In this paper I take up the question of the nature of the doxastic attitudes we entertain while inquiring into some matter. Relying on a distinction between two stages of open inquiry, I urge to acknowledge the existence of a distinctive attitude of cognitive inclination towards a proposition qua answer to the question one is inquiring into. I call this attitude “hypothesis”. Hypothesis, I argue, is a sui generis doxastic attitude which differs, both functionally and normatively, from suspended judgement, full (...) belief, credences, and acceptance. In closing, I point to the epistemological significance of hypothesis. More specifically, I contend that holding an attitude of hypothesis enables us to respond rationally to peer disagreement, and I suggest that such an attitude offers a suitable articulation of the view, originally put forward by Philip Kitcher, that cognitive diversity in inquiry has epistemic benefits. (shrink)

It is a widespread idea that suspended judgement implies a state of doxastic neutrality. Jane Friedman has recently claimed that while inquiring into a given question, one suspends one’s judgement on it. Jointly considered, the previous claims imply that one is in a state of doxastic neutrality about a given question while inquiring into it. In this article, I explore the leading cases against Friedman’s perspective, arguing that it is debatable whether they exhibit inquiries into questions without doxastic neutrality. (...) However, I will propose the possibility of “explorative disconfirmation inquiries” to show that doxastically non-neutral inquiries do exist. (shrink)

Epistemologists often claim that in addition to belief and disbelief there is a third, neutral, doxastic attitude. Various terms are used: ‘suspending judgment’, ‘withholding’, ‘agnosticism’. It is also common to claim that the factors relevant to the justification of these attitudes are epistemic in the narrow sense of being factors that bear on the strength or weakness of one’s epistemic position with respect to the target proposition. This paper addresses two challenges to such traditionalism about doxastic attitudes. The first concerns (...) the relevance of non-epistemic factors we might call "future-comparative" – e.g., that you’ll have more decisive evidence on whether p tomorrow – to the justification of suspending judgment. The second, from Jane Friedman, is to explain the point of the neutral attitude without appealing to inquiry and thus taking goal-related factors, which are not epistemic, such as the value of the goal or the prospects for finding means to achieve it, to bear on the justification of the neutral attitude. My defense of traditionalism relies on distinguishing three ways of being neutral on a question: agnosticism, inquiry and suspension of judgment. Traditionalism is saved because, of these, agnosticism alone is a genuine doxastic attitude. (shrink)

There are scenarios in which letting one’s own views on the question whether p direct one’s inquiry into that question brings about individual and collective epistemic benefits. However, these scenarios are also such that one’s evidence doesn’t support believing one’s own views. So, how to vindicate the epistemic benefits of directing one’s inquiry in such an asymmetric way, without asking one to hold a seemingly irrational doxastic attitude? To answer this question, the paper understands asymmetric inquiry direction in terms of (...) having one’s inquiry be guided by a doxastic attitude, which I call hypothesis, that’s governed by the following norm: One’s hypothesis that p is rational just in case it is a manifestation of the best feasible way to form a doxastic attitude regarding p that promotes the best feasible ways to advance towards settling the question of p’s (or of any other suitably related proposition’s) truth-value. (shrink)

The aim of this paper is to clarify under what conditions a philosophical problem arises. I will describe two ways in which we might perceive a question as a problem. First, when we fnd ourselves inclined to believe in propositions that appear incompatible with each other. Second, when we fnd ourselves inclined to believe in propositions that seem incompatible with our desires. I will discuss both of these cases and articulate a didactic strategy – the Clash Strategy – which can (...) be used in order to plan and facilitate dialogic inquiry in the classroom. This strategy suggests introducing stimuli that will help students appreciate the clash that drives the philosophical question to be explored during the dialogue. In summary, this paper aims to ofer insights to all teachers interested in promoting dialogic inquiry, whether philosophical or otherwise, in education. (shrink)

The paper addresses the nature of seemings in light of their role in inquiry. Seemings are mental states or events with propositional content that have a specific phenomenology often referred to as “felt truth”. In epistemology, seemings are mainly discussed as possible (non-inferential) justifications for belief. Yet, epistemology has recently taken a zetetic turn, that is, a turn toward the study of inquiry. I will argue that the role of seemings in epistemology should be re-assessed from the perspective of (...) inquiry and deliberation. Seemings may fall short of justifying beliefs, but there is an important role for seemings in the zetetic realm, which may also explain why seemings lack direct justificatory force for beliefs. Seemings, so I will suggest, can provide normative reasons for performing certain zetetic tasks, but they cannot provide normative reasons for making up one’s mind about the inquired question. In this sense, the zetetic role of seemings can be described as guiding rational inquiry. I will suggest that a modified version of the so-called Taking Evidence View about the nature of seemings is best fitted to accommodate their zetetic role. Yet, contra existing versions of the Taking Evidence View, my modified account acknowledges that seemings are genuine experiences (of one’s evidential standing during inquiry) that are accompanied by the characteristic phenomenology of what is called a “feeling of truth”. By doing so, my modified version of the Taking Evidence View incorporates the (as I take it) most important feature of the rival Experience View of seemings. (shrink)

The paper deals with the question of the structure of knowledge and the precise relationship between propositional "knowledge that" and dispositional "knowledge how." In the first part of my essay, I provide an analysis of the term 'knowing how' and argue that the usual alternatives in the recent epistemological debate – knowing how is either a form of propositional or dispositional knowledge – are misleading. In fact it depends on the semantic and pragmatic context of the usage of (...) this term whether 'knowing how' refers to a type of dispositional knowledge, to propositional knowledge, or to a hybrid form of both. Only in the first case, can one say that dispositional know how cannot be reduced to any form of propositional knowledge. Yet, this case is the most interesting one to consider in the investigation of the nature of knowledge, if one assumes that knowing that p presupposes "having found out that p." Having found something out, however, presupposes certain acts of epistemic inquiry and corresponding epistemic abilities. Examined more carefully, it is shown that the dispositional knowledge-how is a necessary condition for propositional knowledge-that, hence propositional knowledge-that is a species of the dispositional knowledge-how. Accordingly, dispositional knowledge has to be understood as being at the very core of our notion of knowledge, including propositional knowledge. (shrink)

It is generally thought that suspension of judgment about a proposition p is the doxastic attitude one is rationally compelled to adopt whenever the epistemic reasons for and against p are equipollent or equally credible, that is, whenever the total body of available evidence bearing on p epistemically justifies neither belief nor disbelief in p. However, in a recent contribution to this journal, Jan Wieland proposes “to broaden the conditions for suspension, and argue that it is rational to suspend belief (...) on a certain issue even if one’s current evidence is not neutral (or even close to neutral)”. My aim in this paper is to point to a number of problems in Wieland’s position, some of which in connection with the account of Pyrrhonian skepticism found in the extant works of Sextus Empiricus. (shrink)

