Definition and Revision: A Response to McGee and Martin Anil Gupta Philosophical Issues, Vol. 8, Truth. (1997), pp. 419443. Stable URL: http://links.jstor.org/sici?sici=15336077%281997%298%3C419%3ADARART%3E2.0.CO%3B2N Philosophical Issues is currently published by Ridgeview Publishing Company.
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PHILOSOPHICAL ISSUES, 8 Truth. 1997
Definition and Revision: A Response to McGee and Mart in' Anil Gupta
Vann McGee and Donald A. Martin are critics of the sort that authors wish to have: critics who spare no labor in getting to the bottom of their subject; critics who have the wisdom to distinguish the central from the peripheral; and critics who are perceptive and fair. I believe it was Mark Wilson who proposed McGee and Martin for this session. My first reaction when I heard the proposal was to wonder why Mark was suggesting such tough and penetrating critics when gentler ones were available. I now recognize that tough critics can be good critics, if they are wise and fair. 'This paper is my contribution to the session on Nuel Belnap's and my book held a t the 1996 Central Division Meetings of the American Philosophical Association. Belnap, unfortunately, did not attend the session because of a conflicting commitment in Germany. He and I did discuss in some detail an early outline of this paper, and I want to thank him for his perceptive comments. (Belnap, of course, is not to be blamed for the weaknesses in this paper.) Thanks to AndrC Chapuis, James Hardy, and Allen Hazen for their comments. Thanks to the National Endowment for the Humanities for support. This paper was written while I held an NEH Fellowship for University Teachers.
Most of McGee's and Martin's criticisms and questions are directed at one element of the revision theory an element that is pivotal and deserving of scrutiny. I will begin by locating this element in a barebones account of the theory. Then I will turn to McGee's and Martin's criticisms and questions, and I will make some observations in response.
1 Bare Bones of the Revision Theory The fundamental claims of Nuel Belnap's and my book, The Revision Theory of Truth (RTT), can be summed up in the following two theses.
A. Circular definitions and, more generally, systems of mutually interdependent definitions are logically legitimate. Semantic sense can be made of such definitions.
B. Truth is a circular concept. These theses are motivated by a strong parallelism that obtains between the behavior of concepts defined by circular definitions and the behavior of the concept of truth. Concepts defined by circular definitions exhibit the same types of pathologicality that truth exhibits. Moreover, like truth, these concepts apply unproblematically over a range of objects. Much of R T T is devoted to exploring ways of making sense of circular (and mutually interdependent) definitions, and to applying the lessons learned here to the concept of truth (and some other concepts). The two theses mentioned above are not, I wish to stress, mere frosting on top of a more substantial technical cake. From the philosophical point of view, the two theses are fundamental and the technical machinery of R T T is simply a means of substantiating them. The explanation that the revision theory gives of paradox lies in these theses. Why, for example, is it that a paradox like the Liar arises for truth, reference, and necessity but not for the concept of natural number? Response: Because truth, reference, and necessity are but natural number is not circular. For another example, why is it that the sentence 'This very sentence is not true' is paradoxical, but the parallel selfreferential sentence 'This very sentence is not an expression of English' is not paradoxical? Response: Because the definition of 'true' makes the truth conditions of the former viciously circular, but the definition of 'expression of English' does not have the same effect on the latter.
30. DEFINITION A N D REVISION
421
R T T s ambition is not normative. RTT does not aim to block the paradox or to propose ways of making truth paradoxfree. Instead, R T T s aim is descriptive. Its aim is to construct a good (though idealized) description of our ordinary concept of truth a description that will help explain both the problematic as well as the unproblematic behavior of truth. RTT draws an analogy with understanding eclipses. A good description of the motions of the heavenly bodies removes the mystery surrounding the eclipses by helping us understand why they occur and when they can be expected to occur without, of course, eliminating the eclipses. Similarly with truth: A good description of the concept helps us understand why the paradoxes occur and when they can be expected to occur but it does not eliminate the paradoxes. It eliminates only the logical anxiety and the philosophical confusions that the paradoxes generate. The two theses, then, are important elements of R T T s description of the concept of truth. They suggest that paradoxes are a "natural" occurrence: The paradoxical behavior of truth arises because truth is a circular concept and because it is in the nature of circular concepts to exhibit this sort of behavior (under certain circumstances). ~ McGee and The two theses have attracted some c r i t i ~ i s m .But Martin are not concerned to argue against them. Their criticisms are directed against the specific semantics that RTT proposes for circular definitions. This semantics is outlined in the propositions CF below. C. Circular definitions impart a meaning to the definiendum, but this meaning has a hypothetical character.
A circular definition of, say, a predicate G, does not provide a rule for determining which objects are G. But, given a hypothesis X about the extension of G, the definition fixes what objects count as G under the hypothesis. So, even though the definition may not yield an extension for G, it does yield a rule of revision p , which can be defined as follows. For all objects d, d E p ( X ) H d satisfies A(x, G) when G is assigned the extension X .3 'See, in particular, Koons 1994 for a n objection t o the thesis t h a t t r u t h is circular. See G u p t a and Belnap 1994 for a reply t o t h e objection. 3Here and below I work with simpler and less general notions of definition, hypothesis, rule of revision, etc. than those found in RTT. Note t h a t t h e revision
R T T s semantics for circular predicates is built on the rule of revision:
D. The extensional significance of a circular predicate is captured by the rule of revision. In the terminology of RTT, the signification of a circular predicate is its rule of r e v i ~ i o n . ~
E. Categorical information can be extracted from the rule of revision. To do so we should consider the effects of repeated applications of the rule to arbitrary hypotheses. Repeated applications of a revision rule p generate revision sequences. We start the sequence with an arbitrary hypothesis. Given a stage Z in the sequence, we move to the next stage by applying p to Z , obtaining p ( Z ) . We can continue the process into the transfinite by summing up the effects of earlier revisions at limit stages. We do this by examining the revision sequence obtained up to the limit. Call an object d positively stable in this sequence if there is a stage in the sequence after which d always belongs to the subsequent stages. Call d negatively stable if there is a stage after which d never belongs to the subsequent stages. Call d unstable if it is neither positively nor negatively stable. The hypothesis Z for a limit stage is required to contain all the objects that are positively stable and none that are negatively stable. On unstable objects there are no constraints. They or some of them can be included in Z or they can be excluded from Z. R T T extracts categorical information from a revision rule p by considering the revision process i.e., all the unending revision sequences that p generates. The principle of extraction is:
F. Those objects which are positively (negatively) stable in the revision sequences of G, irrespective of the initial and subsequent choices, categorically satisfy (fail to satisfy) G. Actually, this is the principle underlying one of the two systems presented in RTT, the system S*; a slightly different principle underlies the other system S#. rule for mutually interdependent definitions takes as argument an assignment of hypotheses to all the definienda and yields a revised assignment of hypotheses as value. 4The signification of an expression carries all the extensional information about the expression. The signification of a classical predicate can be identified with extension but, in general, signification is richer. The signification of, e.g., a threevalued predicate needs to carry more information than just the extension of the predicate. See RTT, pp. 3031.
