Ordinary English contains different forms of quantification over objects. In addition to the usual singular quantification, as in 'There is an apple on the table', there is plural quantification, as in 'There are some apples on the table'. Ever since Frege, formal logic has favored the two singular quantifiers ∀x and ∃x over their plural counterparts ∀xx and ∃xx (to be read as for any things xx and there are some things xx). But in recent decades it has been argued (...) that we have good reason to admit among our primitive logical notions also the plural quantifiers ∀xx and ∃xx. More controversially, it has been argued that the resulting formal system with plural as well as singular quantification qualifies as ‘pure logic’; in particular, that it is universally applicable, ontologically innocent, and perfectly well understood. In addition to being interesting in its own right, this thesis will, if correct, make plural quantification available as an innocent but extremely powerful tool in metaphysics, philosophy of mathematics, and philosophical logic. For instance, George Boolos has used plural quantification to interpret monadic second-order logic and has argued on this basis that monadic second-order logic qualifies as “pure logic.” Plural quantification has also been used in attempts to defend logicist ideas, to account for set theory, and to eliminate ontological commitments to mathematical objects and complex objects. (shrink)

In The Principles of Mathematics, Russell writes: Whatever may be an object of thought, or may occur in any true or false proposition, or can be counted as one, I call a term. This, then, is the widest word in the philosophical vocabulary. I shall use as synonymous with it the words unit, individual and entity. The first two emphasize the fact that every term is one, while the third is derived from the fact that every term has being, i.e. (...) is in some sense. A man, a moment, a number, a class, a relation, a chimera, or anything else that can be mentioned, is sure to be a term'. Now in this remark, a certain extremely general, topic-neutral use of ‘object’ is singled out, a use in which the expression is treated as equivalent to (equally neutral) uses of ‘term’, ‘entity’, ‘unit’, ‘individual’, and ‘thing’. In addition, a claim is made to the effect that the content of the expression, with its emphasis on one, is such as to be adequate to comprehend the sum-total of existence, to include whatever there may be. The paper addresses the question of what this claim means, and whether it might be true. (shrink)

Eternalists say that non-present entities (for instance dinosaurs) exist; presentists say that they do not. But some sceptics deny that this debate is genuine, claiming that presentists simply represent eternalists' quantifiers over non-present entities in different notation. This scepticism may be refuted on purely logical grounds: one of the leading candidate ‘presentist quantifiers’ over non-present things has the inferential role of a quantifier. The dispute over whether non-present objects exist is as genuine and non-verbal as the dispute over whether there (...) is life on other planets. (shrink)

A number of authors have recently weighed in on the issue of whether it is coherent to have bound variables that range over absolutely everything. Prima facie, it is difficult, and perhaps impossible, to coherently state the “relativist” position without violating it. For example, the relativist might say, or try to say, that for any quantifier used in a proposition of English, there is something outside of its range. What is the range of this quantifier? Or suppose we ask the (...) relativist if there are some things that cannot appear in the range of any bound variable. The likely response would be along these lines: “No. For each object o, it possible to include o in the range of quantifiers, but one cannot quantify over everything at once.” This sentence contains unrestricted quantifiers, or so it seems, pending some clever move from a relativist. On the other hand, in the context of set theory, the reasoning behind the Burali-Forti paradox strongly suggests that there are well-orderings strictly longer than the collection of all ordinals. And set theorists regularly do transfinite recursions and transfinite reductions along such well-orderings. The relativist simply points out that one can always define new ordinals, and thus expand the range of one’s bound variables. The purpose of this paper is to explore the iterative framework, proposed in Zermelo’s 1930 paper, “Über Grenzzahlen und Mengenbereiche” (“On boundary numbers and domains of sets”), in order to shed light on these issues, and see what is involved in resolving them. (shrink)

