What is the relationship between objects and properties? According to a standard view, there are primitive individuals that ‘instantiate’ or ‘have’ various properties. According to a rival view, objects are mere ‘bundles’ of properties. While there are a number of reasons to be skeptical of primitive individuals, there are also a number of challenges that the bundle theorist faces. The goal of this paper is to formulate a view about the relationship between objects and properties that avoids many of the (...) problems inherent in both of these views. The view I will end up defending implies a particularly radical version of Monism, and it collapses the object-property distinction altogether. (shrink)

The paper has two parts: 1. A brief exposition of proof-theoretic semantics, not necessarily in connection to natural language. 2. A review, with a contrastive flavour, of some of the applications of PTS to NL with an indication of advantages of PTS as a theory of meaning for NL.

Sortal concepts are at the center of certain logical discussions and have played a significant role in solutions to particular problems in philosophy. Apart from logic and philosophy, the study of sortal concepts has found its place in specific fields of psychology, such as the theory of infant cognitive development and the theory of human perception. In this monograph, different formal logics for sortal concepts and sortal-related logical notions are characterized. Most of these logics are intensional in nature and possess, (...) in addition, a bidimensional character. That is, they simultaneously represent two different logical dimensions. In most cases, the dimensions are those of time and natural necessity, and, in other cases, those of time and epistemic necessity. Another feature of the logics in question concerns second-order quantification over sortal concepts, a logical notion that is also represented in the logics. Some of the logics adopt a constant domain interpretation, others a varying domain interpretation of such quantification. Two of the above bidimensional logics are philosophically grounded on predication sortalism, that is, on the philosophical view that predication necessarily requires sortal concepts. Another bidimensional logic constitutes a logic for complex sortal predicates. These three sorts of logics are among the important novelties of this work since logics with similar features have not been developed up to now, and they might be instrumental for the solution of philosophically significant problems regarding sortal predicates. The book assumes a modern variant of conceptualism as a philosophical background. For this reason, the approach to sortal predicates is in terms of sortal concepts. Concepts, in general, are here understood as intersubjective realizable cognitive capacities. The proper features of sortal concepts are determined by an analysis of the main features of sortal predicates. Posterior to this analysis, the sortal-related logical notions represented in the above logics are discussed. There is also a discussion on the extent to which the set-theoretic formal semantic systems of the book capture different aspects of the conceptualist approach to sortals. These different semantic frameworks are also related to realist and nominalist approaches to sortal predicates, and possible modifications to them are considered that might represent those alternative approaches. (shrink)

There are various well-known paradoxes of modal recombination. This paper offers a solution to a variety of such paradoxes in the form of a new conception of metaphysical modality. On the proposed conception, metaphysical modality exhibits a type of indefinite extensibility. Indeed, for any objective modality there will always be some further, broader objective modality; in other terms, modal space will always be open to expansion.

One might well wonder—is there a category under which every thing falls? Offering an informative account of such a category is no easy task. For nothing would distinguish things that fall under it from those that don’t—there being, after all, none of the latter. It seems hard, then, to say much about any fully general category; and it would appear to do no carving or categorizing or dividing at all. Nonetheless there are candidates for such a fully general office, including (...) thing, being, entity, item, existent, and—especially—object.[ It is not obvious that there is any fully general category (whether object or otherwise). Accordingly, not all accounts of object assign it to a fully general category, instead allowing that there are non-objects. On those views, object does indeed divide. Accounts of object, then, differ with respect to whether there are non-objects. And this is not the only fault line. Other dimensions of difference include what objects there are and what objects are. Accordingly, this entry will survey three broad questions about the category object: What, if any, is its contrast or complement? What is its extension? What is its nature? (shrink)

