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)

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)

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)

We develop a theory of necessity operators within a version of higher-order logic that is neutral about how fine-grained reality is. The theory is axiomatized in terms of the primitive of *being a necessity*, and we show how the central notions in the philosophy of modality can be recovered from it. Various questions are formulated and settled within the framework, including questions about the ordering of necessities under strength, the existence of broadest necessities satisfying various logical conditions, and questions about (...) their logical behaviour. We also wield the framework to probe the conditions under which a logicist account of necessities is possible, in which the theory is completely reducible to logic. (shrink)

One thousand fair causally isolated coins will be independently flipped tomorrow morning and you know this fact. I argue that the probability, conditional on your knowledge, that any coin will land tails is almost 1 if that coin in fact lands tails, and almost 0 if it in fact lands heads. I also show that the coin flips are not probabilistically independent given your knowledge. These results are uncomfortable for those, like Timothy Williamson, who take these probabilities to play a (...) central role in their theorizing. (shrink)

We explore the view that Frege's puzzle is a source of straightforward counterexamples to Leibniz's law. Taking this seriously requires us to revise the classical logic of quantifiers and identity; we work out the options, in the context of higher-order logic. The logics we arrive at provide the resources for a straightforward semantics of attitude reports that is consistent with the Millian thesis that the meaning of a name is just the thing it stands for. We provide models to show (...) that some of these logics are non-degenerate. (shrink)

In explaining the notion of a fundamental property or relation, metaphysicians will often draw an analogy with languages. The fundamental properties and relations stand to reality as the primitive predicates and relations stand to a language: the smallest set of vocabulary God would need in order to write the “book of the world.” This paper attempts to make good on this metaphor. To that end, a modality is introduced that, put informally, stands to propositions as logical truth stands to sentences. (...) The resulting theory, formulated in higher-order logic, also vindicates the Humean idea that fundamental properties and relations are freely recombinable and a variant of the structural idea that propositions can be decomposed into their fundamental constituents via logical operations. Indeed, it is seen that, although these ideas are seemingly distinct, they are not independent, and fall out of a natural and general theory about the granularity of reality. (shrink)

The principle of universal instantiation plays a pivotal role both in the derivation of intensional paradoxes such as Prior’s paradox and Kaplan’s paradox and the debate between necessitism and contingentism. We outline a distinctively free logical approach to the intensional paradoxes and note how the free logical outlook allows one to distinguish two different, though allied themes in higher-order necessitism. We examine the costs of this solution and compare it with the more familiar ramificationist approaches to higher-order logic. Our assessment (...) of both approaches is largely pessimistic, and we remain reluctantly inclined to take Prior’s and Kaplan’s derivations at face value. (shrink)

In this paper I present a precise version of Stalnaker's thesis and show that it is both consistent and predicts our intuitive judgments about the probabilities of conditionals. The thesis states that someone whose total evidence is E should have the same credence in the proposition expressed by 'if A then B' in a context where E is salient as they have conditional credence in the proposition B expresses given the proposition A expresses in that context. The thesis is formalised (...) rigorously and two models are provided that demonstrate that the new thesis is indeed tenable within a standard possible world semantics based on selection functions. Unlike the Stalnaker-Lewis semantics the selection functions cannot be understood in terms of similarity. A probabilistic account of selection is defended in its place. -/- I end the paper by suggesting that this approach overcomes some of the objections often leveled at accounts of indicatives based on the notion of similarity. (shrink)

Most work on the semantic paradoxes within classical logic has centered around what this essay calls “linguistic” accounts of the paradoxes: they attribute to sentences or utterances of sentences some property that is supposed to explain their paradoxical or nonparadoxical status. “No proposition” views are paradigm examples of linguistic theories, although practically all accounts of the paradoxes subscribe to some kind of linguistic theory. This essay shows that linguistic accounts of the paradoxes endorsing classical logic are subject to a particularly (...) acute form of the revenge paradox: that there is no exhaustive classification of sentences into “good” and “bad” such that the T-schema holds when restricted to the “good” sentences unless it is also possible to prove some “bad” sentences. The foundations for an alternative classical nonlinguistic approach is outlined that is not subject to the same kinds of problems. Although revenge paradoxes of different strengths can be formulated, they are found to be indeterminate at higher orders and not inconsistent. (shrink)