Actualists routinely characterize their view by means of the slogan, “Everything is actual.” They say that there aren’t any things that exist but do not actually exist—there aren’t any “mere possibilia.” If there are any things that deserve the label ‘possible world’, they are just actually existing entities of some kind—maximally consistent sets of sentences, or maximal uninstantiated properties, or maximal possible states of affairs, or something along those lines. Possibilists, in contrast, do think that there are mere possibilia, that (...) there are things that are not actual. They think that more exists than what actually exists. All I have done so far, though, is rephrase the slogan in various ways. To say that everything is actual is precisely to say that there are no things that do not actually exist, which is precisely to say that there are no mere possibilia, and which is also precisely to say that we cannot sep- arately quantify over what exists and what is actual. These claims all amount to the same thing. But what is that, exactly? What on earth does it mean to say that everything is actual, that there are no mere possibilia, and so on? What does the actualist slogan really come to? I think the literature is far from clear on this point, and that people work themselves into unnecessary muddles because of it. Indeed, certain confusions that I shall discuss in the first half of this article seem to be on the rise. It is high time to lay out the issues and the choice points as clearly as possible. There are two primary choices to be made; there are two axes along which versions of actualism can vary. One choice has to do with how to treat claims about things that merely could exist. The other choice has to do with the modal status of the view and of how we should think about the “actual” in actualism. I make no claim that the positions I will eventually endorse are star- tlingly new. I think that most people will agree with the decisions I make at both choice points and will in fact find some bits of this essay obvious. But not everyone agrees with my decisions, and it has been my experience that people differ remarkably about which bits they find obvious—a fact I find rather telling. My goal, then, is to show that the two axes are there and to clarify the consequences of the choices. (shrink)

Inscrutability arguments threaten to reduce interpretationist metasemantic theories to absurdity. Can we find some way to block the arguments? A highly influential proposal in this regard is David Lewis’ ‘ eligibility ’ response: some theories are better than others, not because they fit the data better, but because they are framed in terms of more natural properties. The purposes of this paper are to outline the nature of the eligibility proposal, making the case that it is not ad hoc, but (...) instead flows naturally from three independently motivated elements; and to show that severe limitations afflict the proposal. In conclusion, I pick out the element of the eligibility response that is responsible for the limitations: future work in this area should therefore concentrate on amending this aspect of the overall theory. (shrink)

There are certain metaphysically interesting arguments ‘from vagueness’, for unrestricted mereological composition and for four-dimensionalism, which involve a claim to the effect that idioms for unrestricted quantification are precise. An elaboration of Lewis’ argument for this claim, which assumes the view of vagueness as semantic indecision, is presented. It is argued that the argument also works according to other views on the nature of vagueness, which also require for an expression to be vague that there are different admissible alternatives of (...) the relevant sort, such as epistemicism, as defended by Williamson. Recent attempts to resist the argument are discussed and rejected. (shrink)

In the substitutional framework, validity is truth under all substitutions of the nonlogical vocabulary. I develop a theory where □ is interpreted as substitutional validity. I show how to prove soundness and completeness for common modal calculi using this definition.

Conspicuously missing some planks, a latter-day Ship of Theseus catches the eye when one picks up this Book of Three Authors. Teasingly, differentiated preposit.

This paper revisits a challenge for contextualist approaches to paradoxes such as the Liar paradox and Russell’s paradox. Contextualists argue that these paradoxes are to be resolved by appeal to context dependence. This can offer some nice and effective ways to avoid paradox. But there is a problem. Context dependence is, at least to begin with, a phenomenon in natural language. Is there really such context dependence as the solutions to paradoxes require, and is it really just a familiar linguistic (...) phenomenon at work? Not so clearly. In earlier work, I argued that the required form of context dependence does not look like our most familiar instances of context dependence in natural language. I called this extraordinary context dependence. In this paper, I shall explore, somewhat tentatively, a way that we can see the context dependence needed to address paradoxes as not so extraordinary. Doing so will also allow us to connect thinking about the context dependence of quantifier domains with some interesting ideas about the distinctive semantic properties of certain quantifiers. (shrink)

In his important book The Construction of Logical Space, Agustín Rayo lays out a distinctive metametaphysical view and applies it fruitfully to disputes concerning ontology and concerning modality. In this article, I present a number of criticisms of the view developed, mostly focusing on the underlying metametaphysics and Rayo’s claims on its behalf.

