The Liar Paradox is traditionally no topic for the faint of heart. The Greek poet and grammarian Philetus of Cos was one of its first victims, as his epitaph records: ‘O Stranger: Philetus of Cos am I/ ‘Twas The Liar who made me die,/ And the bad nights caused thereby.’ Readers of this collection of 15 original papers written by leading researchers in the field may be even worse off than Philetus and his contemporaries: not only do they face the risk of untimely demise, they also need a solid background in formal logic as well as familiarity with the classical texts in the Liar debate (especially by Tarski and Kripke). Those who have overcome these initial hurdles, however, will appreciate this collection of high-quality papers centered around the Revenge problem.
The Revenge problem threatens theories that attempt to say something informative about (seemingly) paradoxical sentences. Take an initial Liar sentence like
L (L) is false.
By familiar reasoning involving the relevant instances of the homophonic T-schema
T ‘p’ is true iff p;
and substitution of the co-referential term ‘(L)’ for ‘ “(L) is false” ’, we arrive at the conclusion that if (L) is true, it is false, and if (L) is false, it is true. Throw in the assumption that (i) no sentence is both true and false and the assumption that (ii) every sentence is either true or false, and you have a contradiction. A natural reaction is to give up assumption (ii): sentences do not divide into the true and the false, but some fall into a truth-value gap – they are neither true nor false (gappy for short). Recognizing this third semantic status a sentence may have blocks the derivation of a contradiction. Rather, we now have an argument to the effect that (L) is gappy. But here is where the Revenge problem strikes. Our semantic theory attempts to deal with (L) by recognizing a third semantic status. But now consider the Revenge Liar sentence
RL (RL) is either false or gappy.
Analogous reasoning, this time employing the assumption that (iii) no sentence is both true and false or both true and gappy and the assumption that (iv) every sentence is either true, false or gappy, yields a contradiction again. It seems that, in the face of (RL), we only have three equally unhappy options: (a) we accept that our semantic theory simply shifts the paradox from (L) to (RL) and, thus, still implies a contradiction, (b) we repeat the move we made earlier, amending our theory to recognize yet another semantic status for sentences to have, which then invites another Revenge Liar sentence, or (c) we declare (RL) to lie outside the scope of our theory, even though it only involves vocabulary that should be acceptable by the theory's own lights. The Revenge worry is that any attempted solution of the initial Liar paradox faces its own Revenge problem. All the papers in this collection deal with it in one way or another. In what follows I give a brief summary.
One of the core papers of the book is certainly Hartry Field's ‘Solving the Paradoxes, Escaping Revenge’. In it, Field gives a systematic account of how non-classical solutions to the paradoxes of a certain type (what he calls G-solutions) avoid Revenge problems while ensuring expressive completeness. Three contributions respond directly to Field's claim of Revenge-immunity: Hannes Leitgeb argues that assessing the merits of Field's theory of truth ultimately depends on the availability of an adequate meta-theory. He sketches a classical and a non-classical candidate theory and argues that only the latter can be adequate for Field's purposes. However, he expresses doubts as to whether its systematic development can be successful. Both Graham Priest and Augustín Rayo and P. D. Welch argue more directly that Field's theory of truth is not Revenge-immune after all. Priest blames its background set theory which, as is also argued in Steward Shapiro's contribution, faces its own Revenge problem, while Rayo and Welch focus on the availability of higher-order resources. They argue that a key semantic notion of Field's theory – that of being an intended interpretation of L+, the language of the theory – is expressible in L+ once higher-order resources are allowed into the language. Since the expressibility of the key notion would generate paradox, they claim that Field's theory is no better off than standard reactions to the paradox: it prevents inconsistency through expressive incompleteness. Only indirectly addressing Field, Greg Restall assesses the costs of non-classical solutions to the semantic paradoxes in general. He argues that even in a non-classical setting it is very easy to define a conditional-like operator ‘⊃’ which leads to Curry's Paradox (we can prove that anything is the case using sentences of the form ‘this sentence is true ⊃ p’). He concludes that non-classical solutions are not the easy way out they may seem to be.
Non-classical theories of truth are not the only ones on the market that promise to avoid paradox. Stephen Read reminds us of how Thomas Bradwardine (an English cleric and mathematician in the 1300s) proposed to deal with Liar sentences. The key to Bradwardine's solution is the claim that every sentence says whatever follows from what it says and that a sentence is true just in case everything it says is the case. Since contradictions follow from sentences that say of themselves that they are not true, Liar sentences are not true. But this is not everything they say – they also say that contradictory things are the case – so they are not also true. Bradwardine avoids the Revenge problem by the observation that we can have two tokens of the same type sentence, one of which says of itself that it is not true while the other says of another sentence (of the same type) that it is not true. In this respect Bradwardine seems to have been right, as far as Keith Simmons is concerned. He argues in a similar vein that a contextualist approach to the paradoxes is suggested by the phenomenon that we can apparently ‘reason past pathology’ using Revenge Liar sentences. According to him, the Revenge phenomenon suggests that ‘reflectiveness’ is one of the contextual parameters determining the content of an utterance. In particular, a Revenge Liar sentence like (RL) may have a pathological content with respect to a non-reflective context (i.e. one in which its pathologicality is not part of the conversants' common ground) but a true content with respect to a reflective context. Roy T. Cook agrees with Simmons that the Revenge phenomenon is not so much of a problem but rather teaches us something important about how language works. Instead of positing additional contextual parameters, however, he takes it to suggest that our concepts of a statement, a language, and a truth-value are indefinitely extendible. An expression of this fact is, according to Cook, that natural languages and their semantics are in some sense stratified such that repeated application of strategy (b) above – introducing more and more semantic statuses the farther up the ladder we go – is the correct reaction to the Revenge phenomenon.
Less optimistic are those contributors who take the Revenge problem to motivate some kind of inconsistency view of truth. Kevin Scharp argues that the Revenge problem should lead us to accept the view that truth is an inconsistent concept that ought to be abandoned. According to Scharp, truth is a ‘confused’ (technical term!) concept, one that combines inconsistent application rules of two separate concepts. He proposes to replace truth with these two other concepts, with what he calls ascending and descending truth. Douglas Patterson seems happy to keep our language (including our truth predicate) as it is, although he also advocates an inconsistency view according to which competent speakers of a natural language like English process that language according to a semantic theory that implies contradictions, and, thus, cannot be true. The main obstacle to such a view is, according to Patterson, that understanding a language is mistakenly conceived to consist in standing in a factive relation (e.g. knowledge) to a semantic theory of that language. He argues that such a view is explanatorily gratuitous and should be given up. Matti Eklund focusses on the tension between a common strategy for dodging Revenge problems, one that denies expressibility of certain semantic properties in a natural language while allowing that they may be expressed in a meta-language, and what he calls the incredulity problem – created by the appearance that such properties are clearly expressible already in natural language. He argues that this tension is exacerbated once we drop the common simplifying assumption that truth predicates in natural language apply only to sentences of one language. Then, interlanguage paradoxes threaten even in languages with very limited expressive resources because they piggyback on the expressive resources of other languages. Eklund concludes that only inexpressibility in all languages provides a way out and that an inconsistency account of truth can be used to resolve the incredulity problem.
Finally, Tim Maudlin and Thomas Hofweber advocate more general lessons from the Revenge problem while keeping the theory of truth fairly simple. Maudlin argues that the Liar paradox can be resolved with recourse to only three semantic statuses for sentences to have: true, false and ungrounded. In order to deal with the Revenge problem, he argues that we have to deny the rule of assertion according to which only true sentences are correctly assertible. Sometimes ungrounded sentences can be correctly asserted as well. Once this is realised there is no tension between taking, e.g. (RL) to be ungrounded and using (RL) in order to state this very fact. Hofweber even suggests that we do not have to deny the pre-paradox view that the inference rules of classical logic and the introduction and elimination rules for the truth predicate are valid nor that those apparent instances of them which generate paradox really are instances. Instead, Hofweber argues, we should give up the ideal of deductive logic that holds that all instances of logical rules are truth-preserving. Rather, we should retreat to a generic conception of deductive logic which says, roughly, that, typically, instances of valid rules are truth-preserving – this being consistent with the fact that some of them aren't. Applications of the inference- and truth-rules to paradox generating sentences will be those instances that are not truth-preserving. Thus, paradox can be avoided while holding on to the (generic) validity of inference- and truth-rules without restrictions on expressibility. The collection is rounded off by an index as well as a useful introduction by the editor, J.C. Beall, which provides some background on the Liar and the Revenge problem debate. An appendix on formal and informal aspects of Kripke's fixed point theory of truth is included because familiarity with it is presupposed in quite a few of the papers. Though helpful, both chapter and appendix should probably be seen as a reminder rather than an introduction to the uninitiated.
If there is anything that can be criticized about this collection it is perhaps in its editing: first, the papers appear in alphabetical order – one cannot help but wonder whether some other way of grouping the papers would have been more helpful. Second, it appears there was some laxity in proof-reading which resulted in a few too many typos in places and inconsistent (and partly incorrect) cross-references to papers in the same volume. These very minor deficiencies notwithstanding, Revenge of the Liar should be a welcome addition to the libraries of everyone doing serious work on the paradoxes and in the philosophy of logic and language in general.
