My research focuses on the notions of truth, property, class, number, and similar notions. It is driven by two sets of questions. Although these questions are closely related, it will be convenient to treat them separately here. On the one hand, I am interested in questions such as: How does language relate to reality? What is the nature of truth? What is the nature of abstract objects and how can we have knowledge of them? On the other hand, I am interested in questions such as: How can we best characterise the logical and semantic paradoxes (i.e. paradoxes involving the notions of truth, property, and class)? How can we avoid the expressive limitations that hierarchical solutions to the paradoxes (such as type theory) impose upon us?
I have attempted to provide answers to the first three questions by clarifying and revising the deflationary account of truth and by expanding it to properties, classes, and numbers. In what follows I will give a sketch of my views on this topic. After that, I will briefly describe my work on the second set of questions, which is of a more technical nature.
The key idea of the deflationary account of truth is that the predicate ‘is true’ is unlike predicates such as ‘is an electron’. Whereas the latter picks out some salient feature of reality, whose nature we can investigate and hope to discover, this is not the case with the former. The truth predicate exists in our language in order to serve a mere logical or quasi-logical function. The exact nature of this function is a bit controversial among deflationists. In joint work with Lavinia Picollo, I have argued that the truth predicate is best understood as a device to emulate quantification into sentence positions of our language in a grammatically conservative way, i.e. without introducing sentential quantifiers. For example, suppose we want to assert all instances of the law of excluded middle in one fell swoop. If we introduce sentential quantifiers, we can do this by saying ‘For all p, p or not-p’. But we can achieve the same effect without introducing sentential quantifiers by saying instead ‘Every instance of the schema “A or not-A” is true’. The truth predicate can play this role in virtue of being governed by the T-Schema:
‘A’ is true if and only if A
The T-schema enables us to replace an occurrence of a sentence, A, with an equivalent sentence, ‘A’ is true, in which the original sentence is mentioned rather than used. The singular term ‘A’ can be replaced with a variable, thereby allowing us -- indirectly -- to quantify into sentence position. According to deflationism, this is the only reason why we have a truth predicate in our language. (Unlike some other deflationists, I don’t take this to be an empirical claim about the use of ‘is true’ in the vernacular, but as a claim about our best scientific theories. The claim, more precisely, is that a deflationary truth predicate is sufficient for all scientific purposes.) This view is attractive because it demystifies the notion of truth. It is also an interesting example of Wittgenstein’s insight that not all parts of language are there to ‘represent’ or ‘describe’ reality, but may serve a rather different function.
It is my contention that property talk can be understood along similar lines. For instance, suppose that we want to assert all instances of Leibniz’ law in one fell swoop. If we introduce predicate quantifiers, we can say ‘For all F, if x=y and F(x), then F(y)’. But we can achieve the same effect without introducing predicate quantifiers by saying instead ‘If x=y and x has some properties, then that y has that property too’. Property talk can play this role in virtue of being governed by the comprehension scheme (C):
x has the property of being F if and only if x is F
(C) allows us to move from a formula, x is F, where a predicate is used, to an equivalent formula, x has the property of being F, in which that predicate has been ‘nominalised’ and therefore can be replaced by a variable. Property talk allows us -- indirectly -- to quantify into predicate positions in our language. In my view, this is the only reason we have property terms in our language -- they are mere ‘shadows of predicates’ which are needed to generalise predicate places in our language. Again, this view is attractive because it deflates various traditional questions in metaphysics such as Are there negative properties? and Are objects bundles of properties? If properties are shadows of predicates, then the answer to the first question is an obvious ‘yes’ and the answer to the second question is an obvious ‘no’.
This analogy between truth and properties can be extended to numbers. To see this, consider what I call the numerical equivalence schema (NES):
The number of Fs is n if and only if there are n Fs
To take a concrete instance:
The number of planets is eight iff there are eight planets.
On the right-hand side, the number word ‘eight’ can be understood as a numerically definite quantifier: ‘There are eight planets’ can be analysed as ‘There are (mutually distinct) x_1, …, x_8 such that all x_i are planets, and if y is a planet then y is one of the x_i’. This sentence does not refer to any abstract objects. On the left-hand side, the number word ‘eight’ functions as a singular term and can therefore be replaced by a variable. Since both sides are equivalent, the NES allows us -- indirectly -- to generalise statements involving numerically definite quantifiers. Thus, ‘two plus three equals five’ can be seen as a generalisation of statements of the form ‘If there are two apples and three bananas then there are five pieces of fruit’. It is my contention that all facts of arithmetic can be explained on the basis of the NES. For example, it can be shown that all true numerical equations and inequalities follow logically from the NES, given suitable definitions. Again, this view is attractive because it can help us answer some traditional questions in the philosophy of mathematics. For instance, it allows us to answer the old Kantian question of how we can reconcile the applicability of arithmetic with its certainty: numerically definite quantifiers are part of logic (which has universal applicability), and the ontologically committing use of number words simply enables us to generalise statements involving such quantifiers.
