Continued from last post.

**2.5 Implicit definitions**

The above viewpoint allows the traditional account to bring within its fold ideas that might at first sight seem contrary to it. It is sometimes suggested that a term X can be introduced axiomatically, that is, by laying down as axioms certain sentences of the expanded language L+. The axioms are then said to implicitly define X. This idea is easily accommodated within the traditional account. Let a theory be a set of sentences of the expanded language L+. Then, to say that a theory T* is an implicit (stipulative) definition of X is to say that X is governed by the definition

φ =Df The True,

where φ is the conjunction of the members of T*. (If T* is infinite then a stipulation of the above form will be needed for each sentence ψ in T*.)[9] The definition is legitimate, according to the traditional account, so long as it meets the Conservativeness and Eliminability criteria. If it does meet these criteria, let us call T* admissible (for a definition of X). So, the traditional account accommodates the idea that theories can stipulatively introduce new terms, but it imposes a strong demand: the theories must be admissible.[10]

Consider, for concreteness, the special case of classical first-order languages. Let the ground language L be one such, and let its interpretations be models of some sentences T. Say that an interpretation M+ of L+ is an expansion of an interpretation M of L iff M and M+ have the same domain and they assign the same semantic values to the non-logical constants in L. Furthermore, let us say that

T* is an implicit semantic definition of X iff, for each interpretation M of L, there is a unique model M+ of T* such that M+ is an expansion of M.

Then the following claim is immediate:

If T* is admissible then T* is an implicit semantic definition of X.

That is, an admissible theory fixes the semantic value of the defined term in each interpretation of the ground language. This observation provides one natural method of showing that a theory is not admissible:

Padoa's method. To show that T* is not admissible, it suffices to construct two models of T* that are expansions of one and the same interpretation of the ground language L.

Here is a simple and philosophically useful application of Padoa's method. Suppose the proof system of L is Peano Arithmetic and that L is expanded by the addition of a unary predicate Tr (for “Gödel number of a true sentence of L”). Let H be the theory consisting of all the sentences (the “Tarski biconditionals”) of the following form:

Tr(s) ↔ ψ,

where ψ is a sentence of L and s is the canonical name for the Gödel number of ψ. Padoa's method implies that H is not admissible for defining Tr. For H does not fix the interpretation of Tr in all interpretations of L. In particular, it does not do so in the standard model, for H places no constraints on the behavior of Tr on those numbers that are not Gödel numbers of sentences. (If the coding renders each natural number a Gödel number of a sentence, then a non-standard model of Peano Arithmetic provides the requisite counterexample: it has infinitely many expansions that are models of H.) A variant of this argument shows that Tarski's theory of truth, as formulated in L+, is not admissible for defining Tr.

What about the converse of Padoa's method? Suppose we can show that in each interpretation of the ground language, a theory T* fixes a unique semantic value for the defined term. Can we conclude that T* is admissible? This question receives a negative answer for some semantical systems, and a positive answer for others. (In contrast, Padoa's method works so long as the semantic system is not highly contrived.) The converse fails for, e.g., classical second-order languages, but it holds for first-order ones:

Beth's Definability Theorem. If T* is an implicit semantic definition of X in a classical first-order language then T* is admissible.

Note that the theorem holds even if T* is an infinite set. For a proof of the theorem, see Boolos, Burgess, and Jeffrey 2002.

The idea of implicit definition is not in conflict, then, with the traditional account. Where conflict arises is in the philosophical applications of the idea. The failure of strict reductionist programs of the late-nineteenth and early-twentieth century prompted philosophers to explore looser kinds of reductionism. For instance, Frege's definition of number proved to be inconsistent, and thus incapable of sustaining the logicist thesis that the principles of arithmetic are analytic. It turns out, however, that the principles of arithmetic can be derived without Frege's definition; all that is needed is one consequence of it, namely, Hume's Principle:

Hume's Principle. The number of Fs = the number of Gs iff there is a one-to-one correspondence between the Fs and Gs.

