The Metasemantics of Indefinite Extensibility
Assuming that any domain of quantification can always be expanded, I discuss how the possibility of expanding domains of quantification is reflected in the semantics of quantified sentences.
Indefinite extensibility is the thesis that any domain of quantification can always be expanded. But how is the possibility of expanding domains of quantification reflected in the semantics of quantified sentences? This paper introduces a three-dimensional metasemantic account which, I maintain, has important advantages over received metasemantics. The view distinguishes between semantic values and assertoric contents. The semantic value of a quantified sentence is a set of possible worlds that is structured by two accessibility relations, one of which models counterfactual necessity and the other one of which models objectivity. The key idea is that indefinite extensibility is modal variance along this second accessibility relation. Assertoric contents however are ordinary possible worlds propositions. Semantic values determine assertoric contents as a function of a third kind of parameter which is neither a context nor an index parameter but a context of assessment. The advantage of this view is that it explains succinctly what’s at issue in the debate between generality-absolutists, who think that quantification over absolutely everything is possible, and generality-relativists. If the box expresses objectivity, this disagreement concerns the Barcan formula ∀xFx→ ∀xFx, which entails that domains do not grow as one moves to objectively-accessible worlds.
I develop a version of ontological expressivism, according to which quantified sentences at least on occasion express non-representational mental states that are in important respects desire-like.
Ethical expressivism, i.e. the view that ethical claims express non-cognitive mental states, is well-known. Ontological expressivism is the following analogous thesis:
OE: Ontological existence claims express non-cognitive mental states.
Ontological existence claims are expressed by utterances of quantified sentences, such as “something is a number”. Ontological expressivism says of these claims that they do not serve to describe aspects of the world, but express states of mind that are in important respects desire-like. I develop a version of this view in more detail that is modeled after Gibbard's (2003) norm-expressive analysis. The view is composed of two theses, one metaphysical and one linguistic. The metaphysical thesis is that there are various versions of reality, similar to how Gibbard thinks that normative structure may be imposed onto worlds in various ways. The linguistic thesis is that ontological existence claims express the speaker's noncognitive preference for some version of reality over others. I analyze this attitude as a disposition to assess the semantic contents of quantified sentences only relative to particular values of a certain nonstandard parameter. This view explains the difference between ontological and ordinary existence claims as a difference in the mental states that they express, and explains a range of seemingly nonfactual disagreements as concerning how given contents should be assessed.
Carnap's Noncognitivism about Ontology
This paper defends a new interpretation of Carnap's 1956 (1950) distinction between so-called "internal" and "external" questions.
Do numbers exist? Carnap (1956 ) famously argues that this question can be understood in an "internal" and an "external" sense, and calls "external" questions "non-cognitive". These remarks express a certain critique of philosophy, since Carnap also says that external questions are raised "only by philosophers" (p. 207). It is however not at all clear how the internal/external distinction and Carnap’s related critique of philosophy are to be understood. This paper provides a new interpretation. A very important but often-overlooked part of Carnap's view is the idea that speakers evaluate "statements" guided by rules which they use in the assessment (p. 208). I argue that "frameworks" are formal systems that make these rules explicit. "Frameworks", in this view, do not merely determine the meaning of 'there are numbers', or whether there are numbers, but rather determine what is required for there to be numbers. This point is critical for understanding Carnap’s metaphilosophy, and for gauging the importance of his views for contemporary metaontology. I argue that Carnap is best understood as proposing a noncognitivist view about ontology, according to which the acceptance of a framework is a noncognitive disposition to follow particular rules of assessment.
Carnap's Defense of Impredicative Definitions
I explain why Carnap (1937 ) thought that accepting impredicative mathematics amounts to choosing a "form of language" and is free from metaphysical implications.
A definition of a property P is impredicative if it quantifies over a domain to which P belongs. Impredicative mathematics is often thought to come along with special ontological commitments, since it seems that an impredicative definition of a property P is meaningless unless P exists independently of its explicit definition. Carnap (1937 , p. 164) however argues that accepting impredicative mathematics amounts to choosing a “form of language” and is free from metaphysical implications. This paper examines the rationale behind this claim. I argue that Carnap’s defense of impredicative definitions essentially relies on a kind of nonfactualism about meaning. A “form of language” is a formal system, such as (for instance) the system of primitive recursive arithmetic. Carnap (1937 ) develops a particular formal system (Language II) that includes meta-linguistic formulas which formally interpret impredicative expressions. The crucial point of Carnap’s view is that these formulas are in turn interpreted by meta-metalinguistic formulas. That impredicative expressions are meaningful in the formal system which is Language II hence follows from how this system is defined in a failsafe manner. In addition to its interpretative contribution, this discussion yields a Carnapian defense of impredicativity that does not rest on ascribing any particular ontological status to impredicatively defined properties.
No "Easy" Answers to Identity Questions (with Katherine Ritchie)
We discuss the metaphysical methodology of "easy ontologists", and argue that even should existence questions be answerable by the means available to them, identity questions are not.
Easy Ontologists, most notably Thomasson (2015), argue that ontological existence questions are in important respects trivial. They think that we can answer these questions by using our ordinary conceptual skills, perhaps together with some empirical investigations or pragmatic decisions. Ontology thus is "easy", requiring no distinctively metaphysical investigation. This paper raises a two-stage objection to Easy Ontology. We first argue that ontological questions concerning which things exist are inextricably bound up with questions concerning the identity of and differences between kinds of things. We then argue that identity questions cannot be answered using only the resources available to Easy Ontologists. The reason is that identity is a first-order relation, and, unlike existence, is not plausibly construed as a second-order property. It is for this reason not in general possible to determine whether identities hold on purely conceptual grounds. We examine several ways in which an Easy Ontologist might try to get around this problem, and find them all wanting. Easy Ontology hence does not constitute a comprehensive ontological methodology. Even if ontological existence questions are trivial, there still is an important range of ontological questions whose answers require genuinely metaphysical inquiry.