It is tempting to think that multi premise closure creates a special class of paradoxes having to do with the accumulation of risks, and that these paradoxes could be escaped by rejecting the principle, while still retaining single premise closure. I argue that single premise deduction is also susceptible to risks. I show that what I take to be the strongest argument for rejecting multi premise closure is also an argument (...) for rejecting single premise closure. Because of the symmetry between the principles, they come as a package: either both will have to be rejected or both will have to be revised. (shrink)

In this paper, I’ll survey a number of closure principles of epistemic justification and find them all wanting. However, it’ll be my contention that there’s a novel closure principle of epistemic justification that has the virtues of its close cousin closure principles, without their vices. This closure principle of epistemic justification can be happily thought of as a multi-premise closure principle and it cannot be used in Cartesian skeptical arguments that (...) employ a closure principle of epistemic justification. In this way, then, it represents marked improvement over other contemporary closure principles of epistemic justification that require both sacrificing multi-premise closure and forcing anti-Cartesian skeptics who reject the closure principle employed in certain Cartesian skeptical arguments to cast aside justification closure. (shrink)

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

Ever since Fred Dretske (1970) questioned closure, a denial of this principle has been among the standard options for a resolution of epistemological paradoxes such as the skeptical paradox (Cohen 1988) and the lottery paradox (Harman 1973). In this article, the author shall argue that all possible solutions of the latter paradox can only be defended if Multi-Premise Closure is rejected. These possible solutions are contextualism and both simple and sensitive moderate invariantism. It will be (...) shown that skepticism and the denial of Single-Premise Closure are not possible solutions of the lottery paradox. The upshot of the discussion here will be that while Single-Premise Closure is beyond reasonable doubt, resolving the lottery paradox forces one to abandon Multi-Premise Closure. (shrink)

I will present a novel account of justification in terms of knowledge on which one is justified in believing p just in case one could know that p. My main aim is to unravel some of the formal properties that justification has in virtue of its connection to knowledge. Assuming that safety is at least a necessary condition for knowledge, I show that justification doesn’t iterate trivially; isn’t a luminous condition; is closed under a certain kind of multi-premise (...) closure principle, but; surprisingly one can nevertheless believe with justification a set of claims that’s jointly inconsistent. This last feature allows for a rather satisfying solution to the preface paradox. Finally, I contrast my account with other knowledge-first accounts of justification. (shrink)

The paper explains how Gettier’s conclusion can be reached on general theoretical grounds within the framework of epistemic logic, without reliance on thought experiments. It extends the argument to permissive conceptions of justification that invalidate principles of multi-premise closure and require neighbourhood semantics rather than semantics of a more standard type. The paper concludes by recommending a robust methodology that aims at convergence in results between thought experimentation and more formal methods. It also warns against conjunctive definitions (...) as sharing several of the drawbacks of disjunctive definitions. (shrink)

It’s often thought that the phenomenon of risk aggregation poses a problem for multi-premise closure but not for single-premise closure. But recently, Lasonen-Aarnio and Schechter have challenged this thought. Lasonen-Aarnio argues that, insofar as risk aggregation poses a problem for multi-premise closure, it poses a similar problem for single-premise closure. For she thinks that, there being such a thing as deductive risk, risk may aggregate over a single premise and (...) the deduction itself. Schechter argues that single-premise closure succumbs to risk aggregation outright. For he thinks that there could be a long sequence of competent single-premise deductions such that, even though we are justified in believing the initial premise of the sequence, intutively, we are not justified in believing the final conclusion. This intuition, Schechter thinks, vitiates single-premise closure. In this paper, I defend single-premise closure against the arguments offered by Lasonen-Aarnio and Schechter. (shrink)

