Work in progress

Disquotation  and Infinite Conjunctions (with Thomas Schindler) | PDF

One of the main logical functions of the truth predicate is to enable us to express so-called infinite conjunctions. Several authors claim that the truth predicate can serve this function only if it is fully disquotational, which leads to triviality in classical logic. As a consequence, many have concluded that classical logic should be rejected. The purpose of this paper is threefold. First, we consider two accounts available in the literature of what it means to express infinite conjunctions with a truth predicate and argue that they fail to support the necessity of transparency for that purpose. Second, we show that, with the aid of some regimentation, many expressive functions of the truth predicate can actually be performed using truth principles that are consistent in classical logic. Finally, we argue that deflationists have no conceptual commitment to any transparency principles and suggest a reconceptualisation of deflationism according to which the meaning of truth is largely irrelevant for the question of what theory of truth we should adopt. Many philosophers think that the paradoxes pose a special problem for deflationists; we will argue, on the contrary, that deflationists are in a much better position to solve the paradoxes than their opponents.

Reference in Arithmetic

Self-reference has played a prominent role in the development of metamathematics in the past century, starting with Gödel's first incompleteness theorem. Given the nature of this and other questions and results in the area, the informal understanding of self-reference in arithmetic has sufficed so far to explain these phenomena. Recently, however, it has been shown that for other related issues in metamathematics and philosophical logic a precise notion of self-reference and, more generally, reference, are actually needed. First, I discuss the conditions a good notion of reference in arithmetic must satisfy. Then, in accordance, I provide two notions of reference for the language of first-order arithmetic, the second of which I show to be fruitful for addressing the aforementioned issues in metamathematics and philosophical logic.

The Expressive Function of Truth (with Thomas Schindler)

It is often said that the truth predicate serves a logico-expressive function, namely, it allows for the expression of so-called 'infinite conjunctions'. This function prompts the formulation of logics or, better, formal theories of truth. We argue that what principles these systems should validate depends on what it means for an infinite conjunction to express or stand in for all its 'conjuncts'. We examine two accounts for this phenomenon that are available in the literature and show them to be substantially flawed. We put forward a new approach and discuss whether classical or non-classical logics are to be preferred as basis for theories of truth. We also propose a reconceptualisation of deflationism, according to which the meaning of truth in natural language is largely irrelevant for the deflationist's choice of a truth theory. Furthermore, we discuss a similar approach to the class-theoretic paradoxes and suggest that we should be deflationists about classes as well.
Lavinia Picollo,
Jan 5, 2016, 6:05 AM