If we add Hume's Principle to second-order logic, then we can analytically derive (second-order) Peano Arithmetic. (The essentials of the argument are found already in Frege 1884.) It is a central thesis of Neo-Fregeanism that Hume's Principle is an implicit definition of the functional expression ‘the number of’ (see Hale and Wright 2001). If this thesis can be defended then logicism about arithmetic can be sustained while foregoing Frege's explicit (and inconsistent) definition. However, the neo-Fregean thesis is in conflict with the traditional account of definitions, for Hume's principle violates both Conservativeness and Eliminability. The principle allows one to prove, for arbitrary n, that there are at least n objects. (A related application aims to sustain the analyticity of a geometry through the idea that the axioms of geometry are implicit definitions of geometrical concepts such as “point” and “line.” Here, too, there is conflict with the traditional account, for Conservativeness and Eliminability are violated.)

Another example: The reductionist program for theoretical concepts (e.g., those of physics) aimed to solve epistemological problems that these concepts pose. The program aimed to reduce theoretical sentences to (classes of) observational sentences. However, the reductions proved difficult, if not impossible, to sustain. Thus arose the suggestion that perhaps the non-observational component of a theory can, without any claim of reduction, be regarded as an implicit definition of theoretical terms. The precise characterization of the non-observational component can vary with the specific epistemological problem at hand. But there is bound to be a violation of one or both of the two criteria, Conservativeness and Eliminability.[11]

A final example: We know by a theorem of Tarski that no theory can be an admissible definition of the truth predicate, Tr, for the language of Peano Arithmetic considered above. Nonetheless, perhaps we can still regard theory H as an implicit definition of Tr. (Paul Horwich has made a closely related proposal for the ordinary notion of truth.) Here, again, pressure is put on the bounds imposed by the traditional account. H meets the Conservativeness criterion, but not that of Eliminability.

In order to assess the challenge these philosophical applications pose for the traditional account, we need to resolve issues that are under current philosophical debate. Some of the issues are the following. (i) It is plain that some violations of Conservativeness are illegitimate: one cannot make it true by a stipulation that, e.g., Mercury is larger than Venus. Now, if a philosophical application requires some violations of Conservativeness to be legitimate, we need an account of the distinction between the two sorts of cases: the legitimate violations of Conservativeness and the non-legitimate ones. And we need to understand what it is that renders the one, but not the other, legitimate. (ii) A similar issue arises for Eliminability. It would appear that not any old theory can be an implicit definition of a term X. (The theory might contain only tautologies.) If so, then again we need a demarcation of theories that can serve to implicitly define a term from those that cannot. And we need a rationale for the distinction. (iii) The philosophical applications rest crucially on the idea that an implicit definition fixes the meaning of the defined term. We need therefore an account of what this meaning is, and how the implicit definition fixes it. Under the traditional account, formulas containing the defined term can be seen as acquiring their meaning from the formulas of the ground language. (In view of the primacy of the sentential, this fixes the meaning of the defined term.) But this move is not available under a liberalized conception of implicit definition. How, then, should we think of the meaning of a formula under the envisioned departure from the traditional account? (iv) Even if the previous three issues are addressed satisfactorily, an important concern remains. Suppose we allow that a theory T, say, of physics can stipulatively define its theoretical terms, and that it endows the terms with particular meanings. The question remains whether the meanings thus endowed are identical to (or similar enough to) the meanings the theoretical terms have in their actual uses in physics. This question must be answered positively if implicit definitions are to serve their philosophical function. The aim of invoking implicit definitions is to account for the rationality, or the aprioricity, or the analyticity of our ordinary judgments, not of some extraordinary judgments that are somehow assigned to ordinary signs.

For further discussion of these issues, see Horwich 1998, especially chapter 6; Hale and Wright 2001, especially chapter 5; and the works cited there.

**2.6 Vicious-Circle Principle**

Another departure from the traditional theory begins with the idea not that the theory is too strict, but that it is too liberal, that it permits definitions that are illegitimate. Thus, the traditional theory allows the following definitions of, respectively, “liar” and the class of natural numbers N:

(16) z is a liar =Df all propositions asserted by z are false;

(17) z belongs to N =Df z belongs to every inductive class, where a class is inductive when it contains 0 and is closed under the successor operation.

Russell argued that such definitions involve a subtle kind of vicious circle. The definiens of the first definition invokes, Russell thought, the totality of all propositions, but the definition, if legitimate, would result in propositions that can only be defined by reference to this totality. Similarly, the second definition attempts to define the class N by reference to all classes, which includes the class N that is being defined. Russell maintained that such definitions are illegitimate. And he imposed the following requirement—called, the “Vicious-Circle Principle”—on definitions and concepts. (Henri Poincaré had also proposed a similar idea.)

