1. How many dictionaries exist? Hundreds if not thousands. It is inaccurate to use phrases like, "The dictionary defines it as...," of "The English definition of concentration is...." This fails to specify which dictionary and address the fact that individual dictionars usually contain multiple defintions. There's nothing definitive about dictionaries. Dictionaries are just glorified thesaruses full of static, self-reflexive synonyms. Meaning, on the other hand, is a pragmatic, dynamic, ongoing process (which dictionarieas can sometimes aid but usually interfere with). Refer again to the WIttgenstein quote.

If you're still confused, read this entry, continued in my next post, from the Stanford Encyclopedia of Philosophy called, "Definitions."

Definitions

Definitions have interested philosophers since ancient times. Plato's early dialogues portray Socrates raising questions about definitions (e.g., in the Euthyphro, “What is piety?”)—questions that seem at once profound and elusive. The key step in Anselm's “Ontological Proof” for the existence of God is the definition of “God,” and the same holds of Descartes's version of the argument in his Meditation V. More recently, the Frege-Russell definition of number and Tarski's definition of truth have exercised a formative influence on a wide range of contemporary philosophical debates. In all these cases—and many others can be cited—not only have particular definitions been debated; the nature of, and demands on, definitions have also been debated. Some of these debates can be settled by making requisite distinctions, for definitions are not all of one kind: definitions serve a variety of functions, and their general character varies with function. Some other debates, however, are not so easily settled, as they involve contentious philosophical ideas such as essence, concept, and meaning.

1. Some varieties of definition

1.1 Real and nominal definitions

1.2 Dictionary definitions and ostensive definitions

1.3 Stipulative definitions

1.4 Descriptive definitions

1.5 Explicative definitions

1.6 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 Implicit definitions

2.6 Vicious-Circle Principle

2.7 Circular definitions

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1. Some varieties of definition

Ordinary discourse recognizes several different kinds of things as possible objects of definition, and it recognizes several kinds of activity as defining a thing. To give a few examples, we speak of a commission as defining the boundary between two nations; of the Supreme Court as defining, through its rulings, “person” and “citizen”; of a chemist as discovering the definition of gold, and the lexicographer, that of ‘cool’; of a participant in a debate as defining the point at issue; and of a mathematician 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 boundary commission may aim to achieve precision; the Supreme Court, fairness; the chemist and the lexicographer, accuracy; the debater, clarity; and the mathematician, fecundity. The standards by which definitions are judged are thus liable to vary from case to case. The different definitions can perhaps be subsumed under the Aristotelian formula that a definition gives the essence of a thing. But this only highlights 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 often in 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 distinctive role: that of solving epistemological problems. For example, the epistemological status of mathematical truths raises a problem. Immanuel Kant thought that these truths are synthetic a priori, and to account for their status, he offered a theory of space and time—namely, of space and time as forms of, respectively, outer and inner sense. Gottlob Frege and Bertrand Russell sought to undermine Kant's theory by arguing that arithmetical truths are analytic. More precisely, they attempted to construct a derivation of arithmetical principles from definitions of arithmetical concepts, using only logical laws. For the Frege-Russell project to succeed, the definitions used must have a special character. They must be conceptual or explicative of meaning; they cannot be synthetic. It is this kind of definition that has aroused, over the past century or so, the most interest and the most controversy. And it is this kind of definition that will be our primary concern. Let us begin by marking some preliminary but important distinctions.

1.1 Real and nominal definitions

John Locke distinguished, in his Essay, “real essence” from “nominal essence.” Nominal essence, according to Locke, is the “abstract Idea to which the Name is annexed (III.vi.2).” Thus, the nominal essence of the name ‘gold’, Locke said, “is that complex Idea the word Gold stands for, let it be, for instance, a Body yellow, of a certain weight, malleable, fusible, and fixed.” In contrast, the real essence of gold is “the constitution of the insensible parts of that Body, on which those Qualities [mentioned in the nominal essence] and all other Properties of Gold depend (III.vi.2).” A rough way of marking the distinction between real and nominal definitions is to say, following Locke, that the former states real essence, while the latter states nominal essence. The chemist aims at real definition, whereas the lexicographer aims at nominal definition.

