This is the first complete English translation of Gottlob Frege's Grundgesetze der Arithmetik, with introduction and annotation. The importance of Frege's ideas within contemporary philosophy would be hard to exaggerate. He was, to all intents and purposes, the inventor of mathematical logic, and the influence exerted on modern philosophy of language and logic, and indeed on general epistemology, by the philosophical framework.
In this paper, we present the results of two surveys that investigate subjects’ judgments about what can be known or justifiably believed about lottery outcomes on the basis of statistical evidence, testimonial evidence, and “mixed” evidence, while considering possible anchoring and priming effects. We discuss these results in light of seven distinct hypotheses that capture various claims made by philosophers about lay people’s lottery judgments. We conclude by summarizing the main findings, pointing to future research, and comparing our findings to (...) recent studies by Turri and Friedman. (shrink)
The notion of risk plays a central role in economics, finance, health, psychology, law and elsewhere, and is prevalent in managing challenges and resources in day-to-day life. In recent work, Duncan Pritchard (2015, 2016) has argued against the orthodox probabilistic conception of risk on which the risk of a hypothetical scenario is determined by how probable it is, and in favour of a modal conception on which the risk of a hypothetical scenario is determined by how modally close it is. (...) In this article, we use Pritchard’s discussion as a springboard for a more wide-ranging discussion of the notion of risk. We introduce three different conceptions of risk: the standard probabilistic conception, Pritchard’s modal conception, and a normalcy conception that is new (though it has some precursors in the psychological literature on risk perception). Ultimately, we argue that the modal conception is ill-suited to the roles that a notion of risk is required to play and explore the prospects for a form of pluralism about risk, embracing both the probabilistic and the normalcy conceptions. (shrink)
A co-authored article with Roy T. Cook forthcoming in a special edition on the Caesar Problem of the journal Dialectica. We argue against the appeal to equivalence classes in resolving the Caesar Problem.
In this paper I will argue that Boghossian's explanation of how we can acquire a priori knowledge of logical principles through implicit definitions commits a transmission of warrant-failure. To this end, I will briefly outline Boghossian's account, followed by an explanation of what a transmission of warrant-failure consists in. I will also show that this charge is independent of the worry of rule-circularity which has been raised concerning the justification of logical principles and of which Boghossian is fully aware. My (...) argument comes in two steps: firstly, I will argue for the insufficiency of Boghossian's template which is meant to explain how a subject can acquire a warrant for logical principles. I will show however that this insufficiency of his template can be remedied by adopting what I call the Disquotational Step. Secondly, I will argue that incorporating this further step makes his template subject to a transmission of warrant-failure, assuming that certain rather basic and individually motivated principles hold. Thus, Boghossian's account faces a dilemma: either he adopts the Disquotational Step and subjects his account to the charge of a transmission of warrant-failure, or he drops this additional step leaving the account confronted with explaining the gap that has previously been highlighted. I will then suggest various rejoinders that Boghossian might adopt but none of which - I will argue - can resolve the dilemma. Lastly, I will raise and briefly discuss the question whether this worry generalizes to other accounts, such as Hale and Wright's that aim to explain our knowledge of logic and/or mathematics in virtue of implicit definitions. (shrink)
In 1885, Georg Cantor published his review of Gottlob Frege's Grundlagen der Arithmetik . In this essay, we provide its first English translation together with an introductory note. We also provide a translation of a note by Ernst Zermelo on Cantor's review, and a new translation of Frege's brief response to Cantor. In recent years, it has become philosophical folklore that Cantor's 1885 review of Frege's Grundlagen already contained a warning to Frege. This warning is said to concern the defectiveness (...) of Frege's notion of extension. The exact scope of such speculations varies and sometimes extends as far as crediting Cantor with an early hunch of the paradoxical nature of Frege's notion of extension. William Tait goes even further and deems Frege 'reckless' for having missed Cantor's explicit warning regarding the notion of extension. As such, Cantor's purported inkling would have predated the discovery of the Russell-Zermelo paradox by almost two decades. In our introductory essay, we discuss this alleged implicit (or even explicit) warning, separating two issues: first, whether the most natural reading of Cantor's criticism provides an indication that the notion of extension is defective; second, whether there are other ways of understanding Cantor that support such an interpretation and can serve as a precisification of Cantor's presumed warning. (shrink)
