Definitions have interested philosophers since ancient times.Plato’s early dialogues portray Socrates raising questions aboutdefinitions (e.g., in theEuthyphro, “What ispiety?”)—questions that seem at once profound and elusive.The key step in Anselm’s “Ontological Proof” for theexistence of God is the definition of “God,” and the sameholds of Descartes’s version of the argument in hisMeditation V. More recently, the Frege-Russell definition ofnumber and Tarski’s definition of truth have exercised aformative influence on a wide range of contemporary philosophicaldebates. In all these cases—and many others can becited—not only have particular definitions been debated; thenature of, and demands on, definitions have also been debated. Some ofthese debates can be settled by making requisite distinctions, fordefinitions are not all of one kind: definitions serve a variety offunctions, and their general character varies with function. Someother debates, however, are not so easily settled, as they involvecontentious philosophical ideas such as essence, concept, andmeaning.

• 1. Some varieties of definition
  • 1.1 Real and nominal definitions
  • 1.2 Dictionary definitions
  • 1.3 Stipulative definitions
  • 1.4 Descriptive definitions
  • 1.5 Explicative definitions
  • 1.6 Ostensive definitions
  • 1.7 A remark
• 2. The logic of definitions
  • 2.1 Two criteria
  • 2.2 Foundations of the traditional account
  • 2.3 Conservativeness and eliminability
  • 2.4 Definitions in normal form
  • 2.5 Conditional definitions
  • 2.6 Implicit definitions
  • 2.7 Vicious-Circle Principle
  • 2.8 Circular definitions
• Bibliography
• Academic Tools
• Other Internet Resources
• Related Entries

1. Some varieties of definition

Ordinary discourse recognizes several different kinds of things aspossible objects of definition, and it recognizes several kinds ofactivity as defining a thing. To give a few examples, we speak of acommission as defining the boundary between two nations; of theSupreme Court as defining, through its rulings, “person”and “citizen”; of a chemist as discovering the definitionof gold, and the lexicographer, that of ‘cool’; of aparticipant in a debate as defining the point at issue; and of amathematician as laying down the definition of “group.”Different kinds of things are objects of definition here: boundary,legal status, substance, word, thesis, and abstract kind. Moreover,the different definitions do not all have the same goal: the boundarycommission may aim to achieve precision; the Supreme Court, fairness;the chemist and the lexicographer, accuracy; the debater, clarity; andthe mathematician, fecundity. The standards by which definitions arejudged are thus liable to vary from case to case. The differentdefinitions can perhaps be subsumed under the Aristotelian formulathat a definition gives the essence of a thing. But this onlyhighlights the fact that “to give the essence of a thing”is not a unitary kind of activity.

In philosophy, too, several different kinds of definitions are oftenin play, and definitions can serve a variety of different functions(e.g., to enhance precision and clarity). But, in philosophy,definitions have also been called in to serve a highly distinctiverole: that of solving epistemological problems. For example, theepistemological status of mathematical truths raises a problem.Immanuel Kant thought that these truths are syntheticapriori, and to account for their status, he offered a theory ofspace and time—namely, of space and time as forms of,respectively, outer and inner sense. Gottlob Frege and BertrandRussell sought to undermine Kant’s theory by arguing thatarithmetical truths are analytic. More precisely, they attempted toconstruct a derivation of arithmetical principles from definitions ofarithmetical concepts, using only logical laws. For the Frege-Russellproject to succeed, the definitions used must have a specialcharacter. They must be conceptual or explicative of meaning; theycannot be synthetic. It is this kind of definition that has aroused,over the past century or so, the most interest and the mostcontroversy. And it is this kind of definition that will be ourprimary concern. Let us begin by marking some preliminary butimportant distinctions.

1.1 Real and nominal definitions

John Locke distinguished, in hisEssay, “realessence” from “nominal essence.” Nominal essence,according to Locke, is the “abstract Ideato whichthe Name is annexed (III.vi.2).” Thus, the nominal essenceof the name ‘gold’, Locke said, “is that complexIdea the wordGold stands for, let it be, forinstance, a Body yellow, of a certain weight, malleable, fusible, andfixed.” In contrast, the real essence of gold is “theconstitution of the insensible parts of that Body, on which thoseQualities [mentioned in the nominal essence] and all other PropertiesofGold depend (III.vi.2).” A rough way of marking thedistinction between real and nominal definitions is to say, followingLocke, that the former states real essence, while the latter statesnominal essence. The chemist aims at real definition, whereas thelexicographer aims at nominal definition.

This characterization of the distinction is rough because azoologist’s definition of “tiger” should count as areal definition, even though it may fail to provide “theconstitution of the insensible parts” of the tiger. Moreover, anaccount of the meaning of a word should count as a nominal definition,even though it may not take the Lockean form of setting out “theabstract idea to which the name is annexed.” Perhaps it ishelpful to indicate the distinction between real and nominaldefinitions thus: to discover the real definition of a term \(X\) oneneeds to investigate the thing or things denoted by \(X\); to discoverthe nominal definition, one needs to investigate the meaning and useof \(X\). Whether the search for an answer to the Socratic question“What is virtue?” is a search for real definition or onefor nominal definition depends upon one’s conception of thisparticular philosophical activity. When we pursue the Socraticquestion, are we trying to gain a clearer view of our uses of the word‘virtue’, or are we trying to give an account of an idealthat is to some extent independent of these uses? Under the formerconception, we are aiming at a nominal definition; under the latter,at a real definition.

See Robinson 1950 for a critical discussion of the differentactivities that have been subsumed under “realdefinition,” and see Charles 2010 for ancient views on thetopic. Fine 1994 defends the conception that a real definition definesan object by specifying what the object is; in other words, a realdefinition spells out the essence of the object defined. Rosen 2015offers an explanation of real definition in terms of grounding: thedefinition provides the ground of the essence of the object. Themeaning of ‘essence’ and of ‘ground’ remainunder active debate, however.

1.2 Dictionary definitions

Nominal definitions—definitions that explain the meaning of aterm—are not all of one kind. A dictionary explains the meaningof a term, in one sense of this phrase. Dictionaries aim to providedefinitions that contain sufficient information to impart anunderstanding of the term. It is a fact about us language users thatwe somehow come to understand and use a potential infinity ofsentences containing a term once we are given a certain small amountof information about the term. Exactly how this happens is a largemystery. But it does happen, and dictionaries exploit the fact. Notethat dictionary entries are not unique. Different dictionaries cangive different bits of information and yet be equally effective inexplaining the meanings of terms.

Definitions sought by philosophers are not of the sort found in adictionary. Frege’s definition of number (1884) and AlfredTarski’s definition of truth (1983, ch. 8) are not offered ascandidates for dictionary entries. When an epistemologist seeks adefinition of “knowledge,” she is not seeking a gooddictionary entry for the word ‘know’. The philosophicalquest for definition can sometimes fruitfully be characterized as asearch for an explanation of meaning. But the sense of‘explanation of meaning’ here is very different from thesense in which a dictionary explains the meaning of a word.

1.3 Stipulative definitions

A stipulative definition imparts a meaning to the defined term, andinvolves no commitment that the assigned meaning agrees with prioruses (if any) of the term. Stipulative definitions areepistemologically special. They yield judgments with epistemologicalcharacteristics that are puzzling elsewhere. If one stipulativelydefines a “raimex” as, say, a rational, imaginative,experiencing being then the judgment “raimexes arerational” is assured of being necessary, certain, andapriori. Philosophers have found it tempting to explain thepuzzling cases of, e.g.,a priori judgments, by anappeal to stipulative definitions.

