TABLE OF CONTENTS
Chart: Three Kinds of Disputes
Chart: A Definition has Two Components
>> Theory of Definition: Arthur Pap
Chart: The Nature of Definitions: Arthur Pap
>> Words: Handbook of Logic: E. R. Emmet
>> Definition: Semantics and Necessary Truth: Arthur Pap
>> Quotes
>> Definitions: Logic: Wesley C. Salmon
>> Explication of Words: Essays on the Intellectual Powers of Man: Thomas Reid
>> Definition and Division: Principles of Logic: George Hayward Joyce
>> Definition, Division, Classification and Analysis: General Logic: Ralph M. Eaton
>> Avoiding Ambiguity: The Elements of Logic: Stephen F. Barker
>> Classification and Definition: An Introduction to Logic and Scientific Method:
>> Morris R. Cohen; Ernest Nagel
>> Summary of Chapter 3: Introduction to Logic: Irving M. Copi
>> Glossary: Elements of Analytic Philosophy: Arthur Pap
>> Definition: Wikipedia
>> Definitions: Anil Gupta
>> More Quotes
>> References

Definition  and Meaning are two different facets of a term.

THEORY OF DEFINITION*
ARTHUR PAP

Definitions can be classified from (at least) two different points of view. We can ask what sort of statements definitions are, how they are to be justified, and what purpose they serve in the process of acquiring scientific knowledge. For lack of a simpler word, let us call a classification of definitions from this point of view epistemological. We can also distinguish different forms of definition; and a classification from this point of view is naturally called formal.

Epistemological classification. The question is often raised and discussed whether a definition can be true or false, or whether it is just an arbitrary stipulation to use a word in a certain way. The obvious answer is that some of the statements that are, in everyday life, and in science, called “definitions” are merely stipulative and others are not. By just looking at the sequence of words, however, one cannot tell whether one is confronted with a stipulation or with a proposition, i.e., something that can be called true or false. For example: “A spinster is an unmarried woman older than 25.” This would be a stipulative definition if it amounted to the proposal, “Let us use the word ‘spinster’ as an abbreviation for ‘unmarried woman older than 25’.” One can accept or reject a proposal; but since to make a proposal is not to assert anything, the question of truth or falsehood is inappropriate. But the same statement may be meant as a report of the actual usage of the word “spinster”: English-speaking people apply the word “spinster” to women of the described sort and to no other objects. In that case the definition is a proposition, and then it is appropriate to ask whether it is true or false.

The first distinction, then, is that between (linguistic) proposals and propositions. Propositional definitions, in turn, can be classified from two important points of view: they may be empirical propositions, or they may be analytic propositions. And they may be about words (verbal usage) or about objects referred to by words, or they may analyze concepts expressed by words. An empirical proposition is a proposition whose truth or falsehood can only be determined by experience (in the broadest sense of “experience”). And even if there are good reasons for accepting it as true, it remains logically conceivable (i.e., does not involve self-contradiction to suppose) that it be false. An analytic proposition, on the other hand, is arrived at by analysis of what one means by the words used. Thus we would not allow that “All mothers are women” could ever be refuted; one may, of course, change the ordinary meanings of the words, but that would be different from finding the proposition now expressed by these words to be false.

Following Copi,¹ we call a definition which is an empirical proposition about verbal usage lexical. But we split Copi’s category of “theoretical” definitions into theoretical in the sense of empirical propositions about scientific objects, and analyses of concepts. To see the difference, compare “Water is a substance composed of molecules consisting of two hydrogen atoms and one oxygen atom (H₂O)” with “A circle is a closed line any two points on which have the same distance from a given point.” The former statement must be justified by reference to experimental results interpreted by a scientific theory (atomic theory of matter). The latter statement, however, expresses a precise analysis of the property connoted by the word “circle.” I can get a person who has learnt the use of the word “circle” by ostensive definition, i.e., by being conditioned to apply the word “circle” to closed lines of a certain shape and only such lines, to formulate that analysis by just inviting him to reflect on what distinguishes a circle from an ellipse, a square, and other closed lines of regular character. But the cited definition of water could not be arrived at in this way; it expresses the empirical generalization that anything which has the qualitative properties connoted by “water” as the term is used in daily life also has that chemical structure, and conversely.

It is easy to confuse a lexical definition with an analysis because one tends to confuse the use of a word with its mention. When I say, “John is a tall boy,” I use the name “John” to talk about a boy; it is therefore inconsistent to write “John is a tall boy” and also “John is a name,” for the same thing cannot be both a boy and a name. The correct way of writing would be: “‘John’ is a name,” the inner quotes serving to name a name. Now, consider the definition “An uncle is a man who has the same parents as some other person who is a parent.” If it is a lexical definition, then it is a statement about the English word “uncle”; it then asserts that what English-speaking people intend to say about a person x when they say “x is an uncle” is that x is a man who has the same parents as some other person who is a parent. But if it is an analysis, then it is a statement about the property connoted by the word “uncle”: it says that the property of being an uncle is the property of being a man having the same parents as some other person who is a parent. If the relevant rules of the English language changed, say, if “uncle” came to be used in the sense in which “cousin” is now used, the dictionary definition would have to be changed, but the analysis would still be correct if it ever was, for the kinship relation of unclehood does not change when its English name changes. Further, a Frenchman who asserts that “Un oncle est un homme qui a les meme parents que quelque autre personne qui est un parent” makes (provided your instructor’s French translation is correct) precisely the same assertion as the American makes by the words “An uncle is a man who has the same parents as some other person who is a parent;” the American and the Frenchman, in other words, assert the same proposition by means of different sentences. But if the American had made an assertion about the way people in America and Britain use the word “uncle,” and the Frenchman about the way people in France use the word “oncle,” they obviously would have asserted different propositions (in fact, it would be conceivable that one were true and the other false, since it is conceivable that “oncle” might not be the French synonym for “uncle”).

