Traditional geometry concerns itself with planimetric and stereometric considerations, which are at the root of the division between plane and solid geometry. To raise the issue of the relation between these two areas brings with it a host of different problems that pertain to mathematical practice, epistemology, semantics, ontology, methodology, and logic. In addition, issues of psychology and pedagogy are also important here. To our knowledge there is no single contribution that studies in detail even one of the aforementioned areas.

Roughly, a proof of a theorem, is “pure” if it draws only on what is “close” or “intrinsic” to that theorem. Mathematicians employ a variety of terms to identify pure proofs, saying that a pure proof is one that avoids what is “extrinsic,” “extraneous,” “distant,” “remote,” “alien,” or “foreign” to the problem or theorem under investigation. In the background of these attributions is the view that there is a distance measure (or a variety of such measures) between mathematical statements and (...) proofs. Mathematicians have paid little attention to specifying such distance measures precisely because in practice certain methods of proof have seemed self- evidently impure by design: think for instance of analytic geometry and analytic number theory. By contrast, mathematicians have paid considerable attention to whether such impurities are a good thing or to be avoided, and some have claimed that they are valuable because generally impure proofs are simpler than pure proofs. This article is an investigation of this claim, formulated more precisely by proof- theoretic means. After assembling evidence from proof theory that may be thought to support this claim, we will argue that on the contrary this evidence does not support the claim. (shrink)

Many mathematicians have cited depth as an important value in their research. However, there is no single widely accepted account of mathematical depth. This article is an attempt to bridge this gap. The strategy is to begin with a discussion of Szemerédi's theorem, which says that each subset of the natural numbers that is sufficiently dense contains an arithmetical progression of arbitrary length. This theorem has been judged deep by many mathematicians, and so makes for a good case on which (...) to focus in analyzing mathematical depth. After introducing the theorem, four accounts of mathematical depth will be considered. (shrink)

This article will consider imagination in mathematics from a historical point of view, noting the key moments in its conception during the ancient, modern and contemporary eras.

Over the years many mathematicians have voiced a preference for proofs that stay “close” to the statements being proved, avoiding “foreign”, “extraneous”, or “remote” considerations. Such proofs have come to be known as “pure”. Purity issues have arisen repeatedly in the practice of arithmetic; a famous instance is the question of complex-analytic considerations in the proof of the prime number theorem. This article surveys several such issues, and discusses ways in which logical considerations shed light on these issues.

Gaisi Takeuti (1926–2017) is one of the most distinguished logicians in proof theory after Hilbert and Gentzen. He extensively extended Hilbert's program in the sense that he formulated Gentzen's sequent calculus, conjectured that cut-elimination holds for it (Takeuti's conjecture), and obtained several stunning results in the 1950–60s towards the solution of his conjecture. Though he has been known chiefly as a great mathematician, he wrote many papers in English and Japanese where he expressed his philosophical thoughts. In particular, he used (...) several keywords such as "active intuition" and "self-reflection" from Nishida's philosophy. In this paper, we aim to describe a general outline of our project to investigate Takeuti's philosophy of mathematics. In particular, after reviewing Takeuti's proof-theoretic results briefly, we describe some key elements in Takeuti's texts. By explaining these texts, we point out the connection between Takeuti's proof theory and Nishida's philosophy and explain the future goals of our project. (shrink)

Why do mathematics often give several proofs of the same theorem? This is the question raised in this article, introducing the notion of an epistemic ideal and discussing two such ideals, the explanatoriness and purity of proof.

Gaisi Takeuti extended Gentzen's work to higher-order case in 1950's–1960's and proved the consistency of impredicative subsystems of analysis. He has been chiefly known as a successor of Hilbert's school, but we pointed out in the previous paper that Takeuti's aimed to investigate the relationships between "minds" by carrying out his proof-theoretic project rather than proving the "reliability" of such impredicative subsystems of analysis. Moreover, as briefly explained there, his philosophical ideas can be traced back to Nishida's philosophy in Kyoto's (...) school. For the proving the consistency of such systems, it is crucial to prove the well-foundedness of ordinals called "ordinal diagrams" developed for it. Takeuti presented such arguments several times in order to show that they are admitted in his stand point. As a starting point of investigating his finitist stand point, we formulate the system of ordinal notations up to ε0 and reconstruct the well-foundedness arguments of them. (shrink)