This characterization of the distinction is rough because a zoologist's definition of “tiger” should count as a real definition, even though it may fail to provide “the constitution of the insensible parts” of the tiger. Moreover, an account of the meaning of a word should count as a nominal definition, even though it may not take the Lockean form of setting out “the abstract idea to which the name is annexed.” Perhaps it is helpful to indicate the distinction between real and nominal definitions thus: to discover the real definition of a term X one needs to investigate the thing or things denoted by X; to discover the nominal definition, one needs to investigate the meaning and use of X. Whether the search for an answer to the Socratic question “What is virtue?” is a search for real definition or one for nominal definition depends upon one's conception of this particular philosophical activity. When we pursue the Socratic question, 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 ideal that is to some extent independent of these uses? Under the former conception, we are aiming at a nominal definition; under the latter, at a real definition.

For a critical discussion of the different activities that have been subsumed under “real definition,” see Robinson 1950.

1.2 Dictionary definitions and ostensive definitions

Nominal definitions—definitions that explain the meaning of a term—are not all of one kind. A dictionary explains the meaning of a term, in one sense of this phrase. Dictionaries aim to provide definitions that contain sufficient information to impart an understanding of the term. It is a fact about us language users that we somehow come to understand and use a potential infinity of sentences containing a term once we are given a certain small amount of information about the term. Exactly how this happens is a large mystery. But it does happen, and dictionaries exploit the fact. Note that dictionary entries are not unique. Different dictionaries can give different bits of information and yet be equally effective in explaining the meanings of terms.

Something similar occurs with ostensive definitions. We can teach a boy a term, say ‘meter’, by giving the ostensive definition “This stick is one meter long,” while showing the boy a meter stick. There is a mystery here, too. Before the definition is given, the boy does not understand the sentence ‘This stick is one meter long’. Yet, by using the sentence in a particular way, the boy is brought to understand one of the constituents of the sentence and the sentence itself. How can a sentence that lacks meaning for the boy impart to him an understanding of a term and, at the same time, an understanding of itself? Ostensive definitions look simple but, as Ludwig Wittgenstein observed, they are effective only because a complex linguistic and conceptual capacity is operative in the background. It is not easy to provide an account of this capacity.

Definitions sought by philosophers are of neither of these kinds, dictionary or ostensive. Frege's definition of number (1884) and Alfred Tarski's definition of truth (1983, ch. 8) are not offered as candidates for dictionary entries. When an epistemologist seeks a definition of “knowledge,” she is not seeking a good way of teaching young children the word ‘know’. The philosophical quest for definition can sometimes fruitfully be characterized as a search for an explanation of meaning. But the sense of ‘explanation of meaning’ here is very different from the sense in which a dictionary or an ostensive definition explains the meaning of a word.

1.3 Stipulative definitions

A stipulative definition imparts a meaning to the defined term, and involves no commitment that the assigned meaning agrees with prior uses (if any) of the term. Stipulative definitions are epistemologically special. One has a right to stipulatively define terms as one sees fit; the constraints here are practical, not epistemological. Yet, stipulative definitions yield judgments with epistemological characteristics that are puzzling elsewhere. If one stipulatively defines a “raimex” as, say, a rational, imaginative, experiencing being then the judgment “raimexes are rational” is assured of being necessary, certain, and a priori. Philosophers have found it tempting to explain the puzzling cases of, e.g., aprioricity by an appeal to stipulative definitions.

