در قرائت لیندا زاگزبسکی از معرفت‌شناسی فضیلت عناصر بسیاری در کنار هم قرار گرفته‌اند. در این نظریۀ ترکیبی -اگر نگوییم التقاطی- زمینه برای طرح ایرادات بسیار است. در این مقاله، پس از ارائه تقریری بسیار مختصر از لبّ نظریۀ معرفت‌شناسی فضیلت زاگزبسکی، چهار ایراد درباره آن مطرح شده که از این قرارند: فضیلت و باورِ برآمده از مسیری که زاگزبسکی ترسیم می‌کند در همه موارد شرط لازم و کافی برای حصول معرفت نیست؛ تفکیک میان فضایل طبیعی و اکتسابی و نقش (...) هر کدام از آنها در کسب معرفت در نظریه زاگزبسکی به روشنی تبیین نشده و قابل مناقشه است؛ جایگاه عامل و باورنده در نظریۀ زاگزبسکی به مثابه فاعل و دارنده شناخت، از منظر اعتمادگرایانه و طبیعی‌گرایانۀ محض قابل مناقشه است؛ از منظر علوم تجربی و همچنین فلسفه مبتنی بر آزمون، فضایل و رذایل و فهرست آنها به شکلی دیگر تبیین می‌شود؛ و نهایتاً این که جایگاه انگیزۀ کسب معرفت و همچنین رانه‌هایی که به حصول معرفت منجر می‌شوند قابل مناقشه است. (shrink)

This volume, The Brazilian Studies in the Philosophy and History of Science, is the first attempt to present to a general audience, works from Brazil on this subject. The included papers are original, covering a remarkable number of relevant topics of philosophy of science, logic and on the history of science. The Brazilian community has increased in the last years in quantity and in quality of the works, most of them being published in respectable international journals on the subject. The (...) chapters of this volume are forwarded by a general introduction, which aims to sketch not only the contents of the chapters, but it is conceived as a historical and conceptual guide to the development of the field in Brazil. The introduction intends to be useful to the reader, and not only to the specialist, helping them to evaluate the increase in production of this country within the international context. (shrink)

What role can intellectual virtues play in an account of knowledge when we interpret those virtues internalistically, i.e., as depending only on internal states of the cognizer? Though it has been argued that internalist virtues are ill suited to play any role in an account of knowledge, I will show that, on the contrary, internalist virtues can play an important role in recent accounts of knowledge developed to utilize externalist virtues. The virtue account of knowledge developed by Linda Zagzebski is (...) intended to be supplemented by her version of the intellectual virtues which require an external success component. John Greco and Wayne Riggs both develop credit accounts of knowledge on which the abilities we use when we get credit for a true belief must be reliable. I examine the similarities between these three accounts of knowledge and demonstrate that internalist virtues fit into these accounts just as well as externalist virtues. Thus, although internalist virtues do not require a reliable connection to truth, they can still play an important role in defining the truth-requiring concept of knowledge. (shrink)

What is the best model of epistemic agency for virtue epistemology? Insofar as the intellectual and moral virtues are similar, it is desirable to develop models of agency that are similar across the two realms. Unlike Aristotle, the Stoics present a model of the virtues on which the moral and intellectual virtues are unified. The Stoics’ materialism and determinism also help to explain how we can be responsible for our beliefs even when we cannot believe otherwise. In this paper I (...) show how a neo-Stoic model of epistemic agency can address common objections to treating epistemic and moral agency similarly and allow a robust explanatory role for character in determining our actions and beliefs. The picture of epistemic responsibility that flows from this model also explains why we often deserve credit for our knowledge, while demonstrating that the truth of our beliefs is not something for which we are epistemically responsible. (shrink)

John Cook Wilson is an important predecessor of contemporary knowledge first epistemologists: among other parallels, he claimed that knowledge is indefinable. We reconstruct four arguments for this claim discernible in his work, three of which find no clear analogues in contemporary discussions of knowledge first epistemology. We pay special attention to Cook Wilson’s view of the relation between knowledge and forms of thinking (like belief). Claims of Cook Wilson’s that support the indefinability of knowledge include: that knowledge, unlike belief, straddles (...) an active/passive divide; that, rather than entailing belief, knowledge excludes belief; and that understanding forms of thinking other than knowledge (such as belief) depends on understanding knowledge. Reflecting on Cook Wilson’s framework highlights underappreciated concerns relevant to any attempt to define knowledge. (shrink)