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

The conservation laws do not establish the central premise within the argument from causal overdetermination – the causal completeness of the physical domain. Contrary to David Papineau, this is true even if there is no non-physical energy. The combination of the conservation laws with the claim that there is no non-physical energy would establish the causal completeness principle only if, at the very least, two further causal claims were accepted. First, the claim that the only way that something (...) non-physical could affect a physical system is by affecting the amount of energy or momentum within it, or redistributing the energy and momentum within it. Second, the claim that redistribution of energy and momentum cannot be brought about without supplying energy or momentum. Both of these claims, however, are exceedingly difficult to defend in the context of the argument. (shrink)

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

John Hawthorne’s recent monograph Knowledge and Lotteries1 is centred on the following puzzle: Suppose you claim to know that you will not be able to afford to summer in the Hamptons next year. Aware of your modest means, we believe you. But suppose you also claim to know that a ticket you recently purchased in a multi-million dollar lottery is a loser. Most of us have the intuition that you do not know that your ticket is a loser. However, (...) your alleged knowledge of not being able to afford to summer in the Hamptons puts you in a position to know that your ticket is a loser. For the proposition that you will not be able to afford to summer in the Hamptons entails the proposition that you will lose the lottery. And the following principle, what Hawthorne calls ‘Single Premise Closure’ ( p. 34), is very plausible: If you know that p, p entails q, and you competently deduce q from p thereby coming to believe that q (all the while retaining your knowledge of p), then you come to know q. (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)

The primary objective of this paper is to introduce a new epistemic paradox that puts pressure on the claim that justification is closed under multi premise deduction. The first part of the paper will consider two well-known paradoxes—the lottery and the preface paradox—and outline two popular strategies for solving the paradoxes without denying closure. The second part will introduce a new, structurally related, paradox that is immune to these closure-preserving solutions. I will call this paradox, The (...) Paradox of the Pill. Seeing that the prominent closure-preserving solutions do not apply to the new paradox, I will argue that it presents a much stronger case against the claim that justification is closed under deduction than its two predecessors. Besides presenting a more robust counterexample to closure, the new paradox also reveals that the strategies that were previously thought to get closure out of trouble are not sufficiently general to achieve this task as they fail to apply to similar closure-threatening paradoxes in the same vicinity. (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)

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)

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

According to a popular closure principle for epistemic justification, if one is justified in believing each of the premises in set Φ and one comes to believe that ψ on the basis of competently deducing ψ from Φ—while retaining justified beliefs in the premises—then one is justified in believing that ψ. This principle is prima facie compelling; it seems to capture the sense in which competent deduction is an epistemically secure means to extend belief. However, even the (...) single-premise version of this closure principle is in conflict with certain seemingly good inferences involving the epistemic possibility modal ♢. According to other compelling principles concerning competent deduction and epistemic justification, one can competently infer ¬♢φ from ¬φ in deliberation even though there are cases in which one can justifiably believe ¬φ but would be unjustified in believing ¬♢φ. Thus, as we argue, philosophers must choose between unrestricted closure for justification and the validity of these other principles. (shrink)

I examine the meaning and merits of a premise in the Exclusion Argument, the causal closure principle that all physical effects have physical causes. I do so by addressing two questions. First, if we grant the other premises, exactly what kind of closure principle is required to make the Exclusion Argument valid? Second, what are the merits of the requisite closure principle? Concerning the first, I argue that the Exclusion Argument requires a strong, (...) “stringently pure” version of closure. The latter employs two qualifications concerning the physical sufficiency and relative proximity of the physical cause required for every physical effect. The second question is addressed in two steps. I begin by challenging the adequacy of the empirical support offered by David Papineau for closure. Then I assess the merits of “level” and “domain” versions of stringently pure closure. I argue that a domain version lacks adequate and non-question-begging support within the context of the Exclusion Argument. And I argue that the level version leads to a puzzling metaphysics of the physical domain. Thus, we have grounds for rejecting the version of closure required for the Exclusion Argument. This means we can resist the Exclusion Argument while avoiding the implausible implications that come with rejecting one of its other premises. That is, because there are grounds to reject causal closure, one can reasonably affirm the non-overdeterminative causal efficacy of conscious mental states while denying that the latter are identical with physical states. (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)

