Work in progress
In my paper "Reference in Arithmetic" I put forward non-trivial and materially adequate definitions of reference, self-reference, and well-foundedness in the language of Peano arithmetic. Surprisingly, these notions cannot be easily extended to an expansion of this language with a truth predicate, the language in which formal truth theories are usually formulated. Having adequate notions of reference for this language is highly desirable, as it is commonly believed that paradoxical expressions can be characterized in terms of their underlying reference patterns. If we could identify the self-referential or the non-well-founded expressions in which the truth predicate occurs, we could exclude them from our truth principles to avoid contradictions. I have recently submitted a paper for publication in which I put forward alethic notions of reference, self-reference, and well-foundedness, especially designed for that job. These notions focus only on reference via the truth predicate, and are an improvement on the definitions published in the Logica Yearbook 2015. Moreover, I provide several axiomatic truth theories given by instances of disquotational principles restricted by means of the new notions. I prove these systems to be consistent and proof-theoretically very strong.
Satisfaction and Higher-Order Quantification (with T. Schindler)
Just as the truth predicate can be seen as a device for mimicking quantification into sentence position, we argue that the satisfaction (or truth-of) predicate allows us to quantify into predicate position, mimicking second-order quantifiers. To do so, we present some relative interpretability results between theories of satisfaction and systems of higher-order quantification. Moreover, we discuss the relative merits of both formalisms as devices for increasing the expressive power of a language. Finally, we use our results to take a fresh look at the issue of how to interpret second-order quantification. This is ongoing joint work with Thomas Schindler.
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, à la Linnebo?
One Notion of Logical Consequence, Modulo Truth (with G. Sagi)
The history of logic has seen various ways of identifying and studying the nature of logical consequence. The two traditions defining contemporary thought on the topic are the proof-theoretic and the model-theoretic traditions. They are usually seen as very different, and often simply as rivals. We attempt to show they are actually two sides of the same coin, by bringing in a third approach: a modern rendering of the good-old substitutional account, by Volker Halbach. On the one hand, if a deflationary approach to truth (in the sense of my previous work) is adopted, the semantic clauses for connectives in the substitutional account would be mere generalisations of the rules that govern the behaviour of these connectives in the proof-theoretic account. On the other hand, the semantic clauses of the substitutional account could be seen, under certain conditions to be investigated, as notational variants of those corresponding to the model-theoretic account. Thus, the substitutional would serve as a bridge that reconciles the proof- and the model-theoretic approaches to logical consequence. To some extent the three accounts of logical consequence at play could turn out to be not only extensionally equivalent but also equivalent to each other in some stronger way.
Deflationism is not an inferentialism
In logic, inferentialism is the view that the meaning of logical terms is given by their inferential role. In turn, according to deflationism the meaning of the truth predicate can be given by a transparency principle, often expressed in terms of introduction and elimination rules, or biconditionals, plus perhaps some compositionality axioms. Thus, it is usually believed that deflationism is a form of inferentialism, and semantic truth theories such as Kripke’s and Field’s are sometimes regarded as non-deflationary. Now Field, for instance, presents himself as a deflationist. Is his position untenable? We conjecture this is not the case. On the one hand, the role of models provided by semantic theories is not comparable to that given by axiomatic ones. Additionally, formal theories of truth playing the same role as the latter could be given semantically, in terms of models as well.