Drawing on a passage from Ramsey's Truth and Probability, we formulate a simple, plausible constraint on evaluating the accuracy of credences: the Calibration Test. We show that any additive, continuous accuracy measure that passes the Calibration Test will be strictly proper. Strictly proper accuracy measures are known to support the touchstone results of accuracy-first epistemology, for example vindications of probabilism and conditionalization. We show that our use of Calibration is an improvement on previous such appeals by showing how it answers (...) or sidesteps problems that have been raised for previous work in this area. (shrink)

There are many reasons we might want to take the opinions of various individuals and pool them to give the opinions of the group they constitute. If all the individuals in the group have probabilistic opinions about the same propositions, there is a host of pooling functions we might deploy, such as linear or geometric pooling. However, there are also cases where different members of the group assign probabilities to different sets of propositions, which might overlap a lot, a little, (...) or not at all. There are far fewer proposals for how to proceed in these cases, and those there are have undesirable features. I begin by considering four proposals and arguing that they don't work. Then I'll describe my own proposal, which is intended to cover the situation in which we want to pool the individual opinions in order to ascribe an opinion to the group considered as an agent in its own right. (shrink)

What we value, like, endorse, want, and prefer changes over the course of our lives. Richard Pettigrew presents a theory of rational decision making for agents who recognise that their values will change over time and whose decisions will affect those future times.

In this paper, we seek a reliabilist account of justified credence. Reliabilism about justified beliefs comes in two varieties: process reliabilism (Goldman, 1979, 2008) and indicator reliabilism (Alston, 1988, 2005). Existing accounts of reliabilism about justified credence comes in the same two varieties: Jeff Dunn (2015) proposes a version of process reliabilism, while Weng Hong Tang (2016) offers a version of indicator reliabilism. As we will see, both face the same objection. If they are right about what justification is, it (...) is mysterious why we care about justification, for neither of the accounts explains how justification is connected to anything of epistemic value. We will call this the Connection Problem. I begin by describing Dunn’s process reliabilism and Tang’s indicator reliabilism. I argue that, understood correctly, they are, in fact, extensionally equivalent. That is, Dunn and Tang reach the top of the same mountain, albeit by different routes. However, I argue that both face the Connection Problem. In response, I offer my own version of reliabilism, which is both process and indicator, and I argue that it solves that problem. Furthermore, I show that it is also extensionally equivalent to Dunn’s reliabilism and Tang’s. Thus, I reach the top of the same mountain as well. (shrink)

According to certain normative theories in epistemology, rationality requires us to be logically omniscient. Yet this prescription clashes with our ordinary judgments of rationality. How should we resolve this tension? In this paper, I focus particularly on the logical omniscience requirement in Bayesian epistemology. Building on a key insight by Hacking :311–325, 1967), I develop a version of Bayesianism that permits logical ignorance. This includes: an account of the synchronic norms that govern a logically ignorant individual at any given time; (...) an account of how we reduce our logical ignorance by learning logical facts and how we should update our credences in response to such evidence; and an account of when logical ignorance is irrational and when it isn’t. At the end, I explain why the requirement of logical omniscience remains true of ideal agents with no computational, processing, or storage limitations. (shrink)

Conditionalization is one of the central norms of Bayesian epistemology. But there are a number of competing formulations, and a number of arguments that purport to establish it. In this paper, I explore which formulations of the norm are supported by which arguments. In their standard formulations, each of the arguments I consider here depends on the same assumption, which I call Deterministic Updating. I will investigate whether it is possible to amend these arguments so that they no longer depend (...) on it. As I show, whether this is possible depends on the formulation of the norm under consideration. (shrink)

Veritism says that the fundamental source of epistemic value for a doxastic state is the extent to which it represents the world correctly: that is, its fundamental epistemic value is deter...

This paper is part of a book symposium for L. A. Paul (2014) Transformative Experience (OUP).

The Dutch Book Argument for Probabilism assumes Ramsey's Thesis (RT), which purports to determine the prices an agent is rationally required to pay for a bet. Recently, a new objection to Ramsey's Thesis has emerged (Hedden 2013, Wronski & Godziszewski 2017, Wronski 2018)--I call this the Expected Utility Objection. According to this objection, it is Maximise Subjective Expected Utility (MSEU) that determines the prices an agent is required to pay for a bet, and this often disagrees with Ramsey's Thesis. I (...) suggest two responses to Hedden's objection. First, we might be permissive: agents are permitted to pay any price that is required or permitted by RT, and they are permitted to pay any price that is required or permitted by MSEU. This allows us to give a revised version of the Dutch Book Argument for Probabilism, which I call the Permissive Dutch Book Argument. Second, I suggest that even the proponent of the Expected Utility Objection should admit that RT gives the correct answer in certain very limited cases, and I show that, together with MSEU, this very restricted version of RT gives a new pragmatic argument for Probabilism, which I call the Bookless Pragmatic Argument. (shrink)

Rescorla (Erkenntnis, 2020) has recently pointed out that the standard arguments for Bayesian Conditionalization assume that whenever I become certain of something, it is true. Most people would reject this assumption. In response, Rescorla offers an improved Dutch Book argument for Bayesian Conditionalization that does not make this assumption. My purpose in this paper is two-fold. First, I want to illuminate Rescorla’s new argument by giving a very general Dutch Book argument that applies to many cases of updating beyond those (...) covered by Conditionalization, and then showing how Rescorla’s version follows as a special case of that. Second, I want to show how to generalise R. A. Briggs and Richard Pettigrew’s Accuracy Dominance argument to avoid the assumption that Rescorla has identified (Briggs and Pettigrew in Noûs, 2018). In both cases, these arguments proceed by first establishing a very general reflection principle. (shrink)

Accuracy arguments for the core tenets of Bayesian epistemology differ mainly in the conditions they place on the legitimate ways of measuring the inaccuracy of our credences. The best existing arguments rely on three conditions: Continuity, Additivity, and Strict Propriety. In this paper, I show how to strengthen the arguments based on these conditions by showing that the central mathematical theorem on which each depends goes through without assuming Additivity.

How does logic relate to rational belief? Is logic normative for belief, as some say? What, if anything, do facts about logical consequence tell us about norms of doxastic rationality? In this paper, we consider a range of putative logic-rationality bridge principles. These purport to relate facts about logical consequence to norms that govern the rationality of our beliefs and credences. To investigate these principles, we deploy a novel approach, namely, epistemic utility theory. That is, we assume that doxastic attitudes (...) have different epistemic value depending on how accurately they represent the world. We then use the principles of decision theory to determine which of the putative logic-rationality bridge principles we can derive from considerations of epistemic utility. (shrink)

We often ask for the opinion of a group of individuals. How strongly does the scientific community believe that the rate at which sea levels are rising has increased over the last 200 years? How likely does the UK Treasury think it is that there will be a recession if the country leaves the European Union? What are these group credences that such questions request? And how do they relate to the individual credences assigned by the members of the particular (...) group in question? According to the credal judgement aggregation principle, linear pooling, the credence function of a group should be a weighted average or linear pool of the credence functions of the individuals in the group. In this chapter, I give an argument for linear pooling based on considerations of accuracy. And I respond to two standard objections to the aggregation principle. (shrink)

In a series of papers over the past twenty years, and in a new book, Igor Douven has argued that Bayesians are too quick to reject versions of inference to the best explanation that cannot be accommodated within their framework. In this paper, I survey their worries and attempt to answer them using a series of pragmatic and purely epistemic arguments that I take to show that Bayes’ Rule really is the only rational way to respond to your evidence.