The logical and semantic paradoxes
I will now turn to my work on the second set of questions, namely How can we best characterise the logical and semantic paradoxes (i.e. paradoxes involving the notions of truth, property, and class)? How can we avoid the expressive limitations that hierarchical solutions to the paradoxes (such as type theory) impose upon us?
In the beginning of the 20th century mathematicians and philosophers uncovered a number of paradoxes involving the notions of truth, definability, property, concept, and class. They caused what some authors have called the ‘third foundational crisis of mathematics’ and prompted the search for a firm foundation of mathematics.
The paradoxes have traditionally been divided into two groups, the semantic and the logical paradoxes. A paradigmatic example of the first group is the liar paradox (a sentence saying of itself that it is false), which indicates that there is something wrong with the T-schema (mentioned above). A paradigmatic example of the second group is the Russell paradox (concerning the property being a property that does not apply to itself), which indicates that there is something wrong with the comprehension schema (C).
In both cases, the paradoxes have been alleged to result from some form of circularity and have been subjected to similar solutions, namely by banning all forms of self-application. In the case of the semantic paradoxes, the solution came with Tarski's hierarchy of language levels; in the case of the logical paradoxes, the solution came with Russell's theory of types. Very roughly, what happens here is that the predicate ‘is true’ is replaced by an infinite number of predicates ‘is true_1’, ‘is true_2’, …, with the proviso that each truth predicate can only be applied to sentences containing truth predicates with a lower index. Similarly, in the case of properties (concepts, classes), properties are stratified into types with the proviso that each property can only be predicated of objects and properties of a lower type.
Many philosophers and logicians have found these hierarchical solutions excessively restrictive: there are many important applications in philosophy, semantics, and logic where we need to allow for some form of self-application. For example, when we say that all sentences of the form ‘A or not-A’ are true, we mean all sentences, including those that contain the truth predicate themselves. Similarly, the property of being self-identical seems to apply to all objects; in particular, the property of being self-identical seems to exemplify itself.
If hierarchical approaches to the paradoxes are rejected, there are essentially two options left: either one rejects certain instances of the T-schema and the comprehension schema, or one rejects classical logic. Both approaches face two tasks. First, in order to block the paradoxes, one needs some kind of analysis of the paradoxes (which would inform us what instances of comprehension or what rules of classical logic need to be dropped). Second, in contrast to hierarchical approaches, non-hierarchical theories of truth, properties, and classes often suffer from deductive weakness, and are therefore of limited use in foundational contexts.
Consequently, most of my work on the paradoxes has been devoted to the following two tasks:
to provide a comprehensive analysis of the semantic and logical paradoxes
to construct deductively strong theories of truth, properties, and classes
In ‘Axioms for grounded truth’ I introduce several new truth theories that are based on the popular thought that a classical truth-value should only be assigned to those sentences that are grounded in non-semantic facts of the world. It is shown that all these theories of grounded truth have at least the same deductive strength as the standard systems of constructive (‘predicative’) analysis, and I also present a theory that has impredicative strength.
In ‘A disquotational theory of truth as strong as Z2-’ I present a novel truth theory that is equivalent to full second-order arithmetic, a system in which one can codify nearly all of actual mathematical reasoning. This theory is far stronger than most other non-hierarchical truth theories currently on the market.
In ‘A graph-theoretic analysis of the semantic paradoxes’ we try to analyse the paradoxes in terms of their ‘referential behaviour’. To every sentence a domain is assigned, the set of objects that the sentence refers to. From that assignment, we extract graphs that depict the referential structure of a sentence. For example, the liar paradox refers to itself, hence its reference graph consists of a loop. It is shown that there are exactly two referential mechanisms underlying every paradox, either the loop or the so-called double path.
In ‘Some notes on truth and comprehension’ I study some systematic connections between truth and comprehension. This allows me to determine the computational complexity of Leitgeb's and Kripke's theories of truth. I also prove the philosophically interesting result that truth is not ontologically innocent as many authors have claimed, but logically implies the existence of infinitely many objects. It was in this paper that the idea occurred to me that truth, properties, and classes are intimately connected.
In ‘Transcending the theory of types’ and ‘Classes, why and how’ I finally turn to type-free theories of properties and classes, respectively. These theories are interesting insofar as they admit the existence of universal properties and classes, i.e. properties that apply to all objects and classes that contain all objects, including themselves. One of these theories is formulated in classical logic, while the other one is formulated in a 3-valued logic. It is shown that the classical theory can recover the deductive strength of full second-order arithmetic, while the non-classical theory can recover the deductive strength of the classical theory of types.