Single-premise epistemic closure is the principle that: if one is in an evidential position to know that P where P entails Q, then one is in an evidential position to know that Q. In this paper, I defend the viability of opposition to closure. A key task for such an opponent is to precisely formulate a restricted closure principle that remains true to the motivations for abandoning unrestricted closure but does not endorse particularly (...) egregious instances of closure violation. I focus on two brands of epistemic theory that naturally incorporate closure restrictions. The first type holds that the truth value of a knowledge ascription is relative to a relevant question. The second holds that the truth value of a knowledge ascription is relative to a relevant topic. For each approach, I offer a formalization of a leading theory from the literature and use this formalization to evaluate the theory’s adequacy in terms of a precise set of desiderata. I conclude that neither theory succeeds in meeting these desiderata, casting doubt on the viability of the underlying approaches. Finally, I offer a novel variant of the topic-sensitive approach that fares better. (shrink)

Closure is the central premise in one of the best arguments for physicalism—the argument from causal overdetermination. According to Closure, at every time at which a physical event has a sufficient cause, it has a sufficient physical cause. This principle is standardly defended by appealing to the fact that it enjoys empirical support from numerous confirming cases (and no disconfirming cases) in physics. However, in recent literature on mental causation, attempts have been made to provide a (...) stronger argument for it. This essay argues that, insofar as these attempts are successful, they actually establish a far stronger closure principle. Worryingly, the acceptance of this stronger principle presents a new problem for the most popular form of physicalism, that of nonreductive physicalism. The problem shall be referred to as the 'Problem of Strong Closure.'. (shrink)

John Hawthorne has two forceful arguments in favour of:Single-Premise Closure Necessarily, if S knows p, competently deduces q from p, and thereby comes to believe q, while retaining knowledge of p throughout, then S knows q.Each of Hawthorne's arguments rests on an intuitively appealing principle which Hawthorne calls the Equivalence Principle. I show, however, that the opponents of SPC with whom he's engaging - namely Fred Dretske and Robert Nozick - have independent reason to reject this (...) principle, and resultantly conclude that Hawthorne's arguments in favour of SPC are not knock-down. (shrink)

Non-reductive physicalism is the view that mental events cause other events in virtue of their mental properties and that mental properties supervene on, without being identical to, physical properties. Jaegwon Kim has presented several much-discussed arguments against this view. But the much simpler causal closure argument, which purports to establish that every mental property is identical to a physical property, has received less attention than Kim’s arguments. This paper aims to show how a non-reductive physicalist should rebut the causal (...) closure argument. A crucial premise in the argument is a principle stating that the physical world is causally closed. It states, roughly, that every physical event has a physical sufficient cause. I argue that when the principle is formulated in this way, the causal closure argument is invalid. Less frequently, the causal closure principle is formulated in terms of a difference-making conception of causation so that it states that every physical effect has a physical difference-making cause. I argue that the principle, so formulated, is false and the causal closure argument unsound. Either way the argument lacks the apodeictic force to compel acceptance of its conclusion. (shrink)

ABSTRACTThe Necessity of Origins is the thesis that, necessarily, if a material object wholly originates from some particular material, then it could not have wholly originated from any significantly non-overlapping material. Several philosophers have argued for this thesis using as a premise a principle that we call ‘Single Origin Necessity’. However, we argue that Single Origin Necessity is false. So any arguments for The Necessity of Origins that rely on Single Origin Necessity are unsound. We also argue that (...) the Necessity of Origins itself is false. Our arguments rely on a thesis in the ontology of art that we find plausible: Multi-Work Materialism. It is the thesis that works of art that have multiple concrete manifestations are co-located with those manifestations. (shrink)

I argue that fallibilism, single-premise epistemic closure, and one formulation of the “knowledge-action principle” are inconsistent. I will consider a possible way to avoid this incompatibility, by advocating a pragmatic constraint on belief in general, rather than just knowledge. But I will conclude that this is not a promising option for defusing the problem. I do not argue here for any one way of resolving the inconsistency.