This paper aims to provide hyperintensional foundations for mathematical platonism. I examine Hale and Wright's (2009) objections to the merits and need, in the defense of mathematical platonism and its epistemology, of the thesis of Necessitism. In response to Hale and Wright's objections to the role of epistemic and metaphysical modalities in providing justification for both the truth of abstraction principles and the success of mathematical predicate reference, I examine the Necessitist commitments of the abundant conception of properties endorsed by (...) Hale and Wright and examined in Hale (2013), and demonstrate how a two-dimensional approach to the epistemology of mathematics is consistent with Hale and Wright's notion of there being non-evidential epistemic entitlement rationally to trust that abstraction principles are true. A choice point that I flag is that between availing of intensional or hyperintensional semantics. The hyperintensional semantic approach that I advance is a topic-sensitive epistemic two-dimensional truthmaker semantics. I countenance a hyperintensional semantics for novel epistemic abstractionist modalities. I suggest that observational type theory can be applied to first-order abstraction principles in order to make abstraction principles recursively enumerable. Epistemic and metaphysical states and possibilities may thus be shown to play a constitutive role in vindicating the reality of mathematical objects and truth, and in providing a conceivability-based route to the truth of abstraction principles as well as other axioms and propositions in mathematics. (shrink)

This paper reflects on metametaphysics and as such develops a metametameta-physical view: that quietist metametaphysics requires dialetheism, and in turn a paraconsistent logic. I demonstrate this using Carnap’s metametaphysical position in his 'Empiricism, Semantics and Ontology' as an example, with regard to how it exhibits self-reference and results in inconsistency. I show how applying Carnap’s position to itself produces a dilemma, both horns of which lead to a contradiction. Such inconsistency commonly arises from meta-theories with global scope, as the 'meta' (...) approach aims to transcend the scope of that which it is theorizing about, whilst the global nature will place itself back within the scope of that which it is theorizing about, which together result in the theory referring to itself whilst refuting itself. I argue that any global metametaphysical theory that draws a limit to thought will face self-reference problems leading to contradictory realms. My conclusion is conditional: If we want to meta-philosophize in such a way and treat quietist meta-theories as being true, then we need to be dialetheist and utilize a paraconsistent logic in order to accommodate the contradictions that result from such theorizing. (shrink)

G. F. Stout is famous as an early twentieth century proselyte for abstract particulars, or tropes as they are now often called. He advanced his version of trope theory to avoid the excesses of nominalism on the one hand and realism on the other. But his arguments for tropes have been widely misconceived as metaphysical, e.g. by Armstrong. In this paper, I argue that Stout’s fundamental arguments for tropes were ideological and epistemological rather than metaphysical. He moulded his scheme to (...) fit what is actually given to us in perception, arguing that our epistemic practices would break down in an environment where only universals were given to us. (shrink)

Attributions of omnipresence, most familiar within the philosophy of religion, typically take the omnipresence of an entity to either consist in that entity's occupation of certain regions or be dependent upon other of that entity's attributes, such as omnipotence or omniscience. This paper defends an alternative conception of omnipresence that is independent of other purported divine attributes and dispenses with occupation. The resulting view repurposes the metaphysics of necessitism and permanentism, taking omnipresent entities to be those entities that exist at (...) all regions. This view is then shown to best accommodate attributions of omnipresence to a diverse range of metaphysical posits, like abstract entities, and a more diverse class of religious posits. (shrink)

This paper presents an account of what it is for a property or relation (or ‘attribute’ for short) to be logically simple. Based on this account, it is shown, among other things, that the logically simple attributes are in at least one important way sparse. This in turn lends support to the view that the concept of a logically simple attribute can be regarded as a promising substitute for Lewis’s concept of a perfectly natural attribute. At least in part, the (...) advantage of using the former concept lies in the fact that it is amenable to analysis, where that analysis—i.e., the account put forward in this paper—requires the adoption neither of an Armstrongian theory of universals nor of a primitive notion of naturalness, fundamentality, or grounding. (shrink)

Let’s call the sentence “why is there something rather than nothing?” the Question. There’s no consensus, of course, regarding which proposed answer to the Question, if any, is correct, but occasionally there’s also controversy regarding the meaning of the Question itself. In this paper I argue that such controversy persists because there just isn’t one unique interpretation of the Question. Rather, the puzzlement expressed by the sentence “why is there something rather than nothing?” varies depending on the ontology implicitly or (...) explicitly endorsed by the speaker. In this paper I do three things. First, I argue that other proposals according to which the Question has one uniquely adequate interpretation are false. Second, I give several examples of the way in which the meaning of the Question can vary depending on the ontology to which it is coupled. Third, I explore the implications of my thesis for the manner in which we should approach future attempts to answer the Question. (shrink)