In this paper, the Carneades argumentation system is extended to represent a procedural view of inquiry in which evidence is marshalled to support or defeat claims to knowledge. The model is a sequence of moves in a collaborative group inquiry in which parties take turns making assertions about what is known or not known, putting forward evidence to support them, and subjecting these moves to criticisms. It is shown how this model of evaluating evidence in an inquiry is based on (...) a defeasible logic using forms of argument that admit exceptions. It is contended that reasoning from absence of knowledge is as important to inquiry as positive reasoning from evidence to knowledge. The philosophical conflict between this view of reasoning about knowledge and the true-belief-plus view is explored by airing objections and replies on both sides. (shrink)

In a recent paper in this journal, Alex Grzankowski argues, contra Howard Sankey, that to believe that p isn’t to believe that p is true. In this short reply, I’ll agree with Grzankowski that to believe that p isn’t to believe that p is true, and I’ll argue that Sankey’s recent response to Grzankowski is inadequate as it stands. However, it’ll be my contention that Grzankowski’s argument doesn’t demonstrate that believing that p doesn’t require taking it to be the case (...) that p. (shrink)

Citing some recent experimental findings, I argue for the surprising claim that in some cases the less time you have the more you know. More specifically, I present some evidence to suggest that our ordinary knowledge ascriptions are sometimes sensitive to facts about an epistemic subject's truth-irrelevant time constraints such that less is more. If knowledge ascriptions are sensitive in this manner, then this is some evidence of pragmatic encroachment. Along the way, I consider comments made by Jonathan Schaffer and (...) Jennifer Nagel to construe a purist contextualist and a strict invariantist explanation of the data respectively, before giving reasons to resist them in favor of an account that indicates pragmatic encroachment. If successful, this may suggest a new way to argue for the controversial thesis that there is pragmatic encroachment on knowledge. (shrink)

That believing truly as a matter of luck does not generally constitute knowing has become epistemic commonplace. Accounts of knowledge incorporating this anti-luck idea frequently rely on one or another of a safety or sensitivity condition. Sensitivity-based accounts of knowledge have a well-known problem with necessary truths, to wit, that any believed necessary truth trivially counts as knowledge on such accounts. In this paper, we argue that safety-based accounts similarly trivialize knowledge of necessary truths and that two ways of responding (...) to this problem for safety, issuing from work by Williamson and Pritchard, are of dubious success. (shrink)

There are good reasons for pursuing a theory of knowledge by way of understanding the connection between knowledge and luck. Not surprisingly, then, there has been a burgeoning of interest in “luck theories” of knowledge as well as in theories of luck in general. Unfortunately, “luck” proves to be as recalcitrant an analysandum as “knows.” While it is well worth pursuing a general theory of luck despite these difficulties, our theory of knowledge might be made more manageable if we could (...) find a more restricted notion that captured the core phenomena of luck that are relevant to whether or not someone knows. This essay makes the attempt to delineate such a notion, called “mere coincidence.”. (shrink)

Elgin has offered us a powerful articulation of an epistemology that does not, contra veritism, have a concern for truth at its core. I contend that the case for Elgin’s alternative epistemological picture trades upon a faulty conception of what a veritistic epistemological outlook involves. In particular, I argue that the right conception of veritism—one that is fundamentally informed by the intellectual virtues—has none of the problematic consequences that Elgin claims. Relatedly, I maintain that we can account for the core (...) role of objectual understanding in inquiry without thereby giving up on truth as the fundamental epistemic good. (shrink)

A commonly expressed worry in the contemporary literature on the problem of epistemological scepticism is that there is something deeply intellectually unsatisfying about the dominant anti-sceptical theories. In this paper I outline the main approaches to scepticism and argue that they each fail to capture what is essential to the sceptical challenge because they fail to fully understand the role that the problem of epistemic luck plays in that challenge. I further argue that scepticism is best thought of not as (...) a quandary directed at our possession of knowledge simpliciter, but rather as concerned with a specific kind of knowledge that is epistemically desirable. On this view, the source of scepticism lies in a peculiarly epistemic form of angst. It is always by favour of Nature that one knows something. [Wittgenstein 1969: §505]. (shrink)