Charles Sanders Peirce has commonly been interpreted as a proponent of an epistemic theory of truth. Such a theory has the apparent advantage of directly undercutting radical skepticism, but the disadvantage of implausibly entailing that there are no truths concerning irretrievably lost facts. Recently Andrew Howat has defended Peirce’s epistemic constraint on truth by recasting Peirce’s claim that all truths would be believed following sufficient inquiry, not as constitutive of truth, but as a Wittgensteinian hinge proposition. I begin with a (...) twofold historical reply to Howat: first, Peirce can at least plausibly be read as claiming that the claim in question is analytic of the concept <truth>; and second, while Peirce did think we must accept without evidence the further claim that there are any truths (so characterized), he regarded this further claim not as a Wittgensteinian hinge but as a reasonable hope. I then substantively defend the latter Peircean proposal, arguing that hopes, as against beliefs, are chiefly warranted pragmatically, and so can be warranted even to a sufficient level to render them properly action-guiding in cases where belief in the same proposition would remain unjustified. I conclude by suggesting, first, that this proposal is superior to Wittgenstein’s own epistemology in evading psychologism, and second, that it attractively vindicates the reasonableness of inquiry on Peircean grounds that don’t require an appeal to an implausible epistemic theory of truth. (shrink)

The current literature on norms of inquiry features two families of norms: norms that focus on an inquirer’s ignorance and norms that focus on the question’s soundness. I argue that, given a factive conception of ignorance, it’s possible to derive a soundness-style norm from a version of the ignorance norm. A crucial lemma in the argument is that just as one can only be ignorant of a proposition if the proposition is true, so one can only be ignorant with respect (...) to a question if the question is sound. (shrink)

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

Several philosophers advance substantive theories of propositions, to deal with several issues they raise in connection with a concern with a long pedigree in philosophy, the problem of the unity of propositions. The qualification ‘substantive’ is meant to contrast with ‘minimal’ or ‘deflationary’ – roughly, views that reject that propositions have a hidden nature, worth investigating. Substantive views appear to create spurious problems by characterizing propositions in ways that make them unfit to perform their theoretical jobs. I will present in (...) this light some critical points against Hanks’ (2015, 2019) act-theoretic view, and Recanati’s (2019) recent elaboration of Hanks’ notion of cancellation. Both Hanks and Recanati, I’ll argue, rely on problematic conceptions of fiction and pretense. (shrink)

This paper argues that the hypothesis proposed in the Meno is the proposition ‘virtue is good’ alone, and that its epistemic nature is essentially insecure. It has been an object of huge scholarly debate which other hypothesis Socrates posited with regard to the relationship between virtue and knowledge. This debate is, however, misleading in the sense of making us believe that the hypothesis that virtue is good is regarded as a truism in the light of the process of positing a (...) higher hypothesis described in the Phaedo and the Republic. I argue that the hypothesis in question is presented as a result of the earlier discussion of Meno's third definition of virtue, which implies the consequence that morally correct actions are beneficial whether or not accompanied by conventional goods such as wealth and honour. The underlying role of presenting it as a hypothesis is therefore to postpone demonstrating the truth of a controversial thesis which needs a substantial justification. In conclusion, I s... (shrink)

Despite some notable similarities between the scientific realisms of Sellars and Peirce (such as both being anti-representationalist, and future-directed), in his mature work Science and Metaphysics Sellars explicitly critiqued Peirce’s account of truth, as lacking “an intelligible foundation” (Sellars 1968: vii). In this paper, I explore Sellars’ proposed remedy to Peirce’s purported lack, in his complex and enigmatic account of picturing – a non-discursive ‘mapping’ of the world. I argue that although Sellars’ development of this idea is largely sound, much (...) of what he charges Peirce’s philosophy with missing is actually present there. By means of his semiotic icon / index / symbol distinction, Peirce manages to develop a philosophy of language which effectively coordinates the real order and the order of signification within the very structure of the proposition as he understands it, by contrast to Sellars’ claim that the two orders are merely connected by ‘analogy’. I also argue that in between Sellars’ ambitious account of veridicality, which appears to anticipate a form of scientific self-mensuration, and the dismissive ‘anti-truth’ quietism of neopragmatists such as Rorty, Peirce’s “contrite fallibilism” (Peirce CP 1.14) charts a wise middle path. (shrink)

Inquiry into the meaning of logical terms in natural language (‘and’, ‘or’, ‘not’, ‘if’) has generally proceeded along two dimensions. On the one hand, semantic theories aim to predict native speaker intuitions about the natural language sentences involving those logical terms. On the other hand, logical theories explore the formal properties of the translations of those terms into formal languages. Sometimes, these two lines of inquiry appear to be in tension: for instance, our best logical investigation into conditional connectives may (...) show that there is no conditional operator that has all the properties native speaker intuitions suggest if has. Indicative conditionals have famously been the source of one such tension, ever since the triviality proofs of both Lewis (1976) and Gibbard (1981) established conclusions which are in prima facie tension with ordinary judgments about natural language indicative conditionals. In a recent series of papers, Branden Fitelson has strengthened both triviality results (Fitelson 2013, 2015, 2016), revealing a common culprit: a logical schema known as IMPORT-EXPORT. Fitelson’s results focus the tension between the logical results and ordinary judgments, since IMPORT-EXPORT seems to be supported by intuitions about natural language. In this paper, we argue that the intuitions which have been taken to support IMPORT-EXPORT are really evidence for a closely related, but subtly different, principle. We show that the two principles are independent by showing how, given a standard assumption about the conditional operator in the formal language in which IMPORT-EXPORT is stated, many existing theories of indicative conditionals validate one, but not the other. Moreover, we argue that once we clearly distinguish these principles, we can use propositional anaphora to show that IMPORT-EXPORT is in fact not valid for natural language indicative conditionals (given this assumption about the formal conditional operator). This gives us a principled and independently motivated way of rejecting a crucial premise in many triviality results, while still making sense of the speaker intuitions which appeared to motivate that premise. We suggest that this strategy has broad application and an important lesson: in theorizing about the logic of natural language, we must pay careful attention to the translation between the formal languages in which logical results are typically proved, and natural languages which are the subject matter of semantic theory. (shrink)