Longtermism is the view that the most urgent global priorities, and those to which we should devote the largest portion of our resources, are those that focus on (i) ensuring a long future for humanity, and perhaps sentient or intelligent life more generally, and (ii) improving the quality of the lives that inhabit that long future. While it is by no means the only one, the argument most commonly given for this conclusion is that these interventions have greater expected goodness (...) per unit of resource devoted to them than each of the other available interventions, including those that focus on the health and well-being of the current population. In this paper, I argue that, even if we grant the consequentialist ethics upon which this argument depends, and even if we grant one of the axiologies that are typically paired with that ethics to give the argument, we are not morally required to choose an option that maximises expected utility; indeed, we might not even be permitted to do so. Instead, I will argue, if the argument's consequentialism is correct, we should choose using a decision theory that is sensitive to risk, and allows us to give greater weight to worse-case outcomes than expected utility theory does. And, I will show, such decision theories do not always recommend longtermist interventions. Indeed, sometimes, they recommend exactly the opposite: sometimes, they recommend hastening human extinction. Many, though not all, will take this as a reductio of the consequentialism or the axiology of the argument. I remain agnostic on the conclusion we should draw. (shrink)

My aim in this paper is to identify the wrong that is done in all cases of testimonial injustice, if there is one. Miranda Fricker (2007) proposes one account of this distinctive wrong, and Gaile Pohlhaus Jr. (2014) offers another. I think neither works. Nor does an account based on giving due respect to the testifier's epistemic competence. Nor does an account based on exposing the testifier to substantial risk of harm. Rachel Fraser (2023) describes a further account, and the (...) proposal I favour is a slight amendment of this. (shrink)

Adaptive preferences give rise to puzzles in ethics, political philosophy, decision theory, and the theory of action. Like our other preferences, adaptive preferences lead us to make choices, take action, and give consent. In 'False Consciousness for Liberals', recently published in The Philosophical Review, David Enoch (2020) proposes a criterion by which to identify when these choices, actions, and acts of consent are less than fully autonomous; that is, when they suffer from what Natalie Stoljar (2014) calls an 'autonomy deficit'. (...) According to Enoch, such actions are not protected in the usual way against interference by others; there is not the same prohibition against trying to prevent someone from acting in a particular way when that action is motivated by such adaptive preferences and is an attempt to satisfy them. In this note, I raise two concerns about Enoch's criterion. (shrink)

In this paper, I describe and motivate a new species of mathematical structuralism, which I call Instrumental Nominalism about Set-Theoretic Structuralism. As the name suggests, this approach takes standard Set-Theoretic Structuralism of the sort championed by Bourbaki and removes its ontological commitments by taking an instrumental nominalist approach to that ontology of the sort described by Joseph Melia and Gideon Rosen. I argue that this avoids all of the problems that plague other versions of structuralism.

In the recent philosophical literature on inquiry, epistemologists point out that their subject has often begun at the point at which you already have your evidence and then focussed on identifying the beliefs for which that evidence provides justification. But we are not mere passive recipients of evidence. While some comes to us unbidden, we often actively collect it. This has long been recognised, but typically epistemologists have taken the norms that govern inquiry to be practical, not epistemic. The recent (...) literature challenges this assumption and uncovers a rich range of questions about the epistemic normativity of inquiry. In this paper, I approach these questions from the formal side of epistemology. Developing out of the philosophy of science, as it did, this branch of epistemology has long discussed inquiry. And, building on the insights of David Blackwell (1951) and I. J. Good (1967), it has produced a reasonably well-developed framework in which to understand norms of inquiry, both epistemic and practical. In the first half of the paper, I will present the pragmatic versions of this framework due to Blackwell and Good, and the epistemic version due to Wayne Myrvold (2012); in the second half of the paper, I put this framework to work, turning to some of the questions from the recent debate about inquiry and asking how the Blackwell-Good-Myrvold approach can help us answer them. Questions will include: Are there purely epistemic norms that govern these actions (Flores and Woodard forthcoming)? When should we initiate an inquiry, when should we continue it, when should we conclude it, and when should we reopen it? How should we understand Julia Staffel's distinction between transitional attitudes and terminal attitudes (Staffel 2021a,b)? How do epistemic norms of inquiry relate to epistemic norms of belief or credence, and can they conflict (Friedman 2020)? And how should we understand the epistemic error that occurs when someone is resistant to evidence (Simion 2023)? (shrink)