30. DEFINITION AND REVISION
423
It will be useful to note here two features of the revision account of definitions. First, the account does not affect our ordinary ways of working with noncircular definitions; these retain their familiar logic and semantics. Second, the revision account makes a sharp distinction between definitional equivalence ('=Df') and the material biconditional ( ' t t ' ) . A definitional equivalence,
does not entitle one to the corresponding biconditional, Vx[Gx ++ A(x, G)]. The entitlement holds for noncircular definitions and for a restricted set of circular definitions, but not for all definitions. This distinction between '=Df' and ' t t ' , I want to stress, is not an artifact of the theory. It is forced on any general theory of definitions that meets the most minimal conditions namely, conservativeness and preservation of classical logic. No logical space exists for circular concepts without the distinction. Most of McGee's and Martin's criticisms and questions are directed to the way the revision theory extracts categorical information from the hypothetical rule (i.e., propositions E and F above). This is a crucial element of the theory and, as I said, it deserves scrutiny. McGee's and Martin's criticisms and questions fall naturally into three sets. The focus of the first set is the treatment of limit stages in revision processes; the focus of the second is how the revision theory compares with its rivals; and the focus of the third is the applicability of the revision theory to natural languages. The next three sections discuss these criticisms and contain some responses to them.
2
Inductive, Implicit, and Circular Definitions
According to the first set of objections, RTT's semantics for definitions is in conflict with some of our ordinary practices. McGee, the author of these objections, writes: A liberalized conception of definition ought properly to incorporate familiar definitional practices, at least inasmuch as these have proven satisfactory. The GuptaBelnap limit policy doesn't incorporate our tried and true practices in utilizing firstorder positive inductive definitions. (McGee 1997, p. 397.)
[Tlhe revisiontheoretic treatment [of definitions] comes into conflict with our ordinary practices in utilizing implicitly defined terms. (McGee 1997, p. 398.) McGee gives the following nice examples to show the divergence between RTT's semantics and the semantics of inductive and implicit definitions. (1) FX=
~ [Xf
(2) HX=
~
= 0 & Fl] V
[X
HO f & [X = 1 V
= 1 & FO].
(X
= 0 & HI)].
The first definition, if read inductively, yields the null set, 0, as the interpretation of F, and validates the corresponding biconditional: (3) V x ( F x

[x = 0 & Fl] v [x = 1 & FO]).
In general, with inductive definitions, the rule of revision p is monotone i.e., for all hypotheses X and Y, As a consequence, p has a complete lattice of fixed points. It has, in particular, a least fixed point, and the inductive definition interprets the definiendum via this fixed point. In McGee's example ( I ) , the rule of revision, p l , has the following properties:
z E pl(X) a ( z = 0 V z
= I ) , for all hypotheses
X;
The least fixed point of p l is 0, and so, on the inductive reading of ( I ) , the interpretation of the definiendum F is 0. However, as McGee points out, the revision process for p l does not converge to 0; it contains the following unending sequence of revisions: (01, (11, (01, (11,. .  (01, (11, (01,.  . .  .. The interpretation of F on the revision theory, then, is not 0 and the biconditional (3) fails. The second of McGee's examples above, if read as an implicit definition, yields (1) as the interpretation of H and validates the corresponding biconditional
30. DEFINITION AND REVISION 425
In general, a set of definitions can be given an "implicit" reading if there is a unique interpretation of the definienda that makes the corresponding biconditionals true. (In other words, with implicit definitions, the rule of revision has a unique fixed point and the definienda are interpreted via this fixed point.) So, (2) can be given an implicit reading because there is exactly one interpretation of H namely, (1) that makes the biconditional (4) true. However, as McGee points out, the interpretation of H on the revision theory is not (1). The revision rule for (2), p2, has the following properties:
z E p,(X)
+ ( z = 0 V z = I ) , for all hypotheses X ;
So, p2 has a unique fixed point (1). But the revision process does not converge to it; p2 generates the following unending revision sequence:
0, (0, I ) , @ (0, , I ) , . . . 0 , {O, 1 ) , 0 . . . . . .. On the revision theory, then, the interpretation of H is not (1) and the biconditional (4) turns out not to be valid. In response to this instructive objection I want to argue first that the demands it places on the theory of definitions should not be accepted. These demands are: (i) If the rule of revision p has a unique fixed point then the definienda should be interpreted via this fixed point (The ImplicitDefinitions Requirement). (ii) If p is monotone then the definienda should be interpreted via the least fixed point of p (The InductiveDefinitions Requirement). To reject these general demands is not to reject McGee's nice examples, however. In the second part of my response I will sketch a version of the revision theory that accommodates these examples. An important desideratum on the semantics of definitions is conservativeness: The addition of definitions to a language should not affect the semantic status (e.g., the truth value) of sentences that do
not contain the new definienda.5 For example, the introduction of the term 'Martin's Axiom' into our language should not affect the status of a sentence such as 'Chicago is a large city'. Conservativeness should be understood in the following strong way. Conservativeness Desideratum. Let definitions Dl be added to a language L, resulting in the extended language L D l . Let further definitions D2 be added to L Dl, resulting in the language L + Dl + D 2 . Then, the semantic value of the expressions of L Dl should remain the same in L+Dl+D2.
