My thanks to Matti Eklund and Gabriel Uzquiano for their thoughtful and challenging critical essays. In these replies I hope to respond to what I took to be their main points. The focus of their essays is different for the most part, but there is overlap in their discussion of the ineffable. I will thus largely reply to their essays separately, with the exception of the discussion of the ineffable, where I will reply to their points jointly. Let’s start, alphabetically, (...) with Eklund. (shrink)

Philosophy and model theory frequently meet one another. Philosophy and Model Theory aims to understand their interactions -/- Model theory is used in every ‘theoretical’ branch of analytic philosophy: in philosophy of mathematics, in philosophy of science, in philosophy of language, in philosophical logic, and in metaphysics. But these wide-ranging appeals to model theory have created a highly fragmented literature. On the one hand, many philosophically significant mathematical results are found only in mathematics textbooks: these are aimed squarely at mathematicians; (...) they typically presuppose that the reader has a significant background in mathematics; and little clue is given as to their philosophical significance. On the other hand, the philosophical applications of these results are then scattered across disconnected pockets of papers, separated by decades or more. -/- The first aim of Philosophy and Model Theory, then, is to consider the philosophical uses of model theory. On a technical level, we try to show how philosophically significant results connect to one another, and also to state the best version of a result for philosophical purposes. On a philosophical level, we show how similar dialectical situations arise repeatedly, across fragmented debates in varied philosophical areas. -/- The second aim of Philosophy and Model Theory, though, is to consider the philosophy of model theory. Model theory itself is rarely taken as the subject matter of philosophising (contrast this, say, with the philosophy of biology, or the philosophy of set theory). But model theory is a beautiful part of pure mathematics, and worthy of philosophical study in its own right. (shrink)

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)

This paper explores the impact of quantification into predicate position on the metaphysics of properties, arguing that two familiar debates about properties are fundamentally altered by recasting them in a second-order setting. Two theories of properties are outlined, differing over whether the existence of properties is expressed using first-order or second-order quantifiers. It is argued that the second-order theory: provides good reason to regard debate about the locations of properties as contentless; resolves debate about whether properties are particulars or universals (...) in favour of universals. (shrink)

This paper is an investigation of the general logic of "identifications", claims such as 'To be a vixen is to be a female fox', 'To be human is to be a rational animal', and 'To be just is to help one's friends and harm one's enemies', many of which are of great importance to philosophers. I advocate understanding such claims as expressing higher-order identity, and discuss a variety of different general laws which they might be thought to obey. [New version: (...) Nov. 4th, 2016]. (shrink)

How are philosophical questions about what kinds of things there are to be understood and how are they to be answered? This paper defends broadly Fregean answers to these questions. Ontological categories—such as object, property, and relation—are explained in terms of a prior logical categorization of expressions, as singular terms, predicates of varying degree and level, etc. Questions about what kinds of object, property, etc., there are are, on this approach, reduce to questions about truth and logical form: for example, (...) the question whether there are numbers is the question whether there are true atomic statements in which expressions function as singular terms which, if they have reference at all, stand for numbers, and the question whether there are properties of a given type is a question about whether there are meaningful predicates of an appropriate degree and level. This approach is defended against the objection that it must be wrong because makes what there depend on us or our language. Some problems confronting the Fregean approach—including Frege’s notorious paradox of the concept horse—are addressed. It is argued that the approach results in a modest and sober deflationary understanding of ontological commitments. (shrink)

The identity predicate can be defined using second-order quantification: a=b =df ∀F(Fa↔Fb). Less familiarly, a dyadic sentential operator analogous to the identity predicate can be defined using third-order quantification: ϕ≡ψ =df ∀X(Xϕ↔Xψ), where X is a variable of the same syntactic type as a monadic sentential operator. With this notion in view, it is natural to ask after general principles governing its application. More grandiosely, how fine-grained is reality? -/- I will argue that reality is not structured in anything like (...) the way that the sentences we use to talk about it are structured. I do so by formulating a higher-order analogue of Russell’s paradox of structured propositions. I then relate this argument to the Frege-Russell correspondence. When confronted with the alleged paradox, Frege agreed that reality was not structured, but maintained that propositions (i.e. thoughts) were structured all the same. Russell replied that his paradox showed Frege’s theory of structured thoughts to be inconsistent, to which Frege replied that Russell’s argument failed to heed the distinction between sense and reference. Most recent commentators have sided with Russell. In defense of Frege, I establish the consistency of one version of his rejoinder. I then consider and reject some ways of resisting the argument against a structured conception of reality. I conclude that, if propositions are structured, this is because they correspond not to distinctions in reality, but rather to ways in which those distinctions can be represented. (shrink)

Translation of Frege's 'Über Begriff und Gegenstand' (1892). Translation by Peter Geach, revised by Max Black.