How much does rationality constrain what we should believe on the basis of our evidence? According to this book, not very much. For most people and most bodies of evidence, there is a wide range of beliefs that rationality permits them to have in response to that evidence. The argument, which takes inspiration from William James' ideas in 'The Will to Believe', proceeds from two premises. The first is a theory about the basis of epistemic rationality. It's called epistemic utility (...) theory, and it says that what it is epistemically rational for you to believe is what it would be rational for you to choose if you were given the chance to pick your beliefs and, when picking them, you were to care only about their epistemic value. So, to say which beliefs are permitted, we must say how to measure epistemic value, and which decision rule to use when picking your beliefs. The second premise is a claim about attitudes to epistemic risk, and it says that rationality permits many different such attitudes. These attitudes can show up in epistemic utility theory in two ways: in the way you measure epistemic value; and in the decision rule you use to pick beliefs. This book explores the latter. The result is permissivism about epistemic rationality: different attitudes to epistemic risk lead to different choices of prior beliefs; given most bodies of evidence, different priors lead to different posteriors; and even once we fix your attitudes to epistemic risk, if they are at all risk-inclined, there is a range of different priors and therefore different posteriors they permit. (shrink)

Epistemologists who study credences have a well-developed account of how you should change them when you learn new evidence; that is, when your body of evidence grows. What's more, they boast a diverse range of epistemic and pragmatic arguments that support that account. But they do not have a satisfactory account of when and how you should change your credences when you become aware of possibilities and propositions you have not entertained before; that is, when your awareness grows. In this (...) paper, I consider the arguments for the credal epistemologist's account of how to respond to evidence, and I ask whether they can help us generate an account of how to respond to awareness growth. The results are surprising: the arguments that all support the same norms for responding to evidence growth support a number of different norms when they are applied to awareness growth. Some of these norms seem too weak, others too strong. I ask what we should conclude from this, and argue that our credal response to awareness growth is considerably less rigorously constrained than our credal response to new evidence. (shrink)

to appear in Lambert, E. and J. Schwenkler (eds.) Transformative Experience (OUP) -/- L. A. Paul (2014, 2015) argues that the possibility of epistemically transformative experiences poses serious and novel problems for the orthodox theory of rational choice, namely, expected utility theory — I call her argument the Utility Ignorance Objection. In a pair of earlier papers, I responded to Paul’s challenge (Pettigrew 2015, 2016), and a number of other philosophers have responded in similar ways (Dougherty, et al. 2015, (...) Harman 2015) — I call our argument the Fine-Graining Response. Paul has her own reply to this response, which we might call the Authenticity Reply. But Sarah Moss has recently offered an alternative reply to the Fine-Graining Response on Paul’s behalf (Moss 2017) — we’ll call it the No Knowledge Reply. This appeals to the knowledge norm of action, together with Moss’ novel and intriguing account of probabilistic knowledge. In this paper, I consider Moss’ reply and argue that it fails. I argue first that it fails as a reply made on Paul’s behalf, since it forces us to abandon many of the features of Paul’s challenge that make it distinctive and with which Paul herself is particularly concerned. Then I argue that it fails as a reply independent of its fidelity to Paul’s intentions. (shrink)

In this paper, I ask three questions of the liberal. In each, I fill in philosophical detail around a certain sort of complaint raised in current public debates about their position. In the first, I probe the limits of the liberal's tolerance for civil disobedience; in the second, I ask how the liberal can adjudicate the most divisive moral disputes of the age; and, in the third, I suggest the liberal faces a problem when there is substantial disagreement about the (...) boundaries of the rational and the reasonable. (shrink)

Our values change. What we value, want, desire, prefer, and how much; for nearly everyone, these will be different at different times in their life. These changes can be gradual or abrupt; they can be long-lasting or short-lived; and they can be induced by forces outside yourself or they can come from within or they can have no specific catalyst at all. Such preference change raises a number of questions for our theorising about rational choice, and these have been discussed (...) at length. In §2 and §3, I’ll outline two of these questions along with some of the putative solutions that have been proposed. But preference change also raises questions for our theorising about autonomy, and these have hardly been considered at all. In §4, I’ll outline three problems for personal autonomy; and in §5, I’ll outline one problem for political autonomy. In §6, I conclude. (shrink)

Many welfarists wish to assign to each possible state of the world a numerical value that measures something like its moral goodness. How are we to determine this quantity? This paper proposes a contractualist approach: a legitimate measure of moral goodness is one that could be justified to each member of the population in question. How do we justify a measure of moral goodness to each individual? Each individual recognises the measure of moral goodness must be a compromise between the (...) different levels of well-being within the population. Some compromises are more reasonable than others; and some are better justifiable to a given member of the population than others. Each member recognises that the social chooser's measure of moral goodness is going to have to deviate from the well-being function of at least some of the members of the population. But we can nonetheless justify it to each of them if it doesn't deviate more than is necessary, and if the deviations from each member are given equal weighting in whatever process we use to determine it. This paper proposes that we begin with a measure of the distance from a proposed compromise to an individual's level of well-being, and then say that the moral goodness is the candidate compromise that minimizes the sum of distances from it to the individuals' levels of well-being. I describe a range of such measures of distance and show that some give utilitarianism, others different versions of prioritarianism. (shrink)

When is it legitimate for a government to ‘nudge’ its citizens, in the sense described by Thaler and Sunstein (2008)? In their original work on the topic, Thaler and Sunstein developed the _‘as judged by themselves’ (or AJBT) test_ to answer this question (Thaler and Sunstein 2008, p. 5). In a recent paper, Paul and Sunstein (2019) raised a concern about this test: it often seems to give the wrong answer in cases in which we are nudged to make a (...) decision that leads to what Paul calls a _personally transformative experience_, that is, one that results in our values changing (Paul 2014). In those cases, the nudgee will judge the nudge to be legitimate after it has taken place, but only because their values have changed as a result of the nudge. In this paper, I take up the challenge of finding an alternative test. I draw on my _aggregate utility account_ of how to choose in the face of what Ullmann-Margalit (2006) calls _big decisions_, that is, decisions that lead to these personally transformative experiences (Pettigrew 2019, Chapters 6 and 7). (shrink)

In formal epistemology, we use mathematical methods to explore the questions of epistemology and rational choice. What can we know? What should we believe and how strongly? How should we act based on our beliefs and values? We begin by modelling phenomena like knowledge, belief, and desire using mathematical machinery, just as a biologist might model the fluctuations of a pair of competing populations, or a physicist might model the turbulence of a fluid passing through a small aperture. Then, we (...) explore, discover, and justify the laws governing those phenomena, using the precision that mathematical machinery affords. For example, we might represent a person by the strengths of their beliefs, and we might measure these using real numbers, which we call credences. Having done this, we might ask what the norms are that govern that person when we represent them in that way. How should those credences hang together? How should the credences change in response to evidence? And how should those credences guide the person’s actions? This is the approach of the first six chapters of this handbook. In the second half, we consider different representations—the set of propositions a person believes; their ranking of propositions by their plausibility. And in each case we ask again what the norms are that govern a person so represented. Or, we might represent them as having both credences and full beliefs, and then ask how those two representations should interact with one another. This handbook is incomplete, as such ventures often are. Formal epistemology is a much wider topic than we present here. One omission, for instance, is social epistemology, where we consider not only individual believers but also the epistemic aspects of their place in a social world. Michael Caie’s entry on doxastic logic touches on one part of this topic, but there is much more. Relatedly, there is no entry on epistemic logic, nor any on knowledge more generally. There are still more gaps. These omissions should not be taken as ideological choices. This material is missing, not because it is any less valuable or interesting, but because we v failed to secure it in time. Rather than delay publication further, we chose to go ahead with what is already a substantial collection. We anticipate a further volume in the future that will cover more ground. Why an open access handbook on this topic? A number of reasons. The topics covered here are large and complex and need the space allowed by the sort of 50 page treatment that many of the authors give. We also wanted to show that, using free and open software, one can overcome a major hurdle facing open access publishing, even on topics with complex typesetting needs. With the right software, one can produce attractive, clear publications at reasonably low cost. Indeed this handbook was created on a budget of exactly £0 (≈ $0). Our thanks to PhilPapers for serving as publisher, and to the authors: we are enormously grateful for the effort they put into their entries. (shrink)