In order to reply to the contemporary skeptic’s argument for the conclusion that we don’t have any empirical knowledge about the external world, several authors have proposed different fallibilist theories of knowledge that reject the epistemic closure principle. Holliday, 1–62 2015a), however, shows that almost all of them suffer from either the problem of containment or the problem of vacuous knowledge or both. Furthermore, Holliday suggests that the fallibilist should allow a proposition to have multiple sets of relevant (...) alternatives, each of which is sufficient while none is necessary, if all its members are eliminated, for knowing that proposition. Not completely satisfied with Holliday’s multi-path reply to the skeptic, the author suggests a new single-path relevant-possibility theory of knowledge and argues that it can avoid both the problem of containment and the problem of vacuous knowledge of a certain sort while rejecting skepticism about the external world. (shrink)

Physicalism is the idea that everything either is physical or is nothing over and above the physical. For this formulation of physicalism to have determinate content, it should be identified what the “physical” refers to; i.e. the body problem. Some other closely related theses, especially the ones employed in the causal arguments for different versions of physicalism, and more especially the causal closure thesis, are also subject to the body problem. In this paper, I do two things. First, I (...) explore the structure of causal arguments for physicalism that represents a general argument. To do this, the premises and the conclusion of the general argument are given exact formulations. Second, drawing on those premises, especially the causal closure thesis, I propose a naturalistic formulation of the physical that satisfies the requirements any formulation of the physical is expected to fulfill. Following this proposal, we also have a recursive algorithm to recognize the set of all physical events. (shrink)

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)

Harman’s lottery paradox, generalized by Vogel to a number of other cases, involves a curious pattern of intuitive knowledge ascriptions: certain propositions seem easier to know than various higher-probability propositions that are recognized to follow from them. For example, it seems easier to judge that someone knows his car is now on Avenue A, where he parked it an hour ago, than to judge that he knows that it is not the case that his car has been stolen and driven (...) away in the last hour. Contextualists have taken this pattern of intuitions as evidence that ‘knows’ does not always denote the same relationship; subject-sensitive invariantists have taken this pattern of intuitions as evidence that non-traditional factors such as practical interests figure in knowledge; still others have argued that the Harman Vogel pattern gives us a reason to abandon the principle that knowledge is closed under known entailment. This paper argues that there is a psychological explanation of the strange pattern of intuitions, grounded in the manner in which we shift between an automatic or heuristic mode of judgment and a controlled or systematic mode. Understanding the psychology behind the pattern of intuitions enables us to see that the pattern gives us no reason to abandon traditional intellectualist invariantism. The psychological account of the paradox also yields new resources for clarifying and defending the single premise closure principle for knowledge ascriptions. (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)

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)

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)

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)

Despite their substantial appeal, closure principles have fallen on hard times. Both anti-luck conditions on knowledge and the defeasibility of knowledge look to be in tension with natural ways of articulating single-premise closure principles. The project of this paper is to show that plausible theses in the epistemology of testimony face problems structurally identical to those faced by closure principles. First I show how Lasonen-Aarnio’s claim that there is a tension between single premise closure (...) and anti-luck constraints on knowledge can be extended to make trouble for transmission theses. Second, I show how Schechter’s claim that there is a tension between single premise closure and the thought that knowledge is defeasible can be extended to make trouble for transmission theses. I end the paper by sketching the consequences of this trouble for the dialectic in the epistemology of testimony. (shrink)

The Knower paradox purports to place surprising a priori limitations on what we can know. According to orthodoxy, it shows that we need to abandon one of three plausible and widely-held ideas: that knowledge is factive, that we can know that knowledge is factive, and that we can use logical/mathematical reasoning to extend our knowledge via very weak single-premise closure principles. I argue that classical logic, not any of these epistemic principles, is the culprit. I develop a consistent (...) theory validating all these principles by combining Hartry Field's theory of truth with a modal enrichment developed for a different purpose by Michael Caie. The only casualty is classical logic: the theory avoids paradox by using a weaker-than-classical K3 logic. I then assess the philosophical merits of this approach. I argue that, unlike the traditional semantic paradoxes involving extensional notions like truth, its plausibility depends on the way in which sentences are referred to--whether in natural languages via direct sentential reference, or in mathematical theories via indirect sentential reference by Gödel coding. In particular, I argue that from the perspective of natural language, my non-classical treatment of knowledge as a predicate is plausible, while from the perspective of mathematical theories, its plausibility depends on unresolved questions about the limits of our idealized deductive capacities. (shrink)