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)

Standard Type Theory, STT, tells us that b^n(a^m) is well-formed iff n=m+1. However, Linnebo and Rayo have advocated the use of Cumulative Type Theory, CTT, has more relaxed type-restrictions: according to CTT, b^β(a^α) is well-formed iff β > α. In this paper, we set ourselves against CTT. We begin our case by arguing against Linnebo and Rayo’s claim that CTT sheds new philosophical light on set theory. We then argue that, while CTT ’s type-restrictions are unjustifiable, the type-restrictions imposed by (...) STT are justified by a Fregean semantics. What is more, this Fregean semantics provides us with a principled way to resist Linnebo and Rayo’s Semantic Argument for CTT. We end by examining an alternative approach to cumulative types due to Florio and Jones; we argue that their theory is best seen as a misleadingly formulated version of STT. (shrink)

The form of nominalism known as 'mathematical fictionalism' is examined and found wanting, mainly on grounds that go back to an early antinominalist work of Rudolf Carnap that has unfortunately not been paid sufficient attention by more recent writers.

This book does three main things. First, it defends mathematical platonism against the main objections to that view (most notably, the epistemological objection and the multiple-reductions objection). Second, it defends anti-platonism (in particular, fictionalism) against the main objections to that view (most notably, the Quine-Putnam indispensability objection and the objection from objectivity). Third, it argues that there is no fact of the matter whether abstract mathematical objects exist and, hence, no fact of the matter whether platonism or anti-platonism is true.

The main task of this paper is to defend anti-platonism by providing an anti-platonist (in particular, a fictionalist) account of the indispensable applications of mathematics to empirical science.

In my reply to michael devitt, It is argued, First, That quine fails to appreciate the force of plato's "one over many" argument for universals. It is argued, Second, That quine's failure springs in part at least from his doctrine of ontological commitment: from the view that predicates need not be treated with ontological seriousness. Finally, An attempt is made to blunt the force of devitt's contention that realists cannot give a coherent explanation of the way that universals stand to (...) particulars. (shrink)

This introduction to mathematical logic starts with propositional calculus and first-order logic. Topics covered include syntax, semantics, soundness, completeness, independence, normal forms, vertical paths through negation normal formulas, compactness, Smullyan's Unifying Principle, natural deduction, cut-elimination, semantic tableaux, Skolemization, Herbrand's Theorem, unification, duality, interpolation, and definability. The last three chapters of the book provide an introduction to type theory (higher-order logic). It is shown how various mathematical concepts can be formalized in this very expressive formal language. This expressive notation facilitates proofs (...) of the classical incompleteness and undecidability theorems which are very elegant and easy to understand. The discussion of semantics makes clear the important distinction between standard and nonstandard models which is so important in understanding puzzling phenomena such as the incompleteness theorems and Skolem's Paradox about countable models of set theory. Some of the numerous exercises require giving formal proofs. A computer program called ETPS which is available from the web facilitates doing and checking such exercises. Audience: This volume will be of interest to mathematicians, computer scientists, and philosophers in universities, as well as to computer scientists in industry who wish to use higher-order logic for hardware and software specification and verification. (shrink)

On reading the last sentence, did you interpret me as saying falsely that everything — everything in the entire universe — was packed into my carry-on baggage? Probably not. In ordinary language, ‘everything’ and other quantifiers (‘something’, ‘nothing’, ‘every dog’, ...) often carry a tacit restriction to a domain of contextually relevant objects, such as the things that I need to take with me on my journey. Thus a sentence of the form ‘Everything Fs’ is true as uttered in a (...) context C if and only if everything that is relevant in C satisfies the predicate ‘F’; ‘everything’ ranges just over the contextually relevant things. Such generality is restricted in a context-relative way. (shrink)

Existence questions have been topics for heated debates in metaphysics, but this book argues that they can often be answered easily, by trivial inferences from uncontroversial premises. This 'easy' approach to ontology leads to realism about disputed entities, and to the view that metaphysical disputes about existence questions are misguided.

I articulate and defend a necessary and sufficient condition for predication. The condition is that a term or term-occurrence stands in the relation of ascription to its designatum, ascription being a fundamental semantic relation that differs from reference. This view has dramatically different semantic consequences from its alternatives. After outlining the alternatives, I draw out these consequences and show how they favour the ascription view. I then develop the view and elicit a number of its virtues.

