# Andre Chapuis and Anil Gupta, eds., Circularity, Definition and Truth.

New Delhi: Indian Council of Philosophical Research, 2000. Pp. vi, 402.This is a collection of eighteen solicited papers on the topics of the title: circularity, definition, and truth. The papers are loosely connected in subject matter, but present a great variety of issues, theories, and approaches. Amongst the many subjects discussed are: the revision theory of truth and applications of revision rules, partiality and fixed point constructions, substitutional quantification, fuzzy logic, negation, belief revision, context dependence, hierarchies, Tarski on truth, deflationism, correspondence theories of truth, and normative aspects of truth. The Liar paradox figures prominently in this collection, but is not alone. Other familiar paradoxes are also discussed, including paradoxes of definability, the Sorites paradox, and the Surprise Exam paradox. So-called paradoxes of rationality, such as the Prisoner's Dilemma, receive attention as well.

As a collection of distinct papers on such a wide range of issues, this volume does not offer itself as a systematic overview of the state of the art in any particular subject. Nonetheless, many of the papers in the volume are excellent, and taken together, they provide a valuable resource for those working in the areas they address. For those with a more casual interest, they provide an interesting snapshot of some active areas of investigation. For the most part, the technical demands made by the papers upon the reader are quite minimal. Most of the collection should be accessible to anyone with a reasonable background in elementary logic, though some of the papers do assume more extensive knowledge.

Instead of attempting to touch upon all of the many issues discussed by the papers in this collection, I shall try to place the book within the current literature, and comment on some of the more prominent themes in it. I shall direct my attention toward four themes. First I shall discuss deflationism and related issues about the nature of truth. Then I shall turn to the Liar paradox, and discuss three themes central to recent work on it: partiality, context dependence, and the revision theory of truth. This will represent a significant portion of the volume, but I regret it will preclude mention of some fine papers in it.

There has been a fairly sharp divide within the literature on truth, certainly since Tarski's (1935) work if not before. On one side of the divide, we find what we might call the traditional problem of the nature of truth, as addressed by correspondence and coherence theories. On the other, we find the problem of the Liar paradox and related problems in developing formal theories of truth. In recent years, much attention on the first side has focused on deflationism about truth, and its relation to the correspondence theory. It has also been noticed in recent years that deflationism provides a natural bridge between the two sides. Deflationism comes in various forms, but among its central ideas is that the T-schema is somehow close enough to a definition of truth. As Tarski observed, this opens the way for the Liar paradox, and will tend to lead to inconsistent theories. Many contemporary deflationists have viewed this as a minor problem, but there is an increasing literature arguing that it poses fundamental problems for deflationism. In this collection, this cause is taken up by Brendel, while a technically sophisticated refinement and defense of deflationism is offered by Halbach. It was observed by Tarski that we should expect a good theory of truth to provide for important generalizations about truth, such as the principles of bivalence and noncontradiction. It has often been pointed out that many versions of deflationism implicate formal theories that cannot do so. Halbach seeks to defend deflationism against this sort of objection, by developing a stronger, yet still deflationist, theory of truth. Other papers in the volume, such as Kovach on normative aspects of truth, and Wilson on correlation and correspondence, engage directly with the traditional problem of truth. Taken as a whole, the collection displays how useful the interaction between the sides of the divide can be.

It is interesting that both sides in the debate between deflationists and correspondence theorists often appeal to Tarski's work. Tarski provided us with the first precise formulation of the T-schema, while also showing how to give a recursive definition of truth. Such a recursive definition often forms the core of modern versions of correspondence theories. Tarski himself occasionally described his work as explicating a classical correspondence conception of truth. This is investigated in some depth in the contribution by Schantz, who argues that this makes much of Tarski's work at odds with contemporary deflationism. The issue is also touched upon briefly by Kremer.

