Reference in ArithmeticSelf-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 semantic paradoxes reveal a conflict between classical reasoning and intuitively correct truth principles such as transparency. For they allow us to derive every sentence of the language of our naïve truth theories, trivialising them. More often than not logicians have advocated the weakening of classical logic to block the paradoxes and avoid triviality. Some hold independent reasons to believe classical logic is incorrect, while others somehow regret the conflict and seek to keep the trimming of classical reasoning to a minimum. This paper is addressed to the latter. To justify their drastic move, logicians of this view often claim that the restrictions imposed on classical inferences need not affect non-semantic reasoning, but just that involving the truth predicate. However, we can see this is not always the case. Halbach & Horsten (2006) compare two different axiomatisations of Kripke's family of fixed-point models with the strong Kleene evaluation scheme: one formulated in classical logic, and another in basic de Morgan logic. They show that the former is stronger than the latter, not only with respect to their truth-theoretic content, but also with respect to their truth-free consequences. A novel and growing field of inquiry in logic is given by a family of paraconsistent logics, that is, systems in which contradictions do not necessarily entail triviality, called 'Logics of Formal Inconsistency' (LFIs). They enrich the language of paraconsistent logics to express consistency or classicality operators, intended to recover consistent reasoning whenever needed. The obvious question is whether an axiomatisation of paraconsistent fixed-point models in a suitable LFI could overcome the weaknesses the other theories suffer from. I show that this is not the case. Therefore, I conclude that the consistency operator is not capable of recovering all classical reasoning after all, contrary to what its promoters maintain. The Expressive Function of Truth (with Thomas Schindler) |