This paper explores the implications of a technological revolution that many in the industry think is likely soon to come to pass: neuromedia. In particular, the paper is interested in how this will constitute an especially persuasive kind of extended cognition, and thereby will facilitate extended epistemic states. This will in turn have ramifications for how we understand the epistemic goals of education. The paper argues that the challenges posed by neuromedia remind us that the overarching epistemic goal of education (...) is not orientated towards facilitating a body of knowledge (or the development of mere cognitive skills) but rather is concerned with the development of intellectual character, where this in turn essentially involves the cultivation of intellectual virtues, character traits that are not amenable to extended cognition. (shrink)

It is argued that a popular way of accounting for the distinctive value of knowledge by appeal to the distinctive value of cognitive achievements fails because it is a mistake to identify knowledge with cognitive achievements. Nevertheless, it is claimed that understanding, properly conceived, is a type of cognitive achievement, and thus that the distinctive value of cognitive achievements can explain why understanding is of special value.

It is argued that it is beneficial to view the debate regarding radical scepticism through the lens of epistemic value. In particular, it is claimed that we should regard radical scepticism as aiming to deprive us of an epistemic standing that is of special value to us, and that this methodological constraint on our dealings with radical scepticism potentially has important ramifications for how we assess the success of an anti-sceptical strategy.

A certain conception of the relevance of virtue epistemology to the philosophy of education is set out. On this conception, while the epistemic goal of education might initially be promoting the pupil's cognitive success, it should ultimately move on to the development of the pupil's cognitive agency. A continuum of cognitive agency is described, on which it is ultimately cognitive achievement, and thus understanding, which is the epistemic goal of education. This is contrasted with a view on which knowledge is (...) the epistemic goal. (shrink)

Our interest is in the possibility of there being a philosophically interesting set of useful false beliefs where the utility in question is specifically epistemic. As we will see, it is hard to delineate plausible candidates in this regard, though several are promising at first blush. We begin with the kind of strictly false claims that are said to be often involved in good scientific practice, such as through the use of idealisations and fictions. The problem is that it is (...) difficult to see that there would be any epistemic utility in believing such claims, as opposed, say, to merely accepting them. Next we turn to the challenge posed by epistemic situationism, which when embedded within a plausible form of virtue epistemology appears to show that sometimes purely situational factors can play a significant explanatory role in one’s cognitive success. But again it is hard to see how the role that these epistemically beneficial situational factors contribute can be cashed out in terms of epistemically useful false beliefs on the part of the subject. Finally, we turn to the Wittgensteinian conception of hinge commitments, commitments that are held to be epistemically useful even if false. While the epistemic utility of these commitments is defended, it is argued that one cannot make sense of these commitments in terms of belief. Support is thus canvassed, albeit in a piecemeal fashion, for the thesis that the prospects for there being philosophically interesting cases of epistemically useful false belief are poor. (shrink)

This paper explores the ramifications of the extended cognition thesis in the philosophy of mind for contemporary epistemology. In particular, it argues that all theories of knowledge need to accommodate the ability intuition that knowledge involves cognitive ability, but that once this requirement is understood correctly there is no reason why one could not have a conception of cognitive ability that was consistent with the extended cognition thesis. There is thus, surprisingly, a straightforward way of developing our current thinking about (...) knowledge such that it incorporates the extended cognition thesis. (shrink)

Two key intuitions regarding knowledge are explored: that knowledge is a kind of cognitive achievement and that knowledge excludes luck. It is claimed that a proper understanding of how these intuitions should inform our conception of knowledge leads to some surprising results, not just as regards the theory of knowledge more generally but also as regards the epistemology of testimonial knowledge. In particular, it is argued that this conception of knowledge motivates a new kind of proposal B quasi-reductionism B that (...) can accommodate the motivations behind both reductionist and anti-reductionist accounts of the epistemology of testimonial knowledge. (shrink)

A certain construal of the Gettier problem is offered, according to which this problem concerns the task of identifying the anti-luck condition on knowledge. A methodology for approaching this construal of the Gettier problem—anti-luck epistemology—is set out, and the utility of such a methodology is demonstrated. It is argued that a range of superficially distinct cases which are meant to pose problems for anti-luck epistemology are in fact related in significant ways. It is claimed that with these cases properly understood, (...) anti-luck epistemology is able to offer a suitable diagnosis of them which doesn’t threaten the necessity of the anti-luck condition for knowledge. (shrink)