Knowledge is good, ignorance is bad. So it seems, anyway. But in this dissertation, I argue that some ignorance is epistemically valuable. Sometimes, we should suspend judgment even though by believing we would achieve knowledge. In this apology for ignorance (ignorance, that is, of a certain kind), I defend the following four theses: 1) Sometimes, we should continue inquiry in ignorance, even though we are in a position to know the answer, in order to achieve more than mere knowledge (e.g. (...) understanding) while minimizing the effects of confirmation bias. 2) It’s false that we should believe every proposition such that we are guaranteed to be right about it (and even such that we are guaranteed to know it) if we believe it. 3) Being in a position to know is the norm of assertion: importantly, this does not require belief or (thereby) knowledge, and so proper assertion can survive speaker-ignorance. 4) It can be permissible and conversationally useful to tell audiences things that it is logically impossible for them to come to know: Proper assertion can survive (necessary) audience-side ignorance. Cumulatively, this project suggests that, properly understood, ignorance has an important role to play in the good epistemic life. (shrink)

What is it to know someone? Epistemologists rarely take up this question, though recent developments make such inquiry possible and desirable. This paper advances an account of how such interpersonal knowledge goes beyond mere propositional and qualitative knowledge about someone, giving a central place to second-personal treatment. It examines what such knowledge requires, and what makes it distinctive within epistemology as well as socially. It assesses its theoretic value for several issues in moral psychology, epistemic injustice, and philosophy of (...) mind. And it offers an account of the complex content in play if interpersonal knowledge is to be understood in terms of its mental states and their functions. (shrink)

Scientists, philosophers, and other researchers commonly assert their theories. This is surprising, as there are good reasons for skepticism about theories in cutting-edge research. I propose a new account of assertion in research contexts that vindicates these assertions. This account appeals to a distinct propositional attitude called endorsement, which is the rational attitude of committed advocacy researchers have to their theories. The account also appeals to a theory of conversational pragmatics known as the Question Under Discussion model, or QUD. (...) Hence, I call the theory the EQUD model. Motivating this account is a recognition that the speech act of assertion has two roles to play in research contexts. The first is the advocacy role, in which researchers assert a theory in order to advocate for it. The second is the evidential role, which is used to add to the common stock of information available to a field of inquiry. The EQUD model provides an account of warranted assertion for both these roles in research contexts. This success provides support for the theory of endorsement. It also provides support for information updating accounts of assertion. (shrink)

According to a traditional interpretation of Kierkegaard, he endorses a strong form of direct doxastic voluntarism on which we can, by brute force of will, make a “leap of faith” to believe propositions that we ourselves take to be improbable and absurd. Yet most leading Kierkegaard scholars now wholly reject this reading, instead interpreting Kierkegaard as holding that the will can affect what we believe only indirectly. This paper argues that Kierkegaard does in fact endorse a restricted, sophisticated, and plausible (...) version of direct doxastic voluntarism. On Kierkegaard’s view, when we take ourselves to be in an epistemically permissive situation, we have the ability to form outright beliefs (but not credences) at will in virtue of our ability to voluntarily 1) open or close inquiry and 2) determine our attitude towards epistemic risk. (shrink)

Kierkegaard's pseudonym Johannes Climacus famously defines faith as a risky “venture” that requires “holding fast” to “objective uncertainty.” Yet puzzlingly, he emphasizes that faith requires resolute conviction and certainty. Moreover, Climacus claims that all beliefs about contingent propositions about the external world “exclude doubt” and “nullify uncertainty,” but also that uncertainty is “continually present” in these very same beliefs. This paper argues that these apparent contradictions can be resolved by interpreting Climacus as a belief‐credence dualist. That is, Climacus holds that (...) beliefs and credences (i.e., degrees of belief) are two irreducibly distinct types of mental states. Beliefs are resolutions that close inquiry, thereby excluding doubt and reflecting subjective certainty by disregarding the possibility of error. Credences, by contrast, reflect assessments of evidential probabilities, thereby encoding a recognition of “objective uncertainty” by acknowledging the possibility of error. In addition to solving a vexing interpretive puzzle and showing how Kierkegaard anticipates contemporary views about the nature of belief and credence, this paper demonstrates that Kierkegaard developed a sophisticated account of the nature of belief, doubt, and certainty that merits serious philosophical consideration. (shrink)

W. V. Quine famously defended two theses that have fallen rather dramatically out of fashion. The first is that intensions are “creatures of darkness” that ultimately have no place in respectable philosophical circles, owing primarily to their lack of rigorous identity conditions. However, although he was thoroughly familiar with Carnap’s foundational studies in what would become known as possible world semantics, it likely wouldn’t yet have been apparent to Quine that he was fighting a losing battle against intensions, due in (...) large measure to developments stemming from Carnap’s studies and culminating in the work of Kripke, Hintikka, and Bayart. These developments undermined Quine’s crusade against intensions on two fronts. First, in the context of possible world semantics, intensions could after all be given rigorous identity conditions by defining them (in the simplest case) as functions from worlds to appropriate extensions, a fact exploited to powerful and influential effect in logic and linguistics by the likes of Kaplan, Montague, Lewis, and Cresswell. Second, the rise of possible world semantics fueled a strong resurgence of metaphysics in contemporary analytic philosophy that saw properties and propositions widely, fruitfully, and unabashedly adopted as ontological primitives in their own right. This resurgence — happily, in my view — continues into the present day. -/- For a time, at any rate, Quine experienced somewhat better success with his second thesis: that higher-order logic is, at worst, confused and, at best, a quirky notational alternative to standard first-order logic. However, Quine notwithstanding, a great deal of recent work in formal metaphysics transpires in a higher-order logical framework in which properties and propositions fall into an infinite hierarchy of types of (at least) every finite order. Initially, the most philosophically compelling reason for embracing such a framework since Russell first proposed his simple theory of types was simply that it provides a relatively natural explanation of the paradoxes. However, since the seminal work of Prior there has been a growing trend to consider higher-order logic to be the most philosophically natural framework for metaphysical inquiry, many of the contributors to this volume being among the most important and influential advocates of this view. Indeed, this is now quite arguably the dominant view among formal metaphysicians. -/- In this paper, and against the current tide, I will argue in §1 that there are still good reasons to think that Quine’s second battle is not yet lost and that the correct framework for logic is first-order and type-free — properties and propositions, logically speaking, are just individuals among others in a single domain of quantification — and that it arises naturally out of our most basic logical and semantical intuitions. The data I will draw upon are not new and are well-known to contemporary higher-order metaphysicians. However, I will try to defend my thesis in what I believe is a novel way by suggesting that these basic intuitions ground a reasonable distinction between “pure” logic and non-logical theory, and that Russell-style semantic paradoxes of truth and exemplification arise only when we move beyond the purely logical and, hence, do not of themselves provide any strong objection to a type-free conception of properties and propositions. -/- Most of my arguments in §1 are largely independent of any specific account of the nature of properties and propositions beyond their type-freedom. However, I will in addition argue that there are good reasons to take propositions, at least, to be very fine-grained. My arguments are thus bolstered significantly if it can be shown that there are in fact well-defined examples of logics that are not only type-free but which comport with such a conception of propositions. It is the purpose of §2 to lay out a logic of this sort in some detail, drawing especially upon work by George Bealer and related work of my own. With the logic in place, it will be possible to generalize the line of argument noted above regarding Russell-style paradoxes and, in §3, apply it to two propositional paradoxes — the Prior-Kaplan paradox and the Russell-Myhill paradox — that are often taken to threaten the sort of account developed here. (shrink)