+
+
+
The Implicit and InductiveDefinitions Requirements can be met only by violating conservativeness. Here is an example that shows this. Let Dl consist of the definition, (5) J x
=
~
Jx.
f
And let D2 consist of the definition,
(6) K x
=
~
( Jf x V  K x ) .
+
The InductiveDefinitions Requirement dictates that in L Dl the interpretation of J should be 0. But the ImplicitDefinitions requirement dictates that in L + Dl + D2 the interpretation of J should be the domain of discourse, for definitions (5) and ( 6 ) have a fixed point only on this interpretation of J . Conservativeness is thus violated: The addition of D2 changes the status of some sentences of L D l , e.g., those that say that something is J . The problem with conservativeness is due entirely to the ImplicitDefinitions Requirement. Suppose we supplement L Dl with D3, where D3 consists of the definition,
+
+
The ImplicitDefinitions Requirement dictates that the interpretation of J in L + Dl + D3 should be 0. So, the status of the sentences of L Dl can be affected by what further definitions are added to the language. One addition makes nothing belong to the extension of J , but another addition makes everything belong to the extension of J . So, a general theory of definitions can satisfy the ImplicitDefinitions Requirement only by violating conservativeness.
+
5I am assuming that these sentences are not about such things as the number of definitions in the language. A more precise version of the conservativeness requirement can be stated. See RTT, 5A.13.
30. DEFINITION A N D REVISION 427
There is a strong reason, then, for not accepting the ImplicitDefinitions Requirement. This does not mean a rejection of implicit definitions, however. It means the rejection only of one direct, but faulty, method of applying the general theory to implicit definitions. The method is faulty because, e.g., formulas (5) and (6) do not state the true definientia of J x and K x on the implicitdefinitions reading. We need to supply J x and K x with their true definientia before the general theory of definitions is applied. And when we do this, we find there is no conflict between the theory and the practice of implicit definition^.^ The true definientia of J x and K x can be constructed as follows. Replace '=Df' by '*', and J and K by the predicate variables Y and 2,in (5) and (6), obtaining: (Yx ++ Yx), (Zx * Yx V ~ Z X ) . Let B(Y, Z ) be the universal closure with respect to x of the conjunction of the resulting formulas. That is, B(Y, Z ) is: Vx[(Yx * Yx) & (Zx * Yx V ~ Z X ) ] . Then, the true definientia for J x and K x are respectively: (3!Y, Z)[B(Y,211 & (VY, Z)[B(Y,2) Yxl, (3!Y, Z ) [ B(Y, Z)] & (V Y, Z ) [ B(Y, Z ) + Zx]. +
These definientia capture the intent of the implicitdefinitions reading: They state that J and K are to be interpreted via the unique fixed point of (5) and (6). Note that the definientia involve no circularity. Hence, the general theory treats them in the expected way. So, there is no conflict between the theory and the practice of implicit definitions. The general theory subsumes implicit definitions, once these definitions are spelled out and are provided with their true definientia. Similar conclusions can be drawn for inductive definitions and also for coinductive definitions, in which the definiendum is interpreted via the largest fixed point. The true definiens of J x on the inductive reading of (5) is VY[Vz(Yz * Yz)
+ Yx].
6 0 r whatever there is of this practice. Implicit definitions, unlike inductive definitions, are not standard instruments in the toolbox of logicians and mathematicians. They are found more often in the toolbox of philosophers (e.g., Carnap's). Implicit definitions provide a way of gaining analyticity by brute force (Quine would say "by theft") where the straight path is difficult or blocked.
On the coinductive reading, the definiens is
3 Y[Vz(Yz++ Yz)& Yx]. As before, the rejection of the demand labeled "InductiveDefinitions Requirement" above and a parallel "CoinductiveDefinitions Requirementn does not entail the rejection of inductive and coinductive definitions. Once the definientia exploited in these definitions are spelled out, there is no conflict between the general theory and the practices involving these definitions. Inductive definitions, coinductive definitions, and R T T s theory of definitions all work with circular specifications, but with different aims. The former are concerned to use circular specifications to define standard, noncircular, concepts. They do so by eliminating circularity the inductive definitions by taking the least fixed point, the coinductive by taking the largest fixed point. Thus, the circular specification ( 5 ) defines on the inductive reading a totally empty concept, and on the coinductive a totally unconstrained concept. R T T s concern, however, is not to use circular specifications as stepping stones to standard, noncircular, concepts. Its concern instead is to make sense of these specifications just as they are, without devices that eliminate circularity. From this point of view, (5) defines a circular concept J that is everywhere pathological; every object behaves like the TruthTeller with respect to J. There is a requirement in the neighborhood of the InductiveDefinitions Requirement that is more plausible and that deserves to be studied. Call definitions D positive if all the occurrences of its definienda in the definiens are positive. Then the requirement is:
The PositiveDefinit ions Requirement. Let definitions D be positive and let p be the revision rule for D. Then, the . ~ is, revision process for p must manifest no i n ~ t a b i l i t ~That each revision sequence should always stabilize at a fixed point.8 If the requirement is met then the material biconditionals corresponding to positive definitions will be deemed valid. Furthermore, positive definitions will generate no pathologicality other than the 7McGee has indicated t h a t this, and not the InductiveDefinitions Requirement, is the one t h a t he favors. Note, however, t h a t the PositiveDefinitions Requirement has no direct connection with the practice of inductive definitions. Whether a theory satisfies this requirement is independent of whether the theory comes into conflict with the practice of inductive definitions. 'A natural strengthening of this requirement is t h a t all revision sequences must stabilize when p is monotone.