Une preuve est pure si, en gros, elle ne réfère dans son développement qu’à ce qui est « proche » de, ou « intrinsèque » à l’énoncé à prouver. L’infinité des nombres premiers, un théorème classique de l’arithmétique, est un cas d’étude particulièrement riche pour les recherches philosophiques sur la pureté. Deux preuves différentes de ce résultat sont ici considérées, à savoir la preuve euclidienne classique et une preuve « topologique » plus récente proposée par Furstenberg. D’un point de vue (...) naïf, il semblerait que la première soit pure et la seconde impure. Des objections à cette vue naïve sont ici considérées et réfutées. Concernant la preuve euclidienne, la question relève de la logique, notamment de la définissabilité arithmétique de l’addition en termes de successeur et de divisibilité telle que démontrée par Julia Robinson, tandis qu’en ce qui concerne la preuve topologique, la question relève de la sémantique, notamment pour tout ce qui touche au problème de savoir ce qui est « inclus » dans le contenu d’énoncés particuliers.A proof is pure, roughly, if it draws only on what is « close » or « intrinsic » to the statement being proved. The infinitude of prime numbers, a classical theorem of arithmetic, is a rich case study for philosophical investigation of purity. Two different proofs of this result are considered, namely the classical Euclidean proof and a more recent « topological » proof by Furstenberg. Naively the former would seem to be pure and the latter to be impure. Objections to these naive views are considered and met. In the case of the former the issue rests on logical matters, specifically the arithmetic definability of addition in terms of successor and divisibility shown by Julia Robinson, while in the case of the latter the issue rests on semantic matters, specifically with respect to what is « contained » in the content of particular statements. (shrink)

In this paper I begin by extending two results of Solovay; the first characterizes the possible Turing degrees of models of True Arithmetic (TA), the complete first-order theory of the standard model of PA, while the second characterizes the possible Turing degrees of arbitrary completions of P. I extend these two results to characterize the possible Turing degrees of m-diagrams of models of TA and of arbitrary complete extensions of PA. I next give a construction showing that the conditions Solovay (...) identified for his characterization of degrees of models of arbitrary completions of PA cannot be dropped (I showed that these conditions cannot be simplified in the paper. (shrink)

When mathematicians think of the philosophy of mathematics, they probably think of endless debates about what numbers are and whether they exist. Since plenty of mathematical progress continues to be made without taking a stance on either of these questions, mathematicians feel confident they can work without much regard for philosophical reflections. In his sharp–toned, sprawling book, David Corfield acknowledges the irrelevance of much contemporary philosophy of mathematics to current mathematical practice, and proposes reforming the subject accordingly.

Review of Dowek, Gilles, Computation, Proof, Machine, Cambridge University Press, Cambridge, 2015. Translation of Les Métamorphoses du calcul, Le Pommier, Paris, 2007. Translation from the French by Pierre Guillot and Marion Roman.

This collection of essays explores what makes modern mathematics ‘modern’, where ‘modern mathematics’ is understood as the mathematics done in the West from roughly 1800 to 1970. This is not the trivial matter of exploring what makes recent mathematics recent. The term ‘modern’ (or ‘modernism’) is used widely in the humanities to describe the era since about 1900, exemplified by Picasso or Kandinsky in the visual arts, Rilke or Pound in poetry, or Le Corbusier or Loos in architecture (a building (...) by the latter graces the cover of this book’s dust jacket). (shrink)

We review Solomon Feferman's 1998 essay collection In The Light of Logic (Oxford University Press).