It is valuable for inquiry to have researchers who are committed advocates of their own theories. However, in light of pervasive disagreement, such a commitment is not well explained by the idea that researchers believe their theories. Instead, this commitment, the rational attitude to take toward one’s favored theory during the course of inquiry, is what I call endorsement. Endorsement is a doxastic attitude, but one which is governed by a different type of epistemic rationality. This inclusive epistemic rationality is (...) sensitive to reasons beyond those to think the particular proposition in question is true. Instead, it includes extrinsic epistemic reasons, which concern the health of inquiry more generally. Such extrinsic reasons include the distribution of cognitive labor that a researcher will contribute to by endorsing a particular theory. Recognizing endorsement and inclusive epistemic rationality thus allows us to smooth a tension between individual rationality and collective rationality. It does so by showing how it can be epistemically rational to endorse a theory on the basis of the way this endorsement will benefit collective inquiry. I provide a decision theoretic treatment for inclusive epistemic rationality and endorsement which illustrates how this can be accomplished. (shrink)

The Lockean Thesis says that you must believe p iff you’re sufficiently confident of it. On some versions, the 'must' asserts a metaphysical connection; on others, it asserts a normative one. On some versions, 'sufficiently confident' refers to a fixed threshold of credence; on others, it varies with proposition and context. Claim: the Lockean Thesis follows from epistemic utility theory—the view that rational requirements are constrained by the norm to promote accuracy. Different versions of this theory generate different versions of (...) Lockeanism; moreover, a plausible version of epistemic utility theory meshes with natural language considerations, yielding a new Lockean picture that helps to model and explain the role of beliefs in inquiry and conversation. Your beliefs are your best guesses in response to the epistemic priorities of your context. Upshot: we have a new approach to the epistemology and semantics of belief. And it has teeth. It implies that the role of beliefs is fundamentally different than many have thought, and in fact supports a metaphysical reduction of belief to credence. (shrink)

This article explores the issues of phenomenon and genesis in Early Buddhist thought through a comparative analysis with the Eleatic tradition, aiming to enrich the understanding and dialogue between these philosophical and religious traditions. By examining the comparability of Buddhist thought and Parmenidean philosophy, the study challenges the notion that these traditions are fundamentally alien to each other. The focus is on the concept of genesis, not as creation from nothingness—rejected by both the Buddha and Parmenides—but as the manifestation of (...) the world to the human observer. The article argues that the world reveals itself in particular forms and appearances, which are intimately linked to the phenomenon and its perception by humans. This process is not solely a domain of rigid logical propositions but can be expressed through mythological and religious narratives. The study posits that the poetic expressiveness found in archaic philosophies of both India and Greece provides a valid medium for engaging in philosophical discourse. By adopting this comparative and dialogical perspective, the article aims to generate new philosophical insights and inspire future philosophical inquiry. The reflection on phenomenon and genesis, framed through this comparative lens, highlights the nuanced ways in which different traditions address the nature of reality and human perception, ultimately advocating for a broader, more inclusive understanding of philosophy that transcends conventional boundaries. (shrink)

Propositional contingentism is the thesis that there might have been propositions which might have not have been something. Serious actualism is the thesis that it is impossible for a property to be exemplified without there being something which exemplifies it. Both are popular. Likewise, the dominant view in the metaphysics of modality is that metaphysical possibility and necessity can be understood – in some sense – in terms of possible worlds, i.e. total ways the world could have been. Here, (...) I argue that, given some minimal assumptions, the conjunction of propositional contingentism and serious actualism entails that worlds are modally fragile – every world is ontologically dependent on every proposition. I then show that such a consequence is inconsistent with the claim that propositions true at all possible worlds are necessary. (shrink)

The problem of closure for the traditional unstructured possible worlds model of attitudinal content is that it treats belief and other cognitive states as closed under entailment, despite apparent counterexamples showing that this is not a necessary property of such states. One solution to this problem, which has been proposed recently by several authors (Schaffer 2005; Yalcin 2018; Hoek forthcoming), is to restrict closure in an unstructured setting by treating propositional attitudes as question-sensitive. Here I argue that this line (...) of response is unsatisfying as it stands because the problem of closure is more general than is typically discussed. A version of the problem recurs for attitudes like wondering, entertaining, considering, and so on, which are directed at questions rather than propositions. For such questioning attitudes, the appeal to question-sensitivity is much less convincing as a solution to the problem of closure. (shrink)

In this article, I cast doubt on an apparent truism, namely, that if evidence is available for gathering and use at a negligible cost, then it’s always instrumentally rational for us to gather that evidence and use it for making decisions. Call this ‘value of information’ (VOI). I show that VOI conflicts with two other plausible theses. The first is the view that an agent’s evidence can entail non-trivial propositions about the external world. The second is the view that epistemic (...) rationality requires us to update our credences by conditionalization. These two theses, given some plausible assumptions, make room for rationally biased inquiries where VOI fails. I go on to argue that this is bad news for defenders of VOI. (shrink)

Propositionalists hold that, fundamentally, all attitudes are propositional attitudes. A number of philosophers have recently called the propositionalist thesis into question. It has been argued, successfully I believe, that there are attitudes that are of or about things but which do not have a propositional content concerning those things. If correct, our theories of mind will include non-propositional attitudes as well as propositional attitudes. In light of this, Sinhababu’s recent attack on anti-propositionalists is noteworthy. The present (...) paper aims to sharpen his worries and show that they fail for a range of reasons. Besides merely offering a reply, considering his challenges provides an opportunity to add clarity to this emerging area of research and it allows one to strengthen the case against propositionalism more generally. (shrink)

I argue for an account of grasping, or understanding that, on which we grasp via a higher‐order mental act of Husserlian fulfillment. Fulfillment is the act of matching up the objects of our phenomenally presentational experiences with those of our phenomenally representational thought. Grasping‐by‐fulfilling is importantly different from standard epistemic aims, in part because it is phenomenal rather than inferential. (I endorse Bourget's (2017) arguments to that effect.) I show that grasping‐by‐fulfilling cannot be a species of propositional knowledge or (...) belief, and that it is not essentially connected to justification. I motivate a revisionary epistemology on which achieving propositional knowledge and coming to grasp are dual epistemic aims. My account makes sense of a common occurrence—that we are often unmoved to act on our beliefs until we come to phenomenally experience them in some way. It also explains puzzling features of human inquiry. (shrink)