This paper formulates some paradoxes of inductive knowledge. Two responses in particular are explored: According to the first sort of theory, one is able to know in advance that certain observations will not be made unless a law exists. According to the other, this sort of knowledge is not available until after the observations have been made. Certain natural assumptions, such as the idea that the observations are just as informative as each other, the idea that they are independent, and (...) that they increase your knowledge monotonically (among others) are given precise formulations. Some surprising consequences of these assumptions are drawn, and their ramifications for the two theories examined. Finally, a simple model of inductive knowledge is offered, and independently derived from other principles concerning the interaction of knowledge and counterfactuals. (shrink)

An increasing amount of twenty-first century metaphysics is couched in explicitly hyperintensional terms. A prerequisite of hyperintensional metaphysics is that reality itself be hyperintensional: at the metaphysical level, propositions, properties, operators, and other elements of the type hierarchy, must be more fine-grained than functions from possible worlds to extensions. In this paper I develop, in the setting of type theory, a general framework for reasoning about the granularity of propositions and properties. The theory takes as primitive the notion of a (...) substitution on a proposition (property, etc.) and, among other things, uses this idea to elucidate a number of theoretically important distinctions. A class of structures are identified which can be used to model a wide range of positions about the granularity of reality; certain of these structures are seen to receive a natural treatment in the category of M-sets. (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)

A number of authors have objected to the application of non-classical logic to problems in philosophy on the basis that these non-classical logics are usually characterised by a classical metatheory. In many cases the problem amounts to more than just a discrepancy; the very phenomena responsible for non-classicality occur in the field of semantics as much as they do elsewhere. The phenomena of higher order vagueness and the revenge liar are just two such examples. The aim of this paper is (...) to show that a large class of non-classical logics are strong enough to formulate their own model theory in a corresponding non-classical set theory. Specifically I show that adequate definitions of validity can be given for the propositional calculus in such a way that the metatheory proves, in the specified logic, that every theorem of the propositional fragment of that logic is validated. It is shown that in some cases it may fail to be a classical matter whether a given sentence is valid or not. One surprising conclusion for non-classical accounts of vagueness is drawn: there can be no axiomatic, and therefore precise, system which is determinately sound and complete. (shrink)

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)

In recent years there has been a revitalised interest in non-classical solutions to the semantic paradoxes. In this paper I show that a number of logics are susceptible to a strengthened version of Curry's paradox. This can be adapted to provide a proof theoretic analysis of the omega-inconsistency in Lukasiewicz's continuum valued logic, allowing us to better evaluate which logics are suitable for a naïve truth theory. On this basis I identify two natural subsystems of Lukasiewicz logic which individually, but (...) not jointly, lack the problematic feature. (shrink)

The result of combining classical quantificational logic with modal logic proves necessitism – the claim that necessarily everything is necessarily identical to something. This problem is reflected in the purely quantificational theory by theorems such as ∃x t=x; it is a theorem, for example, that something is identical to Timothy Williamson. The standard way to avoid these consequences is to weaken the theory of quantification to a certain kind of free logic. However, it has often been noted that in order (...) to specify the truth conditions of certain sentences involving constants or variables that don’t denote, one has to apparently quantify over things that are not identical to anything. In this paper I defend a contingentist, non-Meinongian metaphysics within a positive free logic. I argue that although certain names and free variables do not actually refer to anything, in each case there might have been something they actually refer to, allowing one to interpret the contingentist claims without quantifying over mere possibilia. (shrink)

Those inclined to positions in the philosophy of time that take tense seriously have typically assumed that not all regions of space-time are equal: one special region of space-time corresponds to what is presently happening. When combined with assumptions from modern physics this has the unsettling consequence that the shape of this favored region distinguishes people in certain places or people traveling at certain velocities. In this paper I shall attempt to avoid this result by developing a tensed picture of (...) reality that is nonetheless consistent with ‘hypersurface egalitarianism’—the view that all hypersurfaces are equal. (shrink)

Prior to Kripke's seminal work on the semantics of modal logic, McKinsey offered an alternative interpretation of the necessity operator, inspired by the Bolzano-Tarski notion of logical truth. According to this interpretation, `it is necessary that A' is true just in case every sentence with the same logical form as A is true. In our paper, we investigate this interpretation of the modal operator, resolving some technical questions, and relating it to the logical interpretation of modality and some views in (...) modal metaphysics. In particular, we present an hitherto unpublished solution to problems 41 and 42 from Friedman's 102 problems, which uses a different method of proof from the solution presented in the paper of Tadeusz Prucnal. (shrink)