Saul Kripke (1980) has drawn attention to a special kind of stipulative 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 a stipulation, Kripke pointed out, the description serves only to fix the reference of the new name; the name is not synonymous with the description. For, the judgment

(1) Jack the Ripper is the man who murdered X, Y, and Z, if a unique man committed the murders

is contingent, even though the judgment

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

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

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

1.4 Descriptive definitions

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

It is useful to distinguish three grades of descriptive adequacy of a definition: extensional, intensional, and sense. A definition is extensionally adequate iff there are no actual counterexamples to it; it is intensionally adequate iff there are no possible counterexamples to it; and it is sense adequate (or analytic) iff it endows the defined term with the right sense. (The last grade of adequacy itself subdivides into different notions, for “sense” can be spelled out in several different ways.) The definition “Water is H2O,” for example, is intensionally adequate because the identity of water and H2O is necessary (assuming the Kripke-Putnam view about the rigidity of natural-kind terms); the definition is therefore extensionally adequate also. But it is not sense-adequate, for the sense of ‘water’ is not at all the same as that of ‘H2O’. The definition ‘George Washington is the first President of the United States’ is adequate only extensionally but not in the other two grades, while ‘man is a laughing animal’ fails to be adequate in all three grades. When definitions are put to an epistemological use, intensional adequacy is generally insufficient. For such definitions cannot underwrite the rationality or the aprioricity of a problematic subject matter.

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

1.5 Explicative definitions

Sometimes a definition is offered neither descriptively nor stipulatively but as, what Rudolf Carnap (1956, §2) called, an explication. An explication aims to respect some central uses of a term but is stipulative on others. The explication may be offered as an absolute improvement of an existing, imperfect concept. Or, it may be offered as a “good thing to mean” by the term in a specific context for a particular purpose. (The quoted phrase is due to Alan Ross Anderson; see Belnap 1993, 117.)

A simple illustration of explication is provided by the definition of ordered pair in set theory. Here, the pair <x, y> is defined as the set {{x}, {x, y}}. Viewed as an explication, this definition does not purport to capture all aspects of the antecedent uses of ‘ordered pair’ in mathematics (and in ordinary life); instead, it aims to capture the essential uses. The essential fact about our use of ‘ordered pair’ is that it is governed by the principle that pairs are identical iff their respective components are identical:

<x, y> = <u, v> iff x = u & y = v.

And it can be verified that the above definition satisfies the principle. The definition does have some consequences that do not accord with the ordinary notion. For example, the definition implies that an object x is a member of a member of the pair <x, y>, and this implication is no part of the ordinary notion. But the mismatch is not an objection to the explication. What is important for explication is not antecedent meaning but function. So long as the latter is preserved, the former can be let go. It is this feature of explication that led W. V. O. Quine (1960, §53) to extol its virtues and to uphold the definition of “ordered pair” as a philosophical paradigm.

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

1.6 A remark

The kinds into which we have sorted definitions are not mutually exclusive. A dictionary may offer ostensive definitions of some words (e.g., of color words), and a stipulative definition of a term may, as it happens, be extensionally adequate to the antecedent uses of the term. Furthermore, as we shall see below, there are other kinds of definition than those considered so far.

2. The logic of definitions

Many definitions—stipulative, descriptive, and explicative—can be analyzed into three elements: the term that is defined (X), an expression containing the defined term (… X …), and another expression (- - - - - - - ) that is equated by the definition with this expression. Such definitions can be represented thus:

(2) X: … X … =Df - - - - - - - .

When the defined term is clear from the context, the representation may be simplified to

… X … =Df - - - - - - - .

The expression on the left-hand side of ‘=Df’ (i.e., … X …) is the definiendum of the definition, and the expression on the right-hand side is its definiens—it being assumed that the definiendum and the definiens belong to the same logical category. Note the distinction between defined term and definiendum: the defined term in the present example is X; the definiendum is the unspecified expression on the left-hand side of ‘=Df’, which may or may not be identical to X. (Some authors call the defined term ‘the definiendum’; some others use the expression confusedly, sometimes to refer to the defined term and sometimes to the definiendum proper.) Not all definitions found in the logical and philosophical literature fit under scheme (2); partial definitions, for example, fall outside the scheme. However, definitions that conform to (2) are the most important, and they will be our primary concern.