Saul Kripke (1980) has drawn attention to a special kind ofstipulative definition. We can stipulatively introduce a new name(e.g., ‘Jack the Ripper’) through a description (e.g.,“the man who murdered \(X, Y\), and \(Z\)”). In such astipulation, Kripke pointed out, the description serves only to fixthe reference of the new name; the name is not synonymous with thedescription. For, the judgment

(1)

Jack the Ripper is the man who murdered \(X, Y\), and \(Z\), if aunique man committed the murders

is contingent, even though the judgment

Jack the Ripper is Jack the Ripper, if a unique man committed themurders

is necessary. A name such as ‘Jack the Ripper’, Kripkeargued, is rigid: it picks out the same individual across possibleworlds; the description, on the other hand, is non-rigid. Kripke usedsuch reference-fixing stipulations to argue for the existence ofcontingenta priori truths—(1) being an example.Reference-fixing stipulative definitions can be given not only fornames but also for terms in other categories, e.g., common nouns.

See Frege 1914 for a defense of the austere view that, in mathematicsat least, only stipulative definitions should be countenanced.^[1]

1.4 Descriptive definitions

Descriptive definitions, like stipulative ones, spell out meaning, butthey also aim to be adequate to existing usage. When philosophersoffer definitions of, e.g., ‘know’ and ‘free’,they are not being stipulative: a lack of fit with existing usage isan objection to them.

It is useful to distinguish three grades of descriptive adequacy of adefinition: extensional, intensional, and sense. A definition isextensionally adequate iff there are no actualcounterexamples to it; it isintensionally adequate iff thereare no possible counterexamples to it; and it issenseadequate (oranalytic) iff it endows the defined termwith the right sense. (The last grade of adequacy itself subdividesinto different notions, for “sense” can be spelled out inseveral different ways.) The definition “Water isH₂O,” for example, is intensionally adequate becausethe identity of water and H₂O is necessary (assuming theKripke-Putnam view about the rigidity of natural-kind terms); thedefinition is therefore extensionally adequate also. But it is notsense-adequate, for the sense of ‘water’ is not at all thesame as that of ‘H₂O’. The definition‘George Washington is the first President of the UnitedStates’ is adequate only extensionally but not in the other twogrades, while ‘man is a laughing animal’ fails to beadequate in all three grades. When definitions are put to anepistemological use, intensional adequacy is generally insufficient.For such definitions cannot underwrite the rationality or theapriority of a problematic subject matter.

See Quine 1951 & 1960 for skepticism about analytic definitions;see also the entry on theanalytic/synthetic distinction. Horty 2007 offers some ways of thinking about senses of definedexpressions, especially within a Fregean semantic theory.

1.5 Explicative definitions

Sometimes a definition is offered neither descriptively norstipulatively but as, what Rudolf Carnap (1956, §2) called, anexplication. An explication aims to respect some central usesof a term but is stipulative on others. The explication may be offeredas an absolute improvement of an existing, imperfect concept. Or, itmay be offered as a “good thing to mean” by the term in aspecific context for a particular purpose. (The quoted phrase is dueto Alan Ross Anderson; see Belnap 1993, 117.)

A simple illustration of explication is provided by the definition ofordered pair in set theory. Here, the pair \(\langle x,y\rangle\) isdefined as the set \(\{\{x\}, \{x,y\}\}\). Viewed as an explication,this definition does not purport to capture all aspects of theantecedent uses of ‘ordered pair’ in mathematics (and inordinary life); instead, it aims to capture the essential uses. Theessential fact about our use of ‘ordered pair’ is that itis governed by the principle that pairs are identical iff theirrespective components are identical:

And it can be verified that the above definition satisfies theprinciple. The definition does have some consequences that do notaccord with the ordinary notion. For example, the definition impliesthat an object \(x\) is a member of a member of the pair \(\langle x,y\rangle\), and this implication is no part of the ordinary notion.But the mismatch is not an objection to the explication. What isimportant for explication is not antecedent meaning but function. Solong as the latter is preserved, the former can be let go. It is thisfeature of explication that led W. V. O. Quine (1960, §53) toextol its virtues and to uphold the definition of “orderedpair” as a philosophical paradigm.

The truth-functional conditional provides another illustration ofexplication. This conditional differs from the ordinary conditional insome essential respects. Nevertheless, the truth-functionalconditional can be put forward as an explication of the ordinaryconditionalfor certain purposes in certain contexts. Whetherthe proposal is adequate depends crucially on the purposes andcontexts in question. That the two conditionals differ in important,even essential, respects does not automatically disqualify theproposal.

1.6 Ostensive definitions

Ostensive definitions typically depend on context and on experience.Suppose the conversational context renders one dog salient amongseveral that are visible. Then one can introduce the name‘Freddie’ through the stipulation “let this dog becalled ‘Freddie’.” For another example, suppose youare looking at a branch of a bush and you stipulatively introduce thename ‘Charlie’ thus: “let the insect on that branchbe called ‘Charlie’.” This definition can pin areferent on ‘Charlie’ even if there are many insects onthe branch. If your visual experience presents you with only one ofthese insects (say, because the others are too small to be visible),then that insect is the denotation of your use of the description‘the insect on that branch’. We can think of experience aspresenting the subject with a restricted portion of the world. Thisportion can serve as a point of evaluation for the expressions in anostensive definition.^[2] Consequently, the definition can with the aid of experience pin areferent on the defined term when without this aid it would fail to doso. In the present example, the description ‘the insect on thatbranch’ fails to be denoting when it is evaluated at the worldas a whole, but it is denoting when it is evaluated at that portion ofit that is presented in your visual experience. See Gupta 2019 for anaccount of the contribution of experience to the meaning of anostensively defined term.

An ostensive definition can bring about an essential enrichment of alanguage. The ostensive definition of ‘Charlie’ enrichesthe language with a name of a particular insect, and it could well bethat before the enrichment the language lacked resources to denotethat particular insect. Unlike other familiar definitions, ostensivedefinitions can introduce terms that are ineliminable. (So, ostensivedefinitions can fail to meet the Eliminability criterion explainedbelow; they can fail to meet also the Conservativeness criterion, alsoexplained below.)

The capacity of ostensive definitions to introduce essentially newvocabulary has led some thinkers to view them as the source of allprimitive concepts. Thus, Russell maintains inHumanKnowledge that

all nominal definitions, if pushed back far enough, must leadultimately to terms having only ostensive definitions, and in the caseof an empirical science the empirical terms must depend upon terms ofwhich the ostensive definition is given in perception. (p. 242)

In “Meaning and Ostensive Definition”, C. H. Whiteleytakes it as a premiss that ostensive definitions are “the meanswhereby men learn the meanings of most, if not all, of thoseelementary expressions in their language in terms of which otherexpressions are defined.” (332) It should be noted, however,that nothing in the logic and semantics of ostensive definitionswarrants a foundationalist picture of concepts or oflanguage-learning. Such foundationalist pictures were decisivelycriticized by Ludwig Wittgenstein in hisPhilosophicalInvestigations. Wittgenstein’s positive views on ostensivedefinition remain elusive, however; for an interpretation, see Hacker1975.

Ostensive definitions are important, but our understanding of themremains at a rudimentary level. They deserve greater attention fromlogicians and philosophers.

1.7 A remark

The kinds into which we have sorted definitions are not mutuallyexclusive, nor exhaustive. A stipulative definition of a term may, asit happens, be extensionally adequate to the antecedent uses of theterm. A dictionary may offer ostensive definitions of some words(e.g., of color words). An ostensive definition can also beexplicative. For example, one can offer an improvement of apreexisting concept “one foot” thus: “let one footbe the present length of that rod.” In its preexisting use, theconcept “one foot” may be quite vague; the ostensivelyintroduced explication may, in contrast, be relatively precise.Moreover, as we shall see below, there are other kinds of definitionthan those considered so far.

