What is mathematics about? And if it is about some sort of mathematical reality, how can we have access to it? This is the problem raised by Plato, which still today is the subject of lively philosophical disputes. This book traces the history of the problem, from its origins to its contemporary treatment. It discusses the answers given by Aristotle, Proclus and Kant, through Frege's and Russell's versions of logicism, Hilbert's formalism, Gödel's platonism, up to the the current debate on (...) Benacerraf's dilemma and the indispensability argument. Through the considerations of themes in the philosophy of language, ontology, and the philosophy of science, the book aims at offering an historically-informed introduction to the philosophy of mathematics, approached through the lenses of its most fundamental problem. (shrink)

This is the most comprehensive book ever published on philosophical methodology. A team of thirty-eight of the world's leading philosophers present original essays on various aspects of how philosophy should be and is done. The first part is devoted to broad traditions and approaches to philosophical methodology. The entries in the second part address topics in philosophical methodology, such as intuitions, conceptual analysis, and transcendental arguments. The third part of the book is devoted to essays about the interconnections between philosophy (...) and neighbouring fields, including those of mathematics, psychology, literature and film, and neuroscience. (shrink)

Intuition is sometimes derided as an abstruse or esoteric phenomenon akin to crystal-ball gazing. Such derision appears to be fuelled primarily by the suggestion, evidently endorsed by traditional rationalists such as Plato and Descartes, that intuition is a kind of direct, immediate apprehension akin to perception. This paper suggests that although the perceptual analogy has often been dismissed as encouraging a theoretically useless metaphor, a quasi-perceptualist view of intuition may enable rationalists to begin to meet the challenge of supplying a (...) theoretically satisfying treatment of their favoured epistemic source. It is argued, first, that intuitions and perceptual experiences are at a certain level of abstraction the same type of mental state, presentations, which are distinct from beliefs, hunches, inclinations, attractions, and seemings. The notion of a presentation is given a positive explication, which identifies its characteristic features, accounts for several of its substantive psychological roles, and systematically locates it in a threefold division among types of contentful states. Subsequently, it is argued that presentations, intuitive no less than sensory, are by their nature poised to play a distinctive epistemic role. Specifically, in the case of intuition, we encounter an intellectual state that is so structured as to provide justification without requiring justification in turn—something which may, thus, be thought of as ‘given’. (shrink)

Timothy Williamson offers a reductive account of modal knowledge in terms of knowledge of counterfactual conditionals. The account is developed in a broader context of defending two more general theses regarding the subject matter and methodology of philosophy. My primary focus in this paper is Williamson’s account of modal knowledge. I argue (1) that his account of modal knowledge does not support his more general theses regarding the subject matter and methodology of philosophy; (2) that the two central claims of (...) his account of knowledge of counterfactuals are rooted in unsubstantiated empirical assumptions; and (3) that his argument in support of the conclusion that modal knowledge is a special case of counterfactual knowledge rests on three errors: conflating logical reduction and epistemological reduction, a misguided appeal to cognitive economy, and incorrectly locating what needs to be explained by an account of modal knowledge. (shrink)

According to Brandom’s conceptual role semantics, to grasp a concept involves a commitment to drawing certain inferences. This is a consequence of the inferentialist thesis that the meaning of a term is given by its justification through assertibility conditions. Inferential commitments come out from a material notion of inference which underwrites human rational discourse and activity. In this paper I discuss a problem of Brandom’s semantics allegedly exposed in an argument by Paul Boghossian against Dummett’s and Brandom’s substantive conception of (...) meaning. I contend that Boghossian’s analysis is dubious because it overlooks an important difference between Dummett’s and Brandom’s positions related respectively to a monotonic and a non-monotonic view of the norm underwriting meaning. (shrink)

In Boghossian's 1997 paper, 'Analyticity' he presented an account of a prioriknowledge of basic logical principles as available by inference from knowledge of their role in determining the meaning of the logical constants by implicit definitiontogether with knowledge of the meanings so-determined that we possess through ourprivileged access to meaning. Some commentators (e.g. BonJour (1998), Glüer (2003),Jenkins (2008)) have objected that if the thesis of implicit definition on which he relieswere true, knowledge of the meaning of the constants would presuppose (...) knowledge of the very logical principles knowledge of which the account purports to explain. Aconsequence would seem to be that implicit definition is incompatible with privilegedaccess. I argue that whilst it is possible for Boghossian to defend against theseobjections the form of argument he proposes does exhibit a subtle form of questionbegging such that it exhibits a transmission of warrant-failure. (shrink)

