I argue that absolutism, the view that absolutely unrestricted quantification is possible, is to blame for both the paradoxes that arise in naive set theory and variants of these paradoxes that arise in plural logic and in semantics. The solution is restrictivism, the view that absolutely unrestricted quantification is not possible. -/- It is generally thought that absolutism is true and that restrictivism is not only false, but inexpressible. As a result, the paradoxes are blamed, not on illicit quantification, but (...) on the logical conception of set which motivates naive set theory. The accepted solution is to replace this with the iterative conception of set. -/- I show that this picture is doubly mistaken. After a close examination of the paradoxes in chapters 2--3, I argue in chapters 4 and 5 that it is possible to rescue naive set theory by restricting quantification over sets and that the resulting restrictivist set theory is expressible. In chapters 6 and 7, I argue that it is the iterative conception of set and the thesis of absolutism that should be rejected. (shrink)

Reductionists about knowledge-wh hold that "s knows-wh" (e.g. "John knows who stole his car") is reducible to "there is a proposition p such that s knows that p, and p answers the indirect question of the wh-clause." Anti-reductionists hold that "s knows-wh" is reducible to "s knows that p, as the true answer to the indirect question of the wh-clause." I argue that both of these positions are defective. I then offer a new analysis of knowledge-wh as a special kind (...) of de re knowledge. (shrink)

What is the source of logical and mathematical truth? This book revitalizes conventionalism as an answer to this question. Conventionalism takes logical and mathematical truth to have their source in linguistic conventions. This was an extremely popular view in the early 20th century, but it was never worked out in detail and is now almost universally rejected in mainstream philosophical circles. Shadows of Syntax is the first book-length treatment and defense of a combined conventionalist theory of logic and mathematics. It (...) argues that our conventions, in the form of syntactic rules of language use, are perfectly suited to explain the truth, necessity, and a priority of logical and mathematical claims, as well as our logical and mathematical knowledge. (shrink)

This dissertation concerns the foundations of epistemic modality. I examine 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 dissertation demonstrates how phenomenal consciousness and gradational possible-worlds models in Bayesian perceptual psychology relate to epistemic modal space. The dissertation demonstrates, then, how epistemic modality relates to the computational theory of mind; metaphysical modality; deontic modality; logical modality; the types of mathematical modality; to the (...) epistemic status of undecidable propositions and abstraction principles in the philosophy of mathematics; to the apriori-aposteriori distinction; to the modal profile of rational propositional intuition; and to the types of intention, when the latter is interpreted as a modal mental state. Examining the nature of epistemic logic itself, I develop a novel approach to conditions of self-knowledge in the setting of the modal μ-calculus, as well as novel epistemicist solutions to Curry's, the liar, and the knowability paradoxes. Solutions to previously intransigent issues concerning the first-person concept; the distinction between fundamental and derivative truths; and the unity of intention and its role in decision theory, are developed along the way. (shrink)

When does a developing human being acquire moral status? I outline three different positions based on substance ontology that attempt to solve the question by locating some morally salient event in the process of human development question. In the second section, I consider some specific empirical objections to one of these positions, refute them, and then show how similar objections and responses would generalize to the other substance-based positions on the question. The crucial finding is that all the attempts to (...) answer the question that involve substance ontology need to appeal to dispositions. (shrink)

I formulate a theory of groups based on pluralities and counterparts: roughly put, a group is a plurality of entities at a time. This theory comes with counterpart-theoretic semantics for modal and temporal sentences about groups. So this theory of groups is akin to the stage theory of material objects: both take the items they analyze to exist at a single time, and both use counterparts to satisfy certain conditions relating to the modal properties, temporal properties, and coincidence properties of (...) those items. (shrink)

Although the Evans argument against vague identity has been much discussed, proposah for blocking it have not so far satisfied general conditions which any solution ought to meet. Moreover, the relation between ontically vague identity and ontic vagueness more generally has not yet been satisfactorily addressed. I advocate a way of resisting the Evans argument which satisfies the conditions. To show how this approach can vindicate particular cases of ontically vague identity, I develop a framework for describing ontic vagueness in (...) general in terms of multiple actualities. This provides aprìncipled approach to ontically vague identity which is unaffected by the Evans argument. (shrink)