Let us focus on stipulative definitions and reflect on their logic. Some of the important lessons here, we shall see, carry over to descriptive and explicative definitions. For simplicity, let us consider the case where a single definition stipulatively introduces a term. (Multiple definitions bring notational complexity but raise no new conceptual issues.) Suppose, then, that a language L, the ground language, is expanded through the addition of a new term X to an expanded language L+, where X is stipulatively defined by a definition D of form (2). What logical rules govern D? What requirements must the definition fulfill?

Before we address these questions, let us take note of a distinction that is not marked in logic books but which is useful in thinking about definitions. In one kind of definition—call it homogeneous definition—the defined term and the definiendum belong to the same logical category. So, a singular term is defined via a singular term; a general term via a general term; a sentence via a sentence; and so on. Let us say that a homogenous definition is regular iff its definiendum is identical to the defined term. Here are some examples of regular homogeneous definitions:

(3)

1: 1 =Df the successor of 0,

man: man =Df rational animal,

The True: The True =Df everything is identical to itself.

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

It is sometimes said that definitions are mere recipes for abbreviations. Thus, Alfred North Whitehead and Bertrand Russell say of definitions—in particular, those used in Principia Mathematica—that they are “strictly speaking, typographical conveniences (1925, 11).” This viewpoint has plausibility only for regular homogeneous definitions—though it is not really tenable even here. (Whitehead and Russell's own observations make it plain that their definitions are more than mere “typographical conveniences.”[2]) The idea that definitions are mere abbreviations is not at all plausible for the second kind of definition, to which let us now turn.

In the second kind of definition—call it a heterogenous definition—the defined term and the definiendum belong to different logical categories. So, for example, a general term (e.g., ‘man’) may be defined using a sentential definiendum (e.g., ‘x is a man’). For another example, a singular term (e.g., ‘1’) may be defined using a predicate (e.g., ‘is identical to 1’). Heterogeneous definitions are far more common than homogenous ones. In familiar first-order languages, for instance, it is pointless to define, 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 to be atomic. In a heterogeneous definition, however, the definiens can easily be complex; for example,

(4) Gx =Df x > 3 & x < 10.

If the language has a device for abstraction—e.g., for forming sets—we could give a different sort of heterogenous definition of G:

(5) the set of G's =Df the set of numbers between 3 and 10.

Observe that a heterogenous definition such as (4) is not a mere abbreviation. For, if it were, the expression x in it would not be a genuine variable, and the definition would provide no guidance on the role of G in contexts other than Gx. Moreover, if such definitions were abbreviations, they would be subject to the requirement that the definiendum must be shorter than the definiens, but no such requirement exists. On the other hand, genuine requirements on definitions would make little sense. The following stipulation is not a legitimate definition:

(6) Gx =Df x > y & x < 10.

But if it is viewed as a mere abbreviation, there is nothing illegitimate about it.

Some stipulative definitions are nothing but mere devices of abbreviation (e.g., the definitions governing the omission of parentheses in formulas; see Church 1956, §11). However, many stipulative definitions are not of this kind; they introduce meaningful 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 not a meaningful predicate and expresses no concept of any kind. But what is the source of the difference? Why is (4) legitimate, but not (6)? More generally, when is a definition legitimate? What requirements must the definiens fulfill? And, for that matter, the definiendum? Must the definiendum be, for instance, atomic, as in (3) and (4)? If not, what restrictions (if any) are there on the definiendum?

2.1 Two criteria

Any answer to these questions must respect two intuitive criteria. First, the definition should not enable us to establish essentially new claims—call this the Conservativeness criterion. We should not be able to establish, by means of a mere stipulation, new things about, for example, the moon. It is true that unless this criterion is made precise, it is subject to trivial counterexamples, for the introduction of a definition materially affects some facts. Nonetheless, the criterion can be made precise and defensible, and we shall soon see some ways of doing this.