What is the epistemological structure of situations where many small risks amount to a large one? Lottery and preface paradoxes and puzzles about quantum-mechanical blips threaten the idea that competent deduction is a way of extending our knowledge. Seemingly, everyday knowledge involves small risks, and competently deducing the conjunction of many such truths from them yields a conclusion too risky to constitute knowledge. But the dilemma between scepticism and abandoning MPC is false. In extreme cases, objectively improbable truths are known. (...) Safety is modal, not probabilistic, in structure, with closure and factiveness conditions. It is modelled using closeness of worlds. Safety is analogous to knowledge. It suggests an interpretation of possible worlds semantics for epistemic logic. To avoid logical omniscience, a relation of epistemic counterparthood between formulas is introduced. This supports a safety conception of knowledge and formalizes how extending knowledge by deduction depends on logical competence. (shrink)

We present a semi-formal foundational theory of sorts, akin to sets, named librationism because of its way of dealing with paradoxes. Its semantics is related to Herzberger’s semi inductive approach, it is negation complete and free variables (noemata) name sorts. Librationism deals with paradoxes in a novel way related to paraconsistent dialetheic approaches, but we think of it as bialethic and parasistent. Classical logical theorems are retained, and none contradicted. Novel inferential principles make recourse to theoremhood and failure of theoremhood. (...) Identity is introduced à la Leibniz-Russell, and librationism is highly non-extensional. Π 1 1 -comprehension with ordinary Bar-Induction is accounted for (to be lifted). Power sorts are generally paradoxical, and Cantor’s Theorem is blocked as a camouflaged premise is naturally discarded. (shrink)

According to the principle of Knowledge Counter-Closure , knowledge-yielding single-premise deduction requires a known premise: if S believes q solely on the basis of deduction from p, and S knows q, then S must know p. Although prima facie plausible, widely accepted, and supported by seemingly compelling motivations, KCC has recently been challenged by cases where S arguably knows q solely on the basis of deduction from p, yet p is false or p is true but (...) not known . I explore a view that resolves this tension by abandoning KCC in the light of these challenges, and which acknowledges their force but also their limits. Adopting this view helps identify the epistemic constraints that operate on the premises of knowledge-yielding deduction, clarifies the epistemic role of deduction, and allows us to distil the kernel of truth in the motivations that are standardly taken to support KCC. (shrink)

In this paper we study a reducibility that has been introduced by Klaus Weihrauch or, more precisely, a natural extension for multi-valued functions on represented spaces. We call the corresponding equivalence classes Weihrauch degrees and we show that the corresponding partial order induces a lower semi-lattice. It turns out that parallelization is a closure operator for this semi-lattice and that the parallelized Weihrauch degrees even form a lattice into which the Medvedev lattice and the Turing degrees can be (...) embedded. The importance of Weihrauch degrees is based on the fact that multi-valued functions on represented spaces can be considered as realizers of mathematical theorems in a very natural way and studying the Weihrauch reductions between theorems in this sense means to ask which theorems can be transformed continuously or computably into each other. As crucial corner points of this classification scheme the limited principle of omniscience LPO, the lesser limited principle of omniscience LLPO and their parallelizations are studied. It is proved that parallelized LLPO is equivalent to Weak Kőnig's Lemma and hence to the Hahn—Banach Theorem in this new and very strong sense. We call a multi-valued function weakly computable if it is reducible to the Weihrauch degree of parallelized LLPO and we present a new proof, based on a computational version of Kleene's ternary logic, that the class of weakly computable operations is closed under composition. Moreover, weakly computable operations on computable metric spaces are characterized as operations that admit upper semi-computable compact-valued selectors and it is proved that any single-valued weakly computable operation is already computable in the ordinary sense. (shrink)

