Our project aims at developing a unified solution to the semantic paradoxes: intuitively valid arguments which allow us to ‘prove’, on minimal assumptions, contradictions, and indeed arbitrary sentences. The Liar Paradox, involving a sentence L which says of itself that it is not true, is a case in point. A little reflection shows that L is true if and only if it isn’t—in most logics, a contradiction. Far from being an intellectual curiosity, semantic paradoxes threaten to undermine the coherence of some of our most basic concepts, such as truth, denotation, necessity, and validity. Since the existence of sentences such as L can be proved from basic syntactic principles, it is commonly thought that there are only two main possible ways out of the problem: one might either give up naïve semantic principles, such as the equivalence of A and ‘A is true’, or weaken classical logic, i.e. the commonly held canons of correct reasoning. Since Saul Kripke’s highly influential work on truth in the 70’s, the latter revisionary option has been increasingly popular. However, we aim to show that revisionary approaches are off target, and that appreciating why this is so holds the key for solving the semantic paradoxes.
The first part of our project develops two general revenge arguments against revisionary approaches. The first argument argues that revisionary approaches suffer from critical expressive limitations, even if the revision of classical logic is extended to their meta-theories. As we show, under some natural assumptions, either revisionary meta-theories are too weak to serve as proper meta-theories, or they are subject to revenge paradoxes despite being nonclassical. The second argument purports to show that revisionary approaches are bound to be trivial, i.e. they entail the truth of any sentence. The main reason is that any such approach must distinguish between paradoxical and unparadoxical sentences, in the minimal sense a sentence is paradoxical if reasoning classically with it yields absurdity and unparadoxical if one can reason with it classically. However, such a distinction breeds new paradoxes that, in minimally strong nonclassical theories, cannot be blocked by weakening the logic.
The second part of our project argues in favour of a family of solutions—so-called contextualist solutions—which interpret the semantic paradoxes, including revenge arguments, as consistent arguments involving principles whose application shifts the initial context of reasoning to a new, richer context. We argue for one particular, and novel, such solution: one that validates both classical logic and exceedingly intuitive principles about ‘true’, and that allows absolutely unrestricted quantification in certain important cases (thus overcoming one of the main defects of current contextualist theories). Crucially, the contextualist theory we develop avoids standard and revenge paradoxes in a uniform and principled way, thus offering a coherent and univocal diagnosis and treatment of all semantic paradoxes.
Our project will be the first comprehensive study of revenge arguments, as well as the first systematic study of contextualist approaches to semantic paradox. Our overall aim is to write a monograph on paradox and revenge.