This book concerns the foundations of epistemic modality and hyperintensionality and their applications to the philosophy of mathematics. David Elohim examines the nature of epistemic modality, when the modal operator is interpreted as concerning both apriority and conceivability, as well as states of knowledge and belief. The book demonstrates how epistemic modality and hyperintensionality relate to the computational theory of mind; metaphysical modality and hyperintensionality; the types of mathematical modality and hyperintensionality; to the epistemic status of large cardinal axioms, undecidable (...) propositions, and abstraction principles in the philosophy of mathematics; to the modal and hyperintensional profiles of the logic of rational intuition; and to the types of intention, when the latter is interpreted as a hyperintensional mental state. Chapter \textbf{2} argues for a novel type of expressivism based on the duality between the categories of coalgebras and algebras, and argues that the duality permits of the reconciliation between modal and hyperintensional cognitivism and modal and hyperintensional expressivism. Elohim develops a novel, topic-sensitive truthmaker semantics for dynamic epistemic logic, and develops a novel, dynamic two-dimensional semantics grounded in two-dimensional hyperintensional Turing machines. Chapter \textbf{3} provides an abstraction principle for two-dimensional (hyper-)intensions. Chapter \textbf{4} advances a topic-sensitive two-dimensional truthmaker semantics, and provides three novel interpretations of the framework along with the epistemic and metasemantic. Chapter \textbf{5} applies the fixed points of the modal $\mu$-calculus in order to account for the iteration of epistemic states in a single agent, by contrast to availing of modal axiom 4 (i.e. the KK principle). The fixed point operators in the modal $\mu$-calculus are rendered hyperintensional, which yields the first hyperintensional construal of the modal $\mu$-calculus in the literature and the first application of the calculus to the iteration of epistemic states in a single agent instead of the common knowledge of a group of agents. Chapter \textbf{6} advances a solution to the Julius Caesar problem based on Fine's `criterial' identity conditions which incorporate conditions on essentiality and grounding. Chapter \textbf{7} provides a ground-theoretic regimentation of the proposals in the metaphysics of consciousness and examines its bearing on the two-dimensional conceivability argument against physicalism. The topic-sensitive epistemic two-dimensional truthmaker semantics developed in chapters \textbf{2} and \textbf{4} is availed of in order for epistemic states to be a guide to metaphysical states in the hyperintensional setting. -/- Chapters \textbf{8-12} provide cases demonstrating how the two-dimensional hyperintensions of hyperintensional, i.e. topic-sensitive epistemic two-dimensional truthmaker, semantics, solve the access problem in the epistemology of mathematics. Chapter \textbf{8} examines the interaction between Elohim's hyperintensional semantics and the axioms of epistemic set theory, large cardinal axioms, the Epistemic Church-Turing Thesis, the modal axioms governing the modal profile of $\Omega$-logic, Orey sentences such as the Generalized Continuum Hypothesis, and absolute decidability. These results yield inter alia the first hyperintensional Epistemic Church-Turing Thesis and hyperintensional epistemic set theories in the literature. Chapter \textbf{9} examines the modal and hyperintensional commitments of abstractionism, in particular necessitism, and epistemic hyperintensionality, epistemic utility theory, and the epistemology of abstraction. Elohim countenances a hyperintensional semantics for novel epistemic abstractionist modalities. Elohim suggests, too, that higher observational type theory can be applied to first-order abstraction principles in order to make first-order abstraction principles recursively enumerable, i.e. Turing machine computable, and that the truth of the first-order abstraction principle for two-dimensional hyperintensions is grounded in its being possibly recursively enumerable and the machine being physically implementable. Chapter \textbf{10} examines the philosophical significance of hyperintensional $\Omega$-logic in set theory and discusses the hyperintensionality of metamathematics. Chapter \textbf{11} provides a modal logic for rational intuition and provides a hyperintensional semantics. Chapter \textbf{12} avails of modal coalgebras to interpret the defining properties of indefinite extensibility, and avails of hyperintensional epistemic two-dimensional semantics in order to account for the interaction between interpretational and objective modalities and the truthmakers thereof. This yields the first hyperintensional category theory in the literature. Elohim invents a new mathematical trick in which first-order structures are treated as categories, and Vopenka's principle can be satisfied because of the elementary embeddings between the categories and generate Vopenka cardinals in the category of Set in category theory. Chapter \textbf{13} examines modal responses to the alethic paradoxes. Elohim provides a counter-example to epistemic closure for logical deduction. Chapter \textbf{14} examines, finally, the modal and hyperintensional semantics for the different types of intention and the relation of the latter to evidential decision theory. (shrink)

This book concerns the foundations of epistemic modality and hyperintensionality and their applications to the philosophy of mathematics. David Elohim examines the nature of epistemic modality, when the modal operator is interpreted as concerning both apriority and conceivability, as well as states of knowledge and belief. The book demonstrates how epistemic modality and hyperintensionality relate to the computational theory of mind; metaphysical modality and hyperintensionality; the types of mathematical modality and hyperintensionality; to the epistemic status of large cardinal axioms, undecidable (...) propositions, and abstraction principles in the philosophy of mathematics; to the modal and hyperintensional profiles of the logic of rational intuition; and to the types of intention, when the latter is interpreted as a hyperintensional mental state. Chapter \textbf{2} argues for a novel type of expressivism based on the duality between the categories of coalgebras and algebras, and argues that the duality permits of the reconciliation between modal and hyperintensional cognitivism and modal and hyperintensional expressivism. Elohim develops a novel, topic-sensitive truthmaker semantics for dynamic epistemic logic, and develops a novel, dynamic two-dimensional semantics grounded in two-dimensional hyperintensional Turing machines. Chapter \textbf{3} provides an abstraction principle for two-dimensional (hyper-)intensions. Chapter \textbf{4} advances a topic-sensitive two-dimensional truthmaker semantics, and provides three novel interpretations of the framework along with the epistemic and metasemantic. Chapter \textbf{5} applies the fixed points of the modal $\mu$-calculus in order to account for the iteration of epistemic states in a single agent, by contrast to availing of modal axiom 4 (i.e. the KK principle). The fixed point operators in the modal $\mu$-calculus are rendered hyperintensional, which yields the first hyperintensional construal of the modal $\mu$-calculus in the literature and the first application of the calculus to the iteration of epistemic states in a single agent instead of the common knowledge of a group of agents. Chapter \textbf{6} advances a solution to the Julius Caesar problem based on Fine's `criterial' identity conditions which incorporate conditions on essentiality and grounding. Chapter \textbf{7} provides a ground-theoretic regimentation of the proposals in the metaphysics of consciousness and examines its bearing on the two-dimensional conceivability argument against physicalism. The topic-sensitive epistemic two-dimensional truthmaker semantics developed in chapters \textbf{2} and \textbf{4} is availed of in order for epistemic states to be a guide to metaphysical states in the hyperintensional setting. -/- Chapters \textbf{8-12} provide cases demonstrating how the two-dimensional hyperintensions of hyperintensional, i.e. topic-sensitive epistemic two-dimensional truthmaker, semantics, solve the access problem in the epistemology of mathematics. Chapter \textbf{8} examines the interaction between Elohim's hyperintensional semantics and the axioms of epistemic set theory, large cardinal axioms, the Epistemic Church-Turing Thesis, the modal axioms governing the modal profile of $\Omega$-logic, Orey sentences such as the Generalized Continuum Hypothesis, and absolute decidability. These results yield inter alia the first hyperintensional Epistemic Church-Turing Thesis and hyperintensional epistemic set theories in the literature. Chapter \textbf{9} examines the modal and hyperintensional commitments of abstractionism, in particular necessitism, and epistemic hyperintensionality, epistemic utility theory, and the epistemology of abstraction. Elohim countenances a hyperintensional semantics for novel epistemic abstractionist modalities. Elohim suggests, too, that higher observational type theory can be applied to first-order abstraction principles in order to make first-order abstraction principles recursively enumerable, i.e. Turing machine computable, and that the truth of the first-order abstraction principle for two-dimensional hyperintensions is grounded in its being possibly recursively enumerable and the machine being physically implementable. Chapter \textbf{10} examines the philosophical significance of hyperintensional $\Omega$-logic in set theory and discusses the hyperintensionality of metamathematics. Chapter \textbf{11} provides a modal logic for rational intuition and provides a hyperintensional semantics. Chapter \textbf{12} avails of modal coalgebras to interpret the defining properties of indefinite extensibility, and avails of hyperintensional epistemic two-dimensional semantics in order to account for the interaction between interpretational and objective modalities and the truthmakers thereof. This yields the first hyperintensional category theory in the literature. Elohim invents a new mathematical trick in which first-order structures are treated as categories, and Vopenka's principle can be satisfied because of the elementary embeddings between the categories and generate Vopenka cardinals in the category of Set in category theory. Chapter \textbf{13} examines modal responses to the alethic paradoxes. Elohim provides a counter-example to epistemic closure for logical deduction. Chapter \textbf{14} examines, finally, the modal and hyperintensional semantics for the different types of intention and the relation of the latter to evidential decision theory. (shrink)

This essay clarifies quantifier variance and uses it to provide a theory of indefinite extensibility that I call the variance theory of indefinite extensibility. The indefinite extensibility response to the set-theoretic paradoxes sees each argument for paradox as a demonstration that we have come to a different and more expansive understanding of ‘all sets’. But indefinite extensibility is philosophically puzzling: extant accounts are either metasemantically suspect in requiring mysterious mechanisms of domain expansion, or metaphysically suspect in requiring nonstandard assumptions about (...) mathematical objects. Happily, the view of quantifier meanings that underwrites quantifier variance can be used to provide an account of indefinite extensibility that is both metasemantically and metaphysically satisfying. Section 1 introduces the puzzle of indefinite extensibility; section 2 develops and clarifies the metasemantics of quantifier variance; section 3 solves section 1's puzzle of indefinite extensibility by applying section 2's account of quantifier meanings; and section 4 compares the theory developed in section 3 to several other theories in the literature. (shrink)

The aim of this paper is to explore the view that some ontological disputes are “metalinguistic negotiations”, and to make sense of the significance of these controversies in a way that is still compatible with a broadly deflationist approach. I start by considering the view advocated by Eli Hirsch to the effect that some ontological disputes are verbal. I take the Endurantism–Perdurantusm dispute as a case-study and argue that, while it can be conceded that the dispute is verbal at the (...) object-level, this does not rule out the possibility of a non-verbal disagreement at the metalinguistic level. I then explore the metalinguistic dispute hypothesis by seeing how it can be defended from a first objection playing on the idea of inter-translatability, as well as a second objection raising the question of equal theoretical virtues. (shrink)

This article examines whether it is possible to uphold one form of deflationism towards metaphysics, ontological pluralism, whilst maintaining metaphysical realism. The focus therefore is on one prominent deflationist who fits the definition of an ontological pluralist, Eli Hirsch, and his self-ascription as a realist. The article argues that ontological pluralism is not amenable to the ascription of realism under some basic intuitions as to what a “realist” position is committed to. These basic intuitions include a commitment to more than (...) a stuff-ontology, and a view that realism carries with it more than a rejection of idealism. This issue is more than merely terminological. The ascription of realism is an important classification in order to understand what sorts of entities can be the truthmakers within a given theory. “Realism” is thus an important term to understand the nature of the entities that a given theory accepts into its ontology. (shrink)

Much contemporary ontological inquiry takes place within the so-called ‘Quinean tradition’ but, given that some aspects of Quine’s project have been widely abandoned even by those who consider themselves Quineans, it is unclear what this amounts to. Fortunately recent work in metaontology has produced two relevant results here: a clearer characterisation of the metaontology uniting the aforementioned Quineans, most notably undertaken by Peter van Inwagen, and a raft of criticisms of that metaontology. In this paper I critique van Inwagen’s Quinean (...) metaontology, finding that certain challenges, supplemented by pressure to reflect more closely Quine’s work, should drive Quineans to adopt a stronger metaontology incorporating more of Quine’s radical views. I conclude that while van Inwagen’s Quineanism is problematic there are prospects for a viable, more wholeheartedly Quinean, metaontology. (shrink)

Gideon Rosen and Robert Schwartzkopff have independently suggested (variants of) the following claim, which is a varian of Hume's Principle: -/- When the number of Fs is identical to the number of Gs, this fact is grounded by the fact that there is a one-to-one correspondence between the Fs and Gs. -/- My paper is a detailed critique of the proposal. I don't find any decisive refutation of the proposal. At the same time, it has some consequences which many will (...) find objectionable. (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)

This essay is a study of ontological commitment, focused on the special case of arithmetical discourse. It tries to get clear about what would be involved in a defense of the claim that arithmetical assertions are ontologically innocent and about why ontological innocence matters. The essay proceeds by questioning traditional assumptions about the connection between the objects that are used to specify the truth-conditions of a sentence, on the one hand, and the objects whose existence is required in order for (...) the truth-conditions thereby specified to be satisfied, on the other. This allows one to set forth an assignment of truth-conditions to arithmetical sentences whereby nothing is required of the world in order for the truth-conditions of a truth of pure arithmetic to be satisfied. The essay then argues that such an assignment can be used to account for the a priori knowability of certain arithmetical truths. (shrink)

The neo-Fregean account of arithmetical knowledge is centered around the abstraction principle known as Hume’s Principle: for any concepts X and Y , the number of X ’s is the same as the number of Y ’s just in case there is a 1–1 correspondence between X and Y . The Caesar Problem, originally raised by Frege in §56 of Die Grundlagen der Arithmetik , emerges in the context of the neo-Fregean programme, because, though Hume’s Principle provides a criterion of (...) identity for objects falling under the concept of Number–namely, 1–1 correspondence—the principle fails to deliver a criterion of application. That is, it fails to deliver a criterion that will tell us which objects fall under the concept Number, and so, leaves unanswered the question whether Caesar could be a number. Hale and Wright have recently offered a neo-Fregean solution to this problem. The solution appeals to the notion of a categorical sortal. This paper offers a reconstruction of their solution, which has the advantage over Hale and Wright’s original proposal of making clear what the structure of the background ontology is. In addition, it is shown that the Caesar Problem can be solved in a framework more minimal than that of Hale and Wright, viz . one that dispenses with categorical sortals. The paper ends by discussing an objection to the proposed neo-Fregean solutions, based on the idea that Leibniz’s Law gives a universal criterion of identity. This is an idea that Hale and Wright reject. However, it is shown that a solution very much in keeping with their own proposal is available, even if it is granted that Leibniz’s Law provides a universal criterion of identity. (shrink)

In early analytic philosophy, one of the most central questions concerned the status of arithmetical objects. Frege argued against the popular conception that we arrive at natural numbers with a psychological process of abstraction. Instead, he wanted to show that arithmetical truths can be derived from the truths of logic, thus eliminating all psychological components. Meanwhile, Dedekind and Peano developed axiomatic systems of arithmetic. The differences between the logicist and axiomatic approaches turned out to be philosophical as well as mathematical. (...) In this paper, I will argue that Dedekind’s approach can be seen as a precursor to modern structuralism and as such, it enjoys many advantages over Frege’s logicism. I also show that from a modern perspective, Frege’s criticism of abstraction and psychologism is one-sided and fails against the psychological processes that modern research suggests to be at the heart of numerical cognition. The approach here is twofold. First, through historical analysis, I will try to build a clear image of what Frege’s and Dedekind’s views on arithmetic were. Then, I will consider those views from the perspective of modern philosophy of mathematics, and in particular, the empirical study of arithmetical cognition. I aim to show that there is nothing to suggest that the axiomatic Dedekind approach could not provide a perfectly adequate basis for philosophy of arithmetic. (shrink)

It is commonplace for philosophers to distinguish mere truths from truths that perspicuously represent the world's structure. According to a popular view, the perspicuous truths are supposed to be metaphysically revelatory and to play an important role in the accounts of law-hood, confirmation, and linguistic interpretation. Yet, there is no consensus about how to characterize this distinction. I examine strategies developed by Lewis and by Sider in his Writing the Book of the World which purport to explain this distinction in (...) terms of vocabulary: the truths that represent the world perspicuously have better, joint-carving vocabulary. I argue that the distinction between a perspicuous and mere truth concerns both the vocabulary of the sentence and its grammar. I then show that the collective motivations for distinguishing perspicuous from mere truths do not allow Lewis and Sider to properly impose constraints on grammar. (shrink)