2. The logic of definitions

Many definitions—stipulative, descriptive, andexplicative—can be analyzed into three elements: the term thatis defined \((X)\), an expression containing the defined term\((\ldots X\ldots)\), and another expression \((- - - - - - -)\) thatis equated by the definition with this expression. Such definitionscan be represented thus:

(We are setting aside ostensive definitions, which plainly require aricher representation.) When the defined term is clear from thecontext, the representation may be simplified to

The expression on the left-hand side of ‘\(\eqdf\)’ (i.e.,\(\ldots X\ldots)\) is thedefiniendum of the definition, andthe expression on the right-hand side is itsdefiniens—it being assumed that the definiendum and thedefiniens belong to the same logical category. Note the distinctionbetween defined term and definiendum: the defined term in the presentexample is \(X\); the definiendum is the unspecified expression on theleft-hand side of ‘\(\eqdf\)’, which may or may not beidentical to \(X\). (Some authors call the defined term ‘thedefiniendum’; some others use the expression confusedly,sometimes to refer to the defined term and sometimes to thedefiniendum proper.) Not all definitions found in the logical andphilosophical literature fit under scheme (2). Partial definitions,for example, fall outside the scheme; another example is provided bydefinitions of logical constants in terms of introduction andelimination rules governing them. Nonetheless, definitions thatconform to (2) are the most important, and they will be our primaryconcern.

Let us focus on stipulative definitions and reflect on their logic.Some of the important lessons here carry over, as we shall see, todescriptive and explicative definitions. For simplicity, let usconsider the case where a single definition stipulatively introduces aterm. (Multiple definitions bring notational complexity but raise nonew conceptual issues.) Suppose, then, that a language \(L\), theground language, is expanded through the addition of a newterm \(X\) to anexpanded language \(L^{+}\), where \(X\) isstipulatively defined by a definition \(\mathcal{D}\) of form (2).What logical rules govern \(\mathcal{D}\)? What requirements must thedefinition fulfill?

Before we address these questions, let us take note of a distinctionthat is not marked in logic books but which is useful in thinkingabout definitions. In one kind of definition—call ithomogeneous definition—the defined term and thedefiniendum belong to the same logical category. So, a singular termis defined via a singular term; a general term via a general term; asentence via a sentence; and so on. Let us say that a homogeneousdefinition isregular iff its definiendum is identical to thedefined term. Here are some examples of regular homogeneousdefinitions:

Note that ‘The True’, as defined above, belongs to thecategory of sentence, not that of singular term.

It is sometimes said that definitions are mere recipes forabbreviations. Thus, Alfred North Whitehead and Bertrand Russell sayof definitions—in particular, those used inPrincipiaMathematica—that they are “strictly speaking,typographical conveniences (1925, 11).” This viewpoint hasplausibility only for regular homogeneous definitions—though itis not really tenable even here. (Whitehead and Russell’s ownobservations make it plain that their definitions are more than mere“typographical conveniences.”^[3]) The idea that definitions are mere abbreviations is not at allplausible for the second kind of definition, to which we now turn.

In the second kind of definition—call it aheterogeneous definition—the defined term and thedefiniendum belong to different logical categories. So, for example, ageneral term (e.g., ‘man’) may be defined using asentential definiendum (e.g., ‘\(x\) is a man’). Foranother example, a singular term (e.g., ‘1’) may bedefined using a predicate (e.g., ‘is identical to 1’).Heterogeneous definitions are far more common than homogeneous ones.In familiar first-order languages, for instance, it is pointless todefine, say, a one-place predicate \(G\) by a homogeneous definition.These languages have no resources for forming compound predicates;hence, the definiens of a homogeneous definition of \(G\) is bound tobe atomic. In a heterogeneous definition, however, the definiens caneasily be complex; for example,

If the language has a device for nominalization ofpredicates—e.g., a class abstraction operator—we couldgive a different sort of heterogeneous definition forG:

Observe that a heterogeneous definition such as (4) is not a mereabbreviation. For one thing, we regard the expression \(x\) in it as agenuine variable which admits of substitution and binding. So, thedefiniendumGx is not amere abbreviationfor the definiens. Moreover, if such definitions were abbreviations,they would be subject to the requirement that the definiendum must beshorter than the definiens, but no such requirement exists.On the other hand, genuine requirements on definitions would makelittle sense. The following stipulation is not a legitimatedefinition:

But if it is viewed as a mere abbreviation, there is nothingillegitimate about it. (Indeed, mathematicians routinely make use ofabbreviations of this kind, suppressing variables that are temporarilyuninteresting.)

Some stipulative definitions are nothing but mere devices ofabbreviation (e.g., the definitions governing the omission ofparentheses in formulas; see Church 1956, §11). However, manystipulative definitions are not of this kind; they introducemeaningful items into our discourse. Thus, definition (4) renders\(G\) a meaningful unary predicate: \(G\) expresses, in virtue of (4),a particular concept. In contrast, under stipulation (6), \(G\) is nota meaningful predicate and expresses no concept of any kind. But whatis the source of the difference? Why is (4) legitimate, but not (6)?More generally, when is a definition legitimate? What requirementsmust the definiens fulfill? And, for that matter, the definiendum?Must the definiendum be, for instance, atomic, as in (3) and (4)? Ifnot, what restrictions (if any) are there on the definiendum?

2.1 Two criteria

It is a plausible requirement on any answer to these questions thattwo criteria be respected.^[4] First, a stipulative definition should not enable us to establishessentially new claims—call this theConservativenesscriterion. We should not be able to establish, by means of a merestipulation, new things about, for example, the moon. It is true thatunless this criterion is made precise, it is subject to trivialcounterexamples, for the introduction of a definition materiallyaffects some facts. Nonetheless, the criterion can be made precise anddefensible, and we shall soon see some ways of doing this.

Second, the definition should fix the use of the defined expression\(X\)—call this theUse criterion. This criterion isplausible, since only the definition—and nothing else—isavailable to guide us in the use of \(X\). There are complicationshere, however. What counts as a use of \(X\)? Are occurrences withinthe scope of ‘say’ and ‘know’ included? Whatabout the occurrence of \(X\) within quotation contexts, and thosewithin words, for instance, ‘Xenophanes’? Thelast question should receive, it is clear, the answer,“No.” But the answers to the previous questions are not soclear. There is another complication: even if we can somehow separateout genuine occurrences of \(X\), it may be that some of theseoccurrences are rightfully ignored by the definition. For example, adefinition of quotient may leave some occurrences of the termundefined (e.g., where there is division by 0). The orthodox view isto rule such definitions as illegitimate, but the orthodoxy deservesto be challenged here. Let us leave the challenge to another occasion,however, and proceed to bypass the complications through idealization.Let us confine ourselves to ground languages that possess a clearlydetermined logical structure (e.g., a first-order language) and thatcontain no occurrences of the defined term \(X\). And let us confineourselves to definitions that place no restrictions on legitimateoccurrences of \(X\). The Use criterion now dictates then that thedefinition should fix the use of all expressions in the expandedlanguage in which \(X\) occurs.

A variant formulation of the Use criterion is this: the definitionmust fix the meaning of the definiendum. The new formulation is lessdeterminate and more contentious, for it relies on“meaning,” an ambiguous and theoretically contentiousnotion.

Note that the two criteria govern all stipulative definitions,irrespective of whether they are single or multiple, or of whetherthey are of form (2) or not.