The line between propositional and stipulative definitions is not always perfectly sharp. What Copi calls a precising definition of a vague term cuts across the line, for it is partly propositional and partly stipulative. Suppose, for example, you were to define “wealthy American” as meaning “American whose annual income exceeds $15,000.” This definition can claim to be true in the sense that a great many Americans who are commonly referred to as wealthy do satisfy the proposed definition (i.e., have an annual income exceeding $15,000), and a great many who are commonly referred to as “not wealthy” do not satisfy the definiens. But to say that the defined term is, prior to the precising definition, vague, just means that there are borderline cases with respect to it, i.e., persons who would not uniformly be called “wealthy” and would not uniformly be called “not wealthy” either. The precising definition then amounts to the decision to allot these borderline cases to the extension of “wealthy” or to the extension of “not wealthy.”

Analytic definitions of concepts can give rise to analytic statements. Thus the analytic definition of “uncle” above gives rise to the analytic statement “All uncles are men”; the latter statement may be said to be true by definition but it is not itself a definition. An analytic statement is true by definition in the sense that with the help of a correct definition, i.e., one expressing the meaning with which the defined term is actually used, it is transformable into a logically true statement; and a logically true statement is one which can be seen to be true just by virtue of its form, i.e., the meanings of logical constants, such particles as “all,” “some,” “which,” “or.” To say that all uncles are men, is to say that all men who have the same parents as some other person who is a parent, are men. This statement has the form “All A which are B, are A,” and anybody who understands the logical constants “all,” “which,” “are,” can see that such a statement is true no matter what terms be substituted for the schematic letters “A” and “B ” (provided, of course, that terms are used univocally).

Formal classification. Copi distinguishes definition by example (including ostensive definition as a special case) from connotative definition, i.e., definition specifying the conventional connotation (criterion of application) of a term. But the latter kind of definition can have several forms; it is not restricted to what Copi calls “synonymous” definition and definition “by genus and difference.” One important formal distinction is that between explicit and contextual definition. An explicit definition equates the definiendum with the definiens in such a way that one may be replaced by the other in any context without changing the remainder of the sentence. Thus “A father is a male parent” is an explicit definition, by virtue of which the sentence “My father is poor” may be transformed into the synonymous sentence “My male parent is poor.” Similarly, “A brother is a male sibling” is an explicit definition. These definitions also happen to have genus-difference form, but it will be shown presently that an explicit definition need not have that form.

Now, suppose you were asked to define “brother” in terms of “male” and “parent” (and whatever logical constants may be needed). You could not construct a synonym which could replace “brother” in the sentence “Bill is John’s brother” or “John has no brother.” It is true that “brother” might be equated with “human male who has the same parents as some other human,” but if you were to substitute this expression for “brother” in the sentence “Bill is John’s brother” you would obtain a pretty unintelligible sentence: “Bill is John’s human male who has the same parent as some other human”!

A contextual definition is so called because it is a definition of a term in the context of a sentence (more exactly, statement-form) that contains it. Thus a contextual definition of “brother of”² in terms of “male” and “parent of” looks as follows: x is brother of y = x is a human male distinct from y and the parents of x are the parents of y. In order to apply this definition to the above sentences we must translate the sentences in their entirety; we cannot simply lift the term “brother” out of them and replace it by a synonym: “Bill is John’s brother” (i.e., “Bill is brother of John”) becomes “Bill is a human male distinct from John and the parents of Bill are the parents of John”; similarly “John has no brothers” becomes “There is no human male distinct from John whose parents are the parents of John.”

As our example suggests, contextual definition is appropriate especially for terms connoting a relationship. In general, terms that have no meaning whatever in isolation but only in the context of entire statements (“syncategorematic” terms) can be defined only contextually. To explain what “all” means is to explain what a statement of the form “All A are B” means, to explain what “or” means is to explain what a statement of the form “p or q” (where the letters “p” and “q” represent statements) means, to unfold the ambiguity of “is” is to explain how such statements as “This man is the criminal we were looking for” (identity), “That man is strong” (predication), “There is a cat on the couch” (existence) differ in meaning. Contextual definition of “all”: all A are B = there are no A that are not B. Contextual definition of the exclusive sense of “or”: p or q = not-(not-p and not-q) and not-(p and q).

The following kinds of explicit definition should be distinguished: genus-difference, disjunctive, and quantitative. The word “sibling” may be disjunctively defined as “brother or sister” (provided you don’t define “brother” as “male sibling” and “sister” as “female sibling”!),³ “spouse” as “husband or wife.” This procedure amounts to explaining the connotation of a generic term by enumerating the species that make up the genus. It is a legitimate way of explaining the meaning of an unfamiliar word by means of familiar words, but should not be confused with analysis. Thus one would hardly be giving an analysis of the concept “animal” if one were to enumerate the different species of animals: an animal is either a lion or a mouse or a dog etc. etc. An example of a quantitative explicit definition: the momentum of a body is the product of its mass times its velocity. What is defined here is a term designating a magnitude (measurable property), not a class of objects; therefore the terminology of genus, species, difference, and of extension and intension, is not applicable here. Momentum is not a species of velocity, the way lions are a species of animals. Similarly, the definitions “x³ = x•x•x”, “2 = 1 + 1”, “i = √-1” are explicit, but not of genus-difference nor of disjunctive form. On the other hand, some definitions of mathematical concepts properly have genus-difference form. Example: a prime number is a number which is divisible only by unity and by itself.