On the Liar paradox's side of the divide, recent work has centered around four main areas: partiality, contextual and hierarchical approaches, the revision theory of truth, and paraconsistent approaches. Paraconsistency is not discussed in the collection, so I shall not say anything about it here. The rest have their roots in important work of the 1970s and early 1980s (some of which is collected in Martin 1984). They are my remaining three themes. Let us begin with partiality. The idea that certain paradoxical sentences might be taken to be neither true nor false is no doubt quite old, but it gained new impetus from the seminal work of Kripke (1975). As is well known, this paper showed how to build partial interpretations of the truth predicate that apply to a wide range of sentences containing that very predicate. Contra the claim of Tarski, at least a great deal of semantic closure can be achieved. Perhaps less well known is the extent to which Kripke provided a useful collection of mathematical tools for investigating the semantics of truth predicates. Kripke at the same time had been pursuing the branch of mathematical logic known as definability theory, and his theory of truth makes a powerful connection between these subjects. This sort of material is broached in the papers by Antonelli and McGee. Antonelli shows some examples of Kripke-like fixed point constructions, while McGee shows that the sentences true in the Kripke minimal fixed point are precisely those for which the translation to the language without the truth predicate but with substitutional quantifiers is true.

As Kripke's work made partial predicates technically appealing, it made all the more pressing the question just how we are to understand them. This is taken up in the paper by Kremer, which raises the question of why a three-way partitioning of sentences of some language, such as that provided by a Kripkean truth predicate, is theoretically interesting. It is usually held by proponents that the fixed point property, that a sentence is true in a partial model just in case it is in the extension of the truth predicate, establishes the interest of the construction. But Kremer argues that this is not sufficient. He goes on to propose an alternative understanding of the partially defined truth predicate. He suggests that it does not characterize the notion of truth, which he thinks has deflationist properties. Instead, it gives a metalinguistic notion of correspondence, which is useful, for instance, as it allows us to explain speakers' reliability. Here again, we see a nice example of the interplay possible between traditional concerns about the nature of truth and the investigation of the paradoxes.

To this day, partiality approaches to the paradox have been dogged by the so-called 'Strengthened Liar'. This leads to our next theme: context dependence and hierarchical aspects of truth. The Strengthened Liar observes that if we follow a partiality theorist and declare the Liar sentence neither true nor false (or failing to express a proposition, or suffering from some sort of grave semantic defect), then the paradox is only pushed back. For we can go on to conclude that whatever this status may be, it implies that the Liar sentence is not true. This claim is true, but it is just the Liar sentence again. We are back in paradox. As Gaifman's paper points out nicely, the problem here is first of all how this last conclusion can have a truth value--or express a proposition, or otherwise be semantically viable. After all, it appears to be just the claim we agreed could not have a truth value. If it does, then it simply appears to reinstate the Liar paradox in its ordinary form. Either way, we have a problem.

In this reasoning, we step back and draw a conclusion from the semantic status of the Liar sentence. It is a persistent idea that in doing so, we invoke a truth predicate somehow different from the one that figured in the Liar sentence itself. Thus, we can come to a true conclusion, without reinstating the paradox. Though this need not lead to a fully Tarskian hierarchy, it has elements of the basic hierarchical idea. There are several ongoing research projects to show how this aspect of truth can be captured without the unwanted consequences of Tarski's hierarchy. In this volume, Gaifman, Koons, Simmons, and to some extent Skyrms, develop proposals they have previously made along these lines. One version of the idea, pursued in different forms by Koons and by Simmons, is that the truth predicate itself has an indexical element. Truth is [truth.sub.i], where i is a parameter set by context. Both Koons and Simmons discuss theories of how this parameter is to be set. I think we should ask, though, what grounds we have for positing an index on the truth predicate. In particular, what does this proposal imply about natural-language occurrences of the truth predicate? AS Liar-like phenomena can appear in natural-language discourse (and occasionally do), we would be led to conclude that the natural-language truth predicate has an implicit argument, which is set by context. It is not clear whether we have any evidence in favor of this, outside of the need to resolve the paradox. Understanding of context dependence in natural language has advanced considerably in recent years, and it would be useful to see more clearly how context-dependence approaches to the Liar relate to it.