Clark and Chalmers propose that the mind extends further than skin and skull. If they are right, then we should expect this to have some effect on our way of knowing our own mental states. If the content of my notebook can be part of my belief system, then looking at the notebook seems to be a way to get to know my own beliefs. However, it is at least not obvious whether self-ascribing a belief by looking at my notebook (...) is a case of introspection the same way that knowing my non-extended beliefs is. Traditionally this sort of introspection is thought to be privileged and special in ways that the extended introspection case seems not to be. There is nothing privileged about looking at my notebook. Anyone could do it. The aim of the paper is to find out how to understand extended introspection and whether there is something privileged and special about knowing one’s own extended beliefs. Moreover, the notebook case has close analogs using twenty-first century technology. It seems possible to know our beliefs that are extended to smartphones, wearable technology or a cloud-based data store. First, I present the case of extended introspection. I then discuss whether it should be understood as ordinary introspection or as mind-reading. Both seem to be bad fits, which finally prompts an original account for extended introspection based on epistemic rules. (shrink)

It has been known for a few years that no more than Pi-1-1 comprehension is needed for the proof of "Frege's Theorem". One can at least imagine a view that would regard Pi-1-1 comprehension axioms as logical truths but deny that status to any that are more complex—a view that would, in particular, deny that full second-order logic deserves the name. Such a view would serve the purposes of neo-logicists. It is, in fact, no part of my view that, say, (...) Delta-3-1 comprehension axioms are not logical truths. What I am going to suggest, however, is that there is a special case to be made on behalf of Pi-1-1 comprehension. Making the case involves investigating extensions of first-order logic that do not rely upon the presence of second-order quantifiers. A formal system for so-called "ancestral logic" is developed, and it is then extended to yield what I call "Arché logic". (shrink)

Though the divide between reason-based and causal-explanatory approaches in psychiatry and psychopathology is old and deeply rooted, current trends involving multi-factorial explanatory models and evidence-based approaches to interpersonal psychotherapy, show that it has already been implicitly bridged. These trends require a philosophical reconsideration of how reasons can be causes. This paper contributes to that trajectory by arguing that Donald Davidson’s classic paradigm of 1963 is still a valid option.

The role of possible worlds in philosophy is hard to overestimate. Nevertheless, their nature and existence is very controversial. This is particularly serious, since their standard applications depend on there being sufficiently many of them. The paper develops an account of possible worlds on which it is particularly easy to believe in their existence: an account of possible worlds as pleonastic entities. Pleonastic entities are entities whose existence can be validly inferred from statements that neither refer to nor quantify over (...) them as a matter of conceptual necessity. Definitions are proposed that ensure that this is the case for possible worlds. (shrink)

Wright claims that “the epistemology of good abstraction principles should be assimilated to that of basic principles of logical inference”. In this paper I follow Wright’s recommendation, but I consider a different epistemology of logic, namely anti-exceptionalism. Anti-exceptionalism’s main contention is that logic is not a priori, and that the choice between rival logics should be based on abductive criteria such as simplicity, adequacy to the data, strength, fruitfulness, and consistency. This paper’s goal is to lay down the foundations for (...) an application of the anti-exceptionalist methodology to abstraction principles. In Sect. 1 I outline my strategy. I show that anti-exceptionalism has a bearing on the so-called Bad Company problem, and explore some versions of anti-exceptionalism about abstraction. I then consider new criteria for choosing between rival abstraction principles, and, in particular, between consistent revisions of Frege’s Basic Law V. I finally consider how the abstractionist can compare between competing criteria, and argue that anti-exceptionalists should be pluralists about abstraction. (shrink)

Neo-Fregeans need their stipulation of Hume's Principle — $NxFx=NxGx \leftrightarrow \exists R (Fx \,1\hbox {-}1_R\, Gx)$ — to do two things. First, it must implicitly define the term-forming operator ‘Nx…x…’, and second it must guarantee that Hume's Principle as a whole is true. I distinguish two senses in which the neo-Fregeans might ‘stipulate’ Hume's Principle, and argue that while one sort of stipulation fixes a meaning for ‘Nx…x…’ and the other guarantees the truth of Hume's Principle, neither does both.