The paper challenges a widely held interpretation of Frege's conception of logic on which the constituent clauses of basic law V have the same sense. I argue against this interpretation by first carefully looking at the development of Frege's thoughts in Grundlagen with respect to the status of abstraction principles. In doing so, I put forth a new interpretation of Grundlagen §64 and Frege's idea of ‘recarving of content’. I then argue that there is strong evidence in Grundgesetze that Frege (...) did not hold the relevant sense-identity claim regarding basic law V. (shrink)

One major difficulty confronting attempts to clarify the epistemological and ontological status of abstract objects is determining the sense, if any, in which such entities may be characterised as mind and language independent. Our contention is that the tolerant reductionist position of Michael Dummett can be strengthened by drawing on Husserl's mature account of the constitution of ideal objects and mathematical objectivity. According to the Husserlian position we advocate, abstract singular terms pick out weakly mind-independent sedimented meaning-contents. These meaning-contents serve (...) as the ‘thin’ referents of abstract singular terms, but are ultimately founded in prior acts of meaning-constitution. (shrink)

Adherents of Ockham’s fundamental razor contend that considerations of ontological parsimony pertain primarily to fundamental objects. Derivative objects, on the other hand, are thought to be quite unobjectionable. One way to understand the fundamental vs. derivative distinction is in terms of the Aristotelian distinction between ontologically independent and dependent objects. In this paper I will defend the thesis that every natural number greater than 0 is an ontologically dependent object thereby exempting the natural numbers from Ockham’s fundamental razor.

Frege's Grundgesetze was one of the 19th century forerunners to contemporary set theory which was plagued by the Russell paradox. In recent years, it has been shown that subsystems of the Grundgesetze formed by restricting the comprehension schema are consistent. One aim of this paper is to ascertain how much set theory can be developed within these consistent fragments of the Grundgesetze, and our main theorem shows that there is a model of a fragment of the Grundgesetze which defines a (...) model of all the axioms of Zermelo-Fraenkel set theory with the exception of the power set axiom. The proof of this result appeals to G\"odel's constructible universe of sets, which G\"odel famously used to show the relative consistency of the continuum hypothesis. More specifically, our proofs appeal to Kripke and Platek's idea of the projectum within the constructible universe as well as to a weak version of uniformization (which does not involve knowledge of Jensen's fine structure theory). The axioms of the Grundgesetze are examples of abstraction principles, and the other primary aim of this paper is to articulate a sufficient condition for the consistency of abstraction principles with limited amounts of comprehension. As an application, we resolve an analogue of the joint consistency problem in the predicative setting. (shrink)

I here defend a theory consisting of four claims about ‘property’ and properties, and argue that they form a coherent whole that can solve various serious problems. The claims are (1): ‘property’ is defined by the principles (PR): ‘F-ness/Being F/etc. is a property of x iff F’ and (PA): ‘F-ness/Being F/etc. is a property’; (2) the function of ‘property’ is to increase the expressive power of English, roughly by mimicking quantification into predicate position; (3) property talk should be understood at (...) face value: apparent commitments are real and our apparently literal use of ‘property’ is really literal; (4) there are no properties. In virtue of (1)–(2), this is a deflationist theory and in virtue of (3)–(4), it is an error theory. (1) is fleshed out as a claim about understanding conditions, and it is argued at length, and by going through a number of examples, that it satisfies a crucial constraint on meaning claims: all facts about ‘property’ can be explained, together with auxiliary facts, on its basis. Once claim (1) has been expanded upon, I argue that the combination of (1)–(3) provides the means for handling several problems: they help giving a happy-face solution to what I call the paradox of abstraction , they form part of a plausible account of the correctness of committive sentences, and, most importantly, they help respond to various indispensability arguments against nominalism. (shrink)

Neo-Fregean approaches to set theory, following Frege, have it that sets are the extensions of concepts, where concepts are the values of second-order variables. The idea is that, given a second-order entity $X$, there may be an object $\varepsilon X$, which is the extension of X. Other writers have also claimed a similar relationship between second-order logic and set theory, where sets arise from pluralities. This paper considers two interpretations of second-order logic—as being either extensional or intensional—and whether either is (...) more appropriate for this approach to the foundations of set theory. Although there seems to be a case for the extensional interpretation resulting from modal considerations, I show how there is no obstacle to starting with an intensional second-order logic. I do so by showing how the $\varepsilon$ operator can have the effect of “extensionalizing” intensional second-order entities. (shrink)

The rather unrestrained use of second-order logic in the neo-logicist program is critically examined. It is argued in some detail that it brings with it genuine set-theoretical existence assumptions and that the mathematical power that Hume’s Principle seems to provide, in the derivation of Frege’s Theorem, comes largely from the ‘logic’ assumed rather than from Hume’s Principle. It is shown that Hume’s Principle is in reality not stronger than the very weak Robinson Arithmetic Q. Consequently, only a few rudimentary facts (...) of arithmetic are logically derivable from Hume’s Principle. And that hardly counts as a vindication of logicism. (shrink)

In the 1920s, David Hilbert proposed a research program with the aim of providing mathematics with a secure foundation. This was to be accomplished by first formalizing logic and mathematics in their entirety, and then showing---using only so-called finitistic principles---that these formalizations are free of contradictions. ;In the area of logic, the Hilbert school accomplished major advances both in introducing new systems of logic, and in developing central metalogical notions, such as completeness and decidability. The analysis of unpublished material presented (...) in Chapter 2 shows that a completeness proof for propositional logic was found by Hilbert and his assistant Paul Bernays already in 1917--18, and that Bernays's contribution was much greater than is commonly acknowledged. Aside from logic, the main technical contribution of Hilbert's Program are the development of formal mathematical theories and proof-theoretical investigations thereof, in particular, consistency proofs. In this respect Wilhelm Ackermann's 1924 dissertation is a milestone both in the development of the Program and in proof theory in general. Ackermann gives a consistency proof for a second-order version of primitive recursive arithmetic which, surprisingly, explicitly uses a finitistic version of transfinite induction up to www . He also gave a faulty consistency proof for a system of second-order arithmetic based on Hilbert's &egr;-substitution method. Detailed analyses of both proofs in Chapter 3 shed light on the development of finitism and proof theory in the 1920s as practiced in Hilbert's school. ;In a series of papers, Charles Parsons has attempted to map out a notion of mathematical intuition which he also brings to bear on Hilbert's finitism. According to him, mathematical intuition fails to be able to underwrite the kind of intuitive knowledge Hilbert thought was attainable by the finitist. It is argued in Chapter 4 that the extent of finitistic knowledge which intuition can provide is broader than Parsons supposes. According to another influential analysis of finitism due to W. W. Tait, finitist reasoning coincides with primitive recursive reasoning. The acceptance of non-primitive recursive methods in Ackermann's dissertation presented in Chapter 3, together with additional textual evidence presented in Chapter 4, shows that this identification is untenable as far as Hilbert's conception of finitism is concerned. Tait's conception, however, differs from Hilbert's in important respects, yet it is also open to criticisms leading to the conclusion that finitism encompasses more than just primitive recursive reasoning. (shrink)

A linguistic convention is a principle or norm that has been adopted by a person or linguistic community about how to use, and therefore what the meaning is of, a specific term. Examples of such norms or principles are those expressed by propositions that express the laws of logic or those that express implicit definitions. Arguments about the epistemic status of linguistic conventions, very broadly, fall into two camps: the one holds that the basis of linguistic conventions is objective and (...) that, therefore, such conventions are genuinely truth-bearing. Knowledge of such conventions is therefore knowledge of the objective truth of such conventions. This camp might be labelled ‘realist’. The other holds that there is no known basis to linguistic conventions and that conventions are, to the best of our knowledge, not objective, but invented. Their truth is stipulated by the relevant speaker or linguistic community. Such truth is, consequently, trivial and knowledge of such conventions simply knowledge of the stipulation. This camp might be labelled ‘anti-realist’. These are three standard accounts of the epistemic status of linguistic conventions, which all play into the first camp: (1) knowledge by intuition, (2) inferential a priori knowledge and (3) a posteriori knowledge. I give reasons why these accounts should be rejected. I then argue that linguistic conventions, if conceived of as trivial truths, are knowable non-inferentially a priori. Such an epistemic account provides support for the second camp. In this regard, I marshal support from some recent work by Wright and Hale. (shrink)

This special issue collects together nine new essays on logical consequence :the relation obtaining between the premises and the conclusion of a logically valid argument. The present paper is a partial, and opinionated,introduction to the contemporary debate on the topic. We focus on two influential accounts of consequence, the model-theoretic and the proof-theoretic, and on the seeming platitude that valid arguments necessarilypreserve truth. We briefly discuss the main objections these accounts face, as well as Hartry Field’s contention that such objections (...) show consequenceto be a primitive, indefinable notion, and that we must reject the claim that valid arguments necessarily preserve truth. We suggest that the accountsin question have the resources to meet the objections standardly thought to herald their demise and make two main claims: (i) that consequence, as opposed to logical consequence, is the epistemologically significant relation philosophers should be mainly interested in; and (ii) that consequence is a paradoxical notion if truth is. (shrink)

Do numbers exist? Do properties? Do possible worlds? Do fictional characters? Many metaphysicians spend time and effort trying to answer these and other questions about the existence of various entities. These inquiries have recently encountered opposition: a group of philosophers, drawing inspiration from Aristotle, have argued that many or all of the existence questions debated by metaphysicians can be answered trivially, and so are not worth debating. Our task is to defend existence questions from the neo-Aristotelians' attacks.

In “Properties and the Interpretation of Second-Order Logic” Bob Hale develops and defends a deflationary conception of properties where a property with particular satisfaction conditions actually exists if and only if it is possible that a predicate with those same satisfaction conditions exists. He argues further that, since our languages are finitary, there are at most countably infinitely many properties and, as a result, the account fails to underwrite the standard semantics for second-order logic. Here a more lenient version of (...) the view is explored, which allows for the possibility of countably infinite predicates understood as the product of linguistic supertasks. This enriched deflationist account of properties—the Infinitary Deflationary Conception of Existence—supports the standard semantics for models with countable first-order domains, and allows one to prove the categoricity of the second-order Peano axioms. (shrink)