Support is canvassed for a new approach to epistemology called anti-risk epistemology. It is argued that this proposal is rooted in the motivations for an existing account, known as anti-luck epistemology, but is superior on a number of fronts. In particular, anti-risk epistemology is better placed than anti-luck epistemology to supply the motivation for certain theoretical moves with regard to safety-based approaches to knowledge. Moreover, anti-risk epistemology is more easily extendable to epistemological questions beyond that in play in the theory (...) of knowledge specifically. A key advantage of the view, however, is that anti-risk epistemology fares much better than anti-luck epistemology when it comes to accounting for the phenomenon of negative epistemic dependence. In particular, anti-risk epistemology is ideally placed to explain why such epistemic dependence is incompatible with knowledge, even when the negative epistemic dependence in play is of a purely modal variety. (shrink)

‘Motion’ in the acquisition of knowledge is the focus of this paper. My claim is that the motion in the acquisition of knowledge, albeit abstract, is of two dimensions: one as ‘the mind working towards reality’ and the other as ‘reality walking towards the mind’. Using an epistemic approach, I conceptualize these two dimensions (hereafter as the working and walking narrative) under the Externalism and the Internalism divide and identify that externalism leaves us with more to explore about knowledge acquisition (...) than internalism. Since externalism seems to accommodate the two dimensions better than internalism, I present the working and walking narrative under externalism. Two envisaged objections are addressed, along with considerations of some probable implications of the working and walking narrative for philosophy, specifically epistemology. (shrink)

While there is now considerable anxiety about whether the psychological theory presupposed by virtue ethics is empirically sustainable, analogous issues have received little attention in the virtue epistemology literature. This paper argues that virtue epistemology encounters challenges reminiscent of those recently encountered by virtue ethics: just as seemingly trivial variation in context provokes unsettling variation in patterns of moral behavior, trivial variation in context elicits unsettling variation in patterns of cognitive functioning. Insofar as reliability is a condition on epistemic virtue, (...) we have reason to doubt that human beings possess the cognitive materials required for epistemic virtue, and thereby reason to think that virtue epistemology is threatened by skepticism. We conclude that while virtue epistemology has resources for addressing this challenge, exploiting these resources forces tradeoffs between empirical and normative adequacy. (shrink)

Upon discovering that certain beliefs we hold are contingent on arbitrary features of our background, we often feel uneasy. I defend the proposal that if such cases of contingency anxiety involve defeaters, this is because of the epistemic significance of disagreement. I note two hurdles to our accepting this Disagreement Hypothesis. Firstly, some cases of contingency anxiety apparently involve no disagreement. Secondly, the proposal may seem to make our awareness of the influence of arbitrary background factors irrelevant in determining whether (...) to revise our beliefs. I show that each of these problems can be successfully accommodated by the Disagreement Hypothesis. (shrink)

Upon discovering that certain beliefs we hold are contingent on arbitrary features of our background, we often feel uneasy. I defend the proposal that if such cases of contingency anxiety involve defeaters, this is because of the epistemic significance of disagreement. I note two hurdles to our accepting this Disagreement Hypothesis. Firstly, some cases of contingency anxiety apparently involve no disagreement. Secondly, the proposal may seem to make our awareness of the influence of arbitrary background factors irrelevant in determining whether (...) to revise our beliefs. I show that each of these problems can be successfully accommodated by the Disagreement Hypothesis. (shrink)

Even with the lack of consensus on the nature of an argument, the thesis that explanations and arguments are distinct is near orthodoxy in well-known critical thinking texts and in the more advanced argumentation literature. In this paper, I reconstruct two rationales for distinguishing arguments from explanations. According to one, arguments and explanations are essentially different things because they have different structures. According to the other, while some explanations and arguments may have the same structure, they are different things because (...) explanations are used for different purposes than arguments. I argue that both rationales fail to motivate a distinction between arguments and explanations. Since these are the only rationales for distinguishing arguments from explanations that I am prepared to take seriously, I don’t see why we should exclude explanations from being arguments. (shrink)