Truth by convention, once thought to be the foundation of a uniquely promising approach to explaining our access to the truth in nonempirical domains, is nowadays widely considered an absurdity. Its fall from grace has been due largely to the influence of an argument that can be sketched as follows: our linguistic conventions have the power to make it the case that a sentence expresses a particular proposition, but they can’t by themselves generate truth; whether a given proposition is true—and (...) so whether the sentence that expresses it is true—is a matter of what the world is like, which means it isn’t a matter of convention alone. The consensus is that this argument is decisive against truth by convention. Strikingly, though, it has rarely been formulated with much precision. Here I provide a new rendering of the argument, one that reveals its structure and makes transparent just what assumptions it requires, and then I assess conventionalists’ prospects for resisting each of those assumptions. I conclude that the consensus is mistaken: contrary to what is almost universally thought, there remains a promising way forward for the conventionalist project. Along the way, I clarify conventionalists’ commitments by thinking about what truth by convention would need to be like in order for conventionalism to do the epistemological work it’s intended to do. (shrink)

The original publication of the article contains two formatting errors, the second of which significantly inhibits readability.

According to the most popular account of the a priori, which we might call Analytic Account of the A Priori, we can explain the a priori in terms of the notion of analyticity. According to the least popular account of the a priori, the explanation of the a priori proceeds by appealing to the faculties used in the acquisition of a priori knowledge, such as the faculty of rational intuition – call this Rationalist Account of the A Priori. The main (...) aim of this paper is to challenge the analytic account of the a priori to motivate a return to rationalism. To achieve this aim, I discuss and challenge two very different analytic accounts of the a priori: a concept-based account that relies on meaning-justification links, and an understanding-based account that does not rely on such links. I argue that the former is both extensionally inadequate and explanatorily deficient, and the latter can be made to work but only by employing such a rich notion of understanding that renders it a form of rationalism in disguise. I conclude by motivating a rationalist faculty-based account of the a priori. (shrink)

Neologicists have sought to ground mathematical knowledge in abstraction. One especially obstinate problem for this account is the bad company problem. The leading neologicist strategy for resolving this problem is to attempt to sift the good abstraction principles from the bad. This response faces a dilemma: the system of ‘good’ abstraction principles either falls foul of the Scylla of inconsistency or the Charybdis of being unable to recover a modest portion of Zermelo–Fraenkel set theory with its intended generality. This article (...) argues that the bad company problem is due to the ‘static’ character of abstraction on the neologicist's account and develops a ‘dynamic’ account of abstraction that avoids both horns of the dilemma. 1 Introduction2ion Reconstructed3 From Bad to Worse4 Abstraction Reconceived5 Bad Company Made GoodA AppendixA.1SemanticsA.2LogicA.3ConsistencyA.4Strength. (shrink)

Propositions are a useful tool in philosophical theorizing, even though they are not beyond reasonable nominalistic doubts. Stephen Schiffer’s pleonasticism about propositions is a paradigm example of a realistic account that tries to alleviate such doubts by grounding truths about propositions in ontologically innocent facts. Schiffer maintains two characteristic theses about propositions: first, that they are so-called pleonastic entities whose existence is subject to what he calls something-from-nothing transformations ; and, second, that they are the referents of ‘that’-clauses that function (...) as singular terms in propositional attitude ascriptions. The paper turns the first thesis against the second: if propositions are pleonastic entities, it is argued, we should not take them to be referred to in propositional attitude ascriptions. Rather, propositional attitude ascriptions should be available as bases for propositional something-from-nothing transformations. (shrink)

In virtue of what are we justified in employing the rule of inference Modus Ponens? One tempting approach to answering this question is to claim that we are justified in employing Modus Ponens purely in virtue of facts concerning meaning or concept-possession. In this paper, we argue that such meaning-based accounts cannot be accepted as the fundamental account of our justification.

The paper argues that no second order theory is ontologically commited to anything beyond what its individual variables range over.

I argue that Neo-Fregean accounts of arithmetical language and arithmetical knowledge tacitly rely on a thesis I call [Success by Default]—the thesis that, in the absence of reasons to the contrary, we are justified in thinking that certain stipulations are successful. Since Neo-Fregeans have yet to supply an adequate defense of [Success by Default], I conclude that there is an important gap in Neo-Fregean accounts of arithmetical language and knowledge. I end the paper by offering a naturalistic remedy.

Review of *New Essays on the A Priori*, an excellent collection edited by Paul Boghossian and Christopher Peacocke. Contributors include: Tyler Burge; Quassim Cassam; Philip Kitcher; Penelope Maddy; Hartry Field; Paul Horwich; Peter Railton; Stephen Yablo; Bob Hale; Crispin Wright; Frank Jackson; Stewart Shapiro; Michael Friedman; Martin Davies; Bill Brewer; and Thomas Nagel.

Wright claims that his and Hale’s abstractionist neologicist project is primarily epistemological in aim. Its epistemological aims include establishing the possibility of a priori mathematical knowledge, and establishing the possibility of reference to abstract mathematical objects. But, as Wright acknowledges, there is a question of how neologicist epistemology applies to actual, ordinary mathematical beliefs. I take up this question, focusing on arithmetic. Following a suggestion of Hale and Wright, I consider the possibility that the neologicist account provides an idealised reconstruction (...) of how people acquire cognition of numbers or arithmetical knowledge. I conclude that it is unlikely that this account can be used to reconstruct even idealised answers to pressing questions about our actual epistemology. (shrink)

Once Hilbert asserted that the axioms of a theory `define` theprimitive concepts of its language `implicitly''. Thus whensomeone inquires about the meaning of the set-concept, thestandard response reads that axiomatic set-theory defines itimplicitly and that is the end of it. But can we explainthis assertion in a manner that meets minimum standards ofphilosophical scrutiny? Is Jané (2001) wrong when hesays that implicit definability is ``an obscure notion''''? Doesan explanation of it presuppose any particular view on meaning?Is it not a scandal (...) of the philosophy of mathematics that no answersto these questions are around? We submit affirmative answers to allquestions. We argue that a Wittgensteinian conception of meaninglooms large beneath Hilbert''s conception of implicit definability.Within the specific framework of Horwich''s recent Wittgensteiniantheory of meaning called semantic deflationism, we explain anexplicit conception of implicit definability, and then go on toargue that, indeed, set-theory, defines the set-conceptimplicitly according to this conception. We also defend Horwich''sconception against a recent objection from the Neo-Fregeans Hale and Wright (2001). Further, we employ the philosophicalresources gathered to dissolve all traditional worries about thecoherence of the set-concept, raisedby Frege, Russell and Max Black, and whichrecently have been defended vigorously by Hallett (1984) in hismagisterial monograph Cantorian set-theory and limitationof size. Until this day, scandalously, these worries havebeen ignored too by philosophers of mathematics. (shrink)

According to the species of neo-logicism advanced by Hale and Wright, mathematical knowledge is essentially logical knowledge. Their view is found to be best understood as a set of related though independent theses: (1) neo-fregeanism-a general conception of the relation between language and reality; (2) the method of abstraction-a particular method for introducing concepts into language; (3) the scope of logic-second-order logic is logic. The criticisms of Boolos, Dummett, Field and Quine (amongst others) of these theses are explicated and assessed. (...) The issues discussed include reductionism, rejectionism, the Julius Caesar problem, the Bad Company objections, and the charge that second-order logic is set theory in disguise. (shrink)

Abstraction principles provide implicit definitions of mathematical objects. In this paper, an abstraction principle defining categories is proposed. It is unsatisfiable and inconsistent in the expected ways. Two restricted versions of the principle which are consistent are presented.