In this paper I argue that Frege’s concept horse paradox is not easily avoided. I do so without appealing to Wright’s Reference Principle. I then use this result to show that Hale and Wright’s recent attempts to avoid this paradox by rejecting or otherwise defanging the Reference Principle are unsuccessful.

Beliefs have what I will call ‘propositional content’. A belief is always a belief that so-and-so: a belief that grass is green, or a belief that snow is white, or whatever. Other things have propositional content too, such as sentences, judgments and assertions. The Standard View amongst philosophers is that what it is to have a propositional content is to stand in an appropriate relation to a proposition. Moreover, on this view, propositions are objects, i.e. the kind of thing you (...) can refer to with singular terms. For example, on the Standard View, we should parse the sentence ‘Simon believes that Sharon is funny’ as: [Simon] believes [that Sharon is funny]; ‘Simon’ is a term referring to a thinking subject, ‘that Sharon is funny’ is a term referring to a proposition, and ‘x believes y’ is a dyadic predicate expressing the believing relation. In this paper, I argue against the Standard View. This is how I think we should parse ‘Simon believes that Sharon is funny’: [Simon] believes that [Sharon is funny]; here we have a singular term, ‘Simon’, a sentence ‘Sharon is funny’, and a ‘prenective’ joining them together, ‘x believes that p’. On this Prenective View, we do not get at the propositional content of someone’s belief by referring to a reified proposition with a singular term; we simply use the sentence ‘Sharon is funny’ to express that content for ourselves. I argue for the Prenective View in large part by showing that an initially attractive version of the Standard View is actually vulnerable to the same objection that Wittgenstein used against Russell’s multiple-relation theory of judgment. (shrink)

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)

Chapter One: Constructions. Entities, constructions, and functions When one travels from Los Angeles to New York, going, say, by way of St. Louis, Chicago, ...

Fictionalism has long presented an attractive alternative to both heavy-duty realist and simple eliminativist views about entities such as properties, propositions, numbers, and possible worlds. More recently, a different alternative to these traditional views has been gaining popularity: a form of deflationism that holds that trivial arguments may lead us from uncontroversial premisses to conclude that the relevant entities exist — but where commitment to the entities is a trivial consequence of other claims we accept, not a posit to explain (...) what makes the relevant claims true. The deflationist’s trivial arguments, however, have been attacked by fictionalists, who suggest that the ontological conclusions we get from these arguments should not be taken as serious ontological assertions at all, but rather as implicitly in the context of a fiction or simulation. This paper examines the fictionalist’s criticisms of ‘easy’ arguments for numbers, properties, and other entities, and concludes that they beg the question against the deflationist and so do not undermine the deflationist’s position. Close attention to the argument also reveals a crucial disanalogy between overtly fictional discourse and discourse about numbers, properties, and so on, which undermines the case for fictionalism. Finally, I argue that the motivations for fictionalism (particularly those based in its ability to offer a good account of the discourse) are served as well or better by deflationism. Overall, this gives us reason to think that deflationism may provide a preferable approach for those looking for an alternative to both traditional realism and traditional eliminativism. (shrink)

Timothy Williamson gives an original and provocative treatment of deep metaphysical questions about existence, contingency, and change, using the latest resources of quantified modal logic. Contrary to the widespread assumption that logic and metaphysics are disjoint, he argues that modal logic provides a structural core for metaphysics.

Fictionalist approaches to ontology have been an accepted part of philosophical methodology for some time now. On a fictionalist view, engaging in discourse that involves apparent reference to a realm of problematic entities is best viewed as engaging in a pretense. Although in reality, the problematic entities do not exist, according to the pretense we engage in when using the discourse, they do exist. In the vocabulary of Burgess and Rosen (1997, p. 6), a nominalist construal of a given discourse (...) is revolutionary just in case it involves a “reconstruction or revision” of the original discourse. Revolutionary approaches are therefore prescriptive. In contrast, a nominalist construal of a given discourse is hermeneutic just in case it is a nominalist construal of a discourse that is put forth as a hypothesis about how the discourse is in fact used; that is, hermeneutic approaches are descriptive. I will adopt Burgess and Rosen’s terminology to describe the two different spirits in which a fictionalist hypothesis in ontology might be advanced. Revolutionary fictionalism would involve admitting that while the problematic discourse does in fact involve literal reference to nonexistent entities, we ought to use the discourse in such a way that the reference is simply within the pretense. The hermeneutic fictionalist, in contrast, reads fictionalism into our actual use of the problematic discourse. According to her, normal use of the problematic discourse involves a pretense. According to the pretense, and only according to the pretense, there exist the objects to which the discourse would commit its users, were no pretense involved. My purpose in this paper is to argue that hermeneutic fictionalism is not a viable strategy in ontology. My argument proceeds in two steps. First, I discuss in detail several problematic consequences of any interesting application of hermeneutic fictionalism. Of course, if there is good evidence that hermeneutic fictionalism is correct in some cases, then some of these drastic consequences would have to be accepted.. (shrink)

