# Research

Loosely speaking, my main interest is in the question of how thought and language relate to reality. Most of my research focuses on the fundamental notions of theoretical philosophy, in particular those pertaining to logic, semantics, and metaphysics: truth, meaning, and existence. I have tried to shed new light on these notions by looking at limiting cases that challenge our basic assumptions about them, in particular: the semantic and logical paradoxes (the liar, Russell’s paradox), the nature of abstract objects and how we can have knowledge about them (especially numbers, classes, properties, fictional objects), the problem of absolute generality (whether we can quantify over absolutely everything), the rule-following paradox, and the so-called model-theoretic arguments against realism. My interest in the relation between thought and reality also explains my on and off interest in transcendental idealism and the sense-datum theory of perception.

I often try to bring to bear logical and mathematical tools on the issues I am working on, and I have a general deflationary attitude towards most of the topics mentioned above. My aim for the next few years is to develop a comprehensive deflationary account of truth, properties, classes, numbers, propositions and other abstract objects according to which talk about these things is merely a means to express certain generalisations that otherwise would be very hard or impossible to express.

**Truth**

Deflationists about truth claim that the word "true" does not designate a metaphysically significant property of things in the world, but only serves a certain logical or linguistic role in natural language. For example, it allows us to formulate certain generalisations that otherwise would be very difficult to formulate, such as "All theorems of arithmetic are true" or "Everything Einstein said is true". However, the question what the precise function of the truth predicate consists in has not received much attention in the literature. Some philosophers have claimed that the truth predicate is a device for infinite conjunctions and disjunctions, while others have seen it as a means to finitely axiomatize infinite sets of sentences. In joint work with Lavinia Picollo, we have criticised these approaches and argued that the best way of understanding the function of the truth predicate is as enabling us to quantify into sentence and predicate positions of our language without introducing higher-order quantifiers. In support of this claim we have presented novel formal results showing that (i) languages with impredicative sentential quantifiers can be translated into languages containing a disquotational truth predicate and vice versa, and that (ii) languages with impredicative predicate quantifiers can be translated into languages containing a disquotational truth predicate as well and vice versa. In particular, we have proved the surprising result that the whole simple theory of types (with an axiom of infinity) is reducible to disquotational principles of truth, which means that almost our entire body of mathematics can be reduced to a deflationary theory of truth. More importantly though, our analysis suggests systematic answers to many controversial issues surrounding the interpretation of the deflationiary position. Here is an example:

*The conservativity debate*. Deflationists take truth to be an insubstantial, quasi-logical notion. This is often understood to entail that the notion of truth cannot yield new knowledge about the world or have any explanatory power. A number of philosophers have therefore claimed that adding a truth predicate to a theory must yield a conservative extension of it (which means that every sentence not containing the truth predicate that is derivable in the extended theory must already be derivable in the original theory). This generates a conflict, because most interesting theories of truth do not lead to a conservative extension. Based on our analysis of the function of truth, we argue that deflationists are not committed to the conservativity requirement. Very roughly, since the truth predicate is a tool for mimicking sentential and predicate quantification, we should not expect that the addition of a truth predicate leads to a conservative extension because (as is well known) the addition of higher-order quantifiers does not generally lead to a conservative extension either.

**Properties**

In recent work I have tried to extend the deflationary view of truth to properties (following up some remarks by Charles Parsons). As mentioned above, we have shown that languages with a truth predicate and languages with higher-order quantifiers are intertranslatable. Since many philosophers regard second-order logic as a theory of properties, the intertranslatability results suggests that properties fulfil merely a logical function. For example, consider the infinitely many sentences that are obtained from the following schema by replacing the letter “F” by some meaningful predicate of our language:

(1) If it is required of a great general that he is F then Napoleon is F.

Given the notion of property, we can assert all of the infinitely many instances of this schema by asserting the following generalisation:

(2) Napoleon has all the properties required in a great general.

Our property talk is governed by the comprehension schema: x has the property of being F if and only if x is F. According to my view, the whole point of this schema is simply to ensure that the notion of property can play the logical role it does (e.g. it allows us to derive instances of (1) from (2)).

Philosophers have appealed to properties to do explanatory work in many disciplines such as metaphysics, semantics, and philosophy of mathematics. The features that are ascribed to properties in different areas of philosophy seem to generate a conflict. Philosophers that prioritise applications of properties in metaphysics tend to postulate a sparse ontology of properties, according to which only natural properties exist, e.g. properties posited by physics. However, these can hardly be employed in the foundations of semantics and mathematics. Conversely, philosophers that prioritise applications of properties in semantics and mathematics tend to operate with an abundant ontology of abstract properties, according to which there are many properties, e.g. one property corresponding to every (or most) meaningful predicate(s) of our language. However, it seems implausible that abstract objects can account for the causal powers of objects or be truth-makers for laws of nature. The deflationary account of properties provides a solution (of sorts) to this dilemma. Since there corresponds a property to all/most predicates of our language, properties can indeed be used to do foundational work in semantics and mathematics. On the other hand, properties are important in metaphysics precisely because properties play an important expressive role. However, it is an illusion to think that properties themselves play an important explanatory role there -- the explanatory power lies entirely within the predicates that the properties help to generalise.

**Numbers**

There is an interesting analogy between number talk and talk about truth and properties, which suggests that we extend deflationism to numbers as well. Consider what I call the numerical equivalence schema (NES):

The number of Fs is n if and only if there are n Fs (e.g. the number of planets is eight iff there are eight planets)

On the right-hand side, the number word can be analysed as a numerically definite quantifier and is therefore ontologically non-commital. On the left-hand side, the number word seems to function as a singular term and can therefore be replaced by a variable. Since both sides are equivalent, this allows us to indirectly quantify into the position of the numerical definite quantifier. Compare this to the T-schema. The T-schema enables us to replace an occurrence of a sentence A with an equivalent expression: "A" is true, in which the sentence is mentioned rather than used. The singular term can be replaced with a variable, thereby allowing us, indirectly, to generalise sentence places in our language. In recent work, I have therefore explored the possibility of a minimalist or deflationary account of properties consisting of the following two claims: The introduction of number terms via the NES merely serves a certain logical role, and all arithmetical facts can be explained on this basis.