Frege's concern in GGI §10 is neither with the epistemological issue of how we come to know about value-ranges, nor with the semantic-metaphysical issue of whether we have said enough about such objects in order to ensure that any kind of reference to them is possible. The problem which occupies Frege in GGI §10 is the general problem according to which we ‘cannot yet decide’, for any arbitrary function, what value ‘’ has if ‘ℵ’ is a canonical value-range name. This (...) is a problem with the ‘reference’ of value-range names, but only in the weak sense that, if we do not exercise care, value-range terms might become ‘bedeutungslos’ for purely formal reasons. Frege addresses the general problem only for the primitive function- and object-names he has already introduced into his concept-script. I argue that this methodology was perfectly intentional: his intention for GG in general, on display in GGI §10, is to check, for each primitive function- and object-name, as it is introduced into concept-script, whether it interacts with the other primitive names which have already been introduced in such a way that these atomic combinations of primitive names do not become bedeutungslos. If there is a risk of producing a bedeutungslos combination, Frege will make an arbitrary stipulation to ensure that logical hygiene is maintained. I argue that this interpretation does not violate some of the other principal commitments of GG. (shrink)

The paper argues that the theory of Implicit Definition cannot give an account of knowledge of logical principles. According to this theory, the meanings of certain expressions are determined such that they make certain principles containing them true; this is supposed to explain our knowledge of the principles as derived from our knowledge of what the expressions mean. The paper argues that this explanation succeeds only if Implicit Definition can account for our understanding of the logical constants, and that fully (...) understanding a logical constant in turn requires the ability to apply it correctly in particular cases. It is shown, however, that Implicit Definition cannot account for this ability, even if it draws on introduction rules for the logical constants. In particular, Implicit Definition cannot account for our ability to apply negation in particular cases. Owing to constraints relating to the unique characterisation of logical constants, invoking the notion of rejection does not remedy the situation. Given its failure to explain knowledge of logic, the prospects of Implicit Definition to explain other kinds of a priori knowledge are even worse. (shrink)

According to the sortal conception of the universe of individuals every individual falls under a highest sortal, or category. It is argued here that on this conception the identity relation is defined between individuals a and b if and only if a and b fall under a common category. Identity must therefore be regarded as a relation of the form \, with three arguments x, y, and Z, where Z ranges over categories, and where the range of x and y (...) depends on the value of Z. An identity relation of this kind can be made good sense of in Martin-Löf’s type theory. But identity so construed requires a reformulation of Hume’s Principle that makes this principle unfit for explaining the sortal concept of cardinal number. The Neo-Logicist can therefore not appeal to the sortal conception in tackling the Julius Caesar problem, as proposed by Hale and Wright. (shrink)

Is it possible to characterize the sortal essence of Fs for a sortal concept F solely in terms of a criterion of identity C for F? That is, can the question ‘What sort of thing are Fs?’ be answered by saying that Fs are essentially those things whose identity can be assessed in terms of C? This paper presents a case study supporting a negative answer to these questions by critically examining the neo-Fregean suggestion that cardinal numbers can be fully (...) characterized as those things whose identity can be assessed in terms of one-one correspondence between concepts. (shrink)

The neo-Fregeans have argued that definition by abstraction allows us to introduce abstract concepts such as direction and number in terms of equivalence relations such as parallelism between lines and one-one correspondence between concepts. This paper argues that definition by abstraction suffers from the fact that an equivalence relation may not be sufficient to determine a unique concept. Frege’s original verdict against definition by abstraction is thus reinstated.

The goal of the research programme I describe in this article is a realist epistemology for arithmetic which respects arithmetic's special epistemic status (the status usually described as a prioricity) yet accommodates naturalistic concerns by remaining fundamentally empiricist. I argue that the central claims which would allow us to develop such an epistemology are (i) that arithmetical truths are known through an examination of our arithmetical concepts; (ii) that (at least our basic) arithmetical concepts are accurate mental representations of elements (...) of the arithmetical structure of the independent world; (iii) that (ii) obtains in virtue of the normal functioning of our sensory apparatus. The first of these claims protects arithmetic's special epistemic status relative, for example, to the laws of physics, the second preserves the independence of arithmetical truth, and the third ensures that we remain empiricists. Preliminaries Justifying and grounding concepts Cameras and filters An epistemology for arithmetic Concluding remarks. (shrink)