Our visual experience seems to suggest that no continuous curve can cover every point of the unit square, yet in the late nineteenth century Giuseppe Peano proved that such a curve exists. Examples like this, particularly in analysis (in the sense of the infinitesimal calculus) received much attention in the nineteenth century. They helped instigate what Hans Hahn called a “crisis of intuition”, wherein visual reasoning in mathematics came to be thought to be epistemically problematic. Hahn described this “crisis” as (...) follows: Mathematicians had for a long time made use of supposedly geometric evidence as a means of proof in much too naive and much too uncritical a way, till the unclarities and mistakes that arose as a result forced a turnabout. Geometrical intuition was now declared to be inadmissible as a means of proof... (p. 67) Avoiding geometrical evidence, Hahn continued, mathematicians aware of this crisis pursued what he called “logicization”, “when the discipline requires nothing but purely logical fundamental concepts and propositions for its development.” On this view, an epistemically ideal mathematics would minimize, or avoid altogether, appeals to visual representations. This would be a radical reformation of past practice, necessary, according to its advocates, for avoiding “unclarities and mistakes” like the one exposed by Peano. (shrink)

Is a human more conscious than an octopus? In the science of consciousness, it’s oftentimes assumed that some creatures (or mental states) are more conscious than others. But in recent years, a number of philosophers have argued that the notion of degrees of consciousness is conceptually confused. This paper (1) argues that the most prominent objections to degrees of consciousness are unsustainable, (2) examines the semantics of ‘more conscious than’ expressions, (3) develops an analysis of what it is for a (...) degreed property to count as degrees of consciousness, and (4) applies the analysis to various theories of consciousness. I argue that whether consciousness comes in degrees ultimately depends on which theory of consciousness turns out to be correct. But I also argue that most theories of consciousness entail that consciousness comes in degrees. (shrink)

Empirical work has lately confirmed what many philosophers have taken to be true: people are ‘biased toward the future’. All else being equal, we usually prefer to have positive experiences in the future, and negative experiences in the past. According to one hypothesis, the temporal metaphysics hypothesis, future-bias is explained either by our beliefs about temporal metaphysics—the temporal belief hypothesis—or alternatively by our temporal phenomenology—the temporal phenomenology hypothesis. We empirically investigate a particular version of the temporal belief hypothesis according to (...) which future-bias is explained by the belief that time robustly passes. Our results do not match the apparent predictions of this hypothesis, and so provide evidence against it. But we also find that people give more future-biased responses when asked to simulate a belief in robust passage. We take this to suggest that the phenomenology that attends simulation of that belief may be partially responsible for future-bias, and we examine the implications of these results for debates about the rationality of future-bias. (shrink)

This paper examines the idea of "objective phenomenology," or a way of understanding the phenomenal character of conscious experiences that doesn’t require one to have had the kinds of experiences under consideration. My central thesis is that structural facts about experience—facts that characterize purely how conscious experiences are structured—are objective phenomenal facts. I begin by precisifying the idea of objective phenomenology and diagnosing what makes any given phenomenal fact subjective. Then I defend the view that structural facts about experience are (...) objective. I also argue that structural facts about experience, despite being objective, nevertheless still give rise to an explanatory gap. (shrink)

Empirical evidence suggests that a majority of people believe that time robustly passes, and that many also report that it seems to them, in experience, as though time robustly passes. Non-dynamists deny that time robustly passes, and many contemporary non-dynamists—deflationists—even deny that it seems to us as though time robustly passes. Non-dynamists, then, face the dual challenge of explaining why people have such beliefs and make such reports about their experiences. Several philosophers have suggested the stable-self explanation, according to which (...) what partly explains one or both of the explananda is that people believe themselves to have a stable persisting self which is imagined either as moving through time, or as being stationary, with time moving around it. This paper empirically investigates this explanation. We found, however, no evidential support for the stable-self explanation, leaving the non-dynamist but also, we will argue, the dynamist, with an extant explanatory challenge. (shrink)

This is a discussion of the state of discussion on pragmatic reasons for belief.

Virtue theories of argumentation (VTA) emphasize the roles arguers play in the conduct and evaluation of arguments, and lay particular stress on arguers’ acquired dispositions of character, that is, virtues and vices. The inspiration for VTA lies in virtue epistemology and virtue ethics, the latter being a modern revival of Aristotle’s ethics. Aristotle is also, of course, the father of Western logic and argumentation. This paper asks to what degree Aristotle may thereby be claimed as a forefather by VTA.