This paper argues that reliabilism can handle Gettier cases once it restricts knowledge producing reliable processes to those that involve a suitable causal link between the subject’s belief and the fact it references. Causal tracking reliabilism (as this version of reliabilism is called) also avoids the problems that refuted the causal theory of knowledge, along with problems besetting more contemporary theories (such as virtue reliabilism and the “safety” account of knowledge). Finally, causal tracking reliabilism allows for a response to Linda (...) Zagzebski’s challenge that no theory of knowledge can both eliminate the possibility of Gettier cases while also allowing fully warranted but false beliefs. (shrink)

ABSTRACT: There is an intuitive distinction between knowing someone in a detached manner and knowing someone in a more intimate fashion — personally. The latter seems to involve the specially active participation of the person known in a way that the former does not. In this paper I present a novel, communication account of knowing someone personally that successfully explains this participation. The account also illuminates the propositional and testimonial character of the personal knowledge of persons, the conditions of limited (...) transferability to which such knowledge is subject, and its distinctly meaningful role in everyday life. I conclude by considering a worry about self-knowledge that arises on the communication account. (shrink)

The lottery problem is often regarded as a successful counterexample to reliabilism. The process of forming your true belief that your ticket has lost solely on the basis of considering the odds is, from a purely probabilistic viewpoint, much more reliable than the process of forming a true belief that you have lost by reading the results in a normally reliable newspaper. Reliabilism thus seems forced, counterintuitively, to count the former process as knowledge if it so counts the latter process. (...) -/- I offer a theory of empirical knowledge which, while being recognizably reliabilist, restricts empirical knowledge to cases in which the fact that p and the belief that p are causally connected. I show that this form of reliabilism solves the lottery problem, avoids the problems that beset the causal theory of knowledge, and show how it handles a number of problematic cases in the recent literature. (shrink)

This paper identifies and criticizes certain fundamental commitments of virtue theories in epistemology. A basic question for virtues approaches is whether they represent a ‘third force’––a different source of normativity to internalism and externalism. Virtues approaches so-conceived are opposed. It is argued that virtues theories offer us nothing that can unify the internalist and externalist sub-components of their preferred success-state. Claims that character can unify a virtues-based axiology are overturned. Problems with the pluralism of virtues theories are identified––problems with pluralism (...) and the nature of the self; and problems with pluralism and the goals of epistemology. Moral objections to virtue theory are identified––specifically, both the idea that there can be a radical axiological priority to character and the anti-enlightenment tendencies in virtues approaches. Finally, some strengths to virtue theory are conceded, while the role of epistemic luck is identified as an important topic for future work. (shrink)

The thesis that knowledge is a partly evaluative concept is now a widespread view in epistemology, informing some prominent debates in the field. Typically, the view is embraced on the grounds that justification is a necessary condition for knowledge and a normative concept — a reasonable motivation. However, the view also has counterintuitive implications, which have been neglected. In particular, it implies that J.L. Mackie’s error-theory of value entails global epistemic scepticism and that any true knowledge claim suffices to prove (...) the error-theory is false. In this paper, I elaborate these difficulties and address objections at length. (shrink)

A view of knowledge—what I call the "Deserving Credit View of Knowledge" —found in much of the recent epistemological literature, particularly among so-called virtue epistemologists, centres around the thesis that knowledge is something for which a subject deserves credit. Indeed, this is said to be the central difference between those true beliefs that qualify as knowledge and those that are true merely by luck—the former, unlike the latter, are achievements of the subject and are thereby creditable to her. Moreover, it (...) is often further noted that deserving credit is what explains the additional value that knowledge has over merely lucky true belief. In this paper, I argue that the general conception of knowledge found in the DCVK is fundamentally incorrect. In particular, I show that deserving credit cannot be what distinguishes knowledge from merely lucky true belief since knowledge is not something for which a subject always deserves credit. (shrink)

A popular form of virtue epistemology—defended by such figures as Ernest Sosa, Linda Zagzebski and John Greco—holds that knowledge can be exclusively understood in virtue-theoretic terms. In particular, it holds that there isn't any need for an additional epistemic condition to deal with the problem posed by knowledge-undermining epistemic luck. It is argued that the sustainability of such a proposal is called into question by the possibility of epistemic twin earth cases. In particular, it is argued that such cases demonstrate (...) the need for virtue-theoretic accounts of knowledge to appeal to an independent epistemic condition which excludes knowledge-undermining epistemic luck. (shrink)

This essay defends virtue reliabilism against a line of argument put forward by Duncan Pritchard. In the process, it discusses (1) the motivations for virtue reliabilism, (2) some analogies between epistemic virtue and moral virtue, and (3) the relation between virtue (epistemic and otherwise) and luck (epistemic and otherwise). It argues that considerations about virtue and luck suggest a solution to Gettier problems from the perspective of a virtue theory.

