# Work in progress

### Mysterious Quantifiers

A well-known problem, due to Carnap, concerns how the meanings of logical connectives can possibly arise from the rules governing their use. Much less discussed is an analogous problem for quantifiers. This paper argues that the problem for quantifiers is serious, resists the usual solutions, and (on plausible semantic and metasemantic assumptions) forces us into a radically revisionary view of quantifiers and the truth-conditions of quantified sentences. I go on to explore some logical properties of quantifiers with these "sentential" truth-conditions, and the philosophical implications this new perspective on quantification comes with.

### On Arithmetical Pluralism (with Dan Waxman)

Arithmetical pluralism is the view that every consistent arithmetical theory is true (of some objects) and, therefore, as legitimate as any other, at least from a theoretical standpoint. Pluralist views have recently attracted much interest but have also been the subject of significant criticism, most saliently from Putnam (1979) and Koellner (2009). These critics argue that, due to the possibility of arithmetizing the syntax of arithmetical languages, one cannot coherently be a pluralist about arithmetical truth while holding that claims about consistency are matters of fact. In response, Warren (2015) argued that Putnam's and Koellner's argument relies on a misunderstanding, and that it is in fact coherent to maintain a pluralist conception of arithmetical truth while supposing that consistency is a matter of fact. In this paper we argue that it is not. We put forward a modified version of Putnam's and Koellner's argument that isn't subject to Warren's criticisms.

### The Unity of Logical Consequence

The two traditional approaches to formal validity, i.e. the semantic, in terms of semantic clauses and truth preservation, and the syntactic, in terms of rules of inference and the avilability of proofs, are often conceived of as rivals. I show, to the contrary, that, modulo a deflationary account of truth and satisfaction, semantic clauses and inference rules turn out to be conceptually equivalent (in a sense to be explained). As a result, the two approaches to logical consequence conceptually converge too. I conclude as well that, unlike what is normally believed, deflationism is not incompatible with truth-conditional semantics (for logical terms) and semantic approaches to logical consequence.

### A Theory of Untyped Structured Propositions

I put forward a semantic theory of untyped propositions satisfying the equivalence schema. Propositions are well-founded (in a sense to be specified) but are also allowed to exhibit a non-vicious kind of self-reference. Moreover, they satisfy a fine-grained identity criterion according to which two sentences express the same proposition just in case they say the same about the same objects, i.e. just in case they have the same content. This allows me to define shared content, partial truth, and a grounding relation between propositions.

### Comprehensive Deflationism

Relying on a nominalisation process that generates a name for each sentence, the truth predicate allows us to replace sentences with talk about sentences, without altering the truth conditions of expressions. Moreover, if a nominalisation process for predicates is available as well, together with certain syntactic operations the truth predicate also enables us to define a satisfaction predicate and, thus, to quantify into predicate position. Different but structurally similar nominalisation processes can be employed to introduce talk of properties, classes, and propositions into the language. Coupled with this, predicates like property-exemplification and class-membership, that are governed by principles structurally similar to those that govern the satisfaction predicate, can serve the same expressive purposes as satisfaction. Such structural similarities make deflationism about property exemplification and class membership appealing to deflationists about truth and satisfaction. If so, the main issue to be investigated is the ontological status of the bearers of these predicates, that is, properties, classes, and propositions. More specifically, we would like to answer the following questions: Do names for these entities exist in natural language for the sole purpose of providing truth, property exemplification, and class membership with their corresponding bearers, so that they can perform their logico-expressive functions? If so, should we be realists or nominalists about properties, classes, and propositions, or perhaps hold a Carnapian view, or see them as thin objects, Ă  la Linnebo?