The fact that physical laws often admit certain kinds of space-time symmetries is often thought to be problematic for substantivalism --- the view that space-time is as real as the objects it contains. The most prominent alternative, relationism, avoids these problems but at the cost of giving abstract objects (rather than space-time points) a pivotal role in the fundamental metaphysics. This incurs related problems concerning the relation of the physical to the mathematical. In this paper I will present a version (...) of substantivalism that respects Leibnizian theses about space-time symmetries, and argue that it is superior to both relationism and the more orthodox form of substantivalism. (shrink)

An increasing amount of contemporary philosophy of mathematics posits, and theorizes in terms of special kinds of mathematical modality. The goal of this paper is to bring recent work on higher-order metaphysics to bear on the investigation of these modalities. The main focus of the paper will be views that posit mathematical contingency or indeterminacy about statements that concern the `width' of the set theoretic universe, such as Cantor's continuum hypothesis. Within a higher-order framework I show that contingency about the (...) width of the set-theoretic universe refutes two orthodoxies concerning the structure of modal reality: the view that the broadest necessity has a logic of S5, and the "Leibniz biconditionals" stating that what is possible, in the broadest sense of possible, is what is true in some possible world. Nonetheless, I suggest that the underlying picture of modal set-theory is coherent and has attractions. (shrink)

According to an influential idea in the philosophy of set theory, certain mathematical concepts, such as the notion of a well-order and set, are indefinitely extensible. Following Parsons (1983), this has often been cashed out in modal terms. This paper explores instead an extensional articulation of the idea, formulated in higher-order logic, that flat-footedly formalizes some remarks of Zermelo. The resulting picture is incompatible with the idea that the entire universe can be well-ordered, but entirely consistent with the idea that (...) the sets of any set-theoretic universe can be. (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)

A satisfactory theory of linguistic communication must explain how it is that, through the interpersonal exchange of auditory, visual, and tactile stimuli, the communicative preconditions for the acquisition of testimonial knowledge regularly come to be satisfied. Without an account of knowledge-yielding communication this success condition for linguistic theorizing is left opaque, and we are left with an incomplete understanding of testimony, and communication more generally, as a source of knowledge. This paper argues that knowledge-yielding communication should be modelled on knowledge (...) itself. It is argued that knowledge-yielding communication occurs iff interlocutors coordinate on truth values in a non-lucky and non-deviant way. This account is able to do significant explanatory work: it sheds light on the nature of referential communication, and it allows us to capture, in an informative way, the sense in which interlocutors must entertain similar propositions in order to communicate successfully. (shrink)

In this paper I argue that the enkratic principle in its classic formulation may not be a requirement of rationality. The investigation of whether it is leads to some important methodological insights into the study of rationality. I also consider the possibility that we should consider rational requirements as a subset of a broader category of agential requirements.

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)

Kant holds that in order to have knowledge of an object, a subject must be able to “prove” that the object is really possible—i.e., prove that there is neither logical inconsistency nor “real repugnance” between its properties. This is (usually) easy to do with respect to empirical objects, but (usually) impossible to do with respect to particular things-in-themselves. In the first section of the paper I argue that an important predecessor of Kant’s account of our ignorance of real possibility can (...) be found in Locke’s Essay Concerning Human Understanding. In the middle sections I discuss the way in which our inability to prove the real possibility of things-in-themselves motivates Kant’s famous prohibition on certain kinds of knowledge-claims about them. In the final section I examine Hegel’s attempts to dissolve this problem of real repugnance and thereby remove an inherited obstacle to speculative knowledge of the supersensible. -/- . (shrink)

How is it that a speaker _S_ can at once make it obvious to an audience _A_ that she intends to communicate some proposition _p_, and yet at the same time retain plausible deniability with respect to this intention? The answer is that _S_ can bring it about that _A_ has a high justified credence that ‘_S_ intended _p_’ without putting _A_ in a position to know that ‘_S_ intended _p_’. In order to achieve this _S_ has to exploit a (...) sense in which communication can be lottery-like. After defending this view of deniability I argue that it compares favorably to a rival account recently developed by Dinges and Zakkou (Mind, https://doi.org/10.1093/mind/fzac056, 2023 ). (shrink)

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

This essay considers Kant’s theory of modality in light of a debate in contemporary modal metaphysics and modal logic concerning the Barcan formulas. The comparison provides a new and fruitful perspective on Kant’s complex and sometimes confusing claims about possibility and necessity. Two central Kantian principles provide the starting point for the comparison: that the possible must be grounded in the actual and that existence is not a real predicate. Both are shown to be intimately connected to the Barcan formulas, (...) and Kant’s views on what he distinguishes as three different kinds of modality are then considered in light of this connection. (shrink)

This paper argues that underlying social biases are able to affect the processes underlying linguistic interpretation. The result is a series of harms systematically inflicted on marginalised speakers. It is also argued that the role of biases and stereotypes in interpretation complicates Miranda Fricker's proposed solution to epistemic injustice.