Metaphysics has had a long history of debate over its viability, and substantivity. This thesis explores issues connected to the realism question within the domain of metaphysics, ultimately aiming to defend a realist, substantive metaphysics by responding to so-called deflationary approaches, which have become prominent, and well supported within the recent metametaphysical and metaontological literature. To this end, I begin by examining the changing nature of the realism question. I argue that characterising realism and anti-realism through theories of truth unduly (...) places epistemology prior to ontology, and is unwarranted in assuming a non-neutrality between theories of truth and positions within the realism debate. I therefore propose a characterisation of realism and anti-realism understood through truthmaking. This produces a suitable working characterisation of realism that will be used within the remainder of this project. In the second section, I trace the historical influences upon current deflationary approaches to metaphysics, most prominently those of Carnap, and Putnam. I argue that Quine’s supposed attack on Carnap’s anti-metaphysical thought fails, and show how current deflationary thought, most prominently exhibited by Hirsch, came to focus on linguistically derived concerns over the substantivity of metaphysics. In the third section, I outline a number of issues for the deflationist, and defend the coherency and legitimacy of the unrestricted existential quantifier. Focusing on the linguistic aspect of deflationism, I argue that the conception of language that the deflationist relies upon lacks suitable empirical and theoretical support within linguistics and other related domains. Furthermore, I suggest that linguistic analysis in fact supports the claim that the quantifier carries no inherent restrictions. This restores our ability to suitably posit the unrestricted existential quantifier, as a quantifier wherein the domain is only restricted by metaphysically substantive restrictions. Through this, I argue that metaphysics is a substantive domain of discourse. Lastly, I sketch a positive account of how, under an empirically and theoretically justified conception of language, metaphysics can be coherently held to be a realist, substantive enterprise, contra claims that hold that the nature of language inherently prevents metaphysics being considered to be a substantive domain of discourse. (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.

The Bounds of Logic presents a new philosophical theory of the scope and nature of logic based on critical analysis of the principles underlying modern Tarskian logic and inspired by mathematical and linguistic development. Extracting central philosophical ideas from Tarski’s early work in semantics, Sher questions whether these are fully realized by the standard first-order system. The answer lays the foundation for a new, broader conception of logic. By generally characterizing logical terms, Sher establishes a fundamental result in semantics. Her (...) development of the notion of logicality for quantifiers and her work on branching are of great importance for linguistics. Sher outlines the boundaries of the new logic and points out some of the philosophical ramifications of the new view of logic for such issues as the logicist thesis, ontological commitment, the role of mathematics in logic, and the metaphysical underpinning of logic. She proposes a constructive definition of logical terms, reexamines and extends the notion of branching quantification, and discusses various linguistic issues and applications. (shrink)

A crucial part of the contemporary interest in logicism in the philosophy of mathematics resides in its idea that arithmetical knowledge may be based on logical knowledge. Here an implementation of this idea is considered that holds that knowledge of arithmetical principles may be based on two things: (i) knowledge of logical principles and (ii) knowledge that the arithmetical principles are representable in the logical principles. The notions of representation considered here are related to theory-based and structure-based notions of representation (...) from contemporary mathematical logic. It is argued that the theory-based versions of such logicism are either too liberal (the plethora problem) or are committed to intuitively incorrect closure conditions (the consistency problem). Structure-based versions must on the other hand respond to a charge of begging the question (the circularity problem) or explain how one may have a knowledge of structure in advance of a knowledge of axioms (the signature problem). This discussion is significant because it gives us a better idea of what a notion of representation must look like if it is to aid in realizing some of the traditional epistemic aims of logicism in the philosophy of mathematics. (shrink)

Neo-Fregeans such as Bob Hale and Crispin Wright seek a foundation of mathematics based on abstraction principles. These are sentences involving a relation called the abstraction relation. It is usually assumed that abstraction relations must be equivalence relations, so reflexive, symmetric and transitive. In this article I argue that abstraction relations need not be reflexive. I furthermore give an application of non-reflexive abstraction relations to restricted abstraction principles.

One of the main problems plaguing neo-logicism is the Bad Company challenge: the need for a well-motivated account of which abstraction principles provide legitimate definitions of mathematical concepts. In this article a solution to the Bad Company challenge is provided, based on the idea that definitions ought to be conservative. Although the standard formulation of conservativeness is not sufficient for acceptability, since there are conservative but pairwise incompatible abstraction principles, a stronger conservativeness condition is sufficient: that the class of acceptable (...) abstraction principles be strictly logically symmetrically class conservative . The article concludes with an examination of which classes of abstraction principles meet this criteria. (shrink)

Fitch's basic logic is an untyped illative combinatory logic with unrestricted principles of abstraction effecting a type collapse between properties (or concepts) and individual elements of an abstract syntax. Fitch does not work axiomatically and the abstraction operation is not a primitive feature of the inductive clauses defining the logic. Fitch's proof that basic logic has unlimited abstraction is not clear and his proof contains a number of errors that have so far gone undetected. This paper corrects these errors and (...) presents a reasonably intuitive proof that Fitch's system K supports an implicit abstraction operation. Some general remarks on the philosophical significance of basic logic, especially with respect to neo-logicism, are offered, and the paper concludes that basic logic models a highly intensional form of logicism. (shrink)

Gregory Landini offers a detailed historical account of Frege's notations and the philosophical views that led Frege from Begriffssscrhrift to his mature work Grundgesetze, addressing controversial issues that surround the notations.

This paper presents a new interpretation of Frege's context principle on which it applies primarily to singular terms for abstract objects but not necessarily to singular terms for ordinary objects.

In this paper, a general perspective on criteria of identity of kinds of objects is developed. The question of the admissibility of impredicative or circular identity criteria is investigated in the light of the view that is articulated. It is argued that in and of itself impredicativity does not constitute sufficient grounds for rejecting a putative identity criterion. The view that is presented is applied to Davidson’s criterion of identity for events and to the structuralist criterion of identity of places (...) in a structure. (shrink)