30. DEFINITION A N D REVISION 429
TruthTeller type. I think the PositiveDefinitions Requirement is attractive and plausible but not absolute. Other things being equal, a theory that meets it is to be preferred over another that does not. But, when things aren't equal, factors such as conservativeness and simplicity may make a theory that violates the requirement preferable to one that meets it. The PositiveDefinitions Requirement is not met by the systems S* and S# of RTT, but it can be met by changing a little the treatment of limit stages. In RTT, Belnap and I had chosen to work with the simplest and most natural limit rule. But we had noted that other treatments were possible and had mentioned some of them (see RTT, pp. 168169). One treatment we had mentioned was to require full variation. The idea is that in the course of revision the full range of available choices should be considered. Formally the idea can be implemented thus: Say that a hypothesis X coheres with a sequence S iff X contains all the objects that are positively stable in S and none that are negatively stable. So, for example, the hypotheses that cohere with the sequence, (01, (11, (01, (11,. . . (01, (11, {O}, . . . . . ., are 0, (01, (11, and {0,1}. As it happens, these hypotheses are also the ones that cohere with the other revision sequence considered above: 0,{0,1},0, (0, 11,. . . 0 , { 0 , 1 } , 0 . . ..... A sequence S is said to be fullyvaried iff every hypothesis X that coheres with S is cofinal in S i.e., there is no stage in S after which X does not occur in S (RTT, p. 168, fn. 26). So, for example, neither of the two sequences just considered is fullyvaried. The first fails to be fullyvaried because it does not contain 0 (and ( 0 , l ) ) ; the second because it does not contain (1) (and (0)). Here are some simple examples of sequences that are fullyvaried:
(11, (11, (11,. . . (11, (11,. . . . . .; 0, {1}, 0, {l}, . . .O, {1}, 0 . . ..... Let Sgv be a theory of definitions like S* except that it requires revision sequences to be fullyvaried. It is not hard to show that for monotone rules fullyvaried revision sequences always culminate in a fixed point. It follows, therefore, that Skv satisfies the Positives, yields two additional bonuses. First, Definitions Requirement. ,& as Andrk Chapuis (1993, 1996) has argued, the theory meets many of the desiderata laid down by Aladdin Yaqiib in his book (Yaqiib
1993). Second, the theory yields (1) as the interpretation of H in McGee's example (2).'
3 Alternative Theories of Definitions The second set of criticisms and questions is due to Martin and concerns the relative merits of the revision theory over its rivals. Martin points out that rival theories in particular, Saul Kripke's "least fixed point" theory can adopt the principal philosophical theses of the revision theory.10 They too can assert that circular concepts are logically legitimate and that truth is circular. Is there anything that favors the revision theory once its rivals are modified in this way? More particularly: Why accept a revision semantics for circular concepts over, e.g., a fixedpoint account? Martin writes, [Tlhe technical dimension and the philosophical or conceptual dimensions [of the revision theory] are independent, and that it is possible to be a fixed point theorist in the technical sense while holding philosophical views much like those of the revisionists. For this reason, I will henceforth use the term "fixed point theory" for any theory that is mathematically a fixed point theory, irrespective of its philosophical component. (Martin 1997, p. 410.) Martin sketches a theory LFP of definitions that is modeled on Kripke's theory of truth. L F P reads the logical connectives occurring in a definition in the Strong Kleene way, obtaining thereby a Strong Kleene jump for the definition. This jump has a least fixed point, which L F P takes to be the interpretation of the definiendum. Are there reasons for favoring the revision theory over LFP? And 'Note that on the Herzberger limit rule (Herzberger 1982) the interpretation of H is not (1). The Herzberger rule does have the advantage that it validates some semantic principles e.g., the principle that a conjunction is true iff the conjuncts are true. It has the disadvantage, however, that there are equally plausible semantic principles that it fails to validate e.g., the principle that a conditional is true if its antecedent is untrue. Until we can provide a rationale for this differential treatment, semantic principles do not provide a solid ground for preferring the Herzberger rule. The status of the semantic principles in the theory of truth remains murky. One would have thought that any theory of truth should validate them. But this is called into question by an important theorem of McGee's: Under certain minimal conditions, semantic principles imply winconsistency (McGee 1991). ' O ~ c ~ ealso e observes this; see p. 400 of his paper.
30. DEFINITION AND REVISION
431
are these reasons compelling enough to warrant the additional complexities of the revision theory?'' The core philosophical conceptions underlying the revision theory, it is true, do not by themselves force a revision semantics. Belnap and I had taken notice of this point in RTT, but the notice is buried in a footnote: The theoretical moves that have been made in response to the pathological behavior of truth can all be made with respect to circular concepts. For example: (i) One can put forward an Inconsistency View of such concepts. (ii) One can argue that pathological sentences containing circular predicates do not express propositions. (iii) One can attempt to develop a Hierarchy View by which each circular predicate is viewed as systematically ambiguous. (iv) One can opt for a three or fourvalued approach and interpret these predicates via various fixedpoints. . . . In fact, the entire history of the Liar paradox can be mimicked in the context of circular definitions. (RTT, p. 117, fn. 4) Many perhaps all moves made in response to the Liar can be made with circular definitions. But, many perhaps most of these moves lose plausibility when generalized to circular definitions. For example, the Hierarchy View is a plausible response to the Liar. Natural Tarskian hierarchies exist for truth, and it is plausible to suppose, with Tyler Burge and Charles Parsons, that they can provide a descriptive account of our uses of 'true'. But, the Hierarchy View loses plausibility when it is generalized to arbitrary circular definitions. Sometimes circular definitions yield no natural hierarchies. Sometimes the hierarchies they yield provide a poor description of the definitions' behavior. The core philosophical conceptions underlying the revision theory do not force, and are not forced by, a revision semantics. But I believe that the two mutually support each other. The revision semantics shows that sense can be made of circular concepts and, thus, lends plausibility to the philosophical conceptions. The philosophical conceptions, in turn, make the revision semantics plausible by providing an intuitive rationale for its distinctive claim. According to the revision semantics, the signification of truth is given by a rule of revision. This kind of signification is far removed from the two, three, nvalued, and other interpretations familiar from standard "Martin, I should point out, does not think LFP is the only, or the best, fixedpoint theory. Nor does he regard fixedpoint theories as fully satisfactory. He thinks LFP is an attractive theory, and he argues that it is superior to the revision theory in some (but not all) respects.