According to strong composition as identity, the logical principles of one–one and plural identity can and should be extended to the relation between a whole and its parts. Otherwise, composition would not be legitimately regarded as an identity relation. In particular, several defenders of strong CAI have attempted to extend Leibniz’s Law to composition. However, much less attention has been paid to another, not less important feature of standard identity: a standard identity statement is true iff its terms are coreferential. (...) We contend that, if coreferentiality is dropped, indiscernibility is no help in making composition a genuine identity relation. To this aim, we analyse as a case study Cotnoir’s theory of general identity, in which indiscernibility is obtained thanks to a revisionary semantics and true identity statements are allowed to connect non-coreferential terms. We extend Cotnoir’s strategy for indiscernibility to the relation of comaternity, and we show that, neither in the case of composition nor in that of comaternity, indiscernibility contibutes to show that they are genuine identity relations. Finally, we compare Cotnoir’s approach with other versions of strong CAI endorsed by Wallace, Bøhn, and Hovda, and canvass the extent to which they violate coreferentiality. The comparative analysis shows that, in order to preserve coreferentiality, strong CAI is forced to adopt a non-standard semantic treatment of the singular/plural distinction. (shrink)

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

Languages involving modalities and languages involving vagueness have each been thoroughly studied. On the other hand, virtually nothing has been said about the interaction of modality and vagueness. This paper aims to start filling that gap. Section 1 is a discussion of various possible sources of vague modality. Section 2 puts forward a model theory for a quantified language with operators for modality and vagueness. The model theory is followed by a discussion of the resulting logic. In Section 3, the (...) framework will permit us to address a puzzle raised by Elizabeth Barnes and Robert Williams. (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 recent years, many people writing on set theory have invoked the notion of an indefinitely extensible concept. The notion, it is usually claimed, plays an important role in solving the paradoxes of absolute infinity. It is not clear, however, how the notion should be formulated in a coherent way, since it appears to run into a number of problems concerning, for example, unrestricted quantification. In fact, the notion makes perfectly good sense if one endorses a dialetheic solution to the (...) paradoxes. It then morphs from a supposed solution to the paradoxes into a diagnosis of their structure. In this paper I show how. (shrink)

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

Patrick Grim has put forward a set theoretical argument purporting to prove that omniscience is an inconsistent concept and a model theoretical argument for the claim that we cannot even consistently define omniscience. The former relies on the fact that the class of all truths seems to be an inconsistent multiplicity (or a proper class, a class that is not a set); the latter is based on the difficulty of quantifying over classes that are not sets. We first address the (...) set theoretical argument and make explicit some ways in which it depends on mathematical Platonism. Then we sketch a non Platonistic account of inconsistent multiplicities, based on the notion of indefinite extensibility, and show how Grim’s set theoretical argument could fail to be conclusive in such a context. Finally, we confront Grim’s model theoretical argument suggesting a way to define a being as omniscient without quantifying over any inconsistent multiplicity. (shrink)

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

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

In , Peter van Inwagen asked a good question. (Asking the right question is often the hardest part.) He asked: what do you have to do to some objects to get them to compose something---to bring into existence some further thing made up of those objects? Glue them together or what?1 Some said that you don’t have to do anything.2 No matter what you do to the objects, they’ll always compose something further, no matter how they are arranged. Thus we (...) learned of the fusion of the coins in our pockets and the Eiffel tower. (shrink)

Might it be that world itself, independently of what we know about it or how we represent it, is metaphysically indeterminate? This article tackles in turn a series of questions: In what sorts of cases might we posit metaphysical indeterminacy? What is it for a given case of indefiniteness to be 'metaphysical'? How does the phenomenon relate to 'ontic vagueness', the existence of 'vague objects', 'de re indeterminacy' and the like? How might the logic work? Are there reasons for postulating (...) this distinctive sort of indefiniteness? Conversely, are there reasons for denying that there is indefiniteness of this sort? (shrink)

Forthcoming in Philosophical Compass. I explain why plural quantifiers and predicates have been thought to be philosophically significant.

As Williamson puts it, ‘necessitism’ is the metaphysical view that claims that “necessarily everything is necessarily something”. As that claim involves modal unrestricted quantification, the necessitist must accept it as a part of an intelligible discourse. Here, I present one of the main objections that have been presented against the intelligibility of unrestricted quantification: the objection based on the so-called All-in-One Principle. I then propose possible strategies that the necessitist could adopt to shield themselves from the objection.