We often learn the opinions of others without hearing the evidence on which they're based. The orthodox Bayesian response is to treat the reported opinion as evidence itself and update on it by conditionalizing. But sometimes this isn't feasible. In these situations, a simpler way of combining one's existing opinion with opinions reported by others would be useful, especially if it yields the same results as conditionalization. We will show that one method---upco, also known as multiplicative pooling---is specially suited to (...) this role when the opinions you wish to pool concern hypotheses about chances. The result has interesting consequences: it addresses the problem of disagreement between experts; and it sheds light on the social argument for the uniqueness thesis. (shrink)

A decision theory is self-recommending if, when you ask it which decision theory you should use, it considers itself to be among the permissible options. I show that many alternatives to expected utility theory are not self-recommending, and I argue that this tells against them.

When is a belief justified? There are three families of arguments we typically use to support different accounts of justification: arguments from our intuitive responses to vignettes that involve the concept; arguments from the theoretical role we would like the concept to play in epistemology; and arguments from the practical, moral, and political uses to which we wish to put the concept. I focus particularly on the third sort, and specifically on arguments of this sort offered by Clayton Littlejohn in (...) Justification and the Truth-Connection and Amia Srinivasan in ‘Radical Externalism’ : 395–431, 2018) in favour of externalism. I counter Srinivasan’s argument in two ways: first, I show that the internalist’s concept of justification might figure just as easily in the sorts of structural explanation Srinivasan thinks our political goals require us to give; and I argue that the internalist’s concept is needed for a particular political task, namely, to help us build more effective defences against what I call epistemic weapons. I conclude that we should adopt an Alstonian pluralism about the concept of justification. (shrink)

Much of our information comes to us indirectly, in the form of conclusions others have drawn from evidence they gathered. When we hear these conclusions, how can we modify our own opinions so as to gain the benefit of their evidence? In this paper we study the method known as geometric pooling. We consider two arguments in its favour, raising several objections to one, and proposing an amendment to the other.

(This is for the Cambridge Handbook of Analytic Philosophy, edited by Marcus Rossberg) In this handbook entry, I survey the different ways in which formal mathematical methods have been applied to philosophical questions throughout the history of analytic philosophy. I consider: formalization in symbolic logic, with examples such as Aquinas’ third way and Anselm’s ontological argument; Bayesian confirmation theory, with examples such as the fine-tuning argument for God and the paradox of the ravens; foundations of mathematics, with examples such as (...) Hilbert’s programme and Gödel’s incompleteness theorems; social choice theory, with examples such as Condorcet’s paradox and Arrow’s theorem; ‘how possibly’ results, with examples such as Condorcet’s jury theorem and recent work on intersectionality theory; and the application of advanced mathematics in philosophy, with examples such as accuracy-first epistemology. (shrink)

In the first half of this paper, I argue that group belief ascriptions are highly ambiguous. What's more, in many cases, neither the available contextual factors nor known pragmatic considerations are sufficient to allow the audience to identify which of the many possible meanings is intended. In the second half, I argue that this ambiguity often has bad consequences when a group belief ascription is heard and taken as testimony. And indeed it has these consequences even when the ascription is (...) true on the speaker's intended interpretation, when the speaker does not intend to mislead and indeed intends to cooperatively inform, and when the audience incorporates the evidence from the testimony as they should. I conclude by arguing that these consequences should lead us to stop using such ascriptions. (shrink)

Effective altruists (EAs) seek to persuade the globally wealthy to donate a proportion of their income to do good, and specifically to donate it to those charit.