I develop a novel argument against the claim that ordinals are sets. In contrast to Benacerraf’s antireductionist argument, I make no use of covert epistemic assumptions. Instead, my argument uses considerations of ontological dependence. I draw on the datum that sets depend immediately and asymmetrically on their elements and argue that this datum is incompatible with reductionism, given plausible assumptions about the dependence profile of ordinals. In addition, I show that a structurally similar argument can be made against the claim (...) that cardinals are sets. (shrink)

Truthmaker theory has been used to argue for substantial conclusions about the categorial structure of the world, in particular that states of affairs are needed to play the role of truthmakers. In this paper, I argue that closely considering the role of aboutness in truthmaking, that is considering what truthbearers are about, yields the result that there is no good truthmaker-based reason to think that truthmakers must be states of affairs understood as existing entities, whether complex or simple. First, I (...) introduce an aboutness-based account of truthmaking as a metaphysically modest alternative to the orthodox necessitarian account of truthmaking. Second, I discuss the distinction between states and events that has been made on the basis of linguistic evidence regarding aspectual markers and nominalisation. I argue that the modest approach to truthmaking allows us to accept that there is a real distinction between states and events without requiring that the distinction is ontologically substantial. Specifically, what we are talking about with state-truthbearers really differs from what we are talking about with event-truthbearers, but this difference need not be understood as a difference in kinds of entities. Because of its overall modesty, this is a theoretically virtuous result. (shrink)

We defend the thesis that every necessarily true proposition is always true. Since not every proposition that is always true is necessarily true, our thesis is at odds with theories of modality and time, such as those of Kit Fine and David Kaplan, which posit a fundamental symmetry between modal and tense operators. According to such theories, just as it is a contingent matter what is true at a given time, it is likewise a temporary matter what is true at (...) a given possible world; so a proposition that is now true at all worlds, and thus necessarily true, may yet at some past or future time be false in the actual world, and thus not always true. We reconstruct and criticize several lines of argument in favor of this picture, and then argue against the picture on the grounds that it is inconsistent with certain sorts of contingency in the structure of time. (shrink)

In this paper, we offer a contribution to the discussion of one of the most important objections against a relativist position in the absolute generality debate. The inexpressibility objection accuses the generality-relativist of not being able to coherently express her own position. First, we examine Glanzberg’s attempt to reply to this objection and we show that it fails. Second, we study the prospects of generalizing the relativist position. In particular, we analyze Fine’s and Linnebo’s modal approaches and we argue that, (...) even though they are able to coherently express one of the core ideas of relativism while avoiding the inexpressibility objection, there is an important sense in which they are no longer relativist positions. Third, while strengthening the idea that the inexpressibility objection does succeed, we argue that this is no guarantee of the falsity of relativism. Relativism may be inexpressible but true. However, we stress that even if the inexpressibility objection does not supply a definitive, knock-down objection against relativism, if we want to discuss relativism in a rational way, the objection offers a compelling reason not to embrace generality-relativism. (shrink)

This paper presents a new approach to the class-theoretic paradoxes. In the first part of the paper, I will distinguish classes from sets, describe the function of class talk, and present several reasons for postulating type-free classes. This involves applications to the problem of unrestricted quantification, reduction of properties, natural language semantics, and the epistemology of mathematics. In the second part of the paper, I will present some axioms for type-free classes. My approach is loosely based on the Gödel–Russell idea (...) of limited ranges of significance. It is shown how to derive the second-order Dedekind–Peano axioms within that theory. I conclude by discussing whether the theory can be used as a solution to the problem of unrestricted quantification. In an appendix, I prove the consistency of the class theory relative to Zermelo–Fraenkel set theory. (shrink)

A response to critics, Douglas Patterson and Mark Textor, on Künne's modest theory of truth in *Conceptions of Truth*.

This paper examines "moderate modal skepticism", a form of skepticism about metaphysical modality defended by Peter van Inwagen in order to blunt the force of certain modal arguments in the philosophy of religion. Van Inwagen’s argument for moderate modal skepticism assumes Yablo's (1993) influential world-based epistemology of possibility. We raise two problems for this epistemology of possibility, which undermine van Inwagen's argument. We then consider how one might motivate moderate modal skepticism by relying on a different epistemology of possibility, which (...) does not face these problems: Williamson’s (2007: ch. 5) counterfactual-based epistemology. Two ways of motivating moderate modal skepticism within that framework are found unpromising. Nevertheless, we also find a way of vindicating an epistemological thesis that, while weaker than moderate modal skepticism, is strong enough to support the methodological moral van Inwagen wishes to draw. (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)