In one of his meetings with members of the Vienna Circle, Wittgenstein discusses Hermann Weyl’s brief conversion to intuitionism and criticizes his arguments against applying the law of the excluded middle to generalizations over the natural numbers. Like Weyl, however, Wittgenstein rejects the classical model theoretic conception of generality when it comes to infinite domains. Nonetheless, he disagrees with him about the reasons for doing so. This paper provides an account of Wittgenstein’s criticism of Weyl that is based on his (...) differing understanding of what a general statement over infinite domains consists in. This difference in their conception of generality is argued to be central to the middle Wittgenstein’s overall stance on intuitionism as well. While Weyl (and other intuitionists) reject the law of the excluded middle on grounds of constructivity, Wittgenstein argues that general statements over infinite domains do not express propositions in the first place. The origin of this position as well as its consequences for contemporary debates on generality are further assessed. (shrink)

What is absolutely unrestricted quantification? We distinguish two theoretical roles and identify two conceptions of absolute generality: maximally strong generality and maximally inclusive generality. We also distinguish two corresponding kinds of absolute domain. A maximally strong domain contains every potential counterexample to a generalisation. A maximally inclusive domain is such that no domain extends it. We argue that both conceptions of absolute generality are legitimate and investigate the relations between them. Although these conceptions coincide in standard settings, we show how (...) they diverge under more complex assumptions about the structure of meaningful predication, such as cumulative type theory. We conclude by arguing that maximally strong generality is the more theoretically valuable conception. (shrink)

Perhaps the most pressing challenge for singularism—the predominant view that definite plurals like ‘the students’ singularly refer to a collective entity, such as a mereological sum or set—is that it threatens paradox. Indeed, this serves as a primary motivation for pluralism—the opposing view that definite plurals refer to multiple individuals simultaneously through the primitive relation of plural reference. Groups represent one domain in which this threat is immediate. After all, groups resemble sets in having a kind of membership-relation and iterating: (...) we can have groups of groups, groups of groups of groups, etc. Yet there cannot be a group of all non-self-membered groups. In response, we develop a potentialist theory of groups according to which we always can, but do not have to, form a group from any sum. Modalizing group-formation makes it a species of potential, as opposed to actual or completed, infinity. This allows for a consistent, plausible, and empirically adequate treatment of natural language plurals, one which is motivated by the iterative nature of syntactic and semantic processes more generally. (shrink)

The Russellian argument against the possibility of absolutely unrestricted quantification can be answered by the partisan of that quantification in an apparently easy way, namely, arguing that the objects used in the argument do not exist because they are defined in a viciously circular fashion. We show that taking this contention along as a premise and relying on an extremely intuitive Principle of Determinacy, it is possible to devise a reductio of the possibility of absolutely unrestricted quantification. Therefore, there are (...) intuitive reasons to believe that the counter-argument fails to support the possibility of absolutely unrestricted quantification. (shrink)

Formal semantic theories are generally thought to make contact with pre-theoretic semantic notions of aboutness and reference. The nature of that contact is, however, not always straightforward. This paper addresses two debates where that issue assumes a significant role. I begin with Simchen’s recent argument that Lewisian Interpretationism succumbs to referential indeterminacy. I develop a proposal about the relationship between the theoretical notion of a term’s semantic value and the pre-theoretic notion of reference, and argue that the indeterminacy Simchen identifies (...) does not constitute a form of referential indeterminacy. In the second part, I apply these resources to the debate between singularist and pluralist approaches to the semantics of plural terms, arguing that a certain form of singularism that emerges from the dialectic ends up agreeing with the pluralist at the level of reference, if not semantic value. The paper concludes with some remarks on what a properly referential style of singularism might look like. (shrink)

Call a quantifier ‘unrestricted’ if it ranges over absolutely all objects. Arguably, unrestricted quantification is often presupposed in philosophical inquiry. However, developing a semantic theory that vindicates unrestricted quantification proves rather difficult, at least as long as we formulate our semantic theory within a classical first-order language. It has been argued that using a type theory as framework for our semantic theory provides a resolution of this problem, at least if a broadly Fregean interpretation of type theory is assumed. However, (...) the intelligibility of this interpretation has been questioned. In this paper I introduce a type-free theory of properties that can also be used to vindicate unrestricted quantification. This alternative emerges very naturally by reflecting on the features on which the type-theoretic solution of the problem of unrestricted quantification relies. Although this alternative theory is formulated in a non-classical logic, it preserves the deductive strength of classical strict type theory in a natural way. The ideas developed in this paper make crucial use of Russell’s notion of range of significance. (shrink)