I have claimed that risk-weighted expected utility maximizers are rational, and that their preferences cannot be captured by expected utility theory. Richard Pettigrew and Rachael Briggs have recently challenged these claims. Both authors argue that only EU-maximizers are rational. In addition, Pettigrew argues that the preferences of REU-maximizers can indeed be captured by EU theory, and Briggs argues that REU-maximizers lose a valuable tool for simplifying their decision problems. I hold that their arguments do not succeed and (...) that my original claims still stand. However, their arguments do highlight some costs of REU theory. (shrink)

The proof theory of many-valued systems has not been investigated to an extent comparable to the work done on axiomatizatbility of many-valued logics. Proof theory requires appropriate formalisms, such as sequent calculus, natural deduction, and tableaux for classical (and intuitionistic) logic. One particular method for systematically obtaining calculi for all finite-valued logics was invented independently by several researchers, with slight variations in design and presentation. The main aim of this report is to develop the proof theory of finite-valued first order (...) logics in a general way, and to present some of the more important results in this area. In Systems covered are the resolution calculus, sequent calculus, tableaux, and natural deduction. This report is actually a template, from which all results can be specialized to particular logics. (shrink)

Hilbert’s program was an ambitious and wide-ranging project in the philosophy and foundations of mathematics. In order to “dispose of the foundational questions in mathematics once and for all,” Hilbert proposed a two-pronged approach in 1921: first, classical mathematics should be formalized in axiomatic systems; second, using only restricted, “finitary” means, one should give proofs of the consistency of these axiomatic systems. Although Gödel’s incompleteness theorems show that the program as originally conceived cannot be carried out, it had many partial (...) successes, and generated important advances in logical theory and metatheory, both at the time and since. The article discusses the historical background and development of Hilbert’s program, its philosophical underpinnings and consequences, and its subsequent development and influences since the 1930s. (shrink)

Logic has pride of place in mathematics and its 20th century offshoot, computer science. Modern symbolic logic was developed, in part, as a way to provide a formal framework for mathematics: Frege, Peano, Whitehead and Russell, as well as Hilbert developed systems of logic to formalize mathematics. These systems were meant to serve either as themselves foundational, or at least as formal analogs of mathematical reasoning amenable to mathematical study, e.g., in Hilbert’s consistency program. Similar efforts continue, but have been (...) expanded by the development of sophisticated methods to study the properties of such systems using proof and model theory. In parallel with this evolution of logical formalisms as tools for articulating mathematical theories (broadly speaking), much progress has been made in the quest for a mechanization of logical inference and the investigation of its theoretical limits, culminating recently in the development of new foundational frameworks for mathematics with sophisticated computer-assisted proof systems. In addition, logical formalisms developed by logicians in mathematical and philosophical contexts have proved immensely useful in describing theories and systems of interest to computer scientists, and to some degree, vice versa. Three examples of the influence of logic in computer science are automated reasoning, computer verification, and type systems for programming languages. (shrink)

Methods available for the axiomatization of arbitrary finite-valued logics can be applied to obtain sound and complete intelim rules for all truth-functional connectives of classical logic including the Sheffer stroke and Peirce’s arrow. The restriction to a single conclusion in standard systems of natural deduction requires the introduction of additional rules to make the resulting systems complete; these rules are nevertheless still simple and correspond straightforwardly to the classical absurdity rule. Omitting these rules results in systems for intuitionistic versions of (...) the connectives in question. (shrink)

Textbook on Gödel’s incompleteness theorems and computability theory, based on the Open Logic Project. Covers recursive function theory, arithmetization of syntax, the first and second incompleteness theorem, models of arithmetic, second-order logic, and the lambda calculus.

This paper analyses the phenomenology and epistemology of chatbots such as ChatGPT and Bard. The computational architecture underpinning these chatbots are large language models (LLMs), which are generative AI (Artificial Intelligence) systems trained on a massive dataset of text extracted from the Web. We conceptualise these LLMs as multifunctional computational cognitive artifacts, used for various cognitive tasks such as translating, summarizing, answering questions, information-seeking, and much more. Phenomenologically, LLMs can be experienced as a “quasi-other”; when that happens, users anthropomorphise them. (...) For most users, current LLMs are black boxes, i.e., for the most part, they lack data transparency and algorithmic transparency. They can, however, be phenomenologically and informationally transparent, in which case there is an interactional flow. Anthropomorphising and interactional flow can, in some users, create an attitude of (unwarranted) trust towards the output LLMs generate. We conclude this paper by drawing on the epistemology of trust and testimony to examine the epistemic implications of these dimensions. Whilst LLMs generally generate accurate responses, we observe two epistemic pitfalls. Ideally, users should be able to match the level of trust that they place in LLMs to the degree that LLMs are trustworthy. However, both their data and algorithmic opacity and their phenomenological and informational transparency can make it difficult for users to calibrate their trust correctly. The effects of these limitations are twofold: users may adopt unwarranted attitudes of trust towards the outputs of LLMs (which is particularly problematic when LLMs hallucinate), and the trustworthiness of LLMs may be undermined. (shrink)