semantics. The question naturally arises: Why should truth have this strange sort of signification? The philosophical conceptions provide an answer: Truth has the special sort of signification because it is circular. There are at least three respects, it seems to me, in which the revision semantics is superior to the theory LFP sketched by Martin. Two of these (i) and (iii) below issue directly from the philosophical conceptions underlying the revision theory. (i) The revision semantics preserves but LFP violates an important desideratum on the theory of definitions:
Preservation of Logic. Addition of definitions should not disturb the logic of the language. LFP violates this desideratum because it assigns a threevalued interpretation to some definitions, even when the base language is classical. For example, suppose a classical language is extended with the definition, M X = ~Mf X V Mx. LFP yields that M is totally undefined; the extension and antiextension of M are ruled by LFP to be 0. As a consequence, a classically valid inference such as, 3x(x = x), therefore (M0 v NMO), turns out to be invalid on LFP. This problem, I should note, can be overcome in several ways by theories based on the leastfixedpoint idea.12 Perhaps the most attractive one is to interpret the logical operators using the supervaluation method. This yields a monotone rule whose least fixed point can serve as the interpretation of the definiendum.13 The resulting theory meets the "Preservation of Logic" desideratum but, unfortunately, it is not a solution to the next two problems. 12see McGee 1991, Chapter 4, for some responses to a parallel problem in the theory of truth and some telling criticisms of them. 1 3 ~ h supervaluation e rule, a , for the definition,
is as follows. Let U , V , and X be subsets of the domain of discourse and let U and V be disjoint. Let us say that X extends (U, V ) iff U X and X and V are disjoint. Then, a ( ( U ,V ) ) = ( U ' , V ' ) , where U' ( V ' ) is the set of objects that satisfy (do not satisfy) the definiens A ( x ,G) on all interpretations X of G such that X extends (U, V ) .
30. DEFINITION AND REVISION 433
(ii) LFP and the supervaluation theory make definitions "gappier" than is necessary or desirable. For example, suppose that the domain of discourse is {O,1) and that the predicate P is given the following definition: Both fixedpoint theories imply that P is totally undefined.15 Intuitively and I know of no reason to distrust intuition here, one would have expected P to receive the classical interpretation (0). Note that (0) is the only classical fixed point of the definition and the revision process converges to it. No natural scheme, as far as I know, yields a leastfixedpoint theory that is free from problems of this sort.16 (iii) LFP and the supervaluation theory fail to capture and explain the pathological behavior of circular concepts. The key reason why RTT takes the revision rule to be the signification of a circular concept is that the rule captures and explains both the ordinary and the pathological behavior of these concept. The fact that we find the reasoning in, for example, the Liar paradox compelling is not just a simple error on our part. The source of the paradox lies in the concept of truth itself. The paradox issues from the very "meaning" of truth. The revision theory aims to give an account of "meaning" more precisely, signification that substantiates this idea. LFP and the supervaluation theory assign a standard sort of signification to circular concepts. They, in effect, transform circular definitions into noncircular ones, and they assign to circular definienda the signification of the corresponding noncircular one. The procedure yields a standard semantics, but an important dimension of circular concepts is lost as a result. The history of the Liar paradox is a history of attempts to understand truth within the confines of a classical view of concepts; it is a history of attempts to tailor standard semantics to fit the behavior of truth. Hierarchies, TruthValue Gaps, and TruthValue Gluts are all concessions to the strange behavior of truth, but they are all concessions made from the perspective of the classical view of concepts. The concession required, however, is that truth is circular. And once 1 4 ~ h iiss a notational variant of Example 5A.11 of RTT. 15They yield the same result for the definienda in McGee's two examples given above. 161would point t o phenomena such as the one cited here as a partial response t o Martin's criticism t h a t the revision theory is too complex. I know of no simpler way of accommodating the phenomena. But perhaps further investigations will reveal simpler ways.
this concession is made it is not much of a surprise that truth needs a special semantic treatment. Semantics that fit noncircular concepts do not suit the shape that truth exhibits. The distinctive feature of the revision theory is that it lets the phenomena guide its logical and semantic conceptions. It does not try to force phenomena into a preconceived logical and semantic mold. As a consequence, while other theories deny or resist circularity, the revision theory builds on it and exploits it. Once the resistance to circular concepts is abandoned, the important issue, it seems to me, is not one of signification but of how categorical information should be extracted from the hypothetical rule. Fixedpoint constructions are undeniably useful here. But they are not rivals of revision processes. On the contrary, they can work in tandem with revision processes to reveal properties of revision rules.17 It seems to me that once the resistance to circularity is abandoned, substantive semantic agreement is but a small step away. A little give and take should produce accord in what has been a highly contentious subject.18
4
Truth in Natural Languages
The final set of criticisms concerns the applicability of the revision theory to natural languages. The revision theory is developed in 1 7 ~ h e ris e no rivalry, it seems to me, between revision theory and fixedpoint theory understood a la Martin: "I will henceforth use the term 'fixed point theory' for any theory that is mathematically a fixed point theory, irrespective of its philosophical component (Martin, p. 410)". " ~ e f o r e leaving this section I wish to clarify the intent of a few remarks in R T T (pp. 199200) that are criticized by Martin (pp. 4135). In these remarks Belnap and I contrast the revisiontheoretic claim that the Tbiconditionals are definitional of truth with Kripke's instructions to an imaginary idealized subject learning 'true'. Kripke's instructions tell the subject when predications of truth can be asserted and when denied. Our point was that these instructions are incomplete. They do not fix the use of 'true' in all contexts. For example, they do not fix the use of 'true' in hypothetical contexts. Martin objects that the revision theory does not permit unrestricted use of Tarski biconditionals in all contexts, that the revision theory and Kripke's theory are on par here. This is right, but onlv if the Tarski biconditionals are read as material eauivalences. And our ~ o i n t was not that Kripke's instructions do not allow unrestricted use of the material equivalences, but that Kripke's instructions do not fix the proper use whatever it be of 'true' in all contexts. The instructions are compatible with different (and conflicting) rules for the use of 'true' in nonassertoric contexts. This is evidenced by the fact that Kripke supplements his instructions with a special explanation of the logical operators. Tarski biconditionals, read definitionally, are freely available in all contexts and fix the use of 'true' in, e.g., hypothetical contexts.