This article concerns the ongoing neo-logicist program in the philosophy of mathematics. The enterprise began life, in something close to its present form, with Crispin Wright’s seminal [1983]. It was bolstered when Bob Hale [1987] joined the fray on Wright’s behalf and it continues through many extensions, objections, and replies to objections . The overall plan is to develop branches of established mathematics using abstraction principles in the form: Formula where a and b are variables of a given type , (...) Σ is a higher-order operator denoting a function from items of the given type to objects in the range of the first-order variables, and E is an equivalence relation over items of the given type. In what follows, I will sometimes omit the initial quantifiers.Frege [1884, 1893] himself employed three abstraction principles. One of them, used for illustration, comes from geometry: "The direction of l1 is identical to the direction of l2 if and only if l1 is parallel to l2."Call this the direction principle. The second was dubbed N= in [Wright, 1983] and is now called Hume’s principle: Formula where F≈G is an abbreviation of the second-order statement that there is a one-to-one relation mapping the F’s onto the G’s. In words, states that the number of F is identical to the number of G if and only if F is equinumerous with G. Unlike the direction-principle, the relevant variables, F, G here are second-order. Georg Cantor …. (shrink)

This book is important for philosophy of mathematics and for the study of French philosophy. French philosophers are more concerned than most Anglo-American with mathematical practice outside of foundations. This contradicts the fashionable claim that French intellectuals get science all wrong and we return below to a germane example from Sokal and Bricmont [1999]. The emphasis on practice goes back to mid-20th century French historians of science including those Kuhn cites as sources for his orientation in philosophy of science [Kuhn (...) 1996: p. viii]. And the French often look to the experience, rather than the ideology, of Bourbaki. (shrink)

The basic question of ontology is “What exists?”. The basic question of metaontology is: are there objective answers to the basic question of ontology? Here ontological realists say yes, and ontological anti-realists say no. (Compare: The basic question of ethics is “What is right?”. The basic question of metaethics is: are there objective answers to the basic question of ethics? Here moral realists say yes, and moral anti-realists say no.) For example, the ontologist may ask: Do numbers exist? The Platonist (...) says yes, and the nominalist says no. The metaontologist may ask: is there an objective fact of the matter about whether numbers exist? The ontological realist says yes, and the ontological anti-realist says no. Likewise, the ontologist may ask: Given two distinct entities, when does a mereological sum of those entities exist? The universalist says always, while the nihilist says never. The metaontologist may ask: is there an objective fact of the matter about whether the mereological sum of two distinct entities exists? The ontological realist says yes, and the ontological anti-realist says no. Ontological realism is often traced to Quine (1948), who held that we can determine what exists by seeing which entities are endorsed by our best scientific theory of the world. In recent years, the practice of ontology has often presupposed an ever-stronger ontological realism, and strong versions of ontological realism have received explicit statements by Fine (this volume), Sider (2001; this volume), van Inwagen (1998; this volume), and others. (shrink)

This article is an extended critical study of Kit Fine’s The limits of abstraction, which is a sustained attempt to take the measure of the neo-logicist program in the philosophy and foundations of mathematics, founded on abstraction principles like Hume’s principle. The present article covers the philosophical and technical aspects of Fine’s deep and penetrating study.

Metaontology – the study of the nature of ontological issues – has flourished in recent years. The focus of this summary will be on some views and arguments that are central to today’s debate. One theme will be that of how seriously to take ontology: whether there is reason to take a skeptical or deflationary attitude toward ontological claims, as theorists like Rudolf Carnap, Hilary Putnam, and Eli Hirsch in different ways have urged. The other theme will be that of (...) the viability of the Fregean and Quinean approaches to ontology that have been and still are central to analytic philosophy. (shrink)

Abstract: This paper explores the relevance of themes from Wittgenstein's Tractatus Logico-Philosophicus to the ongoing discussion of metaphysical nihilism. I set out by showing how metaphysical nihilism is of paramount importance for cosmological arguments. Metaphysical nihilism is the position that there might have been nothing. Two conflicting intuitions emerge from a survey of discussions of metaphysical nihilism: Firstly, that metaphysical nihilism is true, and secondly, that formulations of the position are somehow unclear or nonsensical. By considering formalizations of philosophical language, (...) the second intuition is sharpened, while the first intuition is given expression through the Tractarian distinction between what is said and what is shown by our symbolism. I conclude by exploring and rejecting objections to making metaphysical nihilism a scientific, rather than a philosophical question. (shrink)

I start by reconsidering two familiar arguments against modal realism. The argument from epistemology relates to the issue whether we can infer the existence of concrete objects by a priori means. The argument from pragmatics purports to refute the analogy between the indispensability of possible worlds and the indispensability of unobserved entities in physical science and of numbers in mathematics. Then I present two novel objections. One focusses on the obscurity of the notion of isolation required by modal realism. The (...) other stresses the arbitrary nature of the rules governing the behaviour of Lewisean universes. All four objections attack the reductive analysis of modality that is supposed to be the chief merit of modal realism. (shrink)

Needless to say, Charles Parsons’s long awaited book1 is a must-read for anyone with an interest in the philosophy of mathematics. But as Parsons himself says, this has been a very long time in the writing. Its chapters extensively “draw on”, “incorporate material from”, “overlap considerably with”, or “are expanded versions of” papers published over the last twenty-five or so years. What we are reading is thus a multi-layered text with different passages added at different times. And this makes for (...) a rather bumpy read. There is another route Parsons could have taken: he could have reprinted the relevant papers with postscripts, and then top-and-tailed the collection with a preface and added concluding reflections. It must sound very ungrateful, but I rather suspect that that might have worked better. Much of the book is about arithmetic. But Parsons has woven into the discussion claims about mathematics more generally and about set theory in particular. We might well have a basic worry about this structure: for do defensible claims about the ontology and epistemology of arithmetic have to be generalizable to apply to more infinitary mathematics? For example, suppose you are attracted to a Hellman-like modal structuralist account of arithmetic: then should you think it a problem if you suspect that such an account can’t readily be extended to cope e.g. with set theory (since you boggle at the idea of possible worlds free of abstracta but with enough structure to somehow model ZFC)? Parsons himself seems to waver over such questions. So for present purposes, I’ll focus just on arithmetic, and in what follows I’ll revisit two of the most familiar Parsonian themes, his views on structuralism as an account of the ontology of arithmetic, and his exploration of the role of intuition in grounding arithmetical knowledge. (shrink)