Second, the definition should fix the use of the defined expression X—call this the Use criterion. This criterion is plausible, since only the definition—and nothing else—is available to guide us in the use of X. There are, however, complications here. What counts as a use of X? Are occurrences within the scope of ‘say’ and ‘know’ included? What about the occurrence of X within quotation contexts, and those within words, for instance, ‘Xenophanes’? The last question should receive, it is clear, the answer, “No.” But the answer to the previous question is not so clear. There is another complication: even if we can somehow separate out genuine occurrences of X, it may be that some of these occurrences are rightfully ignored by the definition. For example, a definition of quotient may leave some occurrences of the term undefined (e.g., where there is division by 0). The orthodox view is to rule such definitions as illegitimate, but the orthodoxy deserves to 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 clearly determined logical structure (e.g., a first-order language) and that contain no occurrences of the defined term X. And let us confine ourselves to definitions that place no restrictions on legitimate occurrences of X. The Use criterion now dictates then that the definition should fix the use of all expressions in the expanded language in which X occurs.

A variant formulation of the Use criterion is this: the definition must fix the meaning of the definiendum. The new formulation is less determinate and more contentious, however, since it relies on “meaning,” an ambiguous and theoretically contentious notion.

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

2.2 Foundations of the traditional account

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

The second idea—the primacy of the sentential—has its roots in the thought that the fundamental uses of a term are in assertion and argument: if we understand the use of a defined term in assertion and argument then we fully grasp the term. The sentential is, however, primary in argument and assertion. Hence, to explain the use of a defined term X, the second idea maintains, it is necessary and sufficient to explain the use of sentential items that contain X. (Sentential items are here understood to include sentences and sentence-like things with free variables, e.g., the definiens of (4); henceforth, these items will be called formulas.) 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 a formula in the ground language. This idea, when conjoined with the primacy of the sentential, leads to a strong version of the Use criterion, called the Eliminability criterion: the definition must reduce each formula containing the defined term to a formula in the ground language, i.e., one free of the defined term. Eliminability is the distinctive thesis of the traditional account and, as we shall see below, it can be challenged.

Note that the traditional account does not require the reduction of all expressions of the extended language; it requires the reduction only of formulas. The definition of a predicate G, for example, need provide no way of reducing G, taken in isolation, to a predicate of the ground language. The traditional account is thus consistent with the thought that a stipulative definition can add a new conceptual resource to the language, for nothing in the ground language expresses the predicative concept that G expresses in the expanded language. This is not to deny that no new proposition—at least in the sense of truth-condition—is expressed in the expanded language.

2.3 Conservativeness and eliminability

Let us now see how Conservativeness and Eliminability can be made precise. First consider languages that have a precise proof system of the familiar sort. Let the ground language L be one such. The proof system of L may be classical, or three-valued, or modal, or relevant, or some other; and it may or may not contain some non-logical axioms. All we assume is that we have available the notions “theorem of L” and “provably equivalent in L,” and also the notions “theorem of L+” and “provably equivalent in L+” that result when the proof system of L is supplemented with D and the logical rules governing definitions. Now, the Conservativeness criterion can be made precise as 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 D is also provable without using D: the definition does not enable us to prove anything new in L. The Eliminability criterion can be made precise thus:

Eliminability criterion (syntactic formulation): For any formula A of L+, there is a formula of L that is provably equivalent in L+ to A.