Vicious-Circle Principle. “Whatever involves all of a collection must not be one of the collection (Russell 1908, 63).”

Another formulation Russell gave of the Principle is this:

Vicious-Circle Principle (variant formulation). “If, provided a certain collection had a total, it would have members only definable in terms of that total, then the said collection has no total (Russell, 1908, 63).”

In an appended footnote, Russell explained, “When I say that a collection has no total, I mean that statements about all its members are nonsense.”

Russell's primary motivation for the Vicious-Circle Principle were the logical and semantic paradoxes. Notions such as “truth,” “proposition,” and “class” generate, under certain unfavorable conditions, paradoxical conclusions. Thus, the claim “Cheney is a liar,” where “liar” is understood as in (16), yields paradoxical conclusions, if Cheney has asserted that he is a liar, and all other propositions asserted by him are, in fact, false. Russell took the Vicious-Circle Principle to imply that if “Cheney is a liar” expresses a proposition, it cannot be in the scope of the quantifier in the definiens of (16). More generally, Russell held that quantification over all propositions, and over all classes, violates the Vicious-Circle Principle and is thus illegitimate. Furthermore, he maintained that expressions such as ‘true’ and ‘false’ do not express a unique concept—in Russell's terminology, a unique “propositional function”—but one of a hierarchy of propositional functions of different orders. Thus the lesson Russell drew from the paradoxes is that the domain of the meaningful is more restricted than it might ordinarily appear, that the traditional account of concepts and definitions needed to be made more restrictive in order to rule out the likes of (16) and (17).

In application to ordinary, informal definitions, the Vicious-Circle Principle does not provide, it must be said, a clear method of demarcating the meaningful from the meaningless. Definition (16) is supposed to be illegitimate because, in its definiens, the quantifier ranges over the totality of all propositions. And we are told that this is prohibited because, were it allowed, the totality of propositions “would have members only definable in terms of the total.” However, unless we know more about the nature of propositions and of the means available for defining them, it is impossible to determine whether (16) violates the Principle. It may be that a proposition such as “Cheney is a liar”—or, to take a less contentious example, “Either Cheney is a liar or he is not”— can be given a definition that does not appeal to the totality of all propositions. If propositions are sets of possible worlds, such a definition would be appear to be feasible.

The Vicious-Circle Principle serves, nevertheless, as an effective motivation for a particular account of legitimate concepts and definitions, namely that embodied in Russell's Ramified Type Theory. The idea here is that one begins with some unproblematic resources that involve no quantification over propositions, concepts, and such. These resources enable one to define, for example, various unary concepts, which are thereby assured of satisfying the Vicious-Circle Principle. Quantification over these concepts is thus bound to be legitimate, and can be added to the language. The same holds for propositions, and for concepts falling under other types: for each type, a quantifier can be added that ranges over items (of that type) that are definable using the initial unproblematic resources. The new quantificational resources enable the definition of further items of each type; these, too, respect the Principle, and again, quantifiers ranging over the expanded totalities can legitimately be added to the language. The new resources permit the definition of yet further items. And the process repeats. The result is that we have a hierarchy of propositions and of concepts of various orders. Each type in the type hierarchy ramifies into a multiplicity of orders. This ramification ensures that definitions formulated in the resulting language are bound to respect the Vicious-Circle Principle. Concepts and classes that can be defined within the confines of this scheme are said to be predicative (in one sense of this word); the others, impredicative.

For further discussion of the Vicious-Circle Principle, see Russell 1908, Whitehead and Russell 1925, Gödel 1944, and Chihara 1973. For a formal presentation of Ramified Type Theory, see Church 1976; for a more informal presentation, see Hazen 1983. See also the entries on type theory and Principia Mathematica, which contain further references.

**2.7 Circular definitions**

The paradoxes can also be used to motivate a conclusion that is the very opposite to Russell's. Consider the following definition of a one-place predicate G:

(18) Gx =Df x = Socrates v (x = Plato & Gx) v (x = Aristotle & ~Gx).