Business management within the United States of America contains unacknowledged, inherent contradictions that constrain individual and collective action, and form barriers against the development of authenticity and solidarity within organizations. The Emersonian themes of conformity, consistency, and knowledge, as developed in his 1841 essay Self-Reliance, were used as constructive points of philosophical inquiry around which to interrogate the theory and praxis of current American business management. The need for such an examination of management is observable in recent social phenomena. Specifically, (...) the great resignation was explored as a potential representative response to dealing with the contradictions between management as currently practiced within the United States and the existential needs of those subjugated by and to its praxis. Philosophy emerges as an effective way to assess the degree to which organizations can achieve their emancipatory potential using the current theory and praxis of business management. (shrink)

There is some consensus that for S to know that p, it cannot be merely a matter of luck that S’s belief that p is true. This consideration has led Duncan Pritchard and others to propose a safety condition on knowledge. In this paper, we argue that the safety condition is not a proper formulation of the intuition that knowledge excludes luck. We suggest an alternative proposal in the same spirit as safety, and find it lacking as well.

Taking his conceptual cue from Ernest Sosa, John Turri has offered a putative conceptual solution to the Gettier problem: Knowledge is cognitively adept belief, and no Gettiered belief is cognitively adept. At the core of such adeptness is a relation of manifestation. Yet to require that relation within knowing is to reach for what amounts to an infallibilist conception of knowledge. And this clashes with the spirit behind the fallibilism articulated by Gettier when stating his challenge. So, Turri’s form of (...) response is irrelevant to that challenge, which was intended to pose a conceptual problem within fallibilist conceptions of knowledge. (And that failure on Turri’s part needs to be highlighted to remind epistemologists of the need to assess Gettier cases by a fallibilist standard. Although that need was described earlier by Robert Almeder, apparently his advice is being overlooked. This paper develops it anew, in a more general form.). (shrink)

The claim that knowledge is a kind of success from ability has great theoretical power: it explains the nature of epistemic normativity, why knowledge is incompatible with luck, and why knowledge is more valuable than mere true belief. This paper addresses objections to the view by wedding it with two additional ideas: that intellectual abilities display a certain structure, and that the concept of knowledge functions to flag good information, and good sources of information, for use in practical reasoning.

A number of virtue epistemologists endorse the following thesis: Knowledge is true belief resulting from intellectual virtue, where Ss true belief results from intellectual virtue just in case S believes the truth because S is intellectually virtuous. This thesis commits one to a sort of contextualism about knowledge attributions. This is because, in general, sentences of the form X occurred because Y occurred require a contextualist treatment. This sort of contextualism is contrasted with more familiar versions. It is argued that (...) the position: (a) yields a better solution to the lottery problem, and (b) may be grounded in a more general theory of virtue and credit. (shrink)

An account of the nature of knowledge must explain the value of knowledge. I argue that modal conditions, such as safety and sensitivity, do not confer value on a belief and so any account of knowledge that posits a modal condition as a fundamental constituent cannot vindicate widely held claims about the value of knowledge. I explain the implications of this for epistemology: We must either eschew modal conditions as a fundamental constituent of knowledge, or retain the modal conditions but (...) concede that knowledge is not more valuable than that which falls short of knowledge. This second horn—concluding that knowledge has no distinctive value—is unappealing since it renders puzzling why so much epistemological theorising focuses on knowledge, and why knowledge seems so important. (shrink)

In this article, we evaluate various responses to a noteworthy objection, namely, the infinite God objection to the kalām cosmological argument. As regards this objection, the proponents of the kalām argument face a dilemma—either an actual infinite cannot exist or God cannot be infinite. More precisely, this objection claims that God’s omniscience entails the existence of an actual infinite with God knowing an actually infinite number of future events or abstract objects, such as mathematical truths. We argue, however, that the (...) infinite God objection is based on two questionable assumptions, namely, that it is possible for an omniscient being to know an actually infinite number of things and that there exist an actually infinite number of abstract objects for God to know. (shrink)