Hume's Principle requires the existence of the finite cardinals and their cardinal, but these are the only cardinals the Principle requires. Were the Principle an analysis of the concept of cardinal number, it would already be peculiar that it requires the existence of any cardinals; an analysis of bachelor is not expected to yield unmarried men. But that it requires the existence of some cardinals, the countable ones, but not others, the uncountable, makes it seem invidious; it is as if (...) an analysis of people required that there be men but not women, or whites but not blacks. If we deprive the Principle of existential commitments, it will cease to yield Dedekind's axioms for the natural numbers and so fail a good test of material adequacy. But since there are cardinals no second-order theory guarantees, neither can the Principle be beefed up to require all cardinals. (shrink)

Imperatives cannot be true or false, so they are shunned by logicians. And yet imperatives can be combined by logical connectives: "kiss me and hug me" is the conjunction of "kiss me" with "hug me". This example may suggest that declarative and imperative logic are isomorphic: just as the conjunction of two declaratives is true exactly if both conjuncts are true, the conjunction of two imperatives is satisfied exactly if both conjuncts are satisfied—what more is there to say? Much more, (...) I argue. "If you love me, kiss me", a conditional imperative, mixes a declarative antecedent ("you love me") with an imperative consequent ("kiss me"); it is satisfied if you love and kiss me, violated if you love but don't kiss me, and avoided if you don't love me. So we need a logic of three -valued imperatives which mixes declaratives with imperatives. I develop such a logic. (shrink)

We speak of products in two senses: in one, we speak of types of products, in the other we speak of the particular objects that are instances of those types. I argue that types of products have the same ontological status as that of material stuffs, like water and gold, which have a non-particular level of existence. I also argue that the relationship between types of products and their instances is logically similar to the relation of constitution, which holds between, (...) say, gold and a ring made of gold. In my approach, types of products are concrete entities, having spatiotemporal properties. This picture fits our commonplace conception of types of products better than alternative approaches according to which types of products are universal, abstract, or mereological entities. (shrink)

This is a study of an under-developed topic in philosophy of language, namely first-person plural pronouns (‘we’, ‘us’, etc.) Richard Vallée has made very important progress by identifying crucial desiderata and putting forward an ingenious proposal about ‘we’ which addresses them. We contend that, despite this impressive progress, he makes some missteps, both omissions and errors; furthermore, his proposal appears implausible as a personal-level psychological story. We thus sketch an alternative approach to the semantics of the first-person plural indexical which, (...) though it builds on Vallée’s important work, departs substantially from it. (shrink)

Maddy's (1990) arguments against Aggregate Theory were undermined by the shift in her position in 1997. The present paper considers Aggregate Theory in the light of this, and the recent search for `New Axioms for Mathematics'. If Set Theory is the part-whole theory of singletons, then identifying singletons with their single members collapses Set Theory into Aggregate Theory. But if singletons are not identical to their single members, then they are not extensional objects and so are not a basis for (...) Science. Either way, the Continuum Hypothesis has no physical interest. (shrink)

Subverting a once widely held Quinean paradigm, there is a growing consensus among philosophers of logic that higher-order quantifiers (which bind variables in the syntactic position of predicates and sentences) are a perfectly legitimate and useful instrument in the logico-philosophical toolbox, while neither being reducible to nor fully explicable in terms of first-order quantifiers (which bind variables in singular term position). This article discusses the impact of this quantificational paradigm shift on metaphysics, focussing on theories of properties, propositions, and identity, (...) as well as on the metaphysics of modality. (shrink)

There will be a few themes. One to get us going: expansion versus contraction. About an object, o, and the region, R, of space(time) in which o is exactly located,1 we may ask: i) must there exist expansions of o: objects in filled superregions2 of R? ii) must there exist contractions of o: objects in filled subregions of..

The neo-logicist argues tliat standard mathematics can be derived by purely logical means from abstraction principles—such as Hume's Principle— which are held to lie 'epistcmically innocent'. We show that the second-order axiom of comprehension applied to non-instantiated properties and the standard first-order existential instantiation and universal elimination principles are essential for the derivation of key results, specifically a theorem of infinity, but have not been shown to be epistemically innocent. We conclude that the epistemic innocence of mathematics has not been (...) established by the neo-logicist. (shrink)

It is often claimed that nominalistic programmes to reconstruct mathematics fail, since they will at some point involve the notion of logical consequence which is unavailable to the nominalist. In this paper we use an idea of Goodman and Quine to develop a nominalistically acceptable explication of logical consequence.