According to the so-called metaphysical conception of analyticity, analytic truths are true in virtue of meaning (or content) alone and independently of (extralinguistic) facts. Quine and Boghossian have tried to present a conclusive argument against the metaphysical conception of analyticity. In effect, they tried to show that the metaphysical conception inevitably leads into a highly implausible view about the truthmakers of analytic truths. We would like to show that their argument fails, since it relies on an ambiguity of the notion (...) of 'independence of (extralinguistic) facts'. If one distinguishes between variation independence and existence independence, the unwelcome view about the truthmakers of analytic truths no longer follows. Thus, there is at best a challenge, but no conclusive argument. The door to the metaphysical conception of analyticity is still open. (shrink)

In “Double Vision Two Questions about the Neo-Fregean Programme”, John MacFarlane’s raises two main questions: (1) Why is it so important to neo-Fregeans to treat expressions of the form ‘the number of Fs’ as a species of singular term? What would be lost, if anything, if they were analysed instead as a type of quantifier-phrase, as on Russell’s Theory of Definite Descriptions? and (2) Granting—at least for the sake of argument—that Hume’s Principle may be used as a means of implicitly (...) defining the number operator, what advantage, if any, does adopting this course possess over a direct stipulation of the Dedekind-Peano axioms? This paper attempts to answer them. In response to the first, we spell out the links between the recognition of numerical terms as vehicles of singular reference and the conception of numbers as possible objects of singular, or object-directed, thought, and the role of the acknowledgement of numbers as objects in the neo-Fregean attempt to justify the basic laws of arithmetic. In response to the second, we argue that the crucial issue concerns the capacity of either stipulation—of Hume’s Principle, or of the Dedekind-Peano axioms—to found knowledge of the principles involved, and that in this regard there are crucial differences which explain why the former stipulation can, but the latter cannot, play the required foundational role. (shrink)

We may define words or concepts, and we may also, as Aristotle and others have thought, define the things for which words stand and of which concepts are concepts. Definitions of words or concepts may be explicit or implicit, and may seek to report preexisting synonymies, as Quine put it, but they may instead be wholly or partly stipulative. Definition by abstraction, of which Hume’s principle is a much discussed example, seek to define a term-forming operator, such as the number (...) operator, by fixing the truth-conditions of identity-statements featuring terms formed by means of that operator. Such definitions are a species of implicit definition. They are typically at least partly stipulative. Definitions of things, or real definitions, are, by contrast, typically conceived as true or false statements about the nature or essence of their definienda, and so not stipulative. There thus appears to be an obvious and head-on clash between taking Hume’s principle as an implicit and at least partly stipultative definition of the number operator and taking it as a real definition, stating the nature or essence of cardinal numbers. This paper argues that this apparent tension can be resolved, and that resolving it sheds light on part of the epistemology or essence and necessity, showing how some of our knowledge of essence and necessity can be a priori. (shrink)

The paper surveys different notions of implicit definition. In particular, we offer an examination of a kind of definition commonly used in formal axiomatics, which in general terms is understood as providing a definition of the primitive terminology of an axiomatic theory. We argue that such “structural definitions” can be semantically understood in two different ways, namely as specifications of the meaning of the primitive terms of a theory and as definitions of higher-order mathematical concepts or structures. We analyze these (...) two conceptions of structural definition both in the history of modern axiomatics and in contemporary philosophical debates. Based on that, we give a systematic assessment of the underlying semantics of these two ways of understanding the definiens of such definitions, by considering alternative model-theoretic and inferential accounts of meaning. (shrink)

Neo-Fregeanism in the philosophy of mathematics consists of two main parts: the logicist thesis, that mathematics (or at least branches thereof, like arithmetic) all but reduce to logic, and the platonist thesis, that there are abstract, mathematical objects. I will here focus on the ontological thesis, platonism. Neo-Fregeanism has been widely discussed in recent years. Mostly the discussion has focused on issues specific to mathematics. I will here single out for special attention the view on ontology which underlies the neo-Fregeans’ (...) claims about mathematical objects, and discuss this view in a broader setting. (shrink)

