Do numbers exist? Carnap (1956 ) famously argues that this question can be understood in an “internal” and in an “external” sense, and calls “external” questions “non-cognitive”. Carnap also says that external questions are raised “only by philosophers” (p. 207), which means that, in his view, philosophers raise ”non-cognitive” questions. However, it is not clear how the internal/external distinction and Carnap’s related views about philosophy should be understood. This paper provides a new interpretation. I draw attention to Carnap’s distinction between purely external statements, which are independent from all frameworks, and pragmatic external statements, which concern which framework one should adopt, and argue that the latter express noncognitive mental states. Specifically, I argue that “frameworks” are systems of rules for the assessment of statements, which are utterances of ordinary language sentences. Pragmatic external statements express noncognitive dispositions to follow only certain rules of assessment. For instance, “numbers exist” understood as a pragmatic external statement expresses a noncognitive disposition to use only rules of assessment according to which numbers do exist. Carnap can thereby be understood as proposing a distinctive form of noncognitivism about ontology that is in some respects analogous to norm-expressivism in metaethics.
A definition of a property P is impredicative if it quantifies over a domain to which P belongs. Due to influential arguments by Ramsey and Gödel, impredicative mathematics is often thought to possess special metaphysical commitments. It seems that an impredicative definition of a property P does not have the intended meaning unless P already exists, suggesting that the existence of P cannot depend on its explicit definition. Carnap (1937 , p. 164) argues, however, that accepting impredicative definitions amounts to choosing a "form of language" and is free from metaphysical implications. This paper explains this view in its historical context. I discuss the development of Carnap’s thought on the foundations of mathematics from the mid-1920s to the mid-1930s, concluding with an account of Carnap’s (1937 ) non-Platonistic defense of impredicativity. This discussion is also important for understanding Carnap’s influential views on ontology more generally, since Carnap’s (1937 ) view, according to which accepting impredicative definitions amounts to choosing a "form of language", is an early precursor of the view Carnap presents in "Empiricism, Semantics and Ontology" (1956 ), according to which referring to abstract entities amounts to accepting a "linguistic framework".
Contemporary debates in ontology often presuppose a distinction between "ordinary" and "ontological" existence claims. For example, five obviously is a number, which trivially entails that numbers exist. Nominalists nevertheless deny the existence of numbers. If nominalists do not want to deny the obvious truth that five is a number, they need to distinguish between two sorts of existence claims, "ontological" and "ordinary" ones. By drawing this distinction, nominalists can concede that 'Numbers exist', understood in its ordinary sense, is true, and nevertheless maintain that, understood in its ontological sense, this sentence is false. But what is the difference between ordinary and ontological existence claims? I propose an expressivist analysis: ordinary existence claims express beliefs while ontological existence claims express noncognitive mental states. I develop a specific version of ontological expressivism that relies on the notion of a "circumstance of evaluation", due to Kaplan (1989). On this view, when speakers assess whether numbers exist, they consider a circumstance of evaluation which determines the truth-value of the proposition that numbers exist. I argue that “numbers exist”, understood as an ontological existence claim, expresses a noncognitive disposition to assess the truth of propositions by considering only circumstances of evaluation at which numbers exist.
Metaphysics is the study of the fundamental structure of reality. Metaphysical anti-realists think that metaphysical sentences do not describe the fundamental structure of reality. Metaphysical anti-realism is a large and heterogenous group of views which includes, for instance, Wilson’s (2014) view on which there is no capital-G grounding relation, Lewis’ view on which an object’s essence is relative to a counterpart relation (see Lewis (1968, p. 122) and (1986, p. 254)), and Hofweber’s (forthcoming) alethic idealism. In this article, I distinguish between various ways in which versions of metaphysical anti-realism could be developed. In particular, some versions of metaphysical anti-realism primarily are ontological theses to the effect that certain entities do not exist, or (if they exist) are nonfundamental. Other versions of metaphysical anti-realism primarily are linguistic theses, about the features of metaphysical sentences. On some views, these sentences have multiple eligible interpretations, while on others they express semantic contents with a merely relative or contingent truth-value. Each of these versions of metaphysical anti-realism possesses its own set of motivations and challenges.
Generality relativism is the thesis that any domain of quantification can always be expanded. On the dominant way of developing this thesis, it is a linguistic thesis about the meanings of quantifiers. For instance, Studd (forthcoming) argues that domains expand when the meanings of quantifiers change through processes that can be compared with semantic drift, as when the meaning of ‘salad’ changes over time. However, I argue for indefinite extensibility as a thesis in modal metaphysics: domains of quantification can always be expanded because it is necessarily possible for there to be more sets. This approach is compatible with the meanings of quantifiers remaining constant. I argue that it has numerous advantages, for instance when it comes to explaining what’s at stake in the debate between generality-absolutists and generality-relativists. This discussion is of more general interest since the dialectic in current debates on the metasemantics of indefinite extensibility is mirrored by discussions between contextualists and relativists in many other areas of philosophy, and the approach developed in this paper can be transferred with important results.
We discuss the metaphysical methodology of "easy ontologists", who think that all meaningful existence questions can be answered "easily", through trivial inferences from obvious truths, perhaps together with some empirical evidence or pragmatic decision. We argue that Easy Ontology does not provide a comprehensive metaphysical methodology. The problem is that existence questions are inextricably bound up with identity questions. Even should existence questions be answerable by the means available to easy ontologists, identity questions in any case are not.
Sider (2011) argues that ontology concerns what exists in a fundamental sense of 'exists'. Epistemic success as an ontologist, in this view, requires speaking a fundamental language, or to speak of the world "in its terms". I argue that this sort of view gives rise to a peculiar form of skepticism that, unlike traditional skepticism, is compatible with the deceived subject's having a lot of knowledge. The problem is that we cannot know whether we speak a fundamental language and, for this reason, might as ontologists epistemically fail even while possessing a lot of knowledge.
Carnap, in The Logical Syntax of Language (1937 ), provides a defense of impredicative definitions that does not rely on platonistic assumptions. Historically, Carnap's views on impredicativity were deeply shaped by exchanges he had with Gödel. This historical fact raises a question, since Gödel later become famous for his Platonistic defense of impredicative definitions (see Gödel 1944). Why did Gödel not go Carnap's way? In this paper, I discuss what motivated Carnap and Gödel, respectively, and assess the comparative merits of their viewpoints
I have co-edited a conference volume with contributions from both philosophy and the social sciences: