Online research in philosophy

The paradoxes of self reference have to be dealt with by anyone seeking to give a satisfactory account of the logic of truth, of properties, and even of sets of numbers. Unfortunately, there is no widespread agreement as to how to deal with these paradoxes. Some approaches block the paradoxical inferences by rejecting as invalid a move that classical logic counts as valid. In the recent literature, this deviant logic analysis of the paradoxes has been called into question.This disagreement motivates a re-examination of the philosophy of formal logic and the status of logical truths and rules. In this paper I do some of this work, and I show that this gives us the means to defend the deviant approaches against such criticisms. As a result I hope to show that these analyses of the paradoxes are worthy of more serious consideration than they have so far received.

According to ‘paracomplete’ theorists like Hartry Field, there are some sentences (such as sentences that attribute untruth to themselves) about which we should reject the relevant instances of the Law of the Excluded Middle without accepting their negations. The central alleged advantage of this approach over other consistent solutions to the Liar Paradox—for example, the view that Liars have some third truth-value other than ‘true’ or ‘false’—lies in its apparently superior ability to avoid ‘revenge paradoxes’. I argue, however, that this advantage is illusory. The paracomplete theorist faces revenge problems of their own.

Thinking about truth can be more dangerous than it looks. Of course, our concept of truth is the source of one of the most frustrating and impenetrable paradoxes humans have ever contemplated, the liar paradox, but that is just the beginning of its treachery. In an effort to understand why one of the most beloved and revered members of our conceptual repertoire could cause us so much trouble, philosophers have for centuries proposed “solutions” to the liar paradox. However, it seems that our concept of truth takes offense to our efforts to understand it because it appears to retaliate against those who propose “solutions” to the liar. It takes its revenge on us by creating new paradoxes from our own attempts to find resolution. That is, most proposed solutions to the liar paradox give rise to new, more insidious paradoxes—often called revenge paradoxes. For our attempts at understanding, truth rewards us with inconsistent theories, untenable logics, and a deep feeling of bewilderment. It is as if our concept of truth lashes out at us because it wants to remain a mystery. After a few run-ins with truth, many philosophers have the good sense to keep their distance. Far from being the serene, profound concept most people take it to be, those of us who think much about the liar paradox know truth to be a vengeful bully—a conceptual misanthrope.

This enjoyable book presents a potpourri of paradoxes with the purpose of showing how they connect to serious philosophical issues. The main paradoxes are Zeno's, the sorites, Newcomb's problem, the paradoxes of confirmation, the surprise examination, and the paradoxes of self-reference. A final chapter defends the assumption that contradictions are unacceptable and an appendix throws in sixteen minor paradoxes. Along the way, R. M. Sainsbury peppers the reader with helpful queries and provocative asides.

We identify a class of paradoxes that are neither set-theoretical or semantical, but that seem to depend on intensionality. In particular, these paradoxes arise out of plausible properties of propositional attitudes and their objects. We try to explain why logicians have neglected these paradoxes, and to show that, like the Russell Paradox and the direct discourse Liar Paradox, these intensional paradoxes are recalcitrant and challenge logical analysis. Indeed, when we take these paradoxes seriously, we may need to rethink the commonly accepted methods for dealing with the logical paradoxes.

The paper offers a solution to the semantic paradoxes, one in which (1) we keep the unrestricted truth schema True(A)A, and (2) the object language can include its own metalanguage. Because of the first feature, classical logic must be restricted, but full classical reasoning applies in ordinary contexts, including standard set theory. The more general logic that replaces classical logic includes a principle of substitutivity of equivalents, which with the truth schema leads to the general intersubstitutivity of True(A) with A within the language.The logic is also shown to have the resources required to represent the way in which sentences (like the Liar sentence and the Curry sentence) that lead to paradox in classical logic are defective. We can in fact define a hierarchy of defectiveness predicates within the language. Contrary to claims that any solution to the paradoxes just breeds further paradoxes (revenge problems) involving defectiveness predicates, there is a general consistency/conservativeness proof that shows that talk of truth and the various levels of defectiveness can all be made coherent together within a single object language.

The paradoxes of self-reference are genuinely paradoxical. The liar paradox, Russell’s paradox and their cousins pose enormous difficulties to anyone who seeks to give a comprehensive theory of semantics, or of sets, or of any other domain which allows a modicum of self-reference and a modest number of logical principles. One approach to the paradoxes of self-reference takes these paradoxes as motivating a non-classical theory of logical consequence. Similar logical principles are used in each of the paradoxical inferences. If one or other of these problematic inferences are rejected, we may arrive at a consistent (or at least, a coherent) theory. In this paper I will show that such approaches come at a serious cost. The general approach of using the paradoxes to restrict the class of allowable inferences places severe constraints on the domain of possible propositional logics, and on the kind of metatheory that is appropriate in the study of logic itself. Proof-theoretic and model-theoretic analyses of logical consequence make provide different ways for non-classical responses to the paradoxes to be defeated by revenge problems: the redefinition of logical connectives thought to be ruled out on logical grounds. Non-classical solutions are not the “easy way out” of the paradoxes.

The vast majority of approaches to the liar paradox generate new paradoxes that are structurally similar to the liar (often called revenge paradoxes). There is a complex group of issues surrounding revenge paradoxes, the expressive powers of natural languages, and the adequacy of approaches to the liar. My goal is to provide a precise framework against which these issues can be formulated and discussed. The centerpiece of this framework is the notion of internalizability: a semantic theory is internalizable for a language if and only if there exists an extension of the language such that (i) the theory is expressible in that extended language, and (ii) the theory assigns meanings to all the relevant sentences of that extended language. The framework is applied to three examples from the literature: Reinhardt and McGee on theories that require expressively richer metalanguages, Field on revenge-immunity, and Gupta on semantic self-sufficiency.

I express my dissatisfaction with the common ways to treat the semantic paradoxes. Not only do they give rise to revenge paradoxes, they ignore the wisdom contained in the ordinary reaction to paradoxes. I instead propose an account that vindicates the ordinary reaction to paradox by putting the blame on us philosophers. It is the wrong conception of what a valid inference is, one that is central to “the ideal of deductive logic” that gives rise to the problem. The solution outlined gives us a new way to accept defeat in light of the paradoxes: the arguments that lead to them are based on valid forms of reasoning, but their conclusions are nonetheless rationally rejected.