A species of contextual definition which is very important in empirical science is the operational definition. The definiens of such a definition has the form of an implication: if a specified test is performed, then a specified result will be observed. Examples: x is soluble in water = if x is immersed in water then x dissolves; x is magnetic = if a small iron body is placed near x, then it will move towards x; x is revengeful = if x has been hurt, then x thirsts for revenge; x is forgiving = if x has been hurt, then x does not hate the person who hurt him (at least not more than before he got hurt). Concepts which are operationally defined as illustrated are often called disposition concepts. To ascribe a disposition to an object is to predict how it would react to a specific kind of stimulation under specific circumstances.

One more form of definition, which is used especially in mathematics and formal logic, should be mentioned: recursive definition. Thus arithmetical addition can be recursively defined as follows:
(x+y´) = (x+y)´ and (x+0) = x. Here “y´” means “the number which is the immediate successor of y”; the notions of successor and zero are undefined but are used to define (recursively) “plus.” By applying this definition to an expression of the form (x+y), one can eliminate the symbol of addition in a finite number of steps. Thus “2 + 3” can be brought into that form by replacing “3” by its definiens “2´.” The step by step elimination of “plus” then proceeds as follows:
2 + 2´ = (2 + 2)´ = (2 + 1´)´ = (2 + 1)´´= (2 + 0´)´´= (2 + 0)´´´ = 2´´´.
The latter expression may, looking up the explicit definition of “5”, be replaced by “5” (hence it is incidentally evident that we have just formally proved “2 + 3 = 5”– though such a formal proof does not tell us what we might do with the equation in practical life).

——————————

* Several years before Arthur Pap died, he wrote the present paper for use with his classes in introductory logic. It was not originally intended for publication, but the ideas in it have a theoretic interest which, in our opinion, merits wider circulation. The manuscript appears here as Professor Pap wrote it except for minor changes in form and an alteration of the wording in the third paragraph under “Formal classification”; it was prepared for publication by John T. Wilcox, Assistant Professor of Philosophy, Emory University.

1 The reference is to I. M. COPI, Introduction to Logic, 1st ed. (New York, Macmillan, 1953). -JTW.

2 Don’t confuse the property-term “brother” with the relation-term “brother of.” The former is, as shown above, explicitly definable on the basis of “human male” and “parent” but not the latter. It should be noted that once “brother of” has been defined, it is perfectly legitimate to define “brother” in terms of “brother of”: a brother is a person who is brother of some other person.

3 It is true that in a dictionary you are likely to find “sibling” defined in terms of “brother” and “sister,” and also the latter words in terms of the former. When such circular definitions are condemned, it is because “definition” is understood as an explanation of the meaning of a word by means of words whose meaning is already known by the person who requests the explanation. But the dictionary maker cannot easily predict which are the words already understood and which are the words that prospective users of the dictionary will “look up.” To play it safe, he may define “sibling” in terms of “brother” and “sister” for the benefit of those who don’t know the meaning of “sibling” but know the meanings of the latter words, and also define the latter words in terms of “sibling” for the benefit of those who may happen to know the meanings of “sibling,” “male” and “female” but not the meanings of “brother” and “sister.”

1 The Nature of Definitions

2 Epistemological Classification: 
    3 Stipulative Definition:  e.g. A spinster is an unmarried woman older than 25.
        5 Abbreviatory: (wholly arbitrary): Not an assertion; nTnF.
        6 Precising: (partly arbitrary): Linguistic
    4 Propositional Definition: Propositional; T or F; An Assertion.
                English speaking people apply the word “spinster” to women of the described sort and to no other objects.
        7 Empirical Proposition: T or F, can only be known by experience.
            9 Lexical: (about linguistic usage): (real) Use of a word.
            10 Theoretical: (about objects):
        8 Analysis of a Concept: (of a property or relation): Mention of a Word.

11 Formal Classification:
      12 Definition by Example: (Denotative): Extension - All members of the Class.
            14 Ostensive: Non-verbal: Car owner: all those who own cars.
            15 Not Ostensive: Verbal or Nominal - Naming members of the extension.
     13 General Definition: (Connotative): Intension: Characteristics of owning a car. Description of the qualifications for membership of the class. The qualities and attributes of the object.
            16 Explicit: Father is a male parent.
                  20 Disjunctive: (enumeration of species): sibling=brother or sister.
                  21 By Genus and Difference: a prime number is a number which is divisible only by unity and by itself.
                  22 By Simple Synonym
                  23 Quantitative: Designates a magnitude, measurable properties, the momentum of a body is the  product of its mass times the velocity.
           17 Contextual
                 24 Operational: Disposition: “if … then”
                 25 Not Operational
          18 Recursive
          19 Axiomatic: (inductive):

2
WORDS

We have seen that for thoughts to be communicated they must usually be put into words, and that words almost always help us to think more clearly and effectively.

Words are obviously of very great importance to mankind, but they must be kept in their place; they must be always the servants of our thinking and not its masters. We must remember that it is we, men and women, who constructed the words in the first place and who collectively decide, under the guidance to some extent of schoolmasters, linguists and so forth, how they shall be used. We must beware always of thinking that because a word exists the ‘thing’ for which that word is supposed to stand necessarily exists too.

As John Stuart Mill said:
The tendency has always been strong to believe that whatever receives a name must be an entity or being, having an independent existence of its own: and if no real entity answering to the name could be found, men did not for that reason suppose that none existed, but imagined that it was something peculiarly abstruse and mysterious, too high to be an object of sense.