Can one quantify over absolutely everything? Absolutists answer positively, while relativists answer negatively. Here, I focus on the absolutism versus relativism debate in the framework of theories of truth, where relativism becomes a form of contextualism about truth predications. Contextualist theories of truth provide elegant and uniform solutions to the semantic paradoxes while preserving classical logic. However, they interpret harmless generalizations (such as “everything is self-identical”) in less than absolutely comprehensive domains, thus systematically misconstruing them. In this article, I show (...) that contextualism is broadly compatible with absolute generality. More specifically, I develop a bipartite contextualist semantics, or “bicontextualism,” on which sentences are split in two groups: the unproblematic sentences, which are compatible with absolute generality, and the problematic ones, which are given a relativist semantics. I then argue that bicontextualism retains the advantages of (orthodox) contextualism and does not give rise to new revenge paradoxes. (shrink)

There is little doubt that a second-order axiomatization of Zermelo-Fraenkel set theory plus the axiom of choice (ZFC) is desirable. One advantage of such an axiomatization is that it permits us to express the principles underlying the first-order schemata of separation and replacement. Another is its almost-categoricity: M is a model of second-order ZFC if and only if it is isomorphic to a model of the form Vκ, ∈ ∩ (Vκ × Vκ) , for κ a strongly inaccessible ordinal.

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)

This paper extracts some of the main theses in the philosophy of mathematics from my book, The Construction of Logical Space. I show that there are important limits to the availability of nominalistic paraphrase functions for mathematical languages, and suggest a way around the problem by developing a method for specifying nominalistic contents without corresponding nominalistic paraphrases. Although much of the material in this paper is drawn from the book — and from an earlier paper — I hope the present (...) discussion will earn its keep by motivating the ideas in a new way, and by suggesting further applications. (shrink)

This paper presents a slightly modified version of the compositional semantics proposed in Events and Semantic Architecture (OUP 2005). Some readers may find this shorter version, which ignores issues about vagueness and causal constructions, easier to digest. The emphasis is on the treatments of plurality and quantification, and I assume at least some familiarity with more standard approaches.

Sellars [1960, ‘Grammar and existence: A preface to ontology’] argues that the truth of a second-order sentence does not incur commitment to there being any sort of abstract entity. This paper begins by exploring the arguments that Sellars offers for the above claim. It then develops those arguments by pointing out places where Sellars has been unclear or ought to have said more. In particular, Sellars's arguments rely on there being a means by which language users could come to understand (...) sentences of a second-order language wherein the truth of second-order existential sentences do not require there to be abstract entities. In addition to this, as Sellars [1979, Naturalism and Ontology] notes, a formal account of quantification is required that does not make use of the apparatus of sequences. Both such a translation of and a formal account of quantification are provided in this article. (shrink)

From early work of N. Goodman to recent approaches by H. Field and D. Lewis, there have been attempts to combine 2nd order languages with calculi of individuals. This paper is a contribution, containing basic definitions and distinctions and some metatheorems, to the development of a general metatheory of such theories.

A dilemma put forward by Schein (1993) and Rayo (2002) suggests that, in order to characterize the semantics of plurals, we should not use predicate logic, but non-singular logic, a formal language whose terms may refer to several things at once. We show that a similar dilemma applies to mass nouns. If we use predicate logic and sets, we arrive at a Russellian paradox when characterizing the semantics of mass nouns. Likewise, a semantics of mass nouns based upon predicate logic (...) and mereological sums is too weak, since it cannot characterize the “intermediary” construals that sentences containing mass nouns may receive. We then develop an account where mass nouns are treated as non-singular terms, which may refer to several things at once. This semantics is faithful to the intuition that, if there are eight pieces of silverware on a table, the speaker refers to eight things at once when he says: “The silverware that is on the table comes from Italy”. We show that this account provides a satisfactory semantics for a wide range of sentences, including cases often seen as difficult, like “The gold on the table weighs seven ounces” (Bunt 1985) and “All phosphorus is either red or black” (Roeper 1983). (shrink)

It is a commonplace of set theory to say that there is no set of all well-orderings nor a set of all sets. We are implored to accept this due to the threat of paradox and the ensuing descent into unintelligibility. In the absence of promising alternatives, we tend to take up a conservative stance and tow the line: there is no universe. In this paper, I am going to challenge this claim by taking seriously the idea that we can (...) talk about the collection of all the sets and many more collections beyond that. A method of articulating this idea is offered through an indefinitely extending hierarchy of set theories. It is argued that this approach provides a natural extension to ordinary set theory and leaves ordinary mathematical practice untouched. (shrink)