(Folklore credits the Polish logician S. Leśniewski for formulating the criteria of Conservativeness and Eliminability, but this is a mistake; see Urbaniak and Hämäri 2012 for discussion and further references.)[3]

Now let us equip L with a model-theoretic semantics. That is, we associate with L a class of interpretations, and we make available the notions “valid in L in the interpretation M” (a.k.a.: “true in L in M”) and “semantically equivalent in L relative to M.” Let the notions “valid in L+ in M” and “semantically equivalent in L+ relative to M” result when the semantics of L is supplemented with that of definition D. The criteria of Conservativeness and Eliminability can now be made precise thus:

Conservativeness criterion (semantic formulation): For all formulas A of L and all interpretations M, if A is valid in L+ in M then A is also valid in L in M.

Eliminability criterion (semantic formulation): For any formula A of L+, there is a formula B of L such that, relative to all interpretations M, B is semantically equivalent in L+ to A.

The syntactic and semantic formulations of the two criteria are plainly parallel. However, even if we suppose that strong completeness theorems hold for L and L+, the two formulations are not equivalent. Indeed, several different, non-equivalent formulations of the two criteria are possible within each framework, the syntactic and the semantic.

Observe that the satisfaction of Conservativeness and Eliminability criteria, whether in their semantic or their syntactic formulation, is not an absolute property of a definition; the satisfaction is relative to the ground language. Different ground languages can have associated with them different systems of proof and different classes of interpretations. Hence, a definition may satisfy the two criteria when added to one language, but may fail to do so when added to a different language. For further discussion of the criteria, see Suppes 1957 and Belnap 1993.

2.4 Definitions in normal form

For concreteness, let us fix the ground language L to be a classical first-order language with identity. The proof system of L may contain some non-logical axioms T; the interpretations of L are then the classical models of T. As before, L+ is the expanded language that results when a definition D of a non-logical constant X is added to L; hence, X may be a name, a predicate, or a function-symbol. Call two definitions equivalent iff they yield the same theorems in the expanded language. Then, it can be shown that if D meets the criteria of Conservativeness and Eliminability then D is equivalent to a definition in normal form as specified below.[4] Since definitions in normal form meet the demands of Conservativeness and Eliminability, the traditional account implies that we lose nothing essential if we require definitions to be in normal form.

The normal form of definitions can be specified as follows. The definitions of names a, n-ary predicates H, and n-ary function symbols f must be, respectively, of the following forms:

(7) a = x =Df ψ(x),

(8) H(x1, … , xn) =Df φ(x1, … , xn),

(9) f(x1, … , xn) = y =Df χ(x1, … , xn, y),[5]

where the variables x1, … , xn, y are all distinct, and the definiens in each case satisfies conditions that can be separated into a general and a specific part. The general condition on definiens is the same in each case: it must not contain the defined term or any free variables other than those in the definiendum. The general conditions remain the same when the traditional account of definition is applied to non-classical logics (e.g., to many-valued and modal logics). The specific conditions are more variable. In classical logic, the specific condition on the definiens ψ(x) of (7) is that it satisfy an existence and uniqueness condition: that it be provable that something satisfies ψ(x) and that at most one thing satisfies ψ(x).[6] 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

∃yχ(x1, … , xn, y) &

∀u∀v[χ(x1, … , xn, u) & χ(x1, … , xn, v) → u = v]

must be provable.[7]

In a logic that allows for vacuous names, the specific condition on the definiens of (7) would be weaker: the existence condition would be dropped. In contrast, in a modal logic that requires names to be non-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 same object across possible worlds.

Definitions that conform to (7)-(9) are heterogeneous; the definiendum is sentential, but the defined term is not. One source of the specific conditions on (7) and (9) is their heterogeneity. The specific conditions are needed to ensure that the definiens, though not of the logical category of the defined term, imparts the proper logical behavior to it. The conditions thus ensure that the logic of the expanded language is the same as that of the ground language. This is the reason why the specific conditions on normal forms can vary with the logic of the ground language. Observe that, whatever this logic, no specific conditions are needed for regular homogeneous definitions.