2.2 Foundations of the traditional account

The traditional account of definitions is founded on three ideas. Thefirst idea is that definitions are generalized identities; the second,that the sentential is primary; and the third, that of reduction. Thefirst idea—that definitions are generalizedidentities—motivates the traditional account’s inferentialrules for definitions. These are, put crudely, that (i) any occurrenceof the definiendum can be replaced by an occurrence of the definiens(Generalized Definiendum Elimination); and, conversely, (ii) anyoccurrence of the definiens can be replaced by an occurrence of thedefiniendum (Generalized Definiendum Introduction).

The second idea—the primacy of the sentential—has itsroots in the thought that the fundamental uses of a term are inassertion and argument: if we understand the use of a defined term inassertion and argument then we fully grasp the term. The sententialis, however, primary in argument and assertion. Hence, to explain theuse of a defined term \(X\), the second idea maintains, it isnecessary and sufficient to explain the use of sentential items thatcontain \(X\). (Sentential items are here understood to includesentences and sentence-like things with free variables, e.g., thedefiniens of (4); henceforth, these items will be calledformulas.) The issues the second idea raises are, of course,large and important, but they cannot be addressed in a brief survey.Let us accept the idea simply as a given.

The third idea—reduction—is that the use of a formula\(Z\) containing the defined term is explained by reducing \(Z\) to aformula in the ground language. This idea, when conjoined with theprimacy of the sentential, leads to a strong version of the Usecriterion, called theEliminability criterion: the definitionmust reduce each formula containing the defined term to a formula inthe ground language, i.e., one free of the defined term. Eliminabilityis the distinctive thesis of the traditional account and, as we shallsee below, it can be challenged.

Note that the traditional account does not require the reduction ofall expressions of the extended language; it requires thereduction only of formulas. The definition of a predicate \(G\), forexample, need provide no way of reducing \(G\), taken in isolation, toa predicate of the ground language. The traditional account is thusconsistent with the thought that a stipulative definition can add anew conceptual resource to the language, for nothing in the groundlanguage expresses the predicative concept that \(G\) expresses in theexpanded language. This is not to deny that no newproposition—at least in the sense of truth-condition—isexpressed in the expanded language.

2.3 Conservativeness and eliminability

Let us now see how Conservativeness and Eliminability can be madeprecise. First consider languages that have a precise proof system ofthe familiar sort. Let the ground language \(L\) be one such. Theproof system of \(L\) may be classical, or three-valued, or modal, orrelevant, or some other; and it may or may not contain somenon-logical axioms. All we assume is that we have available thenotions “provable in \(L\)” and “provably equivalentin \(L\),” and also the notions “provable in\(L^{+}\)” and “provably equivalent in \(L^{+}\)”that result when the proof system of \(L\) is supplemented with adefinition \(\mathcal{D}\) and the logical rules governingdefinitions. Now, the Conservativeness criterion can be made preciseas follows.

Conservativeness criterion (syntactic formulation):Any formula of \(L\) that is provable in \(L^{+}\) is provable in\(L\).

That is, any formula of \(L\) that is provable using definition\(\mathcal{D}\) is also provable without using \(\mathcal{D}\): thedefinition does not enable us to prove anything new in \(L\). TheEliminability criterion can be made precise thus:

Eliminability criterion (syntactic formulation): Forany formula \(A\) of \(L^{+}\), there is a formula of \(L\) that isprovably equivalent in \(L^{+}\) to \(A\).

(Folklore credits the Polish logician S. Leśniewski forformulating the criteria of Conservativeness and Eliminability, butthis is a mistake; see Dudman 1973, Hodges 2008, Urbaniak andHämäri 2012 for discussion and further references.)^[5]

Now let us equip \(L\) with a model-theoretic semantics. That is, weassociate with \(L\) a class of interpretations, and we make availablethe notions “valid in \(L\) in the interpretation \(M\)”(a.k.a.: “true in \(L\) in \(M\)”) and “semanticallyequivalent in \(L\) relative to \(M\).” Let the notions“valid in \(L^{+}\) in \(M^{+}\)” and “semanticallyequivalent in \(L^{+}\) relative to \(M^{+}\)” result when thesemantics of \(L\) is supplemented with that of definition\(\mathcal{D}\). The criteria of Conservativeness and Eliminabilitycan now be made precise thus:

Conservativeness criterion (semantic formulation):For all formulas \(A\) of \(L\), if \(A\) is valid in \(L^{+}\) in allinterpretations \(M^{+}\), then \(A\) is valid in \(L\) in allinterpretations \(M\).

Eliminability criterion (semantic formulation): Forany formula \(A\) of \(L^{+}\), there is a formula \(B\) of \(L\) suchthat, relative to all interpretations \(M^{+}\), \(B\) issemantically equivalent in \(L^{+}\) to \(A\).

The syntactic and semantic formulations of the two criteria areplainly parallel. However, even if we suppose that strong completenesstheorems hold for \(L\) and \(L^{+}\), the two formulations need notbe equivalent: it depends on our semantics for definition\(\mathcal{D}\). Indeed, several different, non-equivalentformulations of the two criteria are possible within each framework,the syntactic and the semantic.

There is another, more stringent notion of semantic conservativenessthat has been prominent in the literature on truth (Halbach 2014, p.69). Say that an interpretation \(M^{+}\) of \(L^{+}\) is anexpansion of an interpretation \(M\) of \(L\) iff \(M\) and\(M^{+}\) assign the same domain(s) to the quantifier(s) inL,and assign the same semantic values to thenon-logical constants in \(L\). Then we have:

Conservativeness criterion (strong semanticformulation): Every interpretation \(M\) of \(L\) can beexpanded to an interpretation \(M^{+}\) of \(L^{+}\).

In other words, a definition is strongly semantically conservative ifit does not rule out any previously available interpretations of theoriginal language.

Observe that the satisfaction of Conservativeness and Eliminabilitycriteria, whether in their semantic or their syntactic formulation, isnot an absolute property of a definition; the satisfaction is relativeto the ground language. Different ground languages can have associatedwith them different systems of proof and different classes ofinterpretations. Hence, a definition may satisfy the two criteria whenadded to one language, but may fail to do so when added to a differentlanguage. For further discussion of the criteria, see Suppes 1957 andBelnap 1993.

2.4 Definitions in normal form

For concreteness, let us fix the ground language \(L\) to be aclassical first-order language with identity. The proof system of\(L\) may contain some non-logical axioms \(T\); the interpretationsof \(L\) are then the classical models of \(T\). As before, \(L^{+}\)is the expanded language that results when a definition\(\mathcal{D}\) of a non-logical constant \(X\) is added to \(L\);hence, \(X\) may be a name, a predicate, or a function-symbol. Calltwo definitionsequivalent iff they yield the same theoremsin the expanded language. Then, it can be shown that if\(\mathcal{D}\) meets the criteria of Conservativeness andEliminability then \(\mathcal{D}\) is equivalent to a definition innormal form as specified below.^[6] Since definitions in normal form meet the demands of Conservativenessand Eliminability, the traditional account implies that we losenothing essential if we require definitions to be in normal form.