Recent years have seen the rise of a new family of non-probabilistic accounts of epistemic justification. According to these views—we may call them Normalcy Views—a belief in P is justified only if, given the evidence, there exists no normal world in which S falsely beliefs that P. This paper aims to raise some trouble for this new approach to justification by arguing that Normalcy Views, while initially attractive, give rise to problematic accounts of epistemic defeat. As we will see, on (...) Normalcy Views seemingly insignificant pieces of evidence turn out to have considerable defeating powers. This problem—I will call it the Easy-Defeat Problem—gives rise to a two-pronged challenge. First, it shows that the Normalcy View has counterintuitive implications and, second, it opens the door to an uncomfortable skeptical threat. (shrink)

The paper critically discusses two prominent arguments against closure principles for knowledge. The first one is the “argument from aggregation”, claiming that closure under conjunction has the consequence that, if one individually knows i premises, one also knows their i-fold conjunction—yet, every one of the premises might exhibit interesting positive epistemic properties while the i-fold conjunction might fail to do so. The second one is the “argument from concatenation”, claiming that closure under entailment has the consequence that, (...) if one knows a premise, one also knows each of its remote consequences one arrives at—yet, again, the premise might exhibit interesting positive epistemic properties while some of its remote consequences might fail to do so. The paper firstly observes that the ways in which these two arguments try to establish that the relevant closure principle has the relevant problematic consequence are strikingly similar. They both crucially involve showing that, given the features of the case, the relevant closure principle acts in effect as a soritical principle, which is in turn assumed to lead validly to the relevant problematic consequence. There are however nontransitive logics of vagueness where soritical principles do not have any problematic consequence. Assuming that one of these logics is the correct logic of vagueness, the paper secondly observes that both arguments describe situations where knowledge is arguably vague in the relevant respects, so that a tolerant logic should be used in reasoning about it, with the effect that the relevant soritical principle no longer validly leads to the relevant problematic consequence. This shows an interesting respect in which the gap between validity and good inference that arguably arises in a transitive framework can be bridged in a tolerant one, thereby approximating better certain features of our epistemic lives as finite subjects. Moreover, even for those who do not subscribe to tolerant logics, the paper’s two observations jointly indicate that, for all the arguments from aggregation and concatenation show, the status of the relevant closure principles should be no worse than that assigned by one’s favoured theory of vagueness to soritical principles, which only rarely is plain falsity and can indeed get arbitrarily close to full truth. (shrink)

This paper compares two kinds of epistemic principles---an underdetermination principle and a deductive closure principle. It argues that each principle provides the basis for an independently motivated skeptical argument. It examines the logical relations between the premises of the two kinds of skeptical argument and concludes that the deductive closure argument cannot be refuted without refuting the underdetermination argument. The underdetermination argument, however, can be refuted without refuting the deductive closure argument. In this respect, (...) the deductive closure argument is the stronger of the two. (shrink)

This study provides a critical appraisal of Duncan Pritchard’s argument to the effect that ability to preserve certain eminently plausible transmission and/or closure principles for knowledge serves as a powerful adequacy test on alternative accounts of so-called Wittgensteinian certainties or hinge commitments. I argue that Pritchard fails to establish this claim—the transmission test does not favour his favourite conception over alternative conceptions premised on the idea that hinge commitments are not supportable via evidential-cognitive routes.

Despite its intrinsic plausibility, the sensitivity principle has remained deeply unpopular on the grounds that it violates an even more plausible closure principle. Here we show that sensitivity does not, in general, violate closure. Sensitivity only violates closure when combined with further auxiliary premises—regarding which of an agent’s commitments constitute that agent’s beliefs—which are optional for the sensitivity theorist.