The fallacy fallacy is either the misdiagnosis of fallacy or the supposition that the conclusion of a fallacy must be a falsehood. This paper explores the relevance of these and related errors of reasoning for the appraisal of arguments, especially within virtue theories of argumentation. In particular, the fallacy fallacy exemplifies the Owl of Minerva problem, whereby tools devised to understand a norm make possible new ways of violating the norm. Fallacies are such tools and so are vices. Hence a (...) similar problem arises with argumentative vices. Fortunately, both instances of the problem have a common remedy. (shrink)

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)

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.

David Lewis—famously—never tasted vegemite. Did he have any knowledge of what it's like to taste vegemite? Most say 'no'; I say 'yes'. I argue that knowledge of what it’s like varies along a spectrum from more exact to more approximate, and that phenomenal concepts vary along a spectrum in how precisely they characterize what it’s like to undergo their target experiences. This degreed picture contrasts with the standard all-or-nothing picture, where phenomenal concepts and phenomenal knowledge lack any such degreed structure. (...) I motivate the degreed picture by appeal to (1) limits in epistemic abilities such as recognition, imagination, and inference, and (2) the semantics of ‘knows what it’s like’ expressions. I argue that approximate phenomenal knowledge cannot be explained merely via determinable or vague phenomenal concepts. I develop a framework for systematizing approximate knowledge of phenomenal character. And I explain how my view challenges some standard assumptions about the acquisition conditions, requirements for mastery, and referential mechanisms of phenomenal concepts. (shrink)

Disagreements that resist rational resolution, often termed “deep disagreements”, have been the focus of much work in epistemology and informal logic. In this paper, I argue that they also deserve the attention of philosophers of mathematics. I link the question of whether there can be deep disagreements in mathematics to a more familiar debate over whether there can be revolutions in mathematics. I propose an affirmative answer to both questions, using the controversy over Shinichi Mochizuki’s work on the abc conjecture (...) as a potential example of both phenomena. I conclude by investigating the prospects for the resolution of mathematical deep disagreements in virtue-theoretic approaches to informal logic and mathematical practice. (shrink)

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)

I argue that assertoric content functions as a means for us to track the responsibilities undertaken by communicators, and that distinctively assertoric commitments are distinguished by being generated directly in virtue of the words the speaker uses. This raises two questions: (a) Why are speakers responsible for the content thus generated? (b) Why is it important for us to distinguish between commitments in terms of their manner of generation? I answer the first question by developing a novel responsibility based metasemantics. (...) I answer the second by reference to the conflicting pressures governing the resources we have available for appraising speech. (shrink)

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

A famous mathematical theorem says that the sum of an infinite series of numbers can depend on the order in which those numbers occur. Suppose we interpret the numbers in such a series as representing instances of some physical quantity, such as the weights of a collection of items. The mathematics seems to lead to the result that the weight of a collection of items can depend on the order in which those items are weighed. But that is very hard (...) to believe. A puzzle then arises: How do we interpret the metaphysical significance of this mathematical theorem? I first argue that prior solutions to the puzzle lead to implausible consequences. Then I develop my own solution, where the basic idea is that the weight of a collection of items is equal to the limit of the weights of its finite subcollections contained within ever-expanding regions of space. I show how my solution is intuitively plausible and philosophically motivated, how it reveals an underexplored line of metaphysical inquiry about quantities and locations, and how it elucidates some classic puzzles concerning supertasks. (shrink)

I argue that experiences can have microphenomenal structures, where the macrophenomenal properties we introspect are realized by non-introspectible microphenomenal properties. After explaining what it means to ascribe a microstructure to experience, I defend the thesis against its principal philosophical challenge, discuss how the thesis interacts with other philosophical issues about experience, and consider our prospects for investigating the microphenomenal realm.

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)

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)

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)

In this chapter I hope to persuade you that defective contexts are more ubiquitous than we typically assume. In doing, so I will draw attention to a number of pressing social and theoretical issues which arise once we start to consider defective contexts. I will proceed by pointing to a number of ways in which defective contexts can emerge without self-correcting in the manner envisioned by Stalnaker. First I will consider situations in which some, but not all interlocutors recognise that (...) the context is defective. I will then consider cases of opaquely defective contexts, and the question of what it is for a context to be “close enough” to being non-defective. (shrink)