30. DEFINITION A N D REVISION 435
an idealized setting. Its details are worked out only for very simple formal languages languages that contain no complications apart from selfreferential truth. Can the lessons learned in this idealized setting be applied to natural languages? Can the revision theory give an account or, at least, the sketch of an account of the semantics of "true in English"? McGee and Martin give two important arguments in favor of a negative answer. The first argument rests on a feature of RTT's object languages, namely, that they all have a settheoretic interpretation. The second rests on a feature of RTT's metalanguage, namely, that it is invariably richer than the object languages. I will take up these arguments in order.
4.1
TRUTH IN LANGUAGES T H A T CONTAIN SET THEORY
The revision theory, as presented in RTT, applies only to those languages whose domain of discourse is a set. This raises a problem, McGee observes, when one tries to apply the theory to languages such as English that contain set theory. The variables of these languages range over all sets (and, possibly, other things). So, their domains of discourse do not constitute a set. Hence, the methods of revision semantics do not apply to them at least not in any direct way. McGee sums up the problem thus: [Tlake any precisely specified fragment of [English],so long as it contains the language of set theory and the adjective "true". . . Can we get a coherent understanding of truth for that language? Not by using revision theory, because the universe of the language isn't a set. (McGee 1997, p. 402.)
There are two approaches to this problem that I am not inclined to follow. The first what McGee calls "the received viewn holds that 'true in English' expresses an incoherent notion. This view is completely contrary to the spirit of the revision theory. As McGee observes, speakers of English apply 'true' not only to statements of everyday discourse but also to statements of set theory and of semantics. The methodology of the revision theory requires that instead of dismissing these uses we should search for ways of understanding them. The second approach maintains that, irrespective of what the users of a language may claim, the domain of discourse is always a set. If this is right, the theory of R T T applies asis and we have an easy way out of the problem. But I do not wish to follow this approach because, first, the best arguments for the claim that the domain must be a set rely on the paradoxes, and I am skeptical of these
arguments. They belong to the same family as the arguments that purport to show that no language can contain its own truth concept. Second, the methodology of the revision theory dictates that we put greater faith in semantic phenomena than in existing semantic theories. Hence, instead of dismissing talk about all sets as illusory, we should look for theories that make sense of this talk. Before we worry about selfreferential truth in a language containing set theory, we need to worry about the semantics of plain old set theory. How should we think of, and reify, its intended interpretation? .If we accept the intuitive idea that the variables of set theory range over all sets, then its intended interpretation is not one of the structures familiar from model theory. What then is it? The question has an easy answer if we allow ourselves to use proper classes. This solves the local problem, but it leaves a larger parallel problem unsolved: that posed by languages whose variables range over absolutely everything. We need a new approach. The possibility that seems to me worthy of exploration is that a new kind of "aggregate" serves the needs of semantics better than sets and classes. Semantics needs abstract objects. It needs reifications of domains, properties, relations, etc. And for many purposes these needs are well served by set theory. But set theory wasn't designed with the goals of semantics foremost in mind. It was designed to serve the foundational needs of mathematics. It has turned out to be a versatile supplier of abstract objects, one that serves many disciplines well. But set theory achieves its enormous versatility at a price. The richness of its constructions entails that certain kinds of domains, properties, and relations are not easily or adequately modeled within it. Perhaps, the abstract objects required by semantics need not figure in all the varied constructions in which sets figure. If so, some other setlike object an "aggregaten may escape the limitations of settheoretic interpretations and may provide a better model for our talk about all sets and about all objects.lg In whatever way we solve the problem of interpreting set theory, a revisiontheoretic account of truth, it seems to me, will always be available. Suppose that we have settled on a language L that ''Several logicians and mathematicians are exploring alternative set theories (Aczel 1988, Antonelli 1992 & 1994, and Forster 1992; Peter Apostoli has also been working in this area). As far as I can tell, their constructions are not primarily motivated by semantics, and are not particularly suited for it. I am suggesting that we pursue a theory of "aggregates" that is shaped by the needs of semantics. (Property theories such as those developed by Francesco Orilia (1995) are motivated by different semantical issues than those under discussion here.)
30. DEFINITION A N D REVISION 437 contains set theory. Suppose also that we have settled on an interpretation of L perhaps settheoretic; perhaps classtheoretic; perhaps Xtheoretic, for some newfound X . We can extend L, as before, to a language LS that contains "true in L+" ( T ) . T is governed by the Tarski biconditionals under their definitional reading. The signification of T is, therefore, a rule of revision. Observe that this rule can be modeled as a settheoretic object: It can be viewed as an operation on the powerset of the set of sentences of the language.20 Hence, revision sequences can be understood in the usual way and the notions "categorically assertible", "paradoxical", etc. can receive their usual definitions. These definitions, I should note, cannot be given within LS, for the rule of revision will not be noncircularly definable in LS. To object to this, however, is to raise the Problem of Semantic SelfSufficiency, to which I now turn.
Even if the problem of interpreting set theory can be solved, a fundamental argument remains for doubting that the revision theory applies to English (and other natural languages). The argument is as follows. (i) A semantic description of English must be possible. For, natural languages such as English are not mysterious, incomprehensible entities. They are part of the natural order and are as comprehensible as other natural phenomena. (ii) This semantic description of English must be formulable in English itself. In other words, English must be semantically selfsuficient. (iii) The revision semantics for a language can be constructed only in a richer metalanguage. (iv) Revision semantics is therefore not suitable for a semantically selfsufficient language such as English. (v) Consequently, revision semantics fails to explain the notion of truth as it applies to English (and other natural languages). Here is how McGee and Martin sum up the problem: [T]o obtain an understanding of the notion of truth as it applies to English.. . one must solve the problem of how to present the semantics of a language within the language itself. Looking at simplified languages is fine, but if we can only give the semantics of our simplified language within an essentially richer metalanguage, the fundamental and difficult problem of how to give the semantics for a language within the language itself will still remain before us. (McGee 1991, p. 147) "This easy method would not be available with 'Lsatisfaction". Its treatment would depend upon t h e specific semantics given for the base language.