In some cases, there appears to be an asymmetry in the evidential value of statistical and more individualized evidence. For example, while I may accept that Alex is guilty based on eyewitness testimony that is 80% likely to be accurate, it does not seem permissible to do so based on the fact that 80% of a group that Alex is a member of are guilty. In this paper I suggest that rather than reflecting a deep defect in statistical evidence, this (...) asymmetry might arise from a general constraint on rational inquiry. Plausibly the degree of evidential support needed to justify taking a proposition to be true depends on the stakes of error. While relying on statistical evidence plausibly raises the stakes by introducing new kinds of risk to members of the reference class, paradigmatically `individualized' evidence---evidence tracing back to A's voluntary behavior---can lower the stakes. The net result explains the apparent evidential asymmetry without positing a deep difference in the brute justificatory power of different types of evidence. (shrink)

I here discuss two problems facing Russellian act-type theories of propositions, and argue that Fregean act-type theories are better equipped to deal with them. The first relates to complex singular terms like '2+2', which turn out not to pose any special problem for Fregeans at all, whereas Soames' theory currently has no satisfactory way of dealing with them (particularly, with such "mixed" propositions as the proposition that 2+2 is greater than 3). Admittedly, one possibility stands out as the most promising (...) one, but it requires that the Russellian treat complex properties as constituents of propositions. This leads to the second major problem for Russellians: that of proliferating propositions. I show how the most direct solution to this problem, that of rejecting complex predicative propositional constituents is available to Fregeans but very implausible for Russellians, since this virtually means rejecting complex properties. (shrink)

I first quickly outline what I think grasping is, and suggest that it is both among our basic aims of inquiry and not essentially tied to belief, justification, or knowledge. Then, I briefly look at some places in the metaphysics of science in which it looks like our aim of grasping and our aim in knowing—or perhaps more specifically in knowing the explanations for things—might seem to conflict. I will use this conflict to support a broader view: sometimes, we might (...) develop philosophical views or theories—and even endorse them—in order to better grasp them, regardless of whether we genuinely believe them, or are justified in so doing. At other times, we may be aiming at propositional knowledge. These aims can come apart, and perhaps even systematically come apart. A pluralism about the value of those aims doesn’t entail an “anything goes” attitude with respect to how we think about what philosophical views to put forward, defend, or endorse, however—far from it. Instead, it suggests that what counts as a virtue of a philosophical theory depends on our aims in espousing it. If philosophers have distinct aims at the meta-level, it will be hard for them to engage in joint theory evaluation at the first-order level. At least all three of theoretical virtues, evaluative judgments of philosophical views, and what attitudes we ought to have towards the views we ourselves espouse will vary according to our guiding metaphilosophical aims. (shrink)

In this paper, I defend the metalinguistic solution to the problem of mathematical omniscience for the possible-worlds account of propositions by combining it with a computational model of knowledge and belief. The metalinguistic solution states that the objects of belief and ignorance in mathematics are relations between mathematical sentences and what they express. The most pressing problem for the metalinguistic strategy is that it still ascribes too much mathematical knowledge under the standard possible-worlds model of knowledge and belief on which (...) these are closed under entailment. I first argue that Stalnaker's fragmentation strategy is insufficient to solve this problem. I then develop an alternative, computational strategy: I propose a model of mathematical knowledge and belief adapted from the algorithmic model of Halpern et al. which, when combined with the metalinguistic strategy, entails that mathematical knowledge and belief require computational abilities to access metalinguistic information, and thus aren't closed under entailment. As I explain, the computational model generalizes beyond mathematics to a version of the functionalist theory of knowledge and belief that motivates the possible-worlds account in the first place. I conclude that the metalinguistic and computational strategies yield an attractive functionalist, possible-worlds account of mathematical content, knowledge, and inquiry. (shrink)

What is curiosity? An attractive option is that it is a desire to know. This analysis has been recently challenged by what I call interrogativism, the view that inquiring attitudes such as curiosity have questions rather than propositions as contents. In this paper, I defend the desire-to-know view, and make three contributions to the debate. First, I refine the view in a way that avoids the problems of its simplest version. Second, I present a new argument for the desire-to-know view (...) that focuses on ascriptions of the form ‘S is curious to φ’, which, despite their prevalence, have been ignored in the literature. Third, I examine the central motivation for interrogativism—the argument from metacognition, according to which animals can be curious yet do not have the metacognitive capacities required by desires to know—and argue that it rests on questionable assumptions about desires and attitude ascriptions. (shrink)

Can someone who suspends judgement about a certain proposition <p> be in a relational state of disagreement with someone who believes <p> as well as with some- one who disbelieves <p>? This paper argues for an af- firmative answer. It develops an account of the notions of suspended judgement and disagreement that explains how and why the suspender is in a relational state of disagreement with both the believer and the disbeliever about the very same proposition <p>. More specifically, the (...) paper first provides a characterisation of the norma- tive profile associated with the state of suspended judge- ment in terms of the set of normative commitments that it engenders in the context of inquiry. It then provides a characterisation of the notion of disagreement in terms the incompatibility between the sets of normative com- mitments characteristic of the three states in question— belief, disbelief, and suspension. (shrink)

In the Phaedrus, Socreates sympathetically describes the ability “to cut up each kind according to its species along its natural joints, and to try not to splinter any part, as a bad butcher might do.” (265e) In contemporary philosophy, Ted Sider (2009, 2011) defends the same idea. As I shall put it, Plato and Sider’s idea is that limning structure is an epistemic goal. My aim in this paper is to articulate and defend this idea. First, I’ll articulate the notion (...) of a structural proposition (§1), and the notion of an epistemic goal (§2), where I’ll assume that epistemic goals are species of accuracy. Then (§3), I’ll argue against some proposals for understanding the idea that limning structure is an epistemic goal: limning structure is neither an aim of belief (§3.1), nor of inquiry (§3.2), nor of concept possession (§3.3). Importantly, non-structural belief is not thereby inaccurate; belief does not “aim” at being structural. Next (§4), I’ll propose a framework for understanding the idea that limning structure is an epistemic goal, and defend that idea. What is required, to defend the view that limning structure is an epistemic goal, is the notion of (what I call) theorizing – a propositional attitude that, unlike belief, does “aim” at being structural (§4.1). I’ll argue that structural truths constitute a species of “important” truths (§4.2), and that apt theorizing is a species of understanding (§4.3). Finally (§5), I’ll discuss the possibility that there is no structure. (shrink)

