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1. Thanks to an anonymous editor for drawing our attention toFrege’s essay.

2. Note that experience can have the effect of restricting not only thedomain of quantification but also the interpretations of non-logicalconstants.

3. Whitehead and Russell observe that a definition of, e.g., cardinalnumber, “contains an analysis of a common idea, and maytherefore express a notable advance (1925, 12).” A little laterthey add, “it will be found, in what follows, that thedefinitions are what is most important, and what most deserves thereader’s prolonged attention.”

4. See, however, the discussion of implicit definitions below.

5. Dudman notes that an application of both criteria can be foundalready in Frege’sBegriffsschrift (1879). Hodgespoints out that the Eliminability criterion can be traced toPascal’s treatment of definitions in the 17th century and evento Porphyry’s writings dating from the third century. Thanks toEd Zalta for drawing our attention to Dudman’s andHodges’s essays. A previous version of this entry misreportedUrbaniak’s views on this topic. Thanks to Urbaniak andHämäri 2012 for the correction.

6. Recall that we have put no restrictions on \(\mathcal{D}\) other thanthose stated at the outset: that its definiendum and definiens are ofthe same logical category and that the former contains the definedterm. The proof of the claim relies on the Replacement Theorem forequivalent formulas.

7. Notice that in a definition in normal form, the defined term is theonly non-logical constant in the definiendum. Hence, in such adefinition, the defined term need not be specified separately.

8. This requirement is the stumbling point when the Ontological Proof isformalized in classical logic. The definition of “God” as“that than which nothing greater can be thought” doesimply the existence of God. But the definition is legitimate only ifthere is one, and only one, being such that nothing greater can bethought than it. The definition cannot serve, therefore, as the groundfor a proof of the existence of God. (If the Ontological Proof isformalized in a logic that admits vacuous singular terms, then thedefinition may well be legitimate, but it will not imply the existenceof God.)

9. The traditional account allowscontextualdefinitions—that is, definitions that provide a method ofreducing sentences containing the defined terms to sentences of theground language. (Such a definition can be viewed as consisting of aninfinity of instances of form (2), each sentence containing thedefined term serving as the definiendum of one instance.) However, thetraditional account implies that a contextual definition adds no newpower, for its effect can be gained by a definition in normalform.

It is instructive to reflect here on Russell’s theory ofdefinite descriptions. (For an account of this theory, see the entryondescriptions.) Suppose a definite description ‘the \(F\)’ is introducedinto a classical first-order language in the manner prescribed byRussell’s theory. The Conservativeness and Eliminabilitycriteria are, it appears, satisfied. Yet an equivalent definition innormal form may well not exist. Why this incongruity?

The answer is that a definite description, under Russell’stheory, is not a genuine singular term; it is not even a meaningfulunit. When ‘the \(F\)’ is added to the ground language inthe manner of Russell, the resulting language merelylookslike a familiar first-order language. In actuality its logic is quitedifferent. For instance, one cannot existentially generalize on theoccurrences of ‘the \(F\)’. The logical form and characterof the formulas of the expanded language is revealed by theirRussellian analyses, and these contain no constituent corresponding to‘the \(F\)’. (There is also the further fact that, underthe Russellian analysis, formulas containing ‘the \(F\)’are potentially ambiguous. The earlier observation holds, however,even if the ambiguity is somehow legislated away—for instance,by prescribing rules for the scope of the definite description.)

Russell’s theory is best thought of as providing a contextualelimination of the definite description, not a contextualdefinition of it.

10. Not all recursive definitions formulable in the language of PeanoArithmetic have normal forms. For instance, a recursive definition canbe given in this language for truth—more precisely, for“Gödel number of a true sentence of the language of PeanoArithmetic”—but the definition cannot be put in normalform. Recursive definitions in first-order arithmetic enable one todefine \(\Pi^{1}_{1}\) sets of natural numbers, whereas normal formsexist only for those that define arithmetical sets. For a study ofrecursive definability in first-order languages, see Moschovakis1974.

11. Note that we can regard a recursive definition such as (15) as animplicit definition by a theory that consists of the universalclosures of the equations.

12. It is sometimes said that logical constants areimplicitlydefined by the logical laws, or by the logical rules, governingthem. More specifically, it has been claimed that the“introduction and elimination” rules for a logicalconnective areimplicit definitions of the connective. (Theidea has its roots in the work of Gerhard Gentzen.) For example, thesentential connective ‘and’, it is claimed, is defined byfollowing rules:

‘And’-Introduction: From \(\phi\) and\(\psi\), one may infer ‘\(\phi\) and \(\psi\)’;

‘And’-Elimination: From ‘\(\phi\)and \(\psi\)’, one may infer \(\phi\) and one may also infer\(\psi\).

These ideas invoke a notion of implicit definition that is quitedifferent from the one under consideration here. Under the latternotion,non-logical constants are implicitly defined by atheory, and the interpretation of logical constants is held fixed. Theinterpretation of the logical constants provides the scaffolding, soto speak, for making sense of implicit definitions. Under the formernotion, the scaffolding is plainly different. For further discussion,see the entry onlogical constants and the works cited there.

13. If the aim is to explain the rationality of accepting a theory on thebasis ofactual observations, then almost the entire theorywould need to be taken as implicitly defining theoretical terms. Nowboth criteria would be a violated.

If the aim is to sustain the idea that the factual component of atheory is identical to its empirical content, then one can take whathas come to be known as the “Carnap sentence” for thetheory as implicitly defining the theoretical terms. Now there is aviolation only of the Eliminability criterion.

For further discussion, see Ramsey 1931, Carnap 1963, Lewis 1970, andDemopoulos 2003.

14. And also for systems of interdependent definitions. From now on, theexpression ‘circular definition’ will be understoodbroadly to include these systems as well.

15. More precisely, finiteness is defined as follows. Letgroundmodels be interpretations of the ground language. And call ahypothesis \(V\)reflexive iff, for some number \(n \gt 0,n\) applications of the revision rule to \(V\) yields \(V\) again. Adefinition \(\mathcal{D}\) isfinite iff, for all groundmodels \(M\), there is a natural number \(m\) such that for allhypotheses \(V\), the result of \(m\) application to \(V\) of therevision rule for \(\mathcal{D}\) in \(M\) is reflexive.

16. The key features of \(\mathbf{C}_{0}\) are that (i) integer indicesare assigned to each step in a derivation to distinguish revisionstages, and (ii) the rules for definitions, namely, DefiniendumIntroduction and Definiendum Elimination, are weakened. If an instanceof the definiens is available as a premiss and its index is \(j\) thenthe corresponding instance of the definiendum may be inferred but mustbe assigned the index \(j + 1\). And, conversely, if an instance ofthe definiendum with an index \(j + 1\) is available, then thecorresponding instance of the definiens may be inferred but must beassigned the index \(j\). For a full account of \(\mathbf{C}_{0}\),see Gupta and Belnap 1993. For more information about finitedefinitions, see Martinez 2001 and Gupta 2006.

17. Since revision sequences are typically non-monotonic, the extensionis not straightforward. The limit stages in a transfinite revisionsequence can be treated in a variety of ways. This topic has beenstudied by Anil Gupta, Hans Herzberger, Nuel Belnap, AladdinYaqūb, André Chapuis, Philip Welch, and others. See theentry on therevision theory of truth for references.