The normal form of definitions can be specified as follows. Thedefinitions of names \(a, n\)-ary predicates \(H\), and \(n\)-aryfunction symbols \(f\) must be, respectively, of the followingforms:

where the variables \(x_{1}\), …, \(x_{n}\), \(y\) are alldistinct, and the definiens in each case satisfies conditions that canbe separated into a general and a specific part.^[7] The general condition on the definiens is the same in each case: itmust not contain the defined term or any free variables other thanthose in the definiendum. The general conditions remain the same whenthe traditional account of definition is applied to non-classicallogics (e.g., to many-valued and modal logics). The specificconditions are more variable. In classical logic, the specificcondition on the definiens \(\psi(x)\) of (7) is that it satisfy anexistence and uniqueness condition: that it be provable that somethingsatisfies \(\psi(x)\) and that at most one thing satisfies \(\psi(x)\).^[8] There are no specific conditions on (8), but the condition on (9)parallels that on (7). An existence and uniqueness claim must hold:the universal closure of the formula

must be provable.^[9]

In a logic that allows for vacuous names, the specific condition onthe definiens of (7) would be weaker: the existence condition would bedropped. In contrast, in a modal logic that requires names to benon-vacuous and rigid, the specific condition would be strengthened:not only must existence and uniqueness be shown to hold necessarily,it must be shown that the definiens is satisfied by one and the sameobject across possible worlds.

Definitions that conform to (7)–(9) are heterogeneous; thedefiniendum is sentential, but the defined term is not. One source ofthe specific conditions on (7) and (9) is their heterogeneity. Thespecific conditions are needed to ensure that the definiens, thoughnot of the logical category of the defined term, imparts the properlogical behavior to it. The conditions thus ensure that the logic ofthe expanded language is the same as that of the ground language. Thisis the reason why the specific conditions on normal forms can varywith the logic of the ground language. Observe that, whatever thislogic, no specific conditions are needed for regular homogeneousdefinitions.

The traditional account makes possible simple logical rules fordefinitions and also a simple semantics for the expanded language.Suppose definition \(\mathcal{D}\) has a sentential definiendum. (Inclassical logic, all definitions can easily be transformed to meetthis condition.) Let \(\mathcal{D}\) be

where \(x_{1}\), …, \(x_{n}\) are all the variables free ineither \(\phi\) or \(\psi\). And let \(\phi(t_{1},\ldots,t_{n})\) and\(\psi(t_{1},\ldots,t_{n})\) result by the simultaneous substitutionof terms \(t_{1}\), …, \(t_{n}\) for \(x_{1}\), …,\(x_{n}\) in, respectively, \(\phi(x_{1},\ldots, x_{n})\) and\(\psi(x_{1},\ldots, x_{n})\); changing bound variables as necessary.Then the rules of inference governing \(\mathcal{D}\) are simplythese:

The semantics for the extended language is also straightforward.Suppose, for instance, \(\mathcal{D}\) is a definition of a name \(a\)and suppose that, when put in normal form, it is equivalent to (7).Then, each classical interpretation \(M\) of \(L\) expands to a uniqueclassical interpretation \(M^{+}\) of the extended language \(L^{+}\).The denotation of \(a\) in \(M^{+}\) is the unique object thatsatisfies \(\psi(x)\) in \(M\); the conditions on \(\psi(x)\) ensurethat such an object exists. The semantics of defined predicates andfunction-symbols is similar. The logic and semantics of definitions innon-classical logics receive, under the traditional account, aparallel treatment.

Note that the inferential force of adding definition (10) to thelanguage is the same as that of adding as an axiom, the universalclosure of

However, this similarity in the logical behavior of (10) and (11)should not obscure the great differences between the biconditional(‘\(\leftrightarrow\)’) and definitional equivalence(‘\(\eqdf\)’). The former is a sentential connective, butthe latter is trans-categorical: not only formulas, but alsopredicates, names, and items of other logical categories can occur onthe two sides of ‘\(\eqdf\)’. Moreover, the biconditionalcan be iterated—e.g., \(((\phi \leftrightarrow \psi)\leftrightarrow \chi)\)—but not definitional equivalence.Finally, a term can be introduced by a stipulative definition into aground language whose logical resources are confined, say, toclassical conjunction and disjunction. This is perfectly feasible,even though the biconditional is not expressible in the language. Insuch cases, the inferential role of the stipulative definition is notmirrored by any formula of the extended language.

The traditional account of definitions should not be viewed asrequiring definitions to be in normal form. The onlyrequirements that it imposes are (i) that the definiendum contain thedefined term; (ii) that the definiendum and the definiens belong tothe same logical category; and (iii) the definition satisfiesConservativeness and Eliminability. So long as these requirements aremet, there are no further restrictions. The definiendum, like thedefiniens, can be complex; and the definiens, like the definiendum,can contain the defined term. So, for example, there is nothingformally wrong if the definition of the functional expression‘the number of’ has as its definiendum the formula‘the number of \(F\)s is the number of \(G\)s’. The roleof normal forms is only to provide an easy way of ensuring thatdefinitions satisfy Conservativeness and Eliminability; they do notprovide the only legitimate format for stipulatively introducing aterm. Thus, the reason why (4) is, but (6) is not, a legitimatedefinition is not that (4) is in normal form and (6) is not.

The reason is that (4) respects, but (6) does not, the two criteria.(The ground language is assumed here to contain ordinary arithmetic;under this assumption, the second definition implies a contradiction.)The following two definitions are also not in normal form:

But both should count as legitimate under the traditional account,since they meet the Conservativeness and Eliminability criteria. Itfollows that the two definitions can be put in normal form. Definition(12) is plainly equivalent to (4), and definition (13) is equivalentto (14):

Observe that the definiens of (13) is not logically equivalent to any\(G\)-free formula. Nevertheless, the definition has a normalform.

Similarly, the traditional account is perfectly compatible withrecursive (a.k.a.:inductive) definitions such asthose found in logic and mathematics. In Peano Arithmetic, forexample, exponentiation can be defined by means of the followingequations:

Here the first equation—called thebaseclause—defines the value of the function when the exponent is 0.And the second clause—called therecursiveclause—uses the value of the function when the exponent is \(n\)to define the value when the exponent is \(n + 1\). This is perfectlylegitimate, according to the traditional account, because a theorem ofPeano Arithmetic establishes that the above definition is equivalentto one in normal form.^[10] Recursive definitions are circular in their format, and indeed it isthis circularity that renders them perspicuous. But the circularity isentirely on the surface, as the existence of normal forms shows. Seethe discussion of circular definitions below.

2.5 Conditional definitions

It is a part of our ordinary practice that we sometimes define termsnot absolutely but conditionally. We sometimes affirm a definition notoutright but within the scope of a condition, which may either be lefttacit or may be set down explicitly. So, for example, we may defineF(x,y) to express the notion “firstcousin once removed” by stipulating that

where it is understood that the variables range over humans. Foranother example, when defining division, we may explicitly set down asa condition on the definition that the divisor not be 0. We maystipulate that

but with the proviso that \( x \neq 0 \). This practice may appear toviolate the Eliminability criterion, for it appears that conditionaldefinitions do not ensure the eliminability of the defined terms inall sentences. Thus (16) does not enable us to prove the equivalenceof

with anyF-free sentence because of the tacit restriction onthe range of variables in (16). Similarly (17) does not enable us toeliminate the defined symbol from

However, if there is a violation of Eliminability here, it is asuperficial one, and it is easily corrected in one of two ways. Thefirst way—the way that conforms best to our ordinarypractices—is to understand the enriched languages that resultfrom adding the definitions to exclude sentences such as (18) and(19). For when we stipulate a definition such as (16), it is not ourintention to speak about the first cousins once removed of numbers; onthe contrary, we wish to exclude all such talk as improper. Similarly,in setting down (17), we wish to exclude talk of division by 0 aslegitimate. So, the first way is to recognize that a conditionaldefinition such as (16) and (17) brings with it restrictions on theenriched language and, consequently, respects the Eliminabilitycriterion once the enriched language is properly demarcated. This ideacan be implemented formally by seeing conditional definitions asformulated within languages with sortal quantification.