An important advance in normativity research over the last decade is an increased understanding of the distinction, and difference, between normativity and rationality. Normativity concerns or picks out a broad set of concepts that have in common that they are, put loosely, guiding. For example, consider two commonly used normative concepts: that of a normative reason and that of ought. To have a normative reason to perform some action is for there to be something that counts in favour of performing (...) that action. Likewise with ought, when there is sufficient evidence for something, one ought to believe it (at least under normal circumstances). Not all guidance need be directed towards a specific state or a specific action. Subject to the requirements of normativity, too, are relations. It is commonly believed, for example, that we ought not to hold contradictory beliefs.1 At least some of the requirements that concern relations amongst an agent’s mental states are, or seem, distinctive. Agents who fail to satisfy these requirements are considered, at least to some degree, irrational. On many current views, being irrational is distinct in some way from not being how one ought to be; rationality is a concept distinct from normativity. Much of the literature on this topic over the last decade stems from attempts to capture the characteristic features of the requirements of rationality. Two influential views in particular did much to set the agenda. The first of these two was put forward John Broome.2 His view, the particulars of which I shall discuss in more detail below, is that the requirements of rationality could be expressed using a normative relation, which he calls a ‘normative requirement’. Normative requirements are conditionals governed by an all-thingsconsidered ought. In the case of rationality, the conditional is made up entirely of mental states.. (shrink)

A list of resources for virtue theories of argumentation. Last updated October 31st, 2023.

In this chapter, I present and explore various arguments for skepticism that are related to memory. My focus will be on the aspects of the arguments that are unique to memory, which are not shared, for example, by the more often explored skeptical arguments related to perception. -/- Here are some interesting upshots. First, a particular problem for justifiably concluding that one's memory is reliable is that any reasoning in favor of this conclusion will either result in epistemically circularity or (...) not be sufficient to justify the conclusion. Second, since many beliefs stored in memory do not appear to be based on evidence, it might also appear to follow that they are not justified. Third, although many stored memory beliefs are not based on stored memory experiences in the same way that perceptual beliefs are based on perceptual experiences, skeptical scenarios loom just as problematically for memory beliefs as they do for perceptual beliefs. (shrink)

The traditional view of evidence in mathematics is that evidence is just proof and proof is just derivation. There are good reasons for thinking that this view should be rejected: it misrepresents both historical and current mathematical practice. Nonetheless, evidence, proof, and derivation are closely intertwined. This paper seeks to tease these concepts apart. It emphasizes the role of argumentation as a context shared by evidence, proofs, and derivations. The utility of argumentation theory, in general, and argumentation schemes, in particular, (...) as a methodology for the study of mathematical practice is thereby demonstrated. Argumentation schemes represent an almost untapped resource for mathematics education. Notably, they provide a consistent treatment of rigorous and non-rigorous argumentation, thereby working to exhibit the continuity of reasoning in mathematics with reasoning in other areas. Moreover, since argumentation schemes are a comparatively mature methodology, there is a substantial body of existing work to draw upon, including some increasingly sophisticated software tools. Such tools have significant potential for the analysis and evaluation of mathematical argumentation. The first four sections of the paper address the relationships of evidence to proof, proof to derivation, argument to proof, and argument to evidence, respectively. The final section directly addresses some of the educational implications of an argumentation scheme account of mathematical reasoning. (shrink)

A number of `no-proposition' approaches to the liar paradox find themselves implicitly committed to a moderate disquotational principle: the principle that if an utterance of the sentence `$P$' says anything at all, it says that $P$ (with suitable restrictions). I show that this principle alone is responsible for the revenge paradoxes that plague this view. I instead propose a view in which there are several closely related language-world relations playing the `semantic expressing' role, none of which is more central to (...) semantic theorizing than any other. I use this thesis about language and the negative result about disquotation to motivate the view that people do say things with utterances of paradoxical sentences, although they do not say the proposition you'd always expect, as articulated with a disquotational principle. (shrink)

This paper develops a theory of analog representation. We first argue that the mark of the analog is to be found in the nature of a representational system’s interpretation function, rather than in its vehicles or contents alone. We then develop the rulebound structure theory of analog representation, according to which analog systems are those that use interpretive rules to map syntactic structural features onto semantic structural features. The theory involves three degree-theoretic measures that capture three independent ways in which (...) a system can be more or less analog. We explain how our theory improves upon prior accounts of analog representation, provides plausible diagnoses for novel challenge cases, extends to hybrid systems that are partially analog and partially symbolic, and accounts for some of the advantages and disadvantages of representing analogically versus symbolically. (shrink)

