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".
Many debates in contemporary ontology appear to have a less than straightforwardly factual subject matter. Do some objects have parts, or is everything simple? Do numbers exist? Do only present objects exist, or are past and future objects equally real? Debates of these and similar questions do not proceed in the same way as scientific debates. Often there appears to be little or no progress, and it is unclear how one should arbitrate between conflicting answers to ontological questions. Philosophers therefore often dismiss ontological debates as pointless or merely verbal. However, in this article, I propose an alternative, expressivist analysis. On my view, utterances of quantified sentences in the context of ontological debates express noncognitive mental states. Starting from this idea, I develop a version of ontological expressivism in more detail that relies on the notion of a "circumstance of evaluation", due to Kaplan (1989). On my view, when speakers assess whether numbers exist, they consider a circumstance of evaluation which determines the truth-value of the proposition that numbers exist. Using this notion, I suggest that "numbers exist", uttered in the context of an ontological debate, expresses a noncognitive disposition to assess the truth of propositions by considering only circumstances of evaluation at which numbers exist.
Metaphysical anti-realism is a large and heterogeneous group of views that do not share a common thesis but only share a certain family resemblance. All metaphysical anti-realists deny that entities of certain sorts are genuine constituents of reality; but this commonality allows for deep differences. In this article, I systematically differentiate between versions of metaphysical anti-realism along two dimensions. First, versions of of metaphysical anti-realism differ in the sort of entities which they concern. Some anti-realist views primarily concern objects; e.g., they may concern the metaphysical status of numbers, of universals, or of composite objects (that have parts), and so on. Other anti-realist views primarily concern facts; e.g., on certain anti-realist views, there is no fact of the matter with regard to the existence of numbers. Second, versions of metaphysical anti-realism also differ in what they say on why certain entities fail to be genuine constituents of reality. For example, some anti-realists simply deny that Fs exist; other anti-realists allow that Fs exist, but deny that they exist fundamentally. I map out the relations between the various forms of metaphysical anti-realism with a particular focus on the role of relativistic ideas in this landscape.
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: