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 do past and future objects exist as well? Debates about 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, according to which 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 "rule of assessment". On this view, when speakers assess whether composite objects exist (for instance), they rely on assumptions with regard to what is required for composition to occur. These assumptions guide their assessment, similar to how the rules of soccer guide a soccer game. Against this backdrop, I argue that "some objects have parts", uttered in the context of an ontological disagreement, expresses a noncognitive disposition to assess the truth of propositions by using only rules according to which the proposition that some objects have parts is to be evaluated as true.
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. Views as different as mathematical nominalism—the view that numbers do not exist—, ontological relativism—the view that what exists depends on a perspective—, and modal conventionalism—-the view that modal facts are conventional—all are versions of metaphysical anti-realism. As the latter two examples suggest, relativist ideas play a starring role in many versions of metaphysical anti-realism. But what does it mean for the existence of something to “depend on” a perspective, or for a modal fact to “depend on” a convention? We can distinguish between various dependence relations, giving rise to an array of drastically different forms of metaphysical anti-realism. This article offers a guided tour. I develop a systematic distinction between various forms of metaphysical anti-realism with a focus on the role of relativist ideas in this landscape.
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Generality relativists think that any domain of quantification can always be expanded. Generality relativists often motivate their view by reflections on Russell’s paradox, which may be taken to show that it is impossible to quantify over absolutely all sets. But generality relativism also raises a host of questions. In particular, how do domains expand? What’s the mechanism? I compare two broad visions. Proponents of linguistic approaches to generality relativism, such as Studd (2019), think that domains expand when language—a tool we use to represent reality—changes. On this approach, generality relativism amounts to the thesis that it is always possible to re-interpret the set-theoretic vocabulary so that, upon re-interpretation, the quantifiers range over more things (in particular, sets). But proponents of metaphysical approaches think that domains expand when reality—what we represent using language—changes. Domains expand when new sets come into existence. I argue that metaphysical approaches have important advantages, in particular when it comes to formally articulating generality relativism in a coherent manner. The upshot of this discussion is that Russell’s paradox does not teach us a lesson about the workings of language, but teaches us a metaphysical lesson, about the modal structure of reality.