Conscious experiences are characterized by mental qualities, such as those involved in seeing red, feeling pain, or smelling cinnamon. The standard framework for modeling mental qualities represents them via points in geometrical spaces, where distances between points inversely correspond to degrees of phenomenal similarity. This paper argues that the standard framework is structurally inadequate and develops a new framework that is more powerful and flexible. The core problem for the standard framework is that it cannot capture precision structure: for example, (...) consider the phenomenal contrast between seeing an object as crimson in foveal vision versus merely as red in peripheral vision. The solution I favor is to model mental qualities using regions, rather than points. I explain how this seemingly simple formal innovation not only provides a natural way of modeling precision, but also yields a variety of further theoretical fruits: it enables us to formulate novel hypotheses about the space and structures of mental qualities, formally differentiate two dimensions of phenomenal similarity, generate a quantitative model of the phenomenal sorites, and define a measure of discriminatory grain. A noteworthy consequence is that the structure of the mental qualities of conscious experiences is fundamentally different from the structure of the perceptible qualities of external objects. (shrink)

This paper argues that new light may be shed on mathematical reasoning in its non-pathological forms by careful observation of its pathologies. The first section explores the application to mathematics of recent work on fallacy theory, specifically the concept of an ‘argumentation scheme’: a characteristic pattern under which many similar inferential steps may be subsumed. Fallacies may then be understood as argumentation schemes used inappropriately. The next section demonstrates how some specific mathematical fallacies may be characterized in terms of argumentation (...) schemes. The third section considers the phenomenon of correct answers which result from incorrect methods. This turns out to pose some deep questions concerning the nature of mathematical knowledge. In particular, it is argued that a satisfactory epistemology for mathematical practice must address the role of luck. (shrink)

I want to say something about the sort of arguments that it is possible to lose, and whether losing arguments can be done well. I shall focus on losing philosophical arguments, and I will be talking about arguments in the sense of acts of arguing. This is the sort of act that one can perform on one’s own or with one other person in private. But in either of these cases it is difficult to win—or to lose. So I shall (...) concentrate on arguments with audiences. We may think of winning or losing such arguments in terms of whether the audience is convinced. Of course, this doesn’t necessarily have anything to do with who is in the right. That means that there are two sorts of loser: real losers, who lose the argument deservedly, because they are in the wrong, and mere losers, who lose the argument undeservedly, because they are in the right. Hence there must also be two sorts of winner: real winners, who win the argument deservedly, because they are in the right, and mere winners, who win the argument undeservedly, because they are in the wrong. An optimal outcome for arguments with losers would be if all the losers are real losers. (shrink)

Kant’s “primacy of the practical” doctrine says that we can form morally justified commitments regarding what exists, even in the absence of sufficient epistemic grounds. In this paper I critically examine three different varieties of Kant’s “moral proof” that can be found in the critical works. My claim is that the third variety—the “moral-psychological argument” based in the need to sustain moral hope and avoid demoralization—has some intriguing advantages over the other two. It starts with a premise that more clearly (...) coheres with Kant’s mature account of moral motivation, and it invokes plausible empirical-psychological theses to motivate a commitment to the full-blown classical deity—the result Kant clearly wanted. From the point of view of its structure, I think this third variety of moral argument also has the most by way of contemporary interest. (shrink)

This three-part chapter explores a higher-order logic we call ‘Classicism’, which extends a minimal classical higher-order logic with further axioms which guarantee that provable coextensiveness is sufficient for identity. The first part presents several different ways of axiomatizing this theory and makes the case for its naturalness. The second part discusses two kinds of extensions of Classicism: some which take the view in the direction of coarseness of grain (whose endpoint is the maximally coarse-grained view that coextensiveness is sufficient for (...) identity), and some which take the view in the direction of fineness of grain (whose endpoint is the maximally fine-grained theory containing all distinctness claims compatible with Classicism). The third part introduces some techniques for constructing models of Classicism, and uses them to prove the consistency of many of the extensions of Classicism introduced in the second part. (shrink)

Much work in MKM depends on the application of formal logic to mathematics. However, much mathematical knowledge is informal. Luckily, formal logic only represents one tradition in logic, specifically the modeling of inference in terms of logical form. Many inferences cannot be captured in this manner. The study of such inferences is still within the domain of logic, and is sometimes called informal logic. This paper explores some of the benefits informal logic may have for the management of informal mathematical (...) knowledge. (shrink)

It is difficult for prioritarians to explain the degree to which justice requires redress for misfortune in a way that avoids imposing unreasonably high costs on more advantaged individuals whilst also economising on intuitionist appeals to judgment. An appeal to hypothetical insurance may be able to solve the problems of cost and judgment more successfully, and can also be defended from critics who claim that resource egalitarianism is best understood to favour the ex post elimination of envy over individual endowments.u.

In their account of theory change in logic, Aberdein and Read distinguish 'glorious' from 'inglorious' revolutions--only the former preserves all 'the key components of a theory' [1]. A widespread view, expressed in these terms, is that empirical science characteristically exhibits inglorious revolutions but that revolutions in mathematics are at most glorious [2]. Here are three possible responses: 0. Accept that empirical science and mathematics are methodologically discontinuous; 1. Argue that mathematics can exhibit inglorious revolutions; 2. Deny that inglorious revolutions are (...) characteristic of science. Where Aberdein and Read take option 1, option 2 is preferred by Mizrahi [3]. This paper seeks to resolve this disagreement through consideration of some putative mathematical revolutions. [1] Andrew Aberdein and Stephen Read, The philosophy of alternative logics, The Development of Modern Logic (Leila Haaparanta, ed.), Oxford University Press, Oxford, 2009, pp. 613-723. [2] Donald Gillies (ed.), Revolutions in Mathematics, Oxford University Press, Oxford, 1992. [3] Moti Mizrahi, Kuhn's incommensurability thesis: What's the argument?, Social Epistemology 29 (2015), no. 4, 361-378. (shrink)

Commentators typically neglect the distinct nature and role of hope in Kant’s system, and simply lump it together with the sort of Belief that arises from the moral proof. Kant himself is not entirely innocent of the conflation. Here I argue, however, that from a conceptual as well as a textual point of view, hope should be regarded as a different kind of attitude. It is an attitude that we can rationally adopt toward some of the doctrines that are not (...) able to be proved from within the bounds of mere reason – either theoretical or practical. This does not mean that hope is unconstrained; there are rational limits, as we shall see. In fact one of my central claims here is that a crucial difference between knowledge, rational Belief, and rational hope is that they are governed by different modal constraints; section II discusses those constraints and the kind of modality involved. In section III, I return to Religion and offer what I take to be Kant’s account of the main objects of rational hope in that text – namely, “alleged outer experiences (miracles)”;a “supposed inner experience(effect of grace)”;and a future collective experience (the construction of a truly ethical society). (shrink)

Douglas Walton’s multitudinous contributions to the study of argumentation seldom, if ever, directly engage with argumentation in mathematics. Nonetheless, several of the innovations with which he is most closely associated lend themselves to improving our understanding of mathematical arguments. I concentrate on two such innovations: dialogue types (§1) and argumentation schemes (§2). I argue that both devices are much more applicable to mathematical reasoning than may be commonly supposed.

This paper considers Kant's views on how it can be rational to hope for God's assistance in becoming morally good. If I am fully responsible for making myself good and can make myself good, then my moral condition depends entirely on me. However, if my moral condition depends entirely on me, then it cannot depend on God, and it is therefore impossible for God to provide me with any assistance. But if it is impossible for God to provide me with (...) any assistance, it is irrational for me to hope for such assistance. I address this conundrum by providing an analysis of one necessary condition of rational hope: hope is rational only if the subject is not in a position to be certain that p is really impossible. I then offer several different strategies on which it might be rational to hope that God provides moral assistance, with the most radical of these strategies suggesting that, given our ignorance of the laws of the intelligible world, for all human beings know it is metaphysically possible that God perform a noumenal miracle on their moral character. -/- . (shrink)