To take a simple case, it is not necessarily true that because the words ‘unicorn’, ‘centaur’ exist, that there are in nature animals for which the words stand. This seems obvious to us now, but it was not always so.

[…]

Primitive men used to think of words as instruments for the control of objects and they often attributed supernatural power to them. As Ogden and Richards tell us in The Meaning of Meaning:
Every ancient Egyptian had two names– one for the world, and another by which he was known to the supernatural powers. The Abyssinian Christian’s second name, given at baptism, is never to be divulged. The guardian deity of Rome had an incommunicable name, and in parts of ancient Greece the holy names of the gods, to ensure against profanation, were engraved on lead tablets and sunk in the sea.

DEFINITION

CIRCULAR - The definiendum occurs in the definiens, or a part of the definiens is defined in terms of the definiendum.
COORDINATIVE - As used by Reichenbach, interpretation of the terms of a formal deductive system by means of expressions denoting observable objects or processes (e.g. definition of “length” in terms of a standard rod).
ELIMINATIVE - Enables elimination of definiendum from any sentence in which it occurs (“to define a term is to show how one can get along without it”).
EXPLICATIVE - Analysis of the meaning of the definiendum.
EXPLICIT - Definiendum can simply be replaced by definiens in any sentence without changing the remainder of the sentence (e.g. father = male parent).
IMPLICIT - A set of postulates (axioms) is said to implicitly define the primitive terms in it, i.e. it delimits their denotations to objects and relations that satisfy it.
IN USE (CONTEXTUAL) - Rule for translating sentences containing definiendum into synonymous sentences that do not contain it; but while being eliminative, it is not explicit (e.g. x is brother of y=x is male and has the same parents as y).
OSTENSIVE - Explaining the meaning of a term by pointing at, or inducing experience of, instances denoted by it.
RECURSIVE - Rule for eliminating definiendum in a finite number of symbolic transformations from expressions in which it occurs together with constant arguments (e.g. “+” from “3+2”).
Arthur Pap; Semantics and Necessary Truth; 1958; p425ff

QUOTES

A definition is a declaration that a certain newly-introduced symbol or combination of symbols is to mean the same as a certain other combination of symbols of which the meaning is already known.
It is to be observed that a definition is, strictly speaking, no part of the subject in which it occurs. For a definition is concerned wholly with symbols, not with what they symbolize. Moreover, it is not true or false, being an expression of a volition, not a proposition.
Whitehead and Russell; Principia Mathematica; 1927

Words have not got — by natural design as it were— senses of which they are the owners. They are instruments by which men give direction to thoughts, nothing more.
I. A. Richards; Multiple Definition; v34; 1934

Are definitions true or false?
When a proposition which is true or false, […] consists of a definition (definiens) and a definiendum, its parts […] are not, therefore, necessarily true or false. […] definitions (being only parts of such propositions) are not true or false.

In general, we do not speak of names as true or false. Brentano says of definitions that they are composite names:
“A name which is composed of several names and which names all logical parts of a logical whole from the highest genus of its range to its lowest species, is called a definition.”

Accordingly definitions are not propositions. But in so far as only propositions may be called true or false, we cannot speak of true or false definitions.
Paul Weingartner; Basic Questions on Truth; 2000; ch5

In any discussion or interpretation of symbols we need a means of identifying referents. The reply to the question what any word or symbol refers to consists in the substitution of a symbol which can be better understood.
Such substitution is Definition.

Definition-
An explicit definition defines one expression (the definiendum) by means of another (the definiens) which can replace the first wherever it occurs.

A contextual definition supplies a replacement for certain longer expressions in which the definiendum occurs but not an equivalent for that expression itself. (If Xs can be contextually defined in terms of Ys, Xs are sometimes said to be logical constructions out of Ys, and ‘X’ to be an incomplete symbol [=contextually defined expression].)

A recursive definition gives a rule for eliminating the definiendum in a finite number of steps. A set of axioms is sometimes said to give an implicit definition of its primitive [=undefined term] terms.

See ch. 3 §1 for the interdefinability of connectives; ch. 4 §3 for Russell’s contextual definition of definite descriptions; ch. 7 §5 for Tarski’s recursive definition of satisfaction; pp, 103-4 for formal conditions on definitions.
Susan Haack; Philosophy of Logics; 2000; p245

It is the object of Definition to determine the nature or meaning or signification of a thing (taking “thing” in its widest application, i.e. as including, not only outward material objects, but also names, notions, mental states, etc.): in other words, definition is the formal attempt to answer the question, “What is it?”

[…] knowledge of a thing is in great measure knowledge of what the thing is not.
William Leslie Davidson; The Logic of Definition; 1885

A concept introduced through NOMINAL DEFINITION is chosen relatively arbitrarily, and can easily be replaced by another. In a  REAL DEFINITION, however, it is the essence of a concept that is analysed, and this procedure cannot be arbitrary.
Dieter Wunderlich; Foundations of Linguistics; 1979; p167

30. DEFINITIONS

A word is a large class of physical things or events such as ink marks, graphite marks, or sound waves. A particular word is used many times; it has many occurrences. The word “language,” for example, occurs frequently in the preceding section. It is the same word in each of these occurrences, and each occurrence is a physical thing. The word is the class of all such occurrences– oral or written– past, present, or future. Words are not, however, merely collections of physical things or events, for words have meaning. Words are symbols.

The meaning of a word is not a natural attribute which man discovers; meaning is given to a word by people who agree to let it have that meaning. For example, there is no intrinsic characteristic of the word “cat” which makes it refer to feline animals; it does so because English-speaking people have adopted a convention to that effect. This is not intended to suggest that people once sat down at a conference table and formally decided the meanings of words. For the most part, these conventions, like many other conventions, have grown gradually and informally over a long period of time. As language continues to grow and develop, these conventions are still subject to change. The important fact is that other conventions could have been adopted without being false or incorrect. Indeed, there are many different languages– English, German, Russian, etc.– all with different conventions. None of these languages is false and none is “the true language.”

A word has meaning if there is a convention establishing its meaning. Definitions express these conventions in the metalanguage. The convention may have been laid down formally by means of a definition, or it may have grown up informally by way of customary usage. In either case, the definition, as a formulation of a convention, is neither true nor false.

Offering a definition is like making a proposal. One may accept it or reject it, but the proposal, itself, is not true or false. If a young man’s proposal, “Let’s get married,” met the response, “That’s false,” it would be a nonsensical reply. Likewise, there may be good reasons for rejecting a proposal to use a certain word in a certain way, but falsity is not one of them. Nor is truth a reason for accepting such a proposal.

Moreover, when the convention governing the meaning of a word has developed informally, a definition may be offered as an explicit formulation of that convention. Again, the definition is neither true nor false; it is more like a rule than like a statement of fact. Rules, like proposals, can be accepted or rejected, complied with or violated. For example, certain conventions of etiquette have developed informally in our culture. These conventions are formulated in rules such as, “Do not eat peas with your knife.” This rule is neither true nor false, but statements which are true or false can be made about the rule. For instance, it is true to say that the foregoing rule expresses a currently accepted convention of etiquette. Similarly, although a particular definition is neither true nor false, the statement that this definition expresses an accepted convention is either true or false. It is important to realize that the statement about the definition is different from the definition itself.

In most cases, the meaning of a word has two aspects. Consider the word “logician.” In the first place, this word refers to various men such as Aristotle, George Boole, Gottlob Frege, Bertrand Russell, Kurt Godel, W. V. Quine, and many others. These people– that is, all people who are logicians– constitute the extension of the word “logician.” The extension of a word consists of the class of all objects to which that word correctly applies. Extension is one aspect of the meaning of a word. In the second place, there are certain properties which distinguish logicians from all other people and things. A logician is a person who is skilled in logic. In order to qualify as a logician, an object must have the properties of being human and being skillful in logic. The intension of the word “logician” consists of these two properties. The intension of a word consists of the properties a thing must have in order to be in the extension of that word. The extension of a word is the class of things to which the word applies; the intension of a word is the collection of properties which determine the things to which the word applies.

There are many ways of specifying the meanings of words; consequently, there are many different types of definitions. To begin with, we may specify the meaning of a word through its extension, or we may specify its meaning through its intension. There is thus a basic distinction between extensional definitions and intensional definitions.

There are two fundamentally different ways of indicating the extensions of words. First, we may simply point to objects in the extension of the word. To give the meaning of the word “dog,” we can point to a variety of dogs. This method of indicating the extension of the word is called “ostensive definition.” Another method of indicating the extension of a word is to name some of the objects in its extension (if the objects in the extension have proper names). Thus, one can mention examples of the extension of the word “dog” by naming various dogs: Fido, Rover, Spot, Rex, Beauregard, etc. An ostensive definition is a nonverbal extensional definition, for the meaning of the word is given not by using other words to explain its meaning, but by pointing to the actual objects. Naming members of the extension, on the other hand, is verbal extensional definition, for the meaning of the word is explained by the use of other words, the names of the members of the extension.

Whether one gives the extension of a word verbally or nonverbally, it is usually impractical or impossible to indicate every member of the extension. It would be impossible to point out each member of the extension of the word “dog,” because this word applies to dogs as yet unborn. Furthermore, it would be impractical to point out every living dog to show the meaning of the word, because there are so many dogs and they are so widely scattered. Very often, then, extensional definitions consist of indicating, either verbally or nonverbally, some members of the extension, assuming that other members of the extension can be recognized on the basis of their similarity to the examples. This process of definition suffers some imprecision, yet the meanings of many words are effectively conveyed by extensional definitions.

A moment’s reflection should be sufficient to realize that some words must be defined nonverbally. If the meaning of a word could be given only by using other words, then it would be impossible to convey the meaning of any word. Unless some words had their meanings given nonverbally, there would be no words with meanings that could be used to explain the meanings of other words. Imagine finding a Sanskrit dictionary in which every Sanskrit word is defined in terms of other Sanskrit words. You could memorize every definition in that dictionary, but you would not know what any of the words mean, for you would not know to what things these words refer. Something like ostensive definition is necessary to relate some of the words to things; it is not sufficient merely to relate all of the words to each other.

Intensional definitions are verbal. One important type of intensional definition is the explicit definition. An explicit definition consists of giving a word or phrase which means the same as the word being defined. For example,

a] “Mendacious” means “deceitful.”
     “Pentagon” means “five-sided plane figure.”
     “Bachelor” means “unmarried adult male.”

In each case, the word being defined appears on the left; it is called the “definiendum.” The word or phrase on the right does the defining; it is called the “definiens.” The definition itself occurs in the metalanguage as a proposal or rule about the use of words in the object language.

A definition is circular if the definiendum occurs in the definiens. For instance,

b] “Pentagon” means “plane figure having the shape of a pentagon.”
is circular because the word to be defined is used in giving the definition. Such definitions are useless. Definitions can also be circular in a less direct manner. For example, the following three definitions taken together are circular:
 
c] “Mendacity” means “lack of veracity.”
“Veracity” means “absence of prevarication.”
“Prevarication” means “mendacity.”
Three words are defined, but each is defined in terms of the other two. Unless a meaning is independently given for one of the three, none of them achieve meaning from this series of definitions.

