The Liar and its Revenge in Context

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 isntin most logics, a contradiction. Far from being an intellectual curiosity, semantic paradox 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 nave 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 Kripkes highly influential work on truth in the 70s, the latter revisionary option has been increasingly popular. However, we aim to show that revisionary approaches are necessarily off target, and that appreciating why this is so holds the key for solving the semantic paradoxes once and for all. The first part of our project develops a novel and general argument against revisionary approaches, to the effect that they 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. However, as we show, such a distinction breeds new paradoxes that cannot be blocked by weakening the logic. We contend that this novel form of revenge argument is at the heart of the semantic paradoxes. The second part of our project argues in favour of a family of solutionsso- called contextualist solutionswhich 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, that is arguably revenge-free, and that allows absolutely unrestricted quantification in certain important cases (thus overcoming one of the main defects of current contextualist theories). 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, to be published by Oxford University Press. The methods used will involve conceptual analysis, formal logic, and model-theory.

Our project's main focus has been the semantic paradoxes: simple arguments requiring only minimal assumptions - the possibility to name sentences, some very modest logical resources, and naive semantic principles such as the interderivability of a sentence 'A' and its truth predication "A' is true' - that allow one to derive contradictions, and indeed to derive any sentence whatsoever. The main aim of our project was to develop a novel solution to the paradoxes. To this end, we have developed a bipartite contextualist theory. The unparadoxical fragment of the language is interpreted in a higher-order meta-theory, which allows the quantifiers of unparadoxical sentences to receive an absolutely unrestricted interpretation. For instance, in the sentence 'Everything is self-identical' the universal quantifier can be interpreted over absolutely everything, as it should. By contrast, standard contextualist theories don't allow an absolutely unrestricted interpretation of the quantifiers - essentially because, by our lights, they overgeneralise a sound lesson to be learned from the semantic paradoxes. A tradition that goes back to Russell interprets the paradoxes as diagonal arguments that can be used to 'diagonalise out' of any given totality: given any totality of objects, one can use a paradoxical argument to find an object that was not included in that totality. So, the argument goes, quantification can never be absolutely unrestricted. We accept Russell's insight, but take it to show that the quantifiers of paradoxical sentences can only be interpreted on a less than absolutely general domain, i.e. over a set. As a result, the paradoxical fragment of the language, the one that includes Liar sentences such as the sentence L that says of itself that it isn't true, receives a restricted interpretation, so as to allow a broadly Russellian interpretation of the paradoxes. But the quantifiers of unparadoxical sentences can be interpreted unrestrictedly - as they should. Unlike virtually any other theory of truth and paradox, our theory validates all everyday inferential uses of 'true'. Moreover, it affords a unified treatment of run-of-the-mill semantic paradoxes such as the Liar, and of so-called revenge paradoxes, i.e. arguments aimed at showing that any non-trivial treatment of the Liar gives rise to new Liar-like paradoxes it is unable to block. Along the way, we formulate a new recipe for generating revenge arguments against any non-classical theory T satisfying the thesis we call Classicality Principles, i.e. that there is some finite number of classically valid principles such that a sentence them if and only if it satisfies all the principles of classical logic. As we show, our recipe threatens to trivialise all the main non-classical approaches to the paradoxes, including the recently much-discussed substructural.