertion be scaled; this means that to each HCP assertion was associated a bound on the deviation from 1 of the conditional probability that is the subject of the assertion. Scaling of HCP assertions is useful for breaking entailment deadlocks. For example, it it is known that the conditional probabilities of C given A and of - C given B are both close to one but the bound on the former's deviation from 1 is much smaller than the latter's, then it may be concluded that in all likelihood the conditional probability of C given A ∧ B is close to one. The resulting logic, called NSL-S, is also nonmonotonic. Despite great differences in their definitions of entailment, entailment in NSL is equivalent to Lehmann and Magidor's rational closure and, disregarding minor differences concerning which premise sets are considered consistent, entailment in NSL-S is equivalent to entailment in Goldszmidt and Pearl's System-Z⁺. Bacchus, Grove, Halpern, and Koller proposed two methods of developing a predicate calculus based on the Criterion of Near Surety. In their randomstructures method, which assumed a prior distribution similar to that of NSL, it appears possible to define an entailment relation equivalent to that of NSL. In their random-worlds method, which assumed a prior distribution dramatically different from that of NSL, it is known that the entailment relation is different from that of NSL. (shrink)

This note calls attention to the fact that the natural extensions of standard logics appear to be an application of the method of inductive generation of logics . One can generalize our observations beyond the range of algebraic logics under suitable conditions on cardinals of certain involved sets.

By using the general investigation framework offered by the cognitive science of religion (CSR), I analyse religion as a necessary condition for the evolutionary path of knowledge. The main argument is the "paradox of the birth of knowledge": in order to get to the meaning of the part, a sense context is needed; but a sense of the whole presupposes the sense (meaning) of the parts. Religion proposes solutions to escape this paradox, based on the imagination of sense (meaning) contexts, (...) respectively closures of these contexts through meta-senses. What is important is the practical effectiveness of solutions proposed by religion, taking into account the costs of faith and the costs of the absence of religious belief. The hypothesis has the following consequences: religion is a necessary condition for the initial evolution of knowledge and the emergence of religion is determined by the evolution of knowledge. The continuation of the solving of paradox is a Bayesian one, using explorations: a sense of the whole allows cognitive arrangements of the parties, which in turn open the possibility of a rearrangement of the whole. The contribution of religion to the emergence of sense (meaning) could be governed by the rule: any map of the world is more useful than no map; any meaning (of life) is better than no meaning. The human mind fills the perceptual and cognitive gaps, some (religious) filling solutions being true vault keys of the entire cognitive construction called the world. Knowledge is conditioned by the existence of an organized context, as the cosmos created by religion by means of explanatory meta-theories supports knowledge by closing the cognitive context and using meaning networks. The proposed analysis is consistent with a redefinition of rationality from the perspective of evolution: the importance and relevance of knowledge is determined by its practical outcome - survival. In the context of useful fictions, it does not matter what God actually does, but what we have done by believing in God. Existence has provided a pragmatic verification of the cognitive solutions that underlie the survival strategies promoted by religions. (shrink)

This review article puts two recent publications —Existence and Heritage: Hermeneutic Explorations in African and Continental Philosophies by Tsenay Serequeberhan and What Fanon Said. A Philosophical Introduction to His Life and Thought by Lewis R. Gordon—in conversation with Achille Mbembe’s renowned On the Postcolony, first published in French in 2000, in English in 2001, and here reviewed in the 2015 Wits University Press edition. The opportunity for such a literary dialogue to take place across fifteen years is occasioned both by (...) the latest South African edition of On the Postcolony, which in itself invites a fresh reading of Mbembe’s manifesto on postcolonial authoritarianism in Africa, as well as by the different potential outcome of the postcolonial condition that Gordon and Serequeberhan envision and propose with respect to Mbembe. I argue that while Mbembe’s pessimistic portrayal of the postcolonial present is one whereby it is difficult to envision transformation, both Serequeberhan and Gordon’s books offer a rich philosophical terrain from which to imagine possible futures, grounded in notions of dialogue and solidarity, that, however, vary significantly for each author. (shrink)