In this paper, I develop a new defense of logicism: one that combines logicism and nominalism. First, I defend the logicist approach from recent criticisms; in particular from the charge that a cruciai principie in the logicist reconstruction of arithmetic, Hume's Principle, is not analytic. In order to do that, I argue, it is crucial to understand the overall logicist approach as a nominalist view. I then indicate a way of extending the nominalist logicist approach beyond arithmetic. Finally, I argue (...) that a nominalist can use the resulting approach to provide a nominalization strategy for mathematics. In this way, mathematical structures can be introduced without ontological costs. And so, if this proposal is correct, we can say that ultimately all the nominalist needs is logic (and, rather loosely, ali the logicist needs is nominalism). (shrink)

The period in the foundations of mathematics that started in 1879 with the publication of Frege's Begriffsschrift and ended in 1931 with Gödel's Über formal unentscheidbare Sätze der Principia Mathematica und verwandter Systeme I can reasonably be called the classical period. It saw the development of three major foundational programmes: the logicism of Frege, Russell and Whitehead, the intuitionism of Brouwer, and Hilbert's formalist and proof-theoretic programme. In this period, there were also lively exchanges between the various schools culminating in (...) the famous Hilbert-Brouwer controversy in the 1920s. -/- The purpose of this anthology is to review the programmes in the foundations of mathematics from the classical period and to assess their possible relevance for contemporary philosophy of mathematics. What can we say, in retrospect, about the various foundational programmes of the classical period and the disputes that took place between them? To what extent do the classical programmes of logicism, intuitionism and formalism represent options that are still alive today? These questions are addressed in this volume by leading mathematical logicians and philosophers of mathematics. (shrink)

This paper defends the view that ontological questions (properly understood) are easy—too easy, in fact, to be subjects of substantive and distinctively philosophical debates. They are easy, roughly, in the sense that they may be resolved straightforwardly—generally by a combination of conceptual and empirical enquiries. After briefly outlining the view and some of its virtues, I turn to examine two central lines of objection. The first is that this ‘easy’ approach is itself committed to substantive ontological views, including an implausibly (...) permissive ontology. The second is that it, like neo-Fregean views, relies on transformation rules that are questionable on both logical and ontological grounds. Ultimately, I will argue, the easy view is not easily assailed by either of these routes, and so remains (thus far) a tenable and attractive approach. (shrink)

The period from 1900 to 1935 was particularly fruitful and important for the development of logic and logical metatheory. This survey is organized along eight "itineraries" concentrating on historically and conceptually linked strands in this development. Itinerary I deals with the evolution of conceptions of axiomatics. Itinerary II centers on the logical work of Bertrand Russell. Itinerary III presents the development of set theory from Zermelo onward. Itinerary IV discusses the contributions of the algebra of logic tradition, in particular, Löwenheim (...) and Skolem. Itinerary V surveys the work in logic connected to the Hilbert school, and itinerary V deals specifically with consistency proofs and metamathematics, including the incompleteness theorems. Itinerary VII traces the development of intuitionistic and many-valued logics. Itinerary VIII surveys the development of semantical notions from the early work on axiomatics up to Tarski's work on truth. (shrink)

The article first rehearses three deflationary theories of reference, (1) disquotationalism, (2) propositionalism (Horwich), and (3) the anaphoric theory (Brandom), and raises a number of objections against them. It turns out that each corresponds to a closely related theory of truth, and that these are subject to analogous criticisms to a surprisingly high extent. I then present a theory of my own, according to which the schema “That S(t) is about t” and the biconditional “S refers to x iff S (...) says something about x” are exhaustive of the notions of aboutness and reference. An account of the usefulness of “about” is then given, which, I argue, is superior to that of Horwich. I close with a few considerations about how the advertised theory relates to well-known issues of reference, the conclusions of which is (1) that the issues concern reference and aboutness only insofar as the words “about” and “refer” serve to generalise over the claims that are really at issue, (2) that the theory of reference will not settle the issues, and (3) that it follows from (2) that the issues do not concern the nature of aboutness or reference. (shrink)

This article treats three aspects of Frege's discussions of definitions. First, I survey Frege's main criticisms of definitions in mathematics. Second, I consider Frege's apparent change of mind on the legitimacy of contextual definitions and its significance for recent neo-Fregean logicism. In the remainder of the article I discuss a critical question about the definitions on which Frege's proofs of the laws of arithmetic depend: do the logical structures of the definientia reflect the understanding of arithmetical terms prevailing prior to (...) Frege's analyses? Unless they do, it is unclear how Frege's proofs demonstrate the analyticity of the arithmetic in use before logicism. Yet, especially in late writings, Frege characterizes definitions as arbitrary stipulations of the senses or references of expressions unrelated to pre-definitional understanding. I conclude by examining some options for conceiving of the status of Frege's logicism in light of this apparent tension, and outline a suggestion for a philosophically fruitful way of resolving this tension. (shrink)

Øystein Linnebo has recently shown that the existence of successors cannot be proven in predicative Frege arithmetic, using Frege’s definitions of arithmetical notions. By contrast, it is shown here that the existence of successor can be proven in ramified predicative Frege arithmetic.