According to the singular conception of reality, there are objects and there are singular properties, i.e. properties that are instantiated by objects separately. It has been argued that semantic considerations about plurals give us reasons to embrace a plural conception of reality. This is the view that, in addition to singular properties, there are plural properties, i.e. properties that are instantiated jointly by many objects. In this article, I propose and defend a novel semantic account of plurals which dispenses with (...) plural properties and thus undermines the semantic argument in favor of the plural conception of reality. (shrink)

In "Two sorts of Naturalism" John McDowell is sketching his own sort of naturalism in ethics as an alternative to "bald naturalism". In this paper I distinguish materialist, idealist and absolute conceptions of nature and of naturalism in order to provide a framework for a clearer understanding of what McDowell’s own naturalism amounts to. I argue that nothing short of an absolute naturalism will do for a number of McDowell's own purposes, but that it is far from obvious that this (...) is his position. If he directly denies that it is, he seems to be left with some rather awkward choices. (shrink)

This paper examines the conceptions of logic from Leibniz, Hume, Kant, Frege, Wittgenstein and Ayer, and regards the six philosophers as the representatives of logical exceptionalism. From their standpoints, this paper refines the tenets of logical exceptionalism as follows: logic is exceptional to all other sciences because of four reasons: (i) logic is formal, neutral to any domain and any entities, and general; (ii) logical truths are made true by the meanings of logical constants they contain or by logicians' rational (...) insight to consequence relations; (iii) logical truths are analytical, a prior and necessary, so not‐revisable; and (iv) logical laws are normative for how to correctly think. However, logical exceptionalism has encountered difficult open problems: What are logical constants? How to justify basic laws of logic? How are logical laws accessible to us? How to explain the reasonability of rival logics and select from them? How to explain the universal applicability of logical laws? How to explain the normativity of logical laws for correct thinking? This paper concludes that logical anti‐exceptionalism is more hopeful to successfully answer these questions than logical exceptionalism. (shrink)

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)

ABSTRACTAccording to a famous argument by Dummett, the concept of set is indefinitely extensible, and the logic appropriate for reasoning about the instances of any such concept is intuitionistic, not classical. But Dummett's argument is widely regarded as obscure. This note explains how the final chapter of Rumfitt's important new book advances our understanding of Dummett's argument, but it also points out some problems and unanswered questions. Finally, Rumfitt's reconstruction of Dummett's argument is contrasted with my own preferred alternative.

In this paper, I motivate a modal account of propositions on the basis of an iterative conception of propositions. As an application, I suggest that the account provides a satisfying solution to the Russell-Myhill paradox. The account is in the spirit of recently developed modal accounts of sets motivated on the basis of the iterative conception of sets.

Recently, some philosophers have argued that we should take quantification of any (finite) order to be a legitimate and irreducible, sui generis kind of quantification. In particular, they hold that a semantic theory for higher-order quantification must itself be couched in higher-order terms. Øystein Linnebo has criticized such views on the grounds that they are committed to general claims about the semantic values of expressions that are by their own lights inexpressible. I show that Linnebo’s objection rests on the assumption (...) of a notion of semantic value or contribution which both applies to expressions of any order, and picks out, for each expression, an extra-linguistic correlate of that expression. I go on to argue that higher-orderists can plausibly reject this assumption, by means of a hierarchy of notions they can use to describe the extra-lingustic correlates of expressions of different orders. (shrink)

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)

Plural logic is widely assumed to have two important virtues: ontological innocence and determinacy. It is claimed to be innocent in the sense that it incurs no ontological commitments beyond those already incurred by the first-order quantifiers. It is claimed to be determinate in the sense that it is immune to the threat of non-standard interpretations that confronts higher-order logics on their more traditional, set-based semantics. We challenge both claims. Our challenge is based on a Henkin-style semantics for plural logic (...) that does not resort to sets or set-like objects to interpret plural variables, but adopts the view that a plural variable has many objects as its values. Using this semantics, we also articulate a generalized notion of ontological commitment which enables us to develop some ideas of earlier critics of the alleged ontological innocence of plural logic. (shrink)