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

The attempt to hold both anti-individualism and privileged self-knowledge may have the absurd consequence that someone could know a priori propositions that are knowable only empirically. This would be so if such an attempt entailed that one could know a priori both the contents of one’s own thoughts and the anti-individualistic entailments from those thought-contents to the world. For then one could also come to know a priori the empirical conditions entailed by one’s thoughts. But I argue that there is (...) no construal of a priori knowledge that could be used to raise an incompatibility problem of this sort. First, I suggest that the incompatibilist a priori must be a stipulative one, since in none of the main philosophical traditions does knowledge of the contents of one’s thoughts count as a priori. Then, I show that under various possible construals of a priori, the incompatibilist argument would be invalid: either a fallacy of equivocation or an argument without a plausible closure principle guaranteeing transmission of epistemic status from premises to conclusion. Finally, I maintain that the only possible construal of the property of being knowable a priori that avoids invalidity is one that fails to generate the intended reductio. (shrink)

The safety conception of knowledge holds that a belief constitutes knowledge iff relevantly similar beliefs—its epistemic counterparts—are true. It promises an instructive account of why certain general principles of knowledge hold. We focus on two such principles that anyone should endorse: the closure principle that knowledge is downward closed under competent conjunction elimination, and the counter-closure principle that knowledge is upward closed under competent conjunction introduction. We argue that anyone endorsing the former must also endorse the (...) latter on pains of an unacceptable form of bootstrapping. We devise new formal models to identify necessary and sufficient conditions for these principles to hold on conceptions that construe knowledge in terms of true counterparts. These conditions state that counterparts of premise and conclusion beliefs are coordinated in certain ways whenever these beliefs stand in the relevant inferential relations. We show that the safety conception faces insuperable problems vindicating these coordination principles, because its epistemic counterpart relation is symmetric. We conclude that it thus proves unable to account for minimal closure properties of knowledge. More generally, our formal results establish parameters within which any conception must operate that construes knowledge in terms of true counterparts. (shrink)

There is an epistemological skepticism that I might be dreaming now, or I might be a brain in a vat (BIV). There is also a demonstration that derives the skeptical conclusion about knowledge of the external world from the premise C1, i.e., I do not know “I am not dreaming (not a BIV) now.” Pessimistic critics (e.g., F. Strawson, B. Stroud) consider that the refutation of C1 is impossible, whereas others have attempted the direct refutation of C1 (e.g., G. (...) E. Moore, H. Putnam, C. Wright), and some (e.g., F. Dretske, R. Nozick) have attempted to refute the closure principle of knowledge used in the demonstration while permitting the validity of C1. Another scholar, M. Williams, maintains that the skeptical demonstration is true only if we presuppose the epistemological premise that we choose to accept or reject at will. It seems that most critics tend to adopt a strategy that allows them to effectively avoid the skeptical consequence, thereby conceding the validity of C1. However, it is difficult to say whether their attempts succeeded, and in my opinion, they are indeed unsuccessful. This is because their concession to Descartes’ argument is insufficient. Their lack of success could also stem from the incompletion of Descartes’ own methodological doubt. The somewhat paradoxical‐sounding aim of this thesis is to show that the skeptical paradox about knowledge can be dissolved only if the Cartesian skepticism is extended far beyond the endpoint of his attempt. The argument of this thesis is based on the important arguments of Wittgenstein's On Certainty （OC）. (shrink)

There are many ordinary propositions we think we know. Almost every ordinary proposition entails some "lottery proposition" which we think we do not know but to which we assign a high probability of being true (for instance: “I will never be a multi-millionaire” entails “I will not win this lottery”). How is this possible - given that some closure principle is true? This problem, also known as “the Lottery puzzle”, has recently provoked a lot of discussion. In (...) this paper I discuss one of the most promising answers to the problem: Stewart Cohen’s contextualist solution which is based on ideas about the salience of chances of error. After presenting some objections to it I sketch an alternative solution which is still contextualist in spirit. (shrink)

This is an encyclopedia article about epistemic closure principles. The article explains what they are, their various philosophical uses, how they are argued for or against, and provides an overview of the related literature.