The second way—the way that conforms best to our actual formalpractices—is to understand the applications of the defined termin cases where the antecedent condition fails as “don’tcare cases” and to make a suitable stipulation concerning suchapplications. So, we may stipulate that nothing other than a human hasfirst cousins once removed, and we may stipulate that the result ofdividing any number by 0 is 0. Thus we may replace (17) by

The resulting definitions satisfy the Eliminability criterion. Thesecond way forces us to exercise care in reading sentences withdefined terms. So, for example, the sentence

though true when division in defined as in (20), does not express aninteresting mathematical truth but one that is merely a byproduct ofour treatment of the “don’t care cases.” Despitethis cost, the gain in simplicity in the notion of proof may wellwarrant, in some contexts, the move to a definition such as (20).

See Suppes 1957 for a different perspective on conditionaldefinitions.

2.6 Implicit definitions

The above viewpoint allows the traditional account to bring within itsfold ideas that might at first sight seem contrary to it. It issometimes suggested that a term \(X\) can be introduced axiomatically,that is, by laying down as axioms certain sentences of the expandedlanguage \(L^{+}\). The axioms are then said toimplicitlydefine \(X\). This idea is easily accommodated within the traditionalaccount. Let atheory be a set of sentences of the expandedlanguage \(L^{+}\). Then, to say that a theory \(T^*\) is animplicit (stipulative)definition of X isto say that \(X\) is governed by the definition

where \(\phi\) is the conjunction of the members of \(T^*\). (If\(T^*\) is infinite then a stipulation of the above form will beneeded for each sentence \(\psi\) in \(T^*\).)^[11] 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 definitionof X). So, thetraditional account accommodates the idea that theories canstipulatively introduce new terms, but it imposes a strong demand: thetheories must be admissible.^[12]

Consider, for concreteness, the special case of classical first-orderlanguages. Let the ground language \(L\) be one such, and let itsinterpretations be models of some sentences \(T\). Let us say that

\(T^*\) is animplicit semantic definition of X iff, for eachinterpretation \(M\) of \(L\), there is a unique model \(M^{+}\) of\(T^*\) such that \(M^{+}\) is an expansion of \(M\).

Then, from the normal form theorem, the following claim isimmediate:

If \(T^*\) is admissible then \(T^*\) is an implicit semanticdefinition of \(X\).

That is, an admissible theory fixes the semantic value of the definedterm in each interpretation of the ground language. This observationprovides one natural method of showing that a theory is notadmissible:

Padoa’s method. To show that \(T^*\) is notadmissible, it suffices to construct two models of \(T^*\) that areexpansions of one and the same interpretation of the ground language\(L\). (Padoa 1900)

Here is a simple and philosophically useful application ofPadoa’s method. Suppose the proof system of \(L\) is PeanoArithmetic and that \(L\) is expanded by the addition of a unarypredicate \(Tr\) (for “Gödel number of a true sentence of\(L\)”). Let \(\mathbf{H}\) be the theory consisting of all thesentences (the “Tarski biconditionals”) of the followingform:

where \(\psi\) is a sentence of \(L\) and \(s\) is the canonical namefor the Gödel number of \(\psi\). Padoa’s method impliesthat \(\mathbf{H}\) is not admissible for defining \(Tr\). For\(\mathbf{H}\) does not fix the interpretation of \(Tr\) in allinterpretations of \(L\). In particular, it does not do so in thestandard model, for \(\mathbf{H}\) places no constraints on thebehavior of \(Tr\) on those numbers that are not Gödel numbers ofsentences. (If the coding renders each natural number a Gödelnumber of a sentence, then a non-standard model of Peano Arithmeticprovides the requisite counterexample: it has infinitely manyexpansions that are models of \(\mathbf{H}\).) A variant of thisargument 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 showthat in each interpretation of the ground language, a theory \(T^*\)fixes a unique semantic value for the defined term. Can we concludethat \(T^*\) is admissible? This question receives a negative answerfor some semantical systems, and a positive answer for others. (Incontrast, Padoa’s method works so long as the semantic system isnot highly contrived.) The converse fails for, e.g., classicalsecond-order languages, but it holds for first-order ones:

Beth’s Definability Theorem. If \(T^*\) is animplicit semantic definition of \(X\) in a classical first-orderlanguage then \(T^*\) is admissible.

Note that the theorem holds even if \(T^*\) is an infinite set. For aproof of the theorem, see Boolos, Burgess, and Jeffrey 2002; see alsoBeth 1953.

The idea of implicit definition is not in conflict, then, with thetraditional account. Where conflict arises is in the philosophicalapplications of the idea. The failure of strict reductionist programsof the late-nineteenth and early-twentieth century promptedphilosophers to explore looser kinds of reductionism. For instance,Frege’s definition of number proved to be inconsistent, and thusincapable of sustaining the logicist thesis that the principles ofarithmetic are analytic. It turns out, however, that the principles ofarithmetic can be derived without Frege’s definition. All thatis needed is one consequence of it, namely, Hume’sPrinciple:

Hume’s Principle. The number of \(F\)s = thenumber of \(G\)s iff there is a one-to-one correspondence between the\(F\)s and \(G\)s.

If we add Hume’s Principle to axiomatic second-order logic, thenwe obtain a consistent theory from which we can analytically derivesecond-order Peano Arithmetic. (The essentials of the argument arefound already in Frege 1884.) It is a central thesis ofNeo-Fregeanism that Hume’s Principle is an implicitdefinition of the functional expression ‘the number of’(see Hale and Wright 2001). If this thesis can be defended, thenlogicism about arithmetic can be sustained. However, the neo-Fregeanthesis is in conflict with the traditional account of definitions, forHume’s Principle violates both Conservativeness andEliminability. The principle allows one to prove, for arbitrary \(n\),that there are at least \(n\) objects.

Another example: The reductionist program for theoretical concepts(e.g., those of physics) aimed to solve epistemological problems thatthese concepts pose. The program aimed to reduce theoretical sentencesto (classes of) observational sentences. However, the reductionsproved difficult, if not impossible, to sustain. Thus arose thesuggestion that perhaps the non-observational component of a theorycan, without any claim of reduction, be regarded as an implicitdefinition of theoretical terms. The precise characterization of thenon-observational component can vary with the specific epistemologicalproblem at hand. But there is bound to be a violation of one or bothof the two criteria, Conservativeness and Eliminability.^[13]

A final example: We know by a theorem of Tarski that no theory can bean admissible definition of the truth predicate, \(Tr\), for thelanguage of Peano Arithmetic considered above. Nonetheless, perhaps wecan still regard theory \(\mathbf{H}\) as an implicit definition of\(Tr\). (Paul Horwich has made a closely related proposal for theordinary notion of truth.) Here, again, pressure is put on the boundsimposed by the traditional account. \(\mathbf{H}\) meets theConservativeness criterion, but not that of Eliminability.

In order to assess the challenge these philosophical applications posefor the traditional account, we need to resolve issues that are undercurrent philosophical debate. Some of the issues are the following.(i) It is plain that some violations of Conservativeness areillegitimate: one cannot make it true by astipulation that,e.g., Mercury is larger than Venus. Now, if a philosophicalapplication requires some violations of Conservativeness to belegitimate, we need an account of the distinction between the twosorts of cases: the legitimate violations of Conservativeness and thenon-legitimate ones. And we need to understand what it is that rendersthe one legitimate, but not the other. (ii) A similar issue arises forEliminability. It would appear that not any old theory can be animplicit definition of a term \(X\). (The theory might contain onlytautologies.) If so, then again we need a demarcation of theories thatcan serve to implicitly define a term from those that cannot. And weneed a rationale for the distinction. (iii) The philosophicalapplications rest crucially on the idea that an implicit definitionfixes the meaning of the defined term. We need therefore an account ofwhat this meaning is, and how the implicit definition fixes it. Underthe traditional account, formulas containing the defined term can beseen as acquiring their meaning from the formulas of the groundlanguage. (In view of the primacy of the sentential, this fixes themeaning of the defined term.) But this move is not available under aliberalized conception of implicit definition. How, then, should wethink of the meaning of a formula under the envisioned departure fromthe traditional account? (iv) Even if the previous three issues areaddressed satisfactorily, an important concern remains. Suppose weallow that a theory \(T\), say, of physics can stipulatively defineits theoretical terms, and that it endows the terms with particularmeanings. The question remains whether the meanings thus endowed areidentical to (or similar enough to) the meanings the theoretical termshave in their actual uses in physics. This question must be answeredpositively if implicit definitions are to serve their philosophicalfunction. The aim of invoking implicit definitions is to account forthe rationality, or the apriority, or the analyticity of our ordinaryjudgments, not of some extraordinary judgments that are somehowassigned to ordinary signs.

The literature on neo-Fregeanism presents an interesting case study inrespect of these issues. Much of the debate concerning the neo-Fregeanthesis can fruitfully be viewed as a debate over the extent andprecise formulation of the criteria of Conservativeness and Use. Forexample, the so-calledJulius Caesar objection (due to Frege1884) urges that Hume’s Principle cannot be a legitimatedefinition of ‘the number of’ because it does notdetermine the use of this expression in mixed identity contexts, suchas ‘\(\text{the number of }F\text{s} = \text{JuliusCaesar}\)’. Other classic objections (Field 1984, Boolos 1997)focus on the non-conservativeness of Hume’s Principle. Boolos1990 raises a particularly sharp point, known as theBad Companyproblem. Definitions of the same kind as Hume’s Principleare known asabstraction principles. Boolos exhibits anabstraction principle that is consistent by itself, but inconsistentin conjunction with Hume’s Principle. This pathologicalsituation never arises with conservative definitions. So, the BadCompany problem illustrates what can go wrong when theConservativeness requirement is violated.

Friends of neo-Fregeanism have responded to these objections invarious ways. Wright 1997 argues that abstraction principles need onlysatisfy a restricted version of Conservativeness, and needn’tsatisfy Eliminability at all. (However, Wright’s proposalsuffered the revenge of the Bad Company problem: see Weir 2003.) Bycontrast, Linnebo 2018 argues for much more stringent requirements onabstraction principles. He countenances onlypredicativeabstraction principles, which satisfy both Conservativeness andEliminability in suitable contexts. Mackereth and Avigad (forthcoming)defend an intermediate position. They hold that abstraction principlesmust satisfy Conservativeness in an unrestricted sense, butneedn’t satisfy Eliminability. Furthermore, Mackereth and Avigadshow that in the absence of Eliminability, the precise formulation ofConservativeness (syntactic vs. semantic, etc.) makes a bigdifference. In particular, it is possible to get impredicativeversions of Hume’s Principle that aresemanticallyconservative, but the same does not appear to be true forsyntactic conservativeness.

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

2.7 Vicious-Circle Principle

Another departure from the traditional theory begins with the idea notthat the theory is too strict, but that it is too liberal, that itpermits definitions that are illegitimate. Thus, the traditionaltheory allows the following definitions of, respectively,“liar” and the class of natural numbers\(\mathbf{N}\):

(22)

\(z\) is a liar \(\eqdf\) all propositions asserted by \(z\) arefalse;

(23)

\(z\) belongs to \(\mathbf{N}\) \(\eqdf\) \(z\) belongs to everyinductive class, where a class is inductive when it contains 0 and isclosed under the successor operation.

Russell argued that such definitions involve a subtle kind of viciouscircle. The definiens of the first definition invokes, Russellthought, the totality of all propositions, but the definition, iflegitimate, would result in propositions that can only be defined byreference to this totality. Similarly, the second definition attemptsto define the class \(\mathbf{N}\) by reference to all classes, whichincludes the class \(\mathbf{N}\) that is being defined. Russellmaintained that such definitions are illegitimate. And he imposed thefollowing requirement—called, the “Vicious-CirclePrinciple”—on definitions and concepts. (HenriPoincaré had also proposed a similar idea.)

Vicious-Circle Principle. “Whatever involvesall 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 havemembers only definable in terms of that total, then the saidcollection has no total (Russell, 1908, 63).”

In an appended footnote, Russell explained, “When I say that acollection has no total, I mean that statements aboutall itsmembers are nonsense.”

Russell’s primary motivation for the Vicious-Circle Principlewere the logical and semantic paradoxes. Notions such as“truth,” “proposition,” and“class” generate, under certain unfavorable conditions,paradoxical conclusions. Thus, the claim “Cheney is aliar,” where “liar” is understood as in (16), yieldsparadoxical conclusions, if Cheney has asserted that he is a liar, andall other propositions asserted by him are, in fact, false. Russelltook the Vicious-Circle Principle to imply that if “Cheney is aliar” expresses a proposition, it cannot be in the scope of thequantifier in the definiens of (16). More generally, Russell held thatquantification over all propositions, and over all classes, violatesthe Vicious-Circle Principle and is thus illegitimate. Furthermore, hemaintained that expressions such as ‘true’ and‘false’ do not express a unique concept—inRussell’s terminology, a unique “propositionalfunction”—but one of a hierarchy of propositionalfunctions of different orders. Thus the lesson Russell drew from theparadoxes is that the domain of the meaningful is more restricted thanit might ordinarily appear, that the traditional account of conceptsand definitions needed to be made more restrictive in order to ruleout the likes of (16) and (17).

In application to ordinary, informal definitions, the Vicious-CirclePrinciple does not provide, it must be said, a clear method ofdemarcating the meaningful from the meaningless. Definition (16) issupposed to be illegitimate because, in its definiens, the quantifierranges over the totality of all propositions. And we are told thatthis is prohibited because, were it allowed, the totality ofpropositions “would have members only definable in terms of thetotal.” However, unless we know more about the nature ofpropositions and of the means available for defining them, it isimpossible to determine whether (16) violates the Principle. It may bethat a proposition such as “Cheney is a liar”—or, totake a less contentious example, “Either Cheney is a liar or heis not”— can be given a definition that does not appeal tothe totality of all propositions. If propositions are sets of possibleworlds, for example, then such a definition would appear to befeasible.

The Vicious-Circle Principle serves, nevertheless, as an effectivemotivation for a particular account of legitimate concepts anddefinitions, namely that embodied in Russell’s Ramified TypeTheory. The idea here is that one begins with some unproblematicresources that involve no quantification over propositions, concepts,and such. These resources enable one to define, for example, variousunary concepts, which are thereby assured of satisfying theVicious-Circle Principle. Quantification over these concepts is thusbound to be legitimate, and can be added to the language. The sameholds for propositions and for concepts falling under other types: foreach type, a quantifier can be added that ranges over items (of thattype) that are definable using the initial unproblematic resources.The new quantificational resources enable the definition of furtheritems of each type; these, too, respect the Principle, and again,quantifiers ranging over the expanded totalities can legitimately beadded to the language. The new resources permit the definition of yetfurther items. And the process repeats. The result is that we have ahierarchy of propositions and of concepts of various orders. Each typein the type hierarchy ramifies into a multiplicity of orders. Thisramification ensures that definitions formulated in the resultinglanguage are bound to respect the Vicious-Circle Principle. Conceptsand classes that can be defined within the confines of this scheme aresaid to bepredicative (in one sense of this word); theothers,impredicative.