The Received View of particles in quantum mechanics is that they are indistinguishable entities within their kinds and that, as a consequence, they are not individuals in the metaphysical sense and self-identity does not meaningfully apply to them. Nevertheless cardinality does apply, in that one can have n> 1 such particles. A number of authors have recently argued that this cluster of claims is internally contradictory: roughly, that having more than one such particle requires that the concepts of distinctness and (...) identity must apply after all. A common thread here is that the notion of identity is too fundamental to forego in any metaphysical account. I argue that this argument fails. I then argue that the failure of individuality and identity applies also to macroscopic physical objects, that the problems cannot be constrained to apply only within the microscopic realm. (shrink)

Necessitism is the controversial thesis that necessarily everything is necessarily something, namely that everything, everywhere, necessarily exists. What is controversial about necessitism is that, at its core, it claims that things could not have failed to exist, while we have a pre-theoretical intuition that not everything necessarily exists. Contingentism, in accordance with common sense, denies necessitism: it claims that some things could have failed to exist. Timothy Williamson is a necessitist and claims that David Lewis is a necessitist too. The (...) paper argues that, granted the assumptions that lead to interpret the Lewisian as a necessitist, she can preserve contingentist intuitions, by genuinely agreeing with the folk that existence is contingent. This is not just the uncontroversial claim that the Lewisian, as a result of the prevalence of restricted quantification in counterpart theoretic regimentations of natural language, can agree with the folk while disagreeing with them in the metaphysical room. Rather, this is the claim that it is in the metaphysical room that the Lewisian can endorse the intuitions lying behind contingentism. (shrink)

Proper names are generally taken to be count nouns. This paper argues that this is mistaken and that at least in some languages, for example German, names divide into mass and count. Making use of Kayne's (2005, 2010) theory of light nouns, this paper argues that light nouns are part of (simple) names and that a mass-count distinction among light nouns explains the behavior of certain types of names in German as mass rather than count. The paper elaborates the role (...) of light nouns with new generalizations regarding their linguistic behavior in quantificational and pronominal NPs, their selection of relative pronouns in German, and a general difference in the support of plural anaphora between English and German. (shrink)

In many different ontological debates, anti-arbitrariness considerations push one towards two opposing extremes. For example, in debates about mereology, one may be pushed towards a maximal ontology (mereological universalism) or a minimal ontology (mereological nihilism), because any intermediate view seems objectionably arbitrary. However, it is usually thought that anti-arbitrariness considerations on their own cannot decide between these maximal or minimal views. I will argue that this is a mistake. Anti-arbitrariness arguments may be used to motivate a certain popular thesis in (...) the philosophy of mathematics that rules out the maximalist view in many different ontological debates. (shrink)

The coherence of omniscience is sometimes challenged using self-referential sentences like, “No omniscient entity knows that which this very sentence expresses,” which suggest that there are truths which no omniscient entity knows. In this paper, I consider two strategies for addressing these challenges: The Common Strategy, which dismisses such self-referential sentences as meaningless, and The Conciliatory Strategy, which discounts them as quirky outliers with no impact on one’s status as being omniscient. I argue that neither strategy succeeds. The Common Strategy (...) fails because it is both unmotivated and impotent. The Conciliatory Strategy fails because it leads to embarrassing situations in which omniscient entities are epistemically inferior to non-omniscient entities: we can, for example, devise trivia-based drinking games that force omniscient entities into an intoxicated state; and, given plausible closure principles for belief, such entities are unable to have the sorts of beliefs that give them reason to refuse to play. (shrink)