The theory of the organism-environment system starts with the proposition that in any functional sense organism and environment are inseparable and form only one unitary system. The organism cannot exist without the environment and the environment has descriptive properties only if it is connected to the organism. Although for practical purposes we do separate organism and environment, this common-sense starting point leads in psychological theory to problems which cannot be solved. Therefore, separation of organism and environment cannot be the basis (...) of any scientific explanation of human behavior. The theory leads to a reinterpretation of basic problems in many fields of inquiry and makes possible the definition of mental phenomena without their reduction either to neural or biological activity or to separate mental functions. According to the theory, mental activity is activity of the whole organism-environment system, and the traditional psychological concepts describe only different aspects of organisation of this system. Therefore, mental activity cannot be separated from the nervous system, but the nervous system is only one part of the organism­environment system. This problem will be dealt with in detail in the second part of the article. (shrink)

This paper offers a defense of veritic epistemic consequentialism, addressing its principal critiques. I argue that the core of epistemological value lies in its conduciveness to truth, rendering true beliefs intrinsically valuable. In response to the criticism that this approach may sacrifice individual truths for a greater aggregate and undervalues autonomous inquiry, I emphasize the well-connectedness of beliefs. Each belief's content is a proposition. Propositions are classified as first-order, second-order, third-order, etc., depending on what they are about. Higher-order propositions are (...) about lower-order ones. I assume that if an epistemic agent believes in p, then she tends to believe in higher-order propositions that are about p. These higher-order propositions are more structural beliefs of the agent. If p is false, then falsity may spread over the whole network through higher-order propositions about p. Thus, sacrificing a single belief may bring more damage to the network, which is not acceptable for veritic epistemic consequentialism. Regarding the issue of autonomy, epistemic acts like reflection, inference, etc. can be represented as higher-order propositions. Therefore, veritic epistemic consequentialism values them. (shrink)

How fine-grained are the contents of our beliefs and other cognitive attitudes? Are the contents of our beliefs individuated solely in terms of the objects, properties, and relations that figure in their truth conditions, or rather in terms of our concepts, or modes of presentation of those objects, properties, and relations? So-called Millians famously maintain the former whereas their Fregean rivals hold the latter. Though much ink was spilled on the question of grain, relatively little was ever achieved by way (...) of consensus. We think the lack of consensus itself cries out for explanation. In this paper, we sketch a pluralist resolution (or, better, a dissolution) of the debate that flows from some extremely minimal commitments regarding the metaphysics of propositions and the attitudinal relations we bear to them. In doing so, we focus on the Act-type conception of propositions of Hanks (2015) and Soames (2010, 2016) and our own (2019) favored deflationary account, Minimalism. (shrink)

The concern of deductive logic is generally viewed as the systematic recognition of logical principles, i.e., of logical truths. This paper presents and analyzes different instantiations of the three main interpretations of logical principles, viz. as ontological principles, as empirical hypotheses, and as true propositions in virtue of meanings. I argue in this paper that logical principles are true propositions in virtue of the meanings of the logical terms within a certain linguistic framework. Since these principles also regulate and control (...) the process of deduction in inquiry, i.e., they are prescriptive for the use of language and thought in inquiry, I argue that logic may, and should, be seen as an instrument or as a way of proceeding (modus procedendi) in inquiry. (shrink)

This article explores the issues of phenomenon and genesis in Early Buddhist thought through a comparative analysis with the Eleatic tradition, aiming to enrich the understanding and dialogue between these philosophical and religious traditions. By examining the comparability of Buddhist thought and Parmenidean philosophy, the study challenges the notion that these traditions are fundamentally alien to each other. The focus is on the concept of genesis, not as creation from nothingness—rejected by both the Buddha and Parmenides—but as the manifestation of (...) the world to the human observer. The article argues that the world reveals itself in particular forms and appearances, which are intimately linked to the phenomenon and its perception by humans. This process is not solely a domain of rigid logical propositions but can be expressed through mythological and religious narratives. The study posits that the poetic expressiveness found in archaic philosophies of both India and Greece provides a valid medium for engaging in philosophical discourse. By adopting this comparative and dialogical perspective, the article aims to generate new philosophical insights and inspire future philosophical inquiry. The reflection on phenomenon and genesis, framed through this comparative lens, highlights the nuanced ways in which different traditions address the nature of reality and human perception, ultimately advocating for a broader, more inclusive understanding of philosophy that transcends conventional boundaries. (shrink)

According to Conciliationism, rationality calls for a removal of dissenting opinions – in the end, the disagreement should lead to skepticism toward the disputed proposition for all the involved parties. However, psychological data regarding group inquiry indicates that groups with dissenting members are more successful in their inquiry with respect to the disputed propositions. So, according to the psychological data, rationality calls for preserving dissent – disagreement should be embraced as a great tool for getting at true beliefs. In this (...) paper I analyze this apparent conflict. (shrink)

[Winner of the American Philosophical Association’s 2024 Journal of Value Inquiry Prize.] My thesis is that certain non-verbal paintings such as Picasso’s GUERNICA make (simple) arguments. If this is correct and the arguments are reasonably good, it would indicate one way that non-literary art can be cognitively valuable, since argument can provide the justification needed for knowledge or understanding. The focus is on painting, but my findings seem applicable to comparable visual art forms (a sculpture is also considered). My approach (...) largely consists of identifying pertinent features of viable literary cognitivism and then showing how they or close analogues can be applied to non-verbal painting. The two main features are the requirements, first, that the relevant knowledge is provided significantly in virtue of the distinctive essential feature of literary fictions, i.e., their fictionality, and second, that the knowledge stems primarily from the content of the work, not from what the auditor brings to the work. Some ways that literary fiction has been taken to be argumentative are explained, and striking similarities are found between argumentative literary fiction and argumentative painting. Potential objections are addressed, and I examine a proposed way to express, in a schematized format, both the power of an argumentative painting and its relatively simple associated propositional content. (shrink)

Sellars was an inferentialist about meaning. He thus effectively accorded modality a categorial function, maintaining that any meaningful assertion involves implicit commitment to rules of material inference, which modal propositions explicitly endorse. But Sellars was also a modal anti-realist, construing modality as “entirely immanent to thought” (LRB §40), not present in the world an sich. These two commitments, Klemick argues, render it impossible in principle for us to describe the world an sich adequately, undermining Sellars’ scientific realism, on which, at (...) the “ideal outcome of scientific inquiry,” “the gulf between appearances and things-in-themselves [... can] be bridged” (SM II §51). Sellars’ attempt to ground his modal anti-realism in an ideal “pure description of the world” (CDCM §79) cannot succeed, given modality’s categorial function. Further, his subsequent deployment of picturing as a naturalistic criterion of ontological commitment founders on an analogue of the Kantian problem of noumenal affection: the descriptions of picturing relations that render this proposal intelligible are emptied of content by the denial that the in-itself is modally characterized. Sellars’ worthy aim of preserving the world’s independence of our conceptual activity, Klemick concludes, doesn’t require denying that it exhibits modal relations isomorphic to the semantic relations within our conceptual scheme. (shrink)