This definition is essentially circular; it is not reducible to one in normal form. Still, intuitively, it provides substantial guidance on the use of G. The definition dictates, for instance, that Socrates falls under G, and that nothing apart from the three ancient philosophers does so. The definition leaves unsettled the status of only two objects, namely, Plato and Aristotle. If we suppose that Plato falls under G, the definition yields that Plato does fall under G (since Plato satisfies the definiens), thus confirming our supposition. The same thing happens if we suppose the opposite, namely, that Plato does not fall under G; again our supposition is confirmed. With Aristotle, any attempt to decide whether he falls under G lands us in an even more precarious situation: if we suppose that Aristotle falls under G, we are led to conclude by the definition that he does not fall under G (since he does not satisfy the definiens); and, conversely, if we suppose that he does not fall under G, we are led to conclude that he does. But even on Plato and Aristotle, the behavior of G is not unfamiliar: G is behaving here in the way the concept of truth behaves on the Truth Teller (“What I am now saying is true”) and the Liar (“What I am now saying is not true”). More generally, there is a strong parallel between the behavior of the concept of truth and concepts defined by circular definitions. Both are typically well defined on a range of cases, and both display a variety of unusual logical behavior on the other cases. Indeed, all the different kinds of perplexing logical behavior found with the concept of truth are found also in concepts defined by circular definitions. This strong parallelism suggests that since truth is manifestly a legitimate concept, so also are concepts defined by circular definitions such as (18). The paradoxes, according to this viewpoint, cast no doubt on the legitimacy of the concept of truth. They show only that the logic and semantics of circular concepts is different from that of non-circular ones. This viewpoint is developed in the revision theory of definitions.

In this theory, a circular definition imparts to the defined term a meaning that is hypothetical in character; the semantic value of the defined term is a rule of revision, not as with non-circular definitions, a rule of application. Consider (18) again. Like any definition, (18) fixes the interpretation of the definiendum if the interpretations of the non-logical constants in the definiens are given. The problem with (18) is that the defined term G occurs in the definiens. But suppose that we arbitrarily assign to G an interpretation—say we let it be the set U of all objects in the universe of discourse (i.e., we suppose that U is the set of objects that satisfy G). Then it is easy to see that the definiens is true precisely of Socrates and Plato. The definition thus dictates that, under our hypothesis, the interpretation of G should be the set {Socrates, Plato}. A similar calculation can be carried out for any hypothesis about the interpretation of G. For example, if the hypothesis is {Xenocrates}, the definition yields the result {Socrates, Aristotle}. In short, even though (18) does not fix sharply what objects fall under G, it does yield a rule or function that, when given a hypothetical interpretation as an input, yields another one as an output. The fundamental idea of the revision theory is to view this rule as a revision rule: the output interpretation is better than the input one (or it is at least as good; this qualification will be taken as read). The semantic value that the definition confers on the defined term is not an extension—a demarcation of the universe of discourse into objects that fall under the defined term, and those that do not. The semantic value is a revision rule.

The revision rule explains the behavior, both ordinary and extraordinary, of a circular concept. Let δ be the revision rule yielded by a definition, and let V be an arbitrary hypothetical interpretation of the defined term. We can attempt to improve our hypothesis V by repeated applications of the rule δ. The resulting sequence,

V, δ(V), δ(δ(V)), δ(δ(δ(V))), … ,

is a revision sequence for δ. The totality of revision sequences for δ, for all possible initial hypotheses, is the revision process generated by δ. For example, the revision rule for (18) generates a revision process that consists of the following revision sequences, among others:

U, {Socrates, Plato}, {Socrates, Plato, Aristotle}, {Socrates, Plato}, …

{Xenocrates}, {Socrates, Aristotle}, {Socrates}, {Socrates, Aristotle}, …

Observe the behavior of our four ancient philosophers in this process. After some initial stages of revision, Socrates always falls in the revised interpretations, and Xenocrates always falls outside. (In this particular example, the behavior of the two is fixed after the initial stage; in other cases, it may take many stages of revision before the status of an object becomes settled.) The revision process yields a categorical verdict on the two philosophers: Socrates categorically falls under G, and Xenocrates categorically falls outside G. Objects on which the process does not yield a categorical verdict are said to be pathological (relative to the revision rule, the definition, or the defined concept). In our example, Plato and Aristotle are pathological relative to (18). The status of Aristotle is not stable in any revision sequence. It is as if the revision process cannot make up its mind about him. Sometimes Aristotle is ruled as falling under G, and then the process reverses itself and declares that he does not fall under G, and then the process reverses itself again. When an object behaves in this way in all revision sequences, it is said to be paradoxical. Plato is also pathological relative to G, but his behavior in the revision process is different. Plato acquires a stable status in each revision sequence, but the status he acquires depends upon the initial hypothesis.