For further discussion of the Vicious-Circle Principle, see Russell1908, 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 entriesontype theory andPrincipia Mathematica, which contain further references.

2.8 Circular definitions

The paradoxes can also be used to motivate a conclusion that is thevery opposite to Russell’s. Consider the following definition ofa one-place predicate \(G\):

This definition is essentially circular; it is not reducible to one innormal form. Still, intuitively, it provides substantial guidance onthe use of \(G\). The definition dictates, for instance, that Socratesfalls under \(G\), and that nothing apart from the three ancientphilosophers mentioned does so. The definition leaves unsettled thestatus of only two objects, namely, Plato and Aristotle. If we supposethat Plato falls under \(G\), the definition yields that Plato doesfall under \(G\) (since Plato satisfies the definiens), thusconfirming our supposition. The same thing happens if we suppose theopposite, namely, that Plato does not fall under \(G\); again oursupposition is confirmed. With Aristotle, any attempt to decidewhether he falls under \(G\) lands us in an even more precarioussituation: if we suppose that Aristotle falls under \(G\), we are ledto conclude by the definition that he does not fall under \(G\) (sincehe does not satisfy the definiens); and, conversely, if we supposethat he does not fall under \(G\), we are led to conclude that hedoes. But even on Plato and Aristotle, the behavior of \(G\) is notunfamiliar: \(G\) is behaving here in the way the concept of truthbehaves on the Truth Teller (“What I am now saying istrue”) and the Liar (“What I am now saying is nottrue”). More generally, there is a strong parallel between thebehavior of the concept of truth and that of concepts defined bycircular definitions. Both are typically well defined on a range ofcases, and both display a variety of unusual logical behavior on theother cases. Indeed, all the different kinds of perplexing logicalbehavior found with the concept of truth are found also in conceptsdefined by circular definitions. This strong parallelism suggests thatsince truth is manifestly a legitimate concept, so also are conceptsdefined by circular definitions such as (18). The paradoxes, accordingto this viewpoint, cast no doubt on the legitimacy of the concept oftruth. They show only that the logic and semantics of circularconcepts is different from that of non-circular ones. This viewpointis developed in therevision theory of definitions.

In this theory, a circular definition imparts to the defined term ameaning that ishypothetical in character; the semantic valueof the defined term is arule of revision, not as withnon-circular definitions, arule of application. Consider(18) again. Like any definition, (18) fixes the interpretation of thedefiniendumif the interpretations of the non-logicalconstants in the definiens are given. The problem with (18) is thatthe defined term \(G\) occurs in the definiens. But suppose that wearbitrarily assign to \(G\) an interpretation—say we let it bethe set \(U\) of all objects in the universe of discourse (i.e., wesuppose that \(U\) is the set of objects that satisfy \(G)\). Then itis easy to see that the definiens is true precisely of Socrates andPlato. The definition thus dictates that, under our hypothesis, theinterpretation of \(G\) should be the set \(\{ \text{Socrates},\text{Plato}\}\). A similar calculation can be carried out for anyhypothesis about the interpretation of \(G\). For example, if thehypothesis is \(\{\text{Xenocrates}\}\), the definition yields theresult \(\{\text{Socrates}, \text{Aristotle}\}\). In short, eventhough (18) does not fix sharply what objects fall under \(G\), itdoes yield a rule or function that, when given a hypotheticalinterpretation as an input, yields another one as an output. Thefundamental idea of the revision theory is to view this rule as arevision rule: the output interpretation is better than theinput one (or it is at least as good; this qualification will be takenas read). The semantic value that the definition confers on thedefined term is not an extension—a demarcation of the universeof discourse into objects that fall under the defined term, and thosethat do not. The semantic value is a revision rule.

The revision rule explains the behavior, both ordinary andextraordinary, of a circular concept. Let \(\delta\) be the revisionrule yielded by a definition, and let \(V\) be an arbitraryhypothetical interpretation of the defined term. We can attempt toimprove our hypothesis \(V\) by repeated applications of the rule\(\delta\). The resulting sequence,

is arevision sequencefor \(\delta\). The totalityof revision sequences for \(\delta\), for all possible initialhypotheses, is therevision process generated by \(\delta\).For example, the revision rule for (18) generates a revision processthat consists of the following revision sequences, among others:

Observe the behavior of our four ancient philosophers in this process.After some initial stages of revision, Socrates always falls in therevised interpretations, and Xenocrates always falls outside. (In thisparticular example, the behavior of the two is fixed after the initialstage; in other cases, it may take many stages of revision before thestatus of an object becomes settled.) The revision process yields acategorical verdict on the two philosophers: Socratescategorically falls under \(G\), and Xenocrates categorically fallsoutside \(G\). Objects on which the process does not yield acategorical verdict are said to bepathological (relativeto the revision rule, the definition, or the defined concept). Inour example, Plato and Aristotle are pathological relative to (18).The status of Aristotle is not stable in any revision sequence. It isas if the revision process cannot make up its mind about him.Sometimes Aristotle is ruled as falling under \(G\), and then theprocess reverses itself and declares that he does not fall under\(G\), and then the process reverses itself again. When an objectbehaves in this way in all revision sequences, it is said to beparadoxical. Plato is also pathological relative to \(G\),but his behavior in the revision process is different. Plato acquiresa stable status in each revision sequence, but the status he acquiresdepends upon the initial hypothesis.

Revision processes help provide a semantics for circular definitions.^[14] They can be used to define semantic notions such as“categorical truth” and logical notions such as“validity.” The characteristics of the logical notions weobtain depend crucially on one aspect of revision: the number ofstages before objects settle down to their regular behavior in therevision process. A definition is said to befinite iff,roughly, its revision process necessarily requires only finitely manysuch stages.^[15] For finite definitions, there is a simple logical calculus,\(\mathbf{C}_{0}\), that is sound and complete for the revision semantics.^[16] With non-finite definitions, the revision process extends into the transfinite.^[17] And these definitions can add considerable expressive power to thelanguage. (When added to first-order arithmetic, these definitionsrender all \(\Pi^{1}_{2}\) sets of natural numbers definable.) Becauseof the expressive power, the general notion of validity for non-finitecircular definitions is not axiomatizable (Kremer 1993). We can giveat best a sound logical calculus, but not a complete one. Thesituation is analogous to that with second-order logic.

Let us observe some general features of the revision theory ofdefinitions. (i) Under this theory, the logic and semantics ofnon-circular definitions—i.e., definitions in normalform—remain the same as in the traditional account. Theintroduction and elimination rules hold unrestrictedly, and revisionstages are dispensable. The deviations from the traditional accountoccur only over circular definitions. (ii) Under the theory, circulardefinitions do not disturb the logic of the ground language. Sentencescontaining defined terms are subject to the same logical laws assentences of the ground language. (iii) Conservativeness holds. Nodefinition, no matter how vicious the circularity in it, entailsanything new in the ground language. Even the utterly paradoxicaldefinition

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

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

The principal motivation for revision theory is descriptive. It hasbeen argued that the theory helps us to understand better our ordinaryconcepts such as truth, necessity, and rational choice. The ordinaryas 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 explicativedefinitions that they be non-circular.

For more detailed treatments of these topics, see Gupta 1988/89, Guptaand Belnap 1993, and Chapuis and Gupta 1999. See also the entry on therevision theory of truth. For critical discussions of the revision theory, see the papers byVann McGee and Donald A. Martin, and the reply by Gupta, in Villanueva1997. See also Shapiro 2006.

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Acknowledgments

The authors would like to thank Edward Zalta and any anonymous editor forhelpful suggestions for improving this entry.