In order to select an area of ignorance and make it the target of inquiry, one first has to be aware of one’s own lack of knowledge in this particular area. In this paper, I explore this connection between ignorance and the aims of research. I emphasize the importance of distinguishing between all the things we don’t know—our total ignorance—and the totality of what we know we don’t know—our conscious ignorance. I argue that while our total ignorance may be conceptualized (...) as a set of true propositions that are unknown to us, our conscious ignorance is best understood as a set of questions. This enables us to identify the preconditions that have to be satisfied in order for a question to be an element of our conscious ignorance. The analysis reveals that conscious ignorance is a complex ability that can often require quite an amount of knowledge and competence. Items of conscious ignorance can, of course, differ in many important respects. To stress one of these differences, I introduce the notion of deep ignorance to signify cases in which we not only fail to know the correct answer to a given question, but cannot even name any plausible candidate answers. I argue that items of deep ignorance should be expected to possess particular attractiveness as targets of research (provided they are not entirely intractable). Finally, I suggest that in light of my analysis, some of the strategies of ignorance-mongers that have engaged agnotologists’ interests over the past years can be understood as attempts to undermine not only knowledge, but also conscious ignorance at the same time—or, in cases where that is not possible, to at least counteract the impression that the ignorance in question is deep. (shrink)

Semantics traditionally focuses on linguistic meaning. In recent years, the Super Linguistics movement has tried to broaden the scope of inquiry in various directions, including an extension of semantics to talk about the meaning of pictures. There are close similarities between the interpretation of language and of pictures. Most fundamentally, pictures, like utterances, can be either true or false of a given state of affairs, and hence both express propositions (Zimmermann, 2016; Greenberg, 2013; Abusch, 2015). Moreover, sequences of pictures, like (...) sequences of utterances, can be used to tell stories. Wordless picture books, comics, and film are cases in point. In this paper I pick up the project of providing a dynamic semantic account of pictorial story-telling, started by Abusch (2012) and continued by Abusch & Rooth (2017); Maier & Bimpikou (2019); Fernando (2020). More specifically, I propose here a semantics of speech and thought bubbles by adding event reference to PicDRT. To get there I first review the projection-based semantics for pictures (section 1), noting the fundamental distinction between symbolic and iconic meaning that makes speech bubbles especially interesting (section 2). I then review the dynamic PicDRT framework for pictorial narratives (section 3), add events (section 4), and propose an account of speech bubbles as quotational event modification (section 5). I end with a brief look at other conventional, symbolic enrichments in comics (section 6). (shrink)

Kant argues in the Critique of the Teleological Power of Judgment that the first stage in resolving the problem of teleology is conceiving it correctly. He explains that the conflict between mechanism and teleology, properly conceived, is an antinomy of the power of judgment in its reflective use regarding regulative maxims, and not an antinomy of the power of judgment in its determining use regarding constitutive principles. The matter in hand does not concern objective propositions regarding the possibility of objects (...) or actual features of certain objects, namely, organisms. It is rather a methodological issue as to the appropriate way to explain the generation, development, and function of organisms. Taken in this manner as subjective maxims guiding the explanation and inquiry of organisms, the principles of mechanism and teleology need not necessarily be seen as contradictorily opposed but instead can be combined in the study of organisms. This, however, is not enough to complete the analysis of the antinomy of the teleological power of judgment. In order to show that there is an antinomy in this case, Kant has to establish that both seemingly conflicting maxims are necessary and natural to the human mind. He does it by grounding them in the ‘special character’ or peculiarity (Eigentümlichkeit) of the human understanding. However, it is not entirely clear just what exactly this peculiarity of the human understanding is. Paul Guyer argues that Kant suggests two different accounts of the peculiarity of the human intellect. According to one account, this peculiarity consists in the fact that our understanding forms general concepts and according to another, in its propensity to proceed from the parts to the whole. I will argue in this paper that Kant puts forward a single account, in which the combination of these two features demonstrate the peculiarity of the human understanding manifested in the encounter with organisms. This account explains the necessity of the regulative maxims of mechanism and teleology, and thus completes Kant’s analysis of the antinomy of the teleological power of judgment. (shrink)

I present a cumulative case for the thesis that we only know propositions that are certain for us. I argue that this thesis can easily explain the truth of eight plausible claims about knowledge: -/- (1) There is a qualitative difference between knowledge and non-knowledge. (2) Knowledge is valuable in a way that non-knowledge is not. (3) Subjects in Gettier cases do not have knowledge. (4) If S knows that P, P is part of S’s evidence. (5) If S knows (...) that P, ~P is epistemically impossible for S. (6) If S knows that P, S can rationally act as if P. (7) If S knows that P, S can rationally stop inquiring whether P. (8) If S knows each of {P1, P2, … Pn}, and competently deduces Q from these propositions, S knows that Q. -/- I then argue that the skeptical costs of this thesis are outweighed by its explanatory power. (shrink)

Assuming a radical stance on embodied cognition, according to which the information ac- quired through basic cognitive processes is not contentful (Hutto and Myin, 2013), and as- suming that perception is a source of rationally grounded knowledge (Pritchard, 2012), a pluralistic account of perceptual knowledge is developed. The paper explains: (i) how the varieties of perceptual knowledge fall under the same broader category; (ii) how they are subject to the same kind of normative constraints; (iii) why there could not be (...) a conflict between the different varieties of perceptual knowledge; and (iv) why the traditional episte- mological inquiry is inclined to overestimate the role of propositional perceptual knowledge. (shrink)