The problem that the semantic paradoxes pose.. . is the problem of understanding our notion of truth. . . . And we have no language beyond our own in which to discuss this problem and in which to formulate our answers. (Martin 1997, pp. 4178) The objection reveals a fundamental disagreement over the very project in which we are engaged, or ought to be engaged. McGee and Martin take the project to be the construction of a language L that can express its own semantic theory (call this the "Semantic SelfSufficiency Project"). It is understood that the construction should be robust in the sense that it should work when L is rich in syntactic and settheoretic resource^.^^ In contrast, the project of RTT is to give the semantics of the predicate "true in L" of L (call this the "Truth Project"). It is understood that the semantics should be general in the sense that it should be applicable to L irrespective of its expressive richness. There are important similarities between the two projects. Both are motivated by the paradoxes. Both begin their exploration with idealized languages, for only then are the projects tractable. Both are directed at natural languages. But there is an important difference: The two projects aim to illuminate different aspects of natural languages. The first project aims to illuminate the semantic selfsufficiency of natural languages. The second project aims to give a systematic account of truth in natural languages. These aims entail that the second project has a certain autonomy that the first project lacks. A construction meets the goal of the first, the Semantic SelfSufficiency Project, only if it works with languages potentially as rich as English. It will not do, if, for example, the construction works only in the absence of negation. Or, for another example, if it works only when the language lacks the concept of truth. But what it is for a language to express truth is revealed by the Truth Project. Hence, the Semantic SelfSufficiency Project must be able to absorb the results of the Truth Project. On the other hand, the Truth Project is independent of the Semantic SelfSufficiency Project. Even if the semantic selfsufficiency of natural languages is illusory, no doubt is cast on the Truth Project. The fact that 'true' is a coherent and useful adjective of English and remains so despite the presence of "1 wish t o draw attention here t o Problem 6B.10 of RTT (p. 214), a problem t h a t has remained open for about fifteen years. (The problem was first put forward in Gupta 1982.) If the problem has a positive solution, then a classical language weak in certain syntactic resources can contain its own semantics (on one conception of semantics).
30. DEFINITION A N D REVISION 439
paradoxes provides a strong reason to pursue the project and to think that it is feasible. McGee and Martin think that the revision semantics for a language can only be given in a richer metalanguage, and they take this to be an argument against the revision theory. They, in effect, make the Truth Project dependent upon the Semantic SelfSufficiency Project. I am suggesting that the dependence goes the other way. In pursuing the Truth Project no demand need be recognized as primary other than that of making sense of the observed behavior of 'true'. Just as considerations of semantic selfsufficiency have little weight when assessing a semantics for 'not', similarly they have little weight when assessing a semantics for 'true'. It is a burden that the Semantic SelfSufficiency Project has assumed to make its constructions applicable even when 'not', 'true', and other words have their English meanings. In aiming to explicate these meanings one does not assume the burden of the Semantic SelfSufficiency Project. Is the Semantic SelfSufficiency Project with all its burdens forced on us by the paradoxes or some other considerations? The idea that motivates the project is that English is semantically selfsufficient. But what exactly is English and what do we know about it that should lead us to think that it is semantically selfsufficient? English is not a static language. It is a language that changes over time. It gains (and also loses) vocabulary. Even its grammatical forms undergo evolution. Is the claim of semantic selfsufficiency supposed to apply to this temporally evolving entity, an entity with a past, a present, and a future? But what does the claim of semantic selfsufficiency mean here? That the semantics of English at one stage of its development can be stated in English at a later stage? This could well be true but provides little motivation for the Semantic SelfSufficiency Project. Does the claim mean that the entire semantic evolution of English can be described at one of the stages? This is implausible and, in any case, far too strong for present purposes. Does the claim mean that the entire evolution of English can be described in a "superEnglish" an English consisting of all its temporal stages? This makes little sense since this "superEnglish" is not a language. Moreover, we know too little about the future evolution of English to make any significant claims about this cookedup entity. The best course, it appears, is to take the claim of semantic selfsufficiency to apply to a particular stage of English, say the present stage. So, let us for the rest of this essay understand by 'English' the present stage of English. A preliminary problem still remains: To attribute semantic selfsufficiency to English we should be clear
about its conceptual resources. But English is very flexible. It has elements whose interpretation can be varied indefinitely. A sentence such as 'This object has that property' can be used to express an indefinite range of propositions (though not all propositions). We are thus faced with a dilemma. Either the semantic selfsufficiency of English is due to the flexible character of English or it is not. If the former, we lose the motivation for the Semantic SelfSufficiency Project, since that project is concerned with languages having fixed conceptual resources. If the latter, then we can view English as having fixed resources. The difficulty now is to provide a good reason for thinking that English is semantically selfsufficient. This reason cannot be empirical, for we have no clear idea of the resources needed to give a semantic description of English. The reason must therefore be a priori. Some very general feature of English must entail its semantic selfsufficiency. What might this be? The feature that is appealed to most often to establish semantic selfsufficiency is the "comprehensibility of English by English speakers". The step from this feature to semantic selfsufficiency faces, however, two major obstacles. The first obstacle is created by an ambiguity in 'comprehensibility'. In one sense, it means simply the ability to understand and use the language. In this sense it is tautological that English is comprehensible by English speakers. And nothing much follows from this triviality. In the other sense, 'comprehensibility' means the ability to give a systematic theory of English. In this sense the claim that English is comprehensible by English speakers is not at all tautological. On the contrary, it is a highly speculative claim.22 Even if we grant that English is a part of 2 2 ~ e i tSimmons h gives an argument for semantic selfsufficiency (in his terminology, "semantic universality") that suffers from a parallel difficulty. [Tlhe claim that natural languages are semantically universal seems far less controversial. A language like English, for example, does contain names of its own expressions, does contain its own semantic predicates, like 'true', 'false', and 'refers', and does seem to have the resources for describing the proper use of these expressions. (Simmons 1993, p. 15) The sense in which one can say with any confidence that English has "the resources for describing the proper use" of its expressions is only that for all (most?) expressions of English, rules can be formulated in English that will guide a speaker to the proper use of those expressions. This does not warrant the claim that English is semantically selfsufficient, that it can contain a complete semantic description of itself. A semantic description may, for instance, need to spell out the notion of "interpretation" for English, but nothing of the sort is needed to guide a speaker to the proper use of expressions.