Anyone who admits the existence of composite objects allows a certain kind of coincidence, coincidence of a thing with its parts. I argue here that a similar sort of coincidence, coincidence of a thing with the stuff that constitutes it, should be equally acceptable. Acknowledgement of this is enough to solve the traditional problem of the coincidence of a statue and the clay or bronze it is made of. In support of this, I offer some principles for the persistence of (...) stuff that are general, not relying on the particular features of a kind of stuff in the way that principles for the persistence for a thing would. This provides a non-arbitrary grounding for stuff that is independent of the conditions on the nature and persistence of things the stuff composes. The principles also provide a general basis for responding to other questions about coincident stuff. (shrink)

There is no preferred reduction of number theory to set theory. Nonetheless, we confidently accept axioms obtained by substituting formulas from the language of set theory into the induction axiom schema. This is only possible, it is argued, because our acceptance of the induction axioms depends solely on the meanings of aritlunetical and logical terms, which is only possible if our 'intended models' of number theory are standard. Similarly, our acceptance of the second-order natural deduction rules depends solely on the (...) meanings of the logical terms, which implies, it is argued, that our second-order quantifiers have to be standard. (shrink)

George Boolos (1984, 1985) has extensively investigated plural quantifi- cation, as found in such locutions as the Geach-Kaplan sentence There are critics who admire only one another, and he found that their logic cannot be adequately formalized within the first-order predicate calculus. If we try to formalize the sentence by a paraphrase using individual variables that range over critics, or over sets or collections or fusions of critics, we misrepresent its logical structure. To represent plural quantification adequately requires the logical (...) resources of the full second-order predicate calculus. (shrink)

where ‘aa’ is a plural term, and ‘F’ a plural predicate. Following George Boolos (1984) and others, many philosophers and logicians also think that plural expressions should be analysed as not introducing any new ontological commitments to some sort of ‘plural entities’, but rather as involving a new form of reference to objects to which we are already committed (for an overview and further details, see Linnebo 2004). For instance, the plural term ‘aa’ refers to Alice, Bob and Charlie simultaneously, (...) and the plural predicate ‘F’ is true of some things just in case these things cooperate. A natural question that arises is whether the step from the singular to the plural can be iterated. Are there terms that stand to ordinary plural terms the way ordinary plural terms stand to singular terms? Let’s call such terms superplural. A superplural term would thus, loosely speaking, refer to several ‘pluralities’ at once, much as an ordinary plural term refers to several objects at once.1 Further, let’s call a predicate superplural if it can be predicated of superplural terms. It is reasonably straightforward to devise a formal logic of superplural terms, superplural predicates, and even superplural quantifiers (see Rayo 2006). But does this formal logic reflect any features of natural languages? In particular, does ordinary English contain superplural terms and predicates? The purpose of this article is to address these questions. We examine some earlier arguments for the existence of superplural expressions in English and find them to be either.. (shrink)

This paper criticizes George Boolos's famous use of plural quantification to argue that monadic second-order logic is pure logic. I deny that plural quantification qualifies as pure logic and express serious misgivings about its alleged ontological innocence. My argument is based on an examination of what is involved in our understanding of the impredicative plural comprehension schema.

David Lewis has a general recipe for analysis: the Canberra Plan. His analyses of mind, color, and value all proceed according to the plan. What is curious is that his analysis of causation – one of his seminal analyses – doesn't. It doesn't and according to Lewis it can't. Lewis has two objections against using the Canberra Plan to analyze causation. After presenting Lewis' objections I argue that they both fail. I then draw some lessons from their failure.

Despite the renown of ‘On Denoting’, much criticism has ignored or misconstrued Russell's treatment of scope, particularly in intensional, but also in extensional contexts. This has been rectified by more recent commentators, yet it remains largely unnoticed that the examples Russell gives of scope distinctions are questionable or inconsistent with his own philosophy. Nevertheless, Russell is right: scope does matter in intensional contexts. In Principia Mathematica, Russell proves a metatheorem to the effect that the scope of a single occurrence of (...) a description in an extensional context does not matter, provided existence and uniqueness conditions are satisfied. But attempts to eliminate descriptions in more complicated cases may produce an analysis with more occurrences of descriptions than featured in the analysand. Taking alternation and negation to be primitive (as in the first edition of Principia), this can be resolved, although the proof is non-trivial. Taking the Sheffer stroke to be primitive (as proposed by Russell in the second edition), with bad choices of scope the analysis fails to terminate. (shrink)