The relationship between the One and the Multiple in mystic philosophy is a profound and central theme that explores the nature of existence, the cosmos, and the divine. This theme is present in various mystical traditions, including those of the East and West, and it addresses the paradoxical coexistence of the unity and multiplicity of all things. -/- In mystic philosophy, the **One** often represents the ultimate reality, the source from which all things emanate and to which all things return. (...) It is the absolute, the infinite, and the unchanging. The One is beyond all attributes and is often associated with the divine or the absolute truth. -/- The **Multiple**, on the other hand, represents the manifest world, the diversity of forms, and the realm of change and plurality. It is the world we experience through our senses, the domain of time and space, where differentiation and individuality are apparent. -/- The relationship between the One and the Multiple is not one of opposition but of emanation and unity. The Multiple is seen as a reflection, expression, or manifestation of the One. In this sense, the diversity of the world doesn’t contradict the unity of the One but rather demonstrates it in a myriad of forms. -/- Mystics seek to understand and experience this relationship through various practices and insights. They aim to transcend the illusion of separation and duality to experience the non-dual reality where the One and the Multiple are recognized as inseparable. -/- This concept can be illustrated by the metaphor of the ocean and its waves. The ocean represents the One—vast, deep, and all-encompassing—while the waves represent the Multiple—distinct, diverse, and ever-changing. Each wave is unique, yet it is not separate from the ocean. The wave’s existence is dependent on and made of the same substance as the ocean. In the same way, each individual entity in the Multiple is an expression of the One. -/- In summary, the relationship between the One and the Multiple in mystic philosophy is a dynamic interplay that challenges the conventional understanding of separation. It invites a deeper exploration of reality, where the apparent multiplicity of the world is a direct expression of the singular, underlying essence of all that is. -/- The relationship between the One and the Multiple in the context of mathematical philosophy is a profound topic that touches upon the very foundations of existence and knowledge. It’s a theme that has been explored by philosophers and mathematicians alike, often leading to the contemplation of unity and diversity within the structures of reality. -/- In mathematics, the concept of the One can be seen as the basis of unity from which all numbers derive. It’s the identity element in multiplication, the starting point in counting, and the foundation of dimension in geometry. The One is often associated with the concept of monism in philosophy, which posits that there is a single, underlying substance or principle that constitutes reality. -/- On the other hand, the Multiple represents the infinite variety and diversity of forms and numbers that arise from the One. It’s the embodiment of plurality and the complex interplay of different entities that mathematics seeks to understand and describe. This reflects the philosophical stance of pluralism, which acknowledges the existence of multiple realities or truths. -/- The interplay between the One and the Multiple can be seen in the mathematical concept of sets. A set can be thought of as a unity, a whole composed of distinct elements. Yet, each element within the set also retains its individuality, contributing to the diversity of the set’s composition. This duality is mirrored in the philosophical exploration of the universal and the particular, where the universal represents the One, and the particulars represent the Multiple. -/- In the realm of mathematical philosophy, this relationship often leads to questions about the nature of mathematical objects: Are they discovered as part of an objective reality (the One), or are they constructed by the human mind from a multitude of experiences (the Multiple)? This debate resonates with the philosophical inquiry into the nature of truth and reality. -/- Reflecting on the One and the Multiple can also lead to a deeper understanding of the self and the cosmos. Just as the number one is integral to the existence of all other numbers, the individual self can be seen as a unique expression of the universal whole. Similarly, the cosmos can be viewed as a grand unity composed of a multiplicity of forms and phenomena. -/- Gödel’s incompleteness theorems have a fascinating connection to the philosophical concepts of the One and the Multiple. These theorems, which are pivotal in mathematical logic and philosophy of mathematics, articulate the inherent limitations of formal axiomatic systems, particularly those sufficient to express the arithmetic of natural numbers. -/- Gödel’s incompleteness theorems have a fascinating connection to the philosophical concepts of the One and the Multiple. These theorems, which are pivotal in mathematical logic and philosophy of mathematics, articulate the inherent limitations of formal axiomatic systems, particularly those sufficient to express the arithmetic of natural numbers⁴. -/- The **first incompleteness theorem** reveals that within any such consistent system, there are propositions that are true but cannot be proven within the system itself. This reflects the idea of the Multiple in that there is an abundance of mathematical truths, a multiplicity that exceeds the unifying framework of any one system. It suggests that the realm of mathematical truth is more extensive than any single formal system can fully capture. -/- The **second incompleteness theorem** extends this by showing that a system cannot prove its own consistency. This relates to the concept of the One, as it implies that a system’s complete self-understanding, its unity and coherence, is unattainable from within. It must look beyond itself, to an external vantage point, to ascertain its consistency. -/- In the context of the One and the Multiple, Gödel’s theorems imply that the One (a consistent formal system) is inherently incomplete and cannot encompass the Multiple (the totality of mathematical truths). This resonates with philosophical discussions about the relationship between unity and plurality. Just as a single philosophical system cannot capture the entirety of truth, a single formal mathematical system cannot encapsulate all mathematical truths. -/- Moreover, Gödel’s work suggests that the pursuit of a single, unified theory of everything in mathematics—a One that encompasses all Multiples—is an inherently Sisyphean task. There will always be more truths (Multiples) than can be derived from any given set of axioms (the One). -/- In essence, Gödel’s incompleteness theorems provide a formal underpinning to the philosophical notion that the One cannot exist without the Multiple, and vice versa. They are interdependent, with the One giving rise to the Multiple, and the Multiple necessitating the One for its expression and comprehension. This interplay is a dance of limitations and possibilities, where the boundaries of logic, mathematics, and philosophy blur into one another. -/- The relationship between the One and the Multiple in the context of the mathematics of zero is a deeply philosophical inquiry that bridges the abstract world of numbers with the existential questions of being and non-being. -/- Zero, in mathematics, is a symbol of absence, a representation of nothingness, yet it holds a pivotal position as a number. It is the void from which all things emerge and to which they return. In the philosophy of mathematics, zero is the paradoxical junction where the One and the Multiple converge and diverge. -/- From the perspective of the One, zero can be seen as the origin—the singular point that precedes the existence of numbers. It is the empty set, the foundation upon which the edifice of mathematics is constructed. As the identity element in addition, zero maintains the integrity of numbers, for adding zero to any number leaves it unchanged, reflecting the immutable nature of the One. -/- Conversely, when we consider the Multiple, zero represents the infinite potentiality of creation. It is the canvas upon which the integers, both positive and negative, express their multitude. Zero is the balance point, the fulcrum around which the symphony of numbers dances. It embodies the plurality of possibilities, the beginning of the number line that stretches infinitely in both directions. -/- Philosophically, zero challenges our understanding of existence. It is both something and nothing—a number that quantifies the absence of quantity. This duality echoes the philosophical struggle to comprehend how the One gives rise to the Multiple. How does the unity of being manifest the diversity of the cosmos? Zero offers a mathematical metaphor for this mystery, as it encapsulates the transition from non-existence to existence, from the undifferentiated One to the differentiated Multiple. -/- In the realm of set theory, zero corresponds to the empty set—a set with no elements. This set is unique in that it is the only set that contains nothing, yet it is the foundation upon which all other sets are built. The empty set is the mathematical embodiment of the One, and all other sets, containing multiple elements, arise from it. -/- Reflecting on zero in the context of Gödel’s incompleteness theorems, we find a resonance with the idea that the One (a consistent formal system) cannot capture the entirety of the Multiple (the totality of mathematical truths). Zero, as the foundation of numbers, similarly suggests that from the nothingness of the One, the infinite complexity of the Multiple emerges—yet it can never be fully encompassed or expressed by any single system. -/- In conclusion, the mathematics of zero offers a profound reflection on the relationship between the One and the Multiple. It serves as a bridge between the abstract and the concrete, the known and the unknowable, challenging us to ponder the origins of existence and the nature of reality itself. -/- . (shrink)