Some philosophers think that life is worth living not merely because of the goods and the bads within it, but also because life itself is good. I explain how this idea can be formalized by associating each version of such of a view with a function from length of life to the value generated by life itself. Then I argue that every version of the view that life itself is good faces some version of the following dilemma: either (1) good (...) human lives are worse than very long lives wholly devoid of pleasure, desire-satisfaction, knowledge, or any other goods, or (2) very short lives containing nothing but suffering are worth living. Since neither result is plausible, we ought to reject the view that life itself is good. On the view I favor, any given life may be worth living because of the goods that it contains, but life itself is neutral. (shrink)

In this paper the logic of broad necessity is explored. Definitions of what it means for one modality to be broader than another are formulated, and it is proven, in the context of higher-order logic, that there is a broadest necessity, settling one of the central questions of this investigation. It is shown, moreover, that it is possible to give a reductive analysis of this necessity in extensional language. This relates more generally to a conjecture that it is not possible (...) to define intensional connectives from extensional notions. This conjecture is formulated precisely in higher-order logic, and concrete cases in which it fails are examined. The paper ends with a discussion of the logic of broad necessity. It is shown that the logic of broad necessity is a normal modal logic between S4 and Triv, and that it is consistent with a natural axiomatic system of higher-order logic that it is exactly S4. Some philosophical reasons to think that the logic of broad necessity does not include the S5 principle are given. (shrink)

I intend to bring recent work applying virtue theory to the study of argument to bear on a much older problem, that of disagreements that resist rational resolution, sometimes termed "deep disagreements". Just as some virtue epistemologists have lately shifted focus onto epistemic vices, I shall argue that a renewed focus on the vices of argument can help to illuminate deep disagreements. In particular, I address the role of arrogance, both as a factor in the diagnosis of deep disagreements and (...) as an obstacle to their mutually acceptable resolution. Arrogant arguers are likely to make any disagreements to which they are party seem deeper than they really are and arrogance impedes the strategies that we might adopt to resolve deep disagreements. As a case in point, since arrogant or otherwise vicious arguers cannot be trusted not to exploit such strategies for untoward ends, any policy for deep disagreement amelioration must require particularly close attention to the vices of argument, lest they be exploited by the unscrupulous. (shrink)

This paper argues that the theory of structured propositions is not undermined by the Russell-Myhill paradox. I develop a theory of structured propositions in which the Russell-Myhill paradox doesn't arise: the theory does not involve ramification or compromises to the underlying logic, but rather rejects common assumptions, encoded in the notation of the $\lambda$-calculus, about what properties and relations can be built. I argue that the structuralist had independent reasons to reject these underlying assumptions. The theory is given both a (...) diagrammatic representation, and a logical representation in a novel language. In the latter half of the paper I turn to some technical questions concerning the treatment of quantification, and demonstrate various equivalences between the diagrammatic and logical representations, and a fragment of the $\lambda$-calculus. (shrink)

Andrews argues for a pluralistic folk psychology that employs different kinds of practices and different kinds of cognitive tools (including personality trait attribution, stereotype activation, inductive reasoning about past behavior, and ...

Higher-order logic augments first-order logic with devices that let us generalize into grammatical positions other than that of a singular term. Some recent metaphysicians have advocated for using these devices to raise and answer questions that bear on many traditional issues in philosophy. In contrast to these 'higher-order metaphysicians', traditional metaphysics has often focused on parallel, but importantly different, questions concerning special sorts of abstract objects: propositions, properties and relations. The answers to the higher-order and the property-theoretic questions may coincide (...) sometimes but will often come apart. I argue that when they do, the higher-order questions are closer to the metaphysical action and so it would be better for these debates to proceed in higher-order terms. (shrink)

There has been little overt discussion of the experimental philosophy of logic or mathematics. So it may be tempting to assume that application of the methods of experimental philosophy to these areas is impractical or unavailing. This assumption is undercut by three trends in recent research: a renewed interest in historical antecedents of experimental philosophy in philosophical logic; a “practice turn” in the philosophies of mathematics and logic; and philosophical interest in a substantial body of work in adjacent disciplines, such (...) as the psychology of reasoning and mathematics education. This introduction offers a snapshot of each trend and addresses how they intersect with some of the standard criticisms of experimental philosophy. It also briefly summarizes the specific contribution of the other chapters of this book. (shrink)