Revision processes help provide a semantics for circular definitions.[12] They can be used to define semantic notions such as “categorical truth” and logical notions such as “validity.” The characteristics of the logical notions we obtain depend crucially on one aspect of revision: the number of stages before objects settle down to their regular behavior in the revision process. A definition is said to be finite iff, roughly, its revision process necessarily requires only finitely many such stages.[13] For finite definitions, there is a simple logical calculus, C0, that is sound and complete for the revision semantics.[14] With non-finite definitions, the revision process extends into the transfinite.[15] And these definitions can add considerable expressive power to the language. (When added to first-order arithmetic, these definitions render all Π12 sets of natural numbers definable.) Because of the expressive power, the general notion of validity for non-finite circular definitions is not axiomatizable (Kremer 1993). We can give at best a sound logical calculus, but not a complete one. The situation is analogous to that with second-order logic.

Let us observe some general features of the revision theory of definitions. (i) Under this theory, the logic and semantics of non-circular definitions—i.e., definitions in normal form—remain the same as in the traditional account. The introduction and elimination rules hold unrestrictedly, and revision stages are dispensable. The deviations from the traditional account occur only over circular definitions. (ii) Under the theory, circular definitions do not disturb the logic of the ground language. Sentences containing defined terms are subject to the same logical laws as sentences of the ground language. (iii) Conservativeness holds. No definition, no matter how vicious the circularity in it, entails anything new in the ground language. Even the utterly paradoxical definition,

Gx =Df ~Gx,

respects the Conservativeness requirement. (iv) Eliminability fails to hold. Sentences of the expanded language are not, in general, reducible to those of the ground language. This failure has two sources. First, revision theory fixes the use, in assertion and argument, of sentences of the expanded language but without reducing the sentences to those of the ground language. The theory thus meets the Use criterion, but not the stronger one of Eliminability. Second, in this theory, a definition can add logical and expressive power to a ground language. The addition of a circular definition can result in the definability of new sets. This is another reason why Eliminability fails.

It may be objected that every concept must have an extension, that there must be a definite totality of objects that fall under the concept. If this is right then a predicate is meaningful—it expresses a concept—only if the predicate necessarily demarcates the world sharply into those objects to which it applies and those to which it does not apply. Hence, the objection concludes, no predicate with an essentially circular definition can be meaningful. The objection is plainly not decisive, for it rests on a premiss that rules out many ordinary and apparently meaningful predicates (e.g., ‘bald’). Nonetheless, it is noteworthy because it illustrates how general issues about meaning and concepts enter the debate on the requirements on legitimate definitions.

The principal motivation for revision theory is descriptive. It has been argued that the theory helps us to understand better our ordinary concepts such as truth, necessity, and rational choice. The ordinary as well as the perplexing behavior of these concepts, it is argued, has its roots in the circularity of the concepts. If this is correct, then there is no logical requirement on descriptive and explicative definitions that they be non-circular. For more detailed treatments of these topics, see Gupta 1988/89, Gupta and Belnap 1993, and Chapuis and Gupta 1999. See also the entry on the revision theory of truth. For critical discussions of the revision theory, see and the papers by Vann McGee and Donald A. Martin, and the reply by Gupta, in Villanueva 1997. See also Shapiro 2006.

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Other Internet Resources

Definitions, Dictionaries, and Meanings, notes by Norman Swartz, Simon Fraser University

Related Entries

analytic/synthetic distinction | Aristotle, General Topics: logic | descriptions | Frege, Gottlob: logic, theorem, and foundations for arithmetic | logical constants | Tarski, Alfred: truth definitions | truth: revision theory of | type theory

Acknowledgments

The author would like to thank the subject editors of the SEP for their helpful suggestions on the first draft of this entry.