Many words, like “dog,” “run,” and “red,” refer to objects, events, or properties. Such words have extensions and intensions. Other words have meaning only as they function in a linguistic context. Words like “if” “unless,” “the,” “only,” “is,” “not,” and “or” do not refer to anything. For instance, there is no such thing as an “unless,” there is no such event as “unlessing,” and there is no such property as being “unless.” Words of this sort have neither intension nor extension; in fact, they have no meaning in isolation. They have purely grammatical functions, and their meanings come from their function in providing structure for the statements in which they occur. We give the meanings or these words by showing how they function in a context. This method of specifying meanings is called “contextual definition.”

Since logic is primarily concerned with form or structure, many of the most important logical words are defined contextually. We have already encountered many such definitions. For example, in discussing categorical statements we had occasion to note that statements of type A, “All F are G,” were equivalent in meaning to statements or the form “Only G are F. “ This is a contextual definition or the word “only.” There is no single word or phrase that is equated in meaning to the word “only”; instead, the context in which the word “only” occurs has the same meaning as a statement that does not contain the word “only” The truth tables, for another example, provide contextual definitions of the truth-functional connectives.

Contextual definitions may be contrasted with explicit definitions on the following grounds. If a word that is explicitly defined occurs in a statement, then we may replace the defined word by its definiens without changing the meaning of the statement.

d] In the statement “Fred Smith is a bachelor,” we may replace the word “bachelor” by the phrase “unmarried adult male,” with the result that the statement “Fred Smith is an unmarried adult male” means the same as the original statement.
By contrast,

e] In the statement “Only mammals are whales,” our contextual definition does not provide any word or phrase with which to replace the word “only.” Our definition does permit us to replace the whole statement in which the word “only” occurs with another statement, “All whales are mammals,” which has the same meaning as the original statement. The meaning of the word “only” is specified by the definition, because the definition enables us to express, without using the word “only,” what was originally expressed with the help of the word “only.”

We have described some of the different types of definitions; now we must discuss some of the purposes definitions are designed to fulfill. Although definitions are not true or false, their adequacy can be judged in terms of their ability to fulfill certain functions.

1. Some definitions are designed to characterize the customary usage of a word. These definitions attempt to make explicit the conventions followed by people who speak the language, or perhaps those who speak it correctly. Definitions given in dictionaries have this function.

When we look up a word in the dictionary to find out what it means, it might be tempting to say that we find the true definition. A dictionary gives a large number of definitions, and these definitions purport to be conventions to which speakers of the language conform. A previous point applies here. For purposes of logical clarity, it is essential to distinguish carefully between definitions and statements about definitions. A definition itself has the force of a proposal to use a certain word with a certain meaning; as such, it is neither true nor false. The statement that a particular definition is the accepted one is a statement about the definition, not the definition itself. This statement is either true or false.

2. Sometimes we define a new word because there is no established way of briefly expressing an important meaning. For example, we might wish to make repeated reference to those months of the year which have fewer than thirty-one days. As a convenient abbreviation, we might coin the word “monette” and define it as “month having fewer than thirty-one days.”

3. A word is vague if there are objects that are neither definitely included in nor definitely excluded from its extension. Definitions often have the purpose of making vague words more precise. For example, the word “rich” is vague. Some people have very little money; they are definitely not rich. Others have millions; they are definitely rich. Some people have quite a lot of money, but they are not fabulously wealthy. Even if we know how much money such a person has, we cannot say whether he is rich or not because of the vagueness of the word “rich.” We might wish to make this word more precise by a definition such as the following: “rich” means “has a fortune of at least half a million dollars.”

4. Sometimes we seek an intensional definition for a word whose extension is quite well known. For instance, we have very little trouble in applying the word “human”; when we encounter an object we can almost always say definitely whether or not it is human. Still, we might have considerable difficulty in saying what properties distinguish humans from nonhumans. The problem is to find an intensional definition which will provide the extension we already accept for the word.

Although we use the word “human” quite adequately in most contexts, its extension is not precisely determined. The case is typical. There are many objects which definitely belong to the extension of the word; there are many other objects which definitely fall outside its extension; and there are some objects which are borderline cases– neither definitely within the extension nor definitely outside the extension. To find an adequate intensional definition of “human” requires that we find a set of properties which are shared by all the objects definitely within the extension of the word but not shared by any of the objects definitely outside its extension. The borderline cases can be dealt with as we see fit.

If an intensional definition is proposed, it must not be too broad or too narrow. The definition will be too broad if it admits into the extension some objects which were definitely outside the extension. The definition will be too narrow if it excludes from the extension objects which were definitely within the extension. Notice that a definition could be both too broad and too narrow. For example, the definition ‘‘‘human’ means ‘rational animal’“ has been proposed. There is reason to suppose that this definition is too broad in some respects and too narrow in others. We would normally regard tiny infants, Mongolian idiots, and insane persons as humans. However, it is doubtful that such beings are rational, so the proposed definition would seem to exclude them from the extension of “human.” Thus, the definition is too narrow. At the same time, certain apes seem to be quite intelligent and capable of elementary reasoning. Such creatures, which are clearly excluded from the extension of “human” as we understand the word, would be included under the proposed definition. In this respect the intensional definition is too broad.