There has been very little discussion of the appropriate principles to govern a modal logic of plurals. What debate there has been has accepted a principle I call (Necinc); informally if this is one of those then, necessarily: this is one of those. On this basis Williamson has criticised the Boolosian plural interpretation of monadic second-order logic. I argue against (Necinc), noting that it isn't a theorem of any logic resulting from adding modal axioms to the plural logic PFO+, and (...) showing that the most obvious formal argument in its favour is question begging. I go on to discuss the behaviour of natural language plurals, motivating a case against (Necinc) by developing a case that natural language plural terms are not de jure rigid designators. The paper concludes by developing a model theory for modal PFO-f which does not validate (Necinc). An Appendix discusses (Necinc) in relation to counterpart theory. Of course, it would be a mistake to think that the rules for "multiple pointing" follow automatically from the rules for pointing proper. Max Black—The Elusiveness of Sets In some influential articles during the 1980s George Boolos proposed an interpretation of monadic second-order logic in terms of plural quantification [4, 5]. One objection to this proposal, pressed by Williamson [22, 456-7], focuses on the modal behaviour of plural variables, arguing that the proposed interpretation yields the wrong results in respect of the modal status of atomic predications. In the present paper I will present this objection and argue against it. In the course of developing the argument, I will have cause to consider the under-investigated question of how a logic for plurals should be extended to incorporate modal operators. (shrink)

Some contemporary semantic views defend an asymmetry thesis concerning defi-nite descriptions and indexicals. Semantically, indexicals are devices of singular refer-ence; they contribute objects to the contents of the speech acts made with utterances including them. Definite descriptions, on the other hand, are generalized quantifiers, behaving roughly the way Russell envisaged in “On Denoting”. The asymmetry thesis depends on the existence of a sufficiently clear-cut distinction between semantics and pragmatics, because indexicals and descriptions are often used in ways that apparently contradict (...) the asymmetry thesis; the semantics/pragmatics distinction is invoked to see behind the appearances. The paper critically examines arguments by Schiffer against the asymmetry thesis, based on referential uses of incomplete descriptions. (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)

It is common for contemporary metaphysical realists to adopt Quine's criterion of ontological commitment while at the same time repudiating his ontological pragmatism. 2 Drawing heavily from the work of others—especially Joseph Melia and Stephen Yablo—I will argue that the resulting approach to meta-ontology is unstable. In particular, if we are metaphysical realists, we need not accept ontological commitment to whatever is quantified over by our best first-order theories.

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

In a recent article, Hofweber presents a new, and surprising, argument for idealism. His argument is surprising because it starts with an apparently innocent premiss from the philosophy of language: that ‘that’-clauses do not refer. I do not think that Hofweber's argument works, and my first aim in this paper is to explain why. However, I agree with Hofweber that what we say about ‘that’-clauses has important metaphysical consequences. My second aim is to argue that, far from leading us into (...) idealism, denying that ‘that’-clauses refer is the first step toward a kind of direct realism about belief. (shrink)

Plural Logic is an extension of First-Order Logic with plural terms and quantifiers. When its plural terms are interpreted as denoting more than one object at once, Plural Logic is usually taken to be ontologically innocent: plural quantifiers do not require a domain of their own, but range plurally over the first-order domain of quantification. Given that Plural Logic is equi-interpretable with Monadic Second-Order Logic, it gives us its expressive power at the low ontological cost of a first-order language. This (...) makes it a valuable tool in various areas of philosophy. Some authors believe that Plural Logic can be extended into an even more expressive logic, Higher-Level Plural Logic, by adding higher-level plural terms and quantifiers to it. The basic idea is that second-level plurals stand to plurals like plurals stand to singulars. Allegedly, Higher-Level Plural Logic enjoys the expressive power of type theory while, again, committing us only to the austere ontology of a first-order language. Were this really the case, Higher-Level Plural Logic would be a very useful tool, extending and strengthening some of the applications of Plural Logic. However, while the notions of plural reference and quantification enjoy widespread acceptance today, their higher-level counterparts have been received with scepticism. The main objection raised against them is that higher-level plural reference is unintelligible. This has been argued, among others, on the grounds that there are no higher-level plurals in natural language and that, if there were any, they could be eliminated. In this thesis, after introducing the debate on plurals in Chapters 1 and 2, I turn to defending the legitimacy of the notion of higher-level plural reference. To this end, in Chapter 3, I present and elucidate the notion. Next, in Chapter 4, I show that some natural languages clearly contain these expressions and that they do so in an ineliminable manner. Finally, in Chapters 5 and 6, I develop a semantics for higher-level plurals that employs only devices previously well-understood by English speakers. To finish, in Chapter 7, I describe an application of Higher-level Plural Logic: a strengthening of the neo-Fregean programme. After describing my proposal, I turn to the issue of the logical status of this formalism and defend an optimistic take on the matter. (shrink)