30. DEFINITION AND REVISION 441
nature and that a naturalistic semantics can, in principle, be given for it, the claim remains highly speculative. For, naturalism about the semantics of English does not imply that its semantics is within the grasp of English speakers.23 The first obstacle, then, is to establish the claim that English is comprehensible in the requisite strong sense to English speakers. Suppose this obstacle is overcome. A second major obstacle remains on the path to semantic selfsufficiency. This is the gulf between the comprehensibility of English by English speakers and the expressibility of the semantics in the frozen presentstageofEnglish. The gulf is created by the fact that English speakers have the capacity to enrich their logical and mathematical resources, and the fact that this capacity may play an essential role in their ability to discover the semantics of English. It may be objected: "How is this logical and mathematical enrichment possible? How can speakers arrive at notions that are not expressible in their language?" The questions raised are good, but note that they will serve as an objection only if one is prepared to say that all our mathematical notions sets, Heyting algebras, cardinal numbers, and the rest were expressible in the language of prehistoric man. Moreover, there is a simple model available of how we might arrive at richer abstractions: We arrive at them by accepting new, stronger axioms. For example, the model explains our gradual acquisition of the iterative conception of set by pointing to our gradual discovery of the axioms of ZF. No doubt this model is too simple. But it does show that what seems actual may well be possible: that we have the capacity to enrich our mathematical conceptions. The issues here are complex and I have been able to treat them only in a cursory manner. I hope nonetheless that the above considerations are sufficient to warrant the following modest conclusion: 2 3 ~ o n t r a sthis t with McGee's attitude:
A central tendency of twentieth century thought has been what we may call naturalism, the belief that human beings are products of nature and the lives of human beings are amenable t o scientific understanding no less than geological or astronomical phenomena. To accept the restriction Tarski proposes [= "no semantically selfsufficient languages"], we must repudiate naturalism, for we reject the possibility of a comprehensive scientific understanding of human thought and language. If this price seems too dear, we must find a way to overcome Tarski's restriction, so that we give a theory of the very language we speak, even though we have no recourse to an essentially richer metalanguage. (McGee 1994, pp. 628629)
The philosophical underpinnings of semantic selfsufficiency need to be carefully considered before it is used as a criterion of adequacy on theories of truth.
5 Concluding Remarks For much of this essay I have been engaged in defensive activities: deflecting criticisms, blocking the course of various arguments, weakening the force of others. I have been anxious to defend the revision theory against McGee's and Martin's penetrating criticisms. When I cast aside the defensive work (and attitude) and reflect on McGee's and Martin's observations, I find I agree with much of what they say. I agree with McGee that the treatment of limit stages in the revision theory needs improvement. I agree with Martin that fixedpoint theories can adopt the framework of circular definitions. And I too find fixedpoint theories attractive when they are situated in this framework. Finally, I agree with McGee and Martin that revision theory, if it is to fulfill its own goals, has to be applicable to English. And difficult obstacles stand in its way: We need a semantics that makes sense of our talk of "all sets" and of "absolutely everything". We need a better understanding of the enormous flexibility of natural languages. We need an account of how our mathematical thought evolves and becomes richer. These and other obstacles remain to be tackled. Fortunately, Belnap and I do not need to tackle them alone. We can count on the cooperation of thinkers like McGee and Martin thinkers who aim to understand the workings of our language and thought.
Aczel, Peter. 1988. Non WellFounded Sets. CSLI, Stanford University. Antonelli, Aldo. 1992. Revision Rules: An Investigation into NonMonotonic Inductive Definitions. Doctoral Dissertation, University of Pittsburgh. Antonelli, Aldo. 1994. "Nonwellfounded sets via revision rules". Journal of Philosophical Logic 23: 633679. Chapuis, Andre. 1993. Circularity, Truth, and the Liar Paradox. Doctoral Dissertation, Indiana University, Bloomington. Chapuis, Andrk. 1996. "Alternative revision theories of truth". Journal of Philosophical Logic 25: 399423. Forster, T. E. 1992. Set Theory with a Universal Set. Oxford University Press.
30. DEFINITION AND REVISION 443 Gupta, Anil. 1982. "Truth and paradox". Journal of Philosophical Logic 11: 160. Reprinted in Martin 1984. Gupta, Anil and Nuel Belnap. 1993. The Revision Theory of Truth. The MIT Press. Gupta, Anil and Nuel Belnap. 1994. "Reply to Robert Koons". Notre Dame Journal of Formal Logic 35: 632636. Herzberger, Hans G. 1982. "Notes on naive semantics". Journal of Philosophical Logic 11, 61102. Reprinted in Martin 1984. Koons, Robert. 1994. "Book review: The Revision Theory of Truth". Notre Dame Journal of Formal Logic 35: 606631. Kripke, Saul A. 1975. "Outline of a theory of truth". Journal of Philosophy 72: 690716. Reprinted in Martin 1984. McGee, Vann. 1991. Truth, Vagueness, and Paradox. Hackett. McGee, Vann. 1994. "Afterword: Truth and paradox". In Robert Harnish, ed., Basic Topics in the Philosophy of Language (Harvester Wheatsheaf, 1994): pp. 615633. McGee, Vann. 1997. "Revision". This volume. (Presented in April 1996 a t the Central Division meetings of the American Philosophical Association.) Martin, Donald A. 1997. "Revision and its rivals". This volume. (Presented in April 1996 a t the Central Division meetings of the American Philosophical Association.) Martin, Robert L., ed., 1984. Recent Essays on Truth and the Liar Paradox. Oxford University Press. Orilia, Francesco. 1995. "Knowledge representation, exemplification, and the GuptaBelnap theory of circular definitions". In M. Gori and G. Soda (eds.), Topics in Artificial Intelligence (SpringerVerlag, 1995). Simmons, Keith. 1993. Universality and the Liar. Cambridge University Press. Yaqiib, Aladdin. 1993. The Liar Speaks the Truth. Oxford University Press.
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Bibliography Outline of a Theory of Truth Saul Kripke The Journal of Philosophy, Vol. 72, No. 19, SeventySecond Annual Meeting American Philosophical Association, Eastern Division. (Nov. 6, 1975), pp. 690716. Stable URL: http://links.jstor.org/sici?sici=0022362X%2819751106%2972%3A19%3C690%3AOOATOT%3E2.0.CO%3B27