When we have succeeded in framing an intensional definition which is neither too broad nor too narrow, we must still consider how it disposes of the borderline cases. To pursue our previous example, we shall find borderline cases if we ask when an organism becomes human. Does a person first qualify as a human being at the moment of birth? Is an unborn baby a human being? Does a foetus become a human when the mother first “feels life”? Is the fertilized egg a human being from the moment of fertilization? These are not purely academic questions. Questions of the following sorts are involved. Does an unborn child have any legal rights? Can an unborn child inherit money or be the beneficiary of a life insurance policy? Is abortion murder? (By definition, it is impossible to murder anything that is not human.)

Even if we have succeeded in framing an intensional definition which deals satisfactorily with the borderline cases we have already encountered, we may still wish to consider certain additional borderline cases we have not yet encountered, and may never encounter. For example, suppose a space ship landed on earth carrying beings from another planet who were obviously intelligent and similar to earth people in many other respects. Suppose further that someone killed one of these beings without any provocation. Would this be murder? It would depend upon our definition of “human.”

Until the word “human” is clearly defined, it does not make sense to ask whether such visitors from space are really human, for the answer depends upon the definition of “human.” Whether a given definition is reasonable and useful is, nevertheless, a very important question. There are numerous legal, ethical, biological, sociological, anthropological, and psychological considerations which are relevant to this question. They do not tell us whether a given definition is true, but they do help us to appraise the adequacy of definitions.

5. Some definitions are designed to introduce a word which will have theoretical importance and utility. Such definitions are common in science. Words like “work” and “energy” are given precise definitions in physics, not so much to remove the vagueness of their everyday meanings, but to provide words that can be used to state important physical generalizations. Indeed, the ordinary meanings are deliberately changed to provide useful physical concepts.

In philosophy, too, we seek definitions which will provide theoretically useful concepts. For example, philosophers have tried to define the word “free” (as it occurs in the phrase “free will”) so that it will mark a significant distinction between free and unfree acts. The resulting concept should enable us to state the connection between freedom and responsibility. It should help us to explain what it means to say that a person could have acted differently, and it should help us to clarify the relation, if any, between freedom and causal determination.

6. In addition to intensions, extensions, and grammatical functions, words have emotive force. A book which goes into great detail might be described by one person as thorough and scholarly, but by another person as tiresome and pedantic. It is not so much that the two people are making different statements of fact about the book as that they are expressing different attitudes toward it.

Definitions are often designed to transfer emotive force. This can be done in either of two ways.

First, we may take a word which has a great deal of emotive force and define it so that it will apply to something we wish to applaud or condemn. For instance, one might define “socialistic” as “tending to equalize wealth by government action.” Since the graduated income tax has the effect of equalizing wealth, the word “socialistic” applies to it. Among people for whom the word “socialistic” has negative connotations, this tends to transfer the negative attitudes to the graduated income tax itself. Among people for whom the word “socialistic” has positive connotations, the effect would be just opposite. Definitions of this kind transfer emotive force from the definiendum to the definiens. The definiens clearly applies to the graduated income tax, so the emotive force of the definiendum–”socialistic” – is transferred to the graduated income tax via the definiens.

Second, the process may be reversed. Suppose a certain drama is admittedly naturalistic. Someone might define “naturalistic” as “glorifying the meanness of human nature and the sordidness of human existence.” This definition transfers negative emotive force from the definiens to the word “naturalistic”– the definiendum– and thence to the play itself. Definitions whose main function is the transfer of emotive force are called “persuasive definitions.”

The foregoing examples could easily give the impression that all persuasive definitions are illegitimate. This is not true. We need words with emotive force to express our feelings, emotions, and attitudes; persuasive definitions help to provide the necessary vocabulary. However, persuasive definitions can lead to difficulty if, in the process of transferring emotive force, we also modify the established descriptive meanings of our words. If the modification of descriptive meaning goes unnoticed, confusion can result.

The preceding list of purposes of definitions is not intended to be exhaustive, nor are these purposes mutually exclusive. In discussing the definition of “human,” we were concerned with characterizing customary usage to some extent (purpose 1), making a somewhat vague word more precise (purpose 3), providing an intensional definition for a word whose extension is fairly clear (purpose 4), and providing a word with theoretical importance and utility (purpose 5).

Some of the most important philosophical problems are basically problems of definition. Philosophers ask: What is justice? What is art? What is religion? What is knowledge? What is truth? In each of these cases, the question could be rephrased: How should we define the word “justice”? How should we define the word “art”? etc. Notice that this transformation takes what appears to be a question in the object language and reformulates it as a question in the metalanguage. It will not do to answer that definitions are conventions, so one definition is as good as another. Definitions are conventions, but some conventions achieve their purposes better than others. Finding an adequate definition is often a delicate matter.
Wesley C. Salmon; Logic; 1973; s30

EXPLICATION OF WORDS

There is no greater impediment to the advancement of knowledge than the ambiguity of words.

[…] definitions and axioms are the foundations of all science.

[…] A definition is nothing else but an explication of the meaning of a word, by words whose meaning is already known. Hence it is evident, that every word cannot be defined; for the definition must consist of words; and there could be no definition, if there were not words previously understood without definition.

[…] It may further be observed, that there are many words which, though they may need explication, cannot be logically defined. A logical definition, that is, a strict and proper definition, must express the kind (genus) of the thing defined, and the specific difference by which the species defined is distinguished from every other species belonging to that kind. It is natural to the mind of man to class things under various kinds, and again to subdivide every kind into its various species. A species may often be subdivided into subordinate species, and then it is considered a kind.

From what has been said of logical definition, it is evident that no word can be logically defined which does not denote a species; because such things only can have a specific difference; and a specific difference is essential to a logical definition. On this account there can be no logical definition of individual things, such as London or Paris. Individuals are distinguished either by proper names, or by accidental circumstances of time or place; but they have no specific difference; and therefore, though they may be known by proper names, or may be described by circumstances or relations, they cannot be defined. It is no less evident, that the most general words cannot be logically defined, because there is not a more general term of which they are a species.