In this paper I propose a novel treatment of generic sentences, which proceeds by means of different levels of analysis. According to this account, all generic sentences (I-generics and D-generics alike) are initially treated in a uniform manner, as involving higher-order predication (following the work of George Boolos, James Higginbotham and Barry Schein on plurals). Their non-uniform character, however, re-emerges at subsequent levels of analysis, when the higher-order predications of the first level are cashed out in terms of quantification over (...) individuals: this last step, I suggest, involves knowledge concerning the lexical meaning of the predicates in question. (shrink)

When some objects are the parts of another object, they compose that object and that object is composite. This article is intended as an introduction to the central questions about composition and a highly selective overview of various answers to those questions. In §1, we review some formal features of parthood that are important for understanding the nature of composition. In §2, we consider some answers to the question: which pluralities of objects together compose something? As we will see, the (...) dominant answers are all of them and none of them. In §§3-4, we examine one of the main arguments that has driven philosophers to these extreme answers: the argument from vagueness. In §5, we turn to the question of whether composition is unique: is it sometimes the case that some things compose more than one thing? Finally, in §6, we turn from the question of which composites exist to the question of which composites exist fundamentally. (shrink)

In 1903, in _The Principles of Mathematics_ (_PoM_), Russell endorsed an account of classes whereupon a class fundamentally is to be considered many things, and not one, and used this thesis to explicate his first version of a theory of types, adding that it formed the logical justification for the grammatical distinction between singular and plural. The view, however, was short-lived; rejected before _PoM_ even appeared in print. However, aside from mentions of a few misgivings, there is little evidence about (...) why he abandoned this view. In this paper, I speculate as to what those reasons were, and evaluate them in light of recent work and interest in plural logic. (shrink)

Part 1 sets out the logical/semantical background to ‘On Denoting’, including an exposition of Russell's views in Principles of Mathematics, the role and justification of Frege's notorious Axiom V, and speculation about how the search for a solution to the Contradiction might have motivated a new treatment of denoting. Part 2 consists primarily of an extended analysis of Russell's views on knowledge by acquaintance and knowledge by description, in which I try to show that the discomfiture between Russell's semantical and (...) epistemological commitments begins as far back as 1903. I close with a non-Russellian critique of Russell's views on how we are able to make use of linguistic representations in thought and with the suggestion that a theory of comprehension is needed to supplement semantic theory. (shrink)

Peter Geach proposed a substitutional construal of quantification over thirty years ago. It is not standardly substitutional since it is not tied to those substitution instances currently available to us; rather, it is pegged to possible substitution instances. We argue that (i) quantification over the real numbers can be construed substitutionally following Geach's idea; (ii) a price to be paid, if it is that, is intuitionism; (iii) quantification, thus conceived, does not in itself relieve us of ontological commitment to real (...) numbers. (shrink)

This paper develops a formal system, consisting of a language and semantics, called serial logic ( SL ). In rough outline, SL permits quantification over, and reference to, some finite number of things in an order , in an ordinary everyday sense of the word “order,” and superplural quantification over things thus ordered. Before we discuss SL itself, some mention should be made of an issue in philosophical logic which provides the background to the development of SL , and with (...) respect to which I wish to contend that the system permits progress. (shrink)

Mereology, the theory of parts and wholes, is sometimes used as a framework for categorisation because it is regarded as ontologically innocent in the sense that the mereological fusion of some entities is nothing over and above the entities. In this paper it is argued that an adequate answer to the question of whether the thesis of the ontological innocence of mereology holds relies crucially on the underlying theory of reference. It is then shown that upholding the thesis comes at (...) high costs, viz. at the cost of a quite strong logical background theory or at paradoxical ways of predicating and counting. (shrink)

Draft (Version 1.1, October 2007): (PDF file) A reply to Jason Stanley’s Analysis criticism of my interest-relative view on vagueness.

Jaakko Hintikka has argued that ordinary first-order logic should be replaced byindependence-friendly first-order logic, where essentially branching quantificationcan be represented. One recurring criticism of Hintikka has been that Hintikka'ssupposedly new logic is equivalent to a system of second-order logic, and henceis neither novel nor first-order. A standard reply to this criticism by Hintikka andhis defenders has been to show that given game-theoretic semantics, Hintikka'sbranching quantifiers receive the exact same treatment as the regular first-orderones. We develop a different reply, based around (...) considerations concerning thenature of logic. In particular, we argue that Hintikka's logic is the logic that bestrepresents the language fragment standard first-order logic is meantto represent. Therefore it earns its keep, and is also properly regarded as first-order. (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)