Abstractionism, which is a development of Frege's original Logicism, is a recent and much debated position in the philosophy of mathematics. This volume contains 16 original papers by leading scholars on the philosophical and mathematical aspects of Abstractionism. After an extensive editors' introduction to the topic of abstractionism, the volume is split into 4 sections. The contributions within these sections explore the semantics and meta-ontology of Abstractionism, abstractionist epistemology, the mathematics of Abstractionis, and finally, Frege's application constraint within an abstractionist (...) setting. (shrink)
Mountaineering is a dangerous activity. For many mountaineers, part of its very attraction is the risk, the thrill of danger. Yet mountaineers are often regarded as reckless or even irresponsible for risking their lives. In this paper, we offer a defence of risk-taking in mountaineering. Our discussion is organised around the fact that mountaineers and non-mountaineers often disagree about how risky mountaineering really is. We hope to cast some light on the nature of this disagreement – and to argue that (...) mountaineering may actually be worthwhilebecause ofthe risks it involves. Section 1 introduces the disagreement and, in doing so, separates out several different notions of risk. Sections 2–4 then consider some explanations of the disagreement, showing how a variety of phenomena can skew people's risk judgements. Section 5 then surveys some recent statistics, to see whether these illuminate how risky mountaineering is. In light of these considerations, however, we suggest that the disagreement is best framed not simply in terms ofhow riskymountaineering is but whether the risks it does involve arejustified. The remainder of the paper, sections 6–9, argues that risk-taking in mountaineering oftenisjustified – and, moreover, that mountaineering can itself be justified byandbecause ofthe risks it involves. (shrink)
This is the first complete English translation of Gottlob Frege's Grundgesetze der Arithmetik (1893 and 1903), with introduction and annotation. As the culmination of his ground-breaking work in the philosophy of logic and mathematics, Frege here tried to show how the fundamental laws of arithmetic could be derived from purely logical principles.
In this paper, we present a formal recipe that Frege followed in his magnum opus “Grundgesetze der Arithmetik” when formulating his definitions. This recipe is not explicitly mentioned as such by Frege, but we will offer strong reasons to believe that Frege applied it in developing the formal material of Grundgesetze. We then show that a version of Basic Law V plays a fundamental role in Frege’s recipe and, in what follows, we will explicate what exactly this role is and (...) explain how it differs from the role played by extensions in his earlier book “Die Grundlagen der Arithmetik”. Lastly, we will demonstrate that this hitherto neglected yet foundational aspect of Frege’s use of Basic Law V helps to resolve a number of important interpretative challenges in recent Frege scholarship, while also shedding light on some important differences between Frege’s logicism and recent neo-logicist approaches to the foundations of mathematics. (shrink)
Mountaineering is a dangerous activity. For many mountaineers, part of its very attraction is the risk, the thrill of danger. Yet mountaineers are often regarded as reckless or even irresponsible for risking their lives. In this paper, we offer a defence of risk-taking in mountaineering. Our discussion is organised around the fact that mountaineers and non-mountaineers often disagree about how risky mountaineering really is. We hope to cast some light on the nature of this disagreement – and to argue that (...) mountaineering may actually be worthwhile because of the risks it involves. Section 1 introduces the disagreement and, in doing so, separates out several different notions of risk. Sections 2–4 then consider some explanations of the disagreement, showing how a variety of phenomena can skew people's risk judgements. Section 5 then surveys some recent statistics, to see whether these illuminate how risky mountaineering is. In light of these considerations, however, we suggest that the disagreement is best framed not simply in terms of how risky mountaineering is but whether the risks it does involve are justified. The remainder of the paper, sections 6–9, argues that risk-taking in mountaineering often is justified – and, moreover, that mountaineering can itself be justified by and because of the risks it involves. (shrink)
This paper introduces and evaluates two contemporary approaches of neo-logicism. Our aim is to highlight the diﬀerences between these two neo-logicist programmes and clarify what each projects attempts to achieve. To this end, we ﬁrst introduce the programme of the Scottish school – as defended by Bob Hale and Crispin Wright1 which we believe to be a..