Positing an index on the truth predicate is not the only way to develop a context-dependence approach to the Liar. One example is given by Gaifman's paper. Rather than pursue the question of how to relate the context dependence of the Liar to context dependence in natural language generally, Gaifman appeals to concepts from the semantics of programming languages.

The centerpiece of this collection is the last of my themes: the revision theory of truth (elaborated at length in Gupta and Belnap 1993). This is discussed, in stone way or another, in seven of the papers: those by Antonelli, Chapuis, Gupta, Koons, Lee, Rosenberg, and Skyrms. Many of these papers offer alternative ways of understanding or applying the revision theory. Most striking, I believe, is the proposal in the work of Chapuis and Gupta, that the concept of rational choice is a circular concept.

According to a widespread classical approach to definitions, definitions are required to be noncircular and conservative. Roughly, it is expected that definitions be explicit definitions, with definienda eliminable in any context. As is elegantly described in Antonelli's paper, many definitional procedures fail to meet these requirements while still being perfectly acceptable. Inductive definitions, for instance, are rarely conservative. But the revision theory goes further. It argues that definitions involving unresolvable circularity are still good definitions. To take an example from Gupta and Belnap (1993), of the sort discussed by numerous papers in this collection, consider the 'definition' Gxiff Fx [disjunction] (Hx[and][logical not]Gx). This sets up a well-defined revision process. For a given "hypothesis" about what the extension of G is, the definition allows a revision of that hypothesis. In some cases, the circularity we see with the occurrence of G on the right of the biconditional leads to an infinite loop in the applications of this rule. In this case, any a such that [logical not]Fa and Ha will do so.

One need not hold onto the classical approach to definitions to be resistant to the idea that this is a good definition. At a minimum, it might be suggested, a definition of a predicate should provide a semantic value: an extension or perhaps an extension and an anti-extension (pace Gupta and Belnap, who argue at length for the acceptability of circular definitions). Though we might take the stable elements under a circular definition to determine an extension and an anti-extension, the revision theory does not generally do this. The papers by Chapuis and Gupta offer an application of the revision theory that bypasses this aspect of the theory. They argue that the concept of rational choice is circular: in particular, they present games in which rational choices exhibit the kind of circularity we saw with circular definitions. (A related point is made by Skyrms.) They argue that a more satisfying account of rationality can be provided by formulating the correct rules of revision for rational choices. I am not really qualified to say whether this is a satisfactory account of rational choice. But it holds out an intriguing prospect for those of us who are concerned that circular definitions fail to provide semantic values. Proponents of the revision theory have always said it is the revision process that is important. Here, we see an application that makes this idea clearer. It may well be that the processes involved in rational choice behave in the way the revision theory describes. In some import ways, this may help us to understand the revision theory. However, the game-theoretic applications all involve finite cycles, and so do not yet shed any light on the much discussed question of limit rules.

While it is certainly not exhaustive, a great many of the philosophically central ideas on truth and paradox find some place in this volume. Some important new ideas are presented, as well as some developments of important research programs. It is a fine collection.

References

Gupta, Anil, and Nuel Belnap. 1993. The Revision Theory of Truth. Cambridge: MIT Press.

Kripke, Saul. 1975. Outline of a Theory of Truth. Journal of Philosophy 72: 690-716.

Martin, Robert L., ed. 1984. Recent Essays on Truth and the Liar Paradox. Oxford: Oxford University Press.

Tarski, Alfred. 1935. Der Wahrheitsbegriff in den formalizierten Sprachen. Studia Philosophica 1:261-405. Translated by J. H. Woodger as "The Concept of Truth in Formalized Languages." In Logic, Semantics, Metamathematics, 2d ed., edited by J. Corcoran. Indianapolis: Hackett, 1983.

MICHAEL GLANZBERG

University of Toronto

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Author: | Glanzberg, Michael |
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Publication: | The Philosophical Review |

Article Type: | Book Review |

Date: | Jul 1, 2002 |

Words: | 2420 |

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