Historically, the problem of realism has often been viewed as an extension of the problem of the external world. At the same time, the assumption has often been made that realism is appropriate to some domains, while antirealism is appropriate to others. I here sketch a neutral realism that rejects both of these assumptions and thereby allows us to treat the realism debate independently of the debate over naturalism. The starting point of neutral realism is indeed neutral with respect to (...) any metaphysical commitment to the existence of some single totality of objects or facts, or to any unified all-encompassing domain that might be identified with nature. The resulting view is shown to open up new prospects for the treatment of the actuality of ethical or aesthetic values as well as for the question of the metaphysical relationship between concrete and abstract entities. (shrink)

This paper presents a new argument for necessitism, the claim that necessarily everything is necessarily something. The argument appeals to principles about the metaphysics of quantification and predication which are best seen as constraints on reality’s fineness of grain. I give this argument in section 4; the impatient reader may skip directly there. Sections 1-3 set the stage by surveying three other arguments for necessitism. I argue that none of them are persuasive, but I think it is illuminating to consider (...) my argument in light of the others and vice versa. These interconnections should be of interest even to those who reject necessitism; of particular interest may be the new conception of validity proposed in section 5. (shrink)

This paper uses the resources of higher-order logic to articulate a Fregean conception of predicate reference, and of word-world relations more generally, that is immune to the concept horse problem. The paper then addresses a prominent style of expressibility problem for views of broadly this kind, versions of which are due to Linnebo, Hale, and Wright.

Can Quine’s criterion for ontological commitment be comparatively applied across different logics? If so, how? Cross-logical evaluations of discourses are central to contemporary philosophy of mathematics and metaphysics. The focus here is on the influential and important arguments of George Boolos and David Lewis that second-order logic and plural quantification don’t incur additional ontological commitments over and above those incurred by first-order quantifiers. These arguments are challenged by the exhibition of a technical tool—the truncation-model construction of notational equivalents—that compares the (...) ontological role and increased expressive strength of non-first-order ideology to first-order ideology. (shrink)

This paper argues that ‘that’-clauses are not singular terms (without denying that their semantical values are propositions). In its first part, three arguments are presented to support the thesis, two of which are defended against recent criticism. The two good arguments are based on the observation that substitution of ‘the proposition that p’ for ‘that p’ may result in ungrammaticality. The second part of the paper is devoted to a refutation of the main argument for the claim that ‘that’-clauses are (...) singular terms, namely that this claim is needed in order to account for the possibility of quantification into ‘that’-clause position. It is shown that not all quantification in natural languages is quantification into the position of singular terms, but that there is also so-called ‘non-nominal quantification’. A formal analysis of non-nominal quantification is given, and it is argued that quantification into ‘that’-clause position can be treated as another kind non-nominal quantification. (shrink)

Cameron, Eklund, Hofweber, Linnebo, Russell and Sider have written critical essays on my book, The Construction of Logical Space (Oxford: Oxford University Press, 2013). Here I offer some replies.

From 1905–1908 onward, Russell thought that his new ‘substitutional theory’ provided him with the right framework to resolve the set-theoretic paradoxes. Even if he did not finally retain this resolution, the substitutional strategy was instrumental in the development of his thought. The aim of this paper is not historical, however. It is to show that Russell's substitutional insight can shed new light on current issues in philosophy of mathematics. After having briefly expounded Russell's key notion of a ‘structured variable’, I (...) connect it to two ongoing discussions: the Caesar problem, and absolute generality. (shrink)