A central element in neo-Fregean philosophy of mathematics is the focus on abstraction principles, and the use of abstraction principles to ground various areas of mathematics. But as is well known, not all abstraction principles are in good standing. Various proposals for singling out the acceptable abstraction principles have been presented. Here I investigate what philosophical underpinnings can be provided for these proposals; specifically, underpinnings that fit the neo-Fregean's general outlook. Among the philosophical ideas I consider are: general views on (...) a priori justification; the idea of abstraction as reconceptualization, the idea that truth is prior to reference in the sense associated with Frege's context principle; and various broadly relativistic views. The conclusions are by and large negative. (shrink)

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)

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)

Alberto Coffa used the phrase "the Copernican turn in semantics" to denote a revolutionary transformation of philosophical views about the connection between the meanings of words and the acceptability of sentences and arguments containing those words. According to the new conception resulting from the Copernican turn, here called "the Copernican view", rules of use are constitutive of the meanings of words. This view has been linked with two doctrines: (A) the instances of meaning-constitutive rules are analytically and a priori true (...) or valid; (B) to grasp a meaning is to accept its rules. The pros and cons of different versions of the Copernican view, ascribable to Wittgenstein, Carnap, Gentzen, Dummett, Prawitz, Boghossian and other authors, will be weighed. A new version will be proposed, which implies neither (A) nor (B). (shrink)

A number of formal constraints on acceptable abstraction principles have been proposed, including conservativeness and irenicity. Hume’s Principle, of course, satisfies these constraints. Here, variants of Hume’s Principle that allow us to count concepts instead of objects are examined. It is argued that, prima facie, these principles ought to be no more problematic than HP itself. But, as is shown here, these principles only enjoy the formal properties that have been suggested as indicative of acceptability if certain constraints on the (...) size of the continuum hold. As a result, whether or not these higher-order versions of Hume’s Principle are acceptable seems to be independent of standard (ZFC) set theory. This places the abstractionist in an uncomfortable dilemma: Either there is some inherent difference between counting objects and counting concepts, or new criteria for acceptability will need to be found. It is argued that neither horn looks promising. (shrink)

According to the most popular non-skeptical views about intuition, intuitions justify beliefs because they are based on understanding. More precisely: if intuiting that p justifies you in believing that p it does so because your intuition is based on your understanding of the proposition that p. The aim of this paper is to raise some challenges for accounts of intuitive justification along these lines. I pursue this project from a non-skeptical perspective. I argue that there are cases in which intuiting (...) that p justifies you in believing that p, but such that there is no compelling reason to think this is because your intuition is based on your understanding of the proposition that p. (shrink)

This paper argues for and explores the implications of the following epistemological principle for knowability a priori (with 'Ka' abbreviating 'it is knowable a priori that'). -/- (AK) For all ϕ, ψ such that ϕ semantically presupposes ψ: if Ka(ϕ), Ka(ψ). -/- Well-known arguments for the contingent a priori and a priori knowledge of logical truth founder when the semantic presuppositions of the putative items of knowledge are made explicit. Likewise, certain kinds of analytic truth turn out to carry semantic (...) presuppositions that make them ineligible as items of a priori knowledge. -/- On a happier note, I argue that (AK) offers an appealing, theory-neutral explanation of the a posteriori character of certain necessary identities, as well as an interesting rationalization for a commonplace linguistic maneuver in philosophical work on the a priori. (shrink)

A shift from a metaphysical framework of substance to one of process enables an integrated account of the emergence of normative phenomena. I show how substance assumptions block genuine ontological emergence, especially the emergence of normativity, and how a process framework permits a thermodynamic-based account of normative emergence. The focus is on two foundational forms of normativity, that of normative function and of representation as emergent in a particular kind of function. This process model of representation, called interactivism, compels changes (...) in many related domains. The discussion ends with brief attention to three domains in which changes are induced by the representational model: perception, learning, and language. (shrink)

Kim's argument appears to render causally efficacious emergence impossible: Hume's argument appears to render normative emergence impossible, and, in its general form, it precludes any emergence at all. I argue that both of these barriers can be overcome, and, in fact, that they each constitute reductions of their respective underlying presuppositions. In particular, causally efficacious ontological emergence can be modeled, but only within a process metaphysics, thus avoiding Kim's argument, and making use of non-abbreviatory forms of definition, thus avoiding Hume's (...) argument. I illustrate these points with models of the emergent nature of normative function and of representation. (shrink)