The Modal Predicate Calculus gives rise to issues surrounding the Barcan formulas, their converses, and necessary existence. I examine these issues by means of the Quantified Argument Calculus, a recently developed, powerful formal logic system. Quarc is closer in syntax and logical properties to Natural Language than is the Predicate Calculus, a fact that lends additional interest to this examination, as Quarc might offer a better representation of our modal concepts. The validity of the Barcan formulas and their converses is (...) shown by Quarc to be a result of the specific incorporation of quantification in the Predicate Calculus, and not as reflecting a feature of the interaction of quantification and modality more generally. Necessary existence is shown to follow from the identification, in the Predicate Calculus on its canonical interpretation, of particular quantification, ascription of existence and the ‘there is’ construction, three constructions which are distinguished in both Quarc and Natural Language. The issues surrounding the Barcan formulas, their converses and necessary existence are thus shown to be an artefact of a specific logic system, not an essential feature of our relevant modal concepts or of formal logic. (shrink)

Tarski's analysis of the concept of truth gives rise to a hierarchy of languages. Does this fragment the concept all the way to philosophical unacceptability? I argue it doesn't, drawing on a modification of Kaplan's theory of indexicals.

Plural Logic is an extension of First-Order Logic which has, as well as singular terms and quantifiers, their plural counterparts. Analogously, Higher-Level Plural Logic is an extension of Plural Logic which has, as well as plural terms and quantifiers, higher-level plural ones. Roughly speaking, higher-level plurals stand to plurals like plurals stand to singulars; they are pluralised plurals. Allegedly, Higher-Level Plural Logic enjoys the expressive power of a simple type theory while committing us to nothing more than the austere ontology (...) of First-Order Logic. Were this true, Higher-Level Plural Logic would be a useful tool, with various applications in philosophy and linguistics. However, while the notions of plural reference and quantification enjoy widespread acceptance today, their higher-level counterparts have been received with a lot of scepticism. In this paper, I argue for the legitimacy of Higher-Level Plural Logic by providing evidence to the effect that natural languages contain higher-level plural expressions and showing that it is likely that they do so in an indispensable manner. Since the arguments I put forward are of the same sort advocates of Plural Logic have employed to defend their position, I conclude that the commonly held view that Plural Logic is legitimate, but not so its higher-level plural extensions is untenable. (shrink)

In this article I discuss the 'procedural postulationist' view of mathematics advanced by Kit Fine in a recent paper. I argue that he has not shown that this view provides an avenue to knowledge of mathematical truths, at least if such truths are objective truths. In particular, more needs to be said about the criteria which constrain which types of entities can be postulated. I also argue that his reliance on second-order quantification means that his background logic is not free (...) of ontological commitment and that his doctrine of 'creative expansion' only makes sense from a radically anti-realist perspective. (shrink)

1. Philosophical background: iteration, ineffability, reflection. There are at least two heuristic motivations for the axioms of standard set theory, by which we mean, as usual, first-order Zermelo–Fraenkel set theory with the axiom of choice (ZFC): the iterative conception and limitation of size (see Boolos, 1989). Each strand provides a rather hospitable environment for the hypothesis that the set-theoretic universe is ineffable, which is our target in this paper, although the motivation is different in each case.

This paper outlines a heterodox and largely unexplored conception of objecthood according to which the notion of an individual object is a determinable. §1 outlines the view. §2 argues that the view is incompatible with a natural analysis of kind membership and, as a consequence, undermines the Quinean distinction between ontology and ideology. The view is then used to alleviate one source of Quinean hostility towards non-trivial restrictions on de re possibility in §3, and to elucidate Fine’s neo-Aristoteltian, non-modal conception (...) of essence in §4. §5 concludes. (shrink)

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

Since the linguistic turn, many have taken semantics to guide ontology. Here, I argue that semantics can, at best, serve as a partial guide to ontological commitment. If semantics were to be our guide, semantic data and semantic treatments would need to be taken seriously. Through an examination of plurals and their treatments, I argue that there can be multiple, equally semantically adequate, treatments of a natural language theory. Further, such treatments can attribute different ontological commitments to a theory. Given (...) this, I argue that semantics can fail to deliver determinate ontological commitments and determinate answers to ontological questions, more generally. (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)

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)

At the most general level, the concept of finitism is typically characterized by saying that finitistic mathematics is that part of mathematics which does not appeal to completed infinite totalities and is endowed with some epistemological property that makes it secure or privileged. This paper argues that this characterization can in fact be sharpened in various ways, giving rise to different conceptions of finitism. The paper investigates these conceptions and shows that they sanction different portions of mathematics as finitistic.