In this article, I explore a Bayesian approach to avalanche decision-making. I motivate this perspective by highlighting a version of the base-rate fallacy and show that a similar pattern applies to decision-making in avalanche-terrain. I then draw out three theoretical lessons from adopting a Bayesian approach and discuss these lessons critically. Lastly, I highlight a number of challenges for avalanche educators when incorporating the Bayesian perspective in their curriculum.
In this short letter to Ed Zalta we raise a number of issues with regards to his version of Neo-Logicism. The letter is, in parts, based on a longer manuscript entitled “What Neo-Logicism could not be” which is in preparation. A response by Ed Zalta to our letter can be found on his website: http://mally.stanford.edu/publications.html (entry C3).
This paper raises and then discusses a puzzle concerning the ontological commitments of mathematical principles. The main focus here is Hume's Principle—a statement that, embedded in second-order logic, allows for a deduction of the second-order Peano axioms. The puzzle aims to put pressure on so-called epistemic rejectionism, a position that rejects the analytic status of Hume's Principle. The upshot will be to elicit a new and very basic disagreement between epistemic rejectionism and the neo-Fregeans, defenders of the analytic status of (...) Hume's Principle, which will provide a new angle from which properly to assess and re-evaluate the current debate. (shrink)
The paper challenges a widely held interpretation of Frege's conception of logic on which the constituent clauses of basic law V have the same sense. I argue against this interpretation by first carefully looking at the development of Frege's thoughts in Grundlagen with respect to the status of abstraction principles. In doing so, I put forth a new interpretation of Grundlagen §64 and Frege's idea of ‘recarving of content’. I then argue that there is strong evidence in Grundgesetze that Frege (...) did not hold the relevant sense-identity claim regarding basic law V. (shrink)
This thesis is concerned with explaining how a subject can acquire a priori knowledge of arithmetic. Every account for arithmetical, and in general mathematical knowledge faces Benacerraf's well-known challenge, i.e. how to reconcile the truths of mathematics with what can be known by ordinary human thinkers. I suggest four requirements that jointly make up this challenge and discuss and reject four distinct solutions to it. This will motivate a broadly Fregean approach to our knowledge of arithmetic and mathematics in general. (...) Pursuing this strategy appeals to the context principle which, it is proposed, underwrites a form of Platonism and explains how reference to and object-directed thought about abstract entities is, in principle, possible. I discuss this principle and defend it against different criticisms as put forth in recent literature. Moreover, I will offer a general framework for implicit definitions by means of which - without an appeal to a faculty of intuition or purely pragmatic considerations - a priori and non-inferential knowledge of basic mathematical principles can be acquired. In the course of this discussion, I will argue against various types of opposition to this general approach. Also, I will highlight crucial shortcomings in the explanation of how implicit definitions may underwrite a priori knowledge of basic principles in broadly similar conceptions. In the final part, I will offer a general account of how non-inferential mathematical knowledge resulting from implicit definitions is best conceived which avoids these shortcomings. (shrink)
The volume is the first collection of essays that focuses on Gottlob Frege's Basic Laws of Arithmetic (1893/1903), highlighting both the technical and the philosophical richness of Frege's magnum opus. It brings together twenty-two renowned Frege scholars whose contributions discuss a wide range of topics arising from both volumes of Basic Laws of Arithmetic. The original chapters in this volume make vivid the importance and originality of Frege's masterpiece, not just for Frege scholars but for the study of the history (...) of logic, mathematics, and philosophy. (shrink)