Some of the most important developments of symbolic logic took place in the 1920s. Foremost among them are the distinction between syntax and semantics and the formulation of questions of completeness and decidability of logical systems. David Hilbert and his students played a very important part in these developments. Their contributions can be traced to unpublished lecture notes and other manuscripts by Hilbert and Bernays dating to the period 1917-1923. The aim of this paper is to describe these results, focussing (...) primarily on propositional logic, and to put them in their historical context. It is argued that truth-value semantics, syntactic ("Post-") and semantic completeness, decidability, and other results were first obtained by Hilbert and Bernays in 1918, and that Bernays's role in their discovery and the subsequent development of mathematical logic is much greater than has so far been acknowledged. (shrink)

Effective altruism sounds so innocuous—who could possibly be opposed to doing good more effectively? Yet it has inspired significant backlash in recent years. This paper addresses some common misconceptions and argues that the core “beneficentric” ideas of effective altruism are both excellent and widely neglected. Reasonable people may disagree on details of implementation, but all should share the basic goals or values underlying effective altruism.

Angell's logic of analytic containment AC has been shown to be characterized by a 9-valued matrix NC by Ferguson, and by a 16-valued matrix by Fine. We show that the former is the image of a surjective homomorphism from the latter, i.e., an epimorphic image. The epimorphism was found with the help of MUltlog, which also provides a tableau calculus for NC extended by quantifiers that generalize conjunction and disjunction.

In Mind and World, John McDowell argues against the view that perceptual representation is non-conceptual. The central worry is that this view cannot offer any reasonable account of how perception bears rationally upon belief. I argue that this worry, though sensible, can be met, if we are clear that perceptual representation is, though non-conceptual, still in some sense 'assertoric': Perception, like belief, represents things as being thus and so.

This paper draws on the 'Fitting Attitudes' analysis of value to argue that we should take the concept of fittingness (rather than value) as our normative primitive. I will argue that the fittingness framework enhances the clarity and expressive power of our normative theorising. Along the way, we will see how the fittingness framework illuminates our understanding of various moral theories, and why it casts doubt on the Global Consequentialist idea that acts and (say) eye colours are normatively on a (...) par. We will see why even consequentialists, in taking rightness to be in some sense determined by goodness, should not think that rightness is conceptually reducible to goodness. Finally, I will use the fittingness framework to explicate the distinction between consequentialist and deontological theories, with particular attention to the contentious case of Rule Consequentialism. (shrink)

In The Boundary Stones of Thought, Rumfitt defends classical logic against challenges from intuitionistic mathematics and vagueness, using a semantics of pre-topologies on possibilities, and a topological semantics on predicates, respectively. These semantics are suggestive but the characterizations of negation face difficulties that may undermine their usefulness in Rumfitt’s project.

There is too much that we do not know about COVID-19. The longer we take to find it out, the more lives will be lost. In this paper, we will defend a principle of risk parity: if it is permissible to expose some members of society (e.g. health workers or the economically vulnerable) to a certain level of ex ante risk in order to minimize overall harm from the virus, then it is permissible to expose fully informed volunteers to a (...) comparable level of risk in the context of promising research into the virus. We apply this principle to three examples of risky research: skipping animal trials for promising treatments, human challenge trials to speed up vaccine development, and low-dose controlled infection or “variolation.” We conclude that if volunteers, fully informed about the risks, are willing to help fight the pandemic by aiding promising research, there are strong moral reasons to gratefully accept their help. To refuse it would implicitly subject others to still graver risks. (shrink)