The traditional account makes possible simple logical rules for definitions and also a simple semantics for the expanded language. Suppose definition D has a sentential definiendum. (In classical logic, all definitions can easily be transformed to meet this condition.) Let D be

(10) φ(x1, … , xn) =Df ψ(x1, … , xn),

where x1, … , xn are all the variables free in either φ or ψ. And let φ(t1, … , tn) and ψ(t1, … , tn) result by the simultaneous substitution of terms t1, … , tn for x1, … , xn in, respectively, φ(x1, … , xn) and ψ(x1, … , xn); changing bound variables as necessary. Then the rules of inference governing D are simply these:

Definiendum Elimination Definiendum Introduction

φ(t1, … , tn)

ψ(t1, … , tn)

ψ(t1, … , tn)

φ(t1, … , tn)

The semantics for the extended language is also straightforward. Suppose, for instance, 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 unique classical interpretation M+ of the extended language L+. The denotation of a in M+ is the unique object that satisfies ψ(x) in M; the conditions on ψ(x) ensure that such an object exists. The semantics of defined predicates and function-symbols is similar. The logic and semantics of definitions in non-classical logics receive, under the traditional account, a parallel treatment.

Note that the inferential force of adding definition (10) to the language is the same as that of adding as an axiom, the universal closure of

(11) φ(x1, … , xn) ↔ψ(x1, … , xn).

However, this similarity in the logical behavior of (10) and (11) should not obscure the great differences between the biconditional (‘↔’) and definitional equivalence (‘=Df’). The former is a sentential connective, but the latter is trans-categorical: not only formulas, but also predicates, names, and items of other logical categories can occur on the two sides of ‘=Df’. Moreover, the biconditional can be iterated—e.g., ((φ ↔ ψ) ↔ χ); not so for definitional equivalence. Finally, a term can be introduced by a stipulative definition into a ground language whose logical resources are confined, say, to classical conjunction and disjunction. This is perfectly feasible, even though the biconditional is not expressible in the language. In such cases, the inferential role of the stipulative definition is not mirrored by any formula of the extended language.

The traditional account of definitions should not be viewed as requiring definitions to be in normal form. The only requirements that it imposes are (i) that the definiendum contain the defined term; (ii) that the definiendum and the definiens belong to the same logical category; and (iii) the definition satisfies Conservativeness and Eliminability. So long as these requirements are met, there are no further restrictions. The definiendum, like the definiens, can be complex; and the definiens, like the definiendum, can contain the defined term. So, for example, there is nothing formally wrong if the definition of the functional expression ‘the number of’ has as its definiendum the formula ‘the number of Fs is the number of Gs’. The role of normal forms is only to provide an easy way of ensuring that definitions satisfy Conservativeness and Eliminability; they do not provide the only legitimate format for stipulatively introducing a term. Thus, the reason why (4) is, but (6) is not, a legitimate definition is not that (4) is in normal form and (6) is not.

(4) Gx =Df x > 3 & x < 10.

(6) Gx =Df x > y & x < 10.

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:

(12) Gx =Df (x > 3 & x < 10) & y = y.

(13) Gx =Df [x = 0 & (G0 v G1)] v [x = 1 & (~G0 & ~G1)].

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

(14) Gx =Df x = 0.

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

Similarly, the traditional account is perfectly compatible with recursive (a.k.a.: inductive) definitions such as those found in logic and mathematics. In Peano Arithmetic, for example, exponentiation can be defined by means of the following equations:

(15)

m0 = 1,

mn + 1 = mn · m.

Here the first equation—called the base clause—defines the value of the function when the exponent is 0. And the second clause—called the recursive clause—uses the value of the function when the exponent is n to define the value when the exponent is n + 1. This is perfectly legitimate, according to the traditional account, because a theorem of Peano Arithmetic establishes that the above definition is equivalent to one in normal form.[8] Recursive definitions are circular in their format, and indeed it is this circularity that renders them perspicuous. But the circularity is entirely on the surface, as the existence of normal forms shows. See the discussion of circular definitions below.