Higher-order realists about properties express their view that there are properties with the help of higher-order rather than first-order quantifiers. They claim two types of advantages for this way of formulating property realism. First, certain gridlocked debates about the nature of properties, such as the immanentism versus transcendentalism dispute, are taken to be dissolved. Second, a further such debate, the tropes versus universals dispute, is taken to be resolved. In this paper I first argue that higher-order realism does not in (...) fact resolve the tropes versus universals dispute. In a constructive spirit, I then develop higher-order realism in a way that leads to a dissolution, rather than a resolution, of this dispute too. (shrink)

Stephen Schiffer presents a groundbreaking account of meaning and belief, and shows how it can illuminate a range of crucial problems regarding language, mind, knowledge, and ontology. He introduces the new doctrine of 'pleonastic propositions' to explain what the things we mean and believe are. He discusses the relation between semantic and psychological facts, on the one hand, and physical facts, on the other; vagueness and indeterminacy; moral truth; conditionals; and the role of propositional content in information acquisition and explanation. (...) This radical new treatment of meaning will command the attention of everyone who works on fundamental questions about language, and will attract much interest from other areas of philosophy. (shrink)

In this article I address the Problem of Universals by answering questions about what facts a solution to the Problem of Universals should explain and how the explanation should go. I argue that a solution to the Problem of Universals explains the facts the Problem of Universals is about by giving the truthmakers (as opposed to the conceptual content and the ontological commitments) of the sentences stating those facts. I argue that the sentences stating the relevant facts are those like (...) 'a has the property F', that is, sentences stating that a particular has a certain property. Finally I show how answering these questions in this way transforms the Problem of Universals, traditionally conceived as the One over Many, that is, the problem of explaining how different particulars can have the same properties, into the Many over One, that is, the problem of explaining how the same particular can have different properties. The Problem of Universals is the problem of the Many over One. (shrink)

ABSTRACTFregeans hold that predicates denote things, albeit things different in kind from what singular terms denote. This leads to a familiar problem: it seems impossible to say what any given predicate denotes. One strategy for avoiding this problem reduces the Fregean position to form of nominalism. I develop an alternative strategy that lets the Fregean hold on to the view that predicate denote things by reconceiving the nature of singular denotation and of Fregean objects.

If I say that Alice is everything Oscar hopes to be, I seem to be quantifying over properties. That suggestion faces an immediate difficulty, however: though Alice may be wise, she surely is not the property of being wise. This problem can be framed in terms of a substitution failure: if a predicate like ‘happy’ denoted a property, we would expect pairs like ‘Oscar is happy’ and ‘Oscar is the property of being happy’ to be equivalent, which they clearly are (...) not. I argue that a Fregean response that draws a distinction between objects and concepts faces serious difficulties, and that a syntactic solution to the substitution problem likewise fails. I propose to account for the substitution failure by instead distinguishing different ways that expressions can stand for properties: whereas ‘the property of being happy’ refers to a property, ‘happy’ expresses or ascribes that property. I go on to compare this view to proposals made by Wright and Liebesman, and end by drawing out a consequence my proposal has for a debate about the ontological commitments of predicatively quantified sentences. (shrink)

Divided into two parts, the first concentrates on the logical properties of propositions, their relation to facts and sentences, and the parallel objects of commands and questions. The second part examines theories of intentionality and discusses the relationship between different theories of naming and different accounts of belief.

Frege’s rejection of singular reference to concepts is centrally implicated in his notorious paradox of the concept horse. I distinguish a number of claims in which that rejection might consist and detail the dialectical difficulties confronting the defense of several such claims. Arguably the least problematic such claim—that it is simply nonsense to say that a concept can be referred to with a singular term—has recently received a novel defense due to Robert Trueman. I set out Trueman’s argument for this (...) claim, identifying and remedying some omissions and errors of formulation therein. I then develop a response to the argument by showing, pace Trueman, that it is possible—and how it is possible—to express identities between objects and concepts. (shrink)