Nay, we cannot define every species of things, because it happens sometimes that we have not words to express the specific difference. Thus a scarlet color is, no doubt, a species of color; but how shall we express the specific difference by which scarlet is distinguished from green or blue? The difference between them is immediately perceived by the eye; but we have not words to express it.

Without having recourse to the principles of logic, we may easily be satisfied that words cannot be defined which signify things perfectly simple, and void of all composition.

[…] When men attempt to define things which cannot be defined, their definitions will always be either obscure or false. It was one of the capital defects of Aristotle’s philosophy, that he pretended to define the simplest things, which neither can be nor need to be defined; such as time and motion.

[…] Since, therefore, it is often impossible to define words which we must use on this subject, we must as much as possible use common words in their common acceptation […]

Thomas Reid; Essays on the Intellectual Powers of Man; 1859; ch1

CHAPTER X.
DEFINITION AND DIVISION

§ 1. Definition. We are concerned in this chapter with two processes, both of which belong, not to the Logic of the Judgment, but to that of the Concept. Neither Definition nor Division, however, can be satisfactorily treated, unless the Predicables have previously been explained. This, therefore, seems to be the most convenient place at which to deal with them.

The definition of an object is the declaration of its essential characteristics. Hence, a definition is given in the form of a proposition, in which the object defined stands as the subject, and the essential characteristics form the predicate. It is this predicate which is the definition properly so called. The discussion of the question belongs, as we have said, to the Logic of the Concept: for considered as a mental act, the definition is the concept which expresses the true nature of the thing defined.

It is carefully to be observed that the definition is concerned with the nature of a thing. For by some logicians it is explained as being simply the connotation of the subject term, as it is understood by competent thinkers. Now it is, of course, the case that whenever the essential characteristics of a thing― its true nature,― are known, these will constitute the intension of its name. Thus the intension of the term ‘triangle,’ is ‘a plane figure contained by three straight lines.’ But it will often happen that a name is applied to a group of objects, which we are perfectly able to identify by certain common properties they possess, while at the same time, we are ignorant of their real nature. Thus, for instance, when a new disease, e.g. the sleeping sickness, makes its appearance, doctors recognize it and give it a name, long before they are able to define it. The term in this case, has a connotation, viz.: the symptoms by which the disease is known: but we have not yet found the definition of the thing. The true definition must do more than enable us to recognize it. It must unfold its nature.

Aristotle expressed this, by saying that the definition gives us the ‘why’ of the thing. The definition ‘Man is a rational animal,’ is a case in point. If we are asked what makes Socrates a man, we reply that he possesses these characteristics. It is not because he is ‘a tool using animal,’ that he is a man, nor yet because he is, ‘an animal that cooks his food,’ though these statements are true. He is a man because he is a rational animal. The ideal definition will then contain the essential characteristics. To what extent we are able actually to realize this ideal in our definitions, we shall see when we study the various kinds of definition.

Definition is always of the universal. Nature gives us general classes, and phenomena which occur subject to general laws. The individual members of these classes, the individual instances of the phenomena, are all different: each has accidental characteristics, by which it differs from every other. The aim of definition is to seize on the type, which is constant amid all this variety. One attack, e.g. of sleeping sickness, or of malarial fever, differs from another in a hundred particulars,― in duration, in intensity, in collateral effects, etc., etc. These are of no importance to the definition; for it is concerned alone with what is essential― with the permanent type.

Hence definition is rightly said to be the aim of science. Science has achieved its object, when it has accurately determined the nature of some substance, or the law of some phenomenon.

It is often said that all definition should be by genus and differentia. There are indeed certain cases in which we can assign the genus and differentia, understanding those words as they were employed in connexion with the Predicables. In some cases, however, this is impossible. Thus an eclipse can be defined, but it has not properly speaking a genus or differentia. Hence these terms are here employed with a certain amount of latitude. Genus should be understood as meaning no more than such attributes as are common alike to the class of objects in question, and to other classes: differentia signifies the notes which are proper to the class, and distinguish it from others.

§ 2. Various Kinds of Definition. (i) Real and Nominal. In the last section we shewed in what a Real definition consists. It is an expression which declares the nature of a thing. A Nominal definition on the other hand, is an expression declaring the meaning of a word. Some logicians, as we have already noticed, have maintained that no definitions are intended to do more than this; that one and all they merely unfold the connotation of terms. Aristotle considers this question at length, and distinguishes two kinds of Nominal definitions. In the first place, there are (1) definitions of names which signify imaginary objects, to which nothing either actual or possible corresponds. We may find an example in the definition of a dragon as ‘a serpent breathing flame.’ An impossible self-contradictory concept cannot provide us with a Real definition: for that must state the essence. An essence which contains repugnant characteristics is no essence at all. Indeed the expression, ‘A dragon is a serpent breathing flame,’ is elliptical. Fully stated, the proposition should be ‘A dragon (as imagined by the writers of fables) is a serpent breathing flame’ (Ch. 7, § 4).

Aristotle further reckons as Nominal definitions, (2) those in which we are unable to assign the essential properties of a thing, and are merely able to indicate it by a description. Thus, the definition of thunder, as ‘a noise in the clouds,’ is nominal: nominal, because it describes what is meant, and yet does not unfold the nature of the object ( An. Post. II ., c. 10, § I, c. 8, §7).

All Nominal definitions, therefore, do not deny existence to their objects. In this latter class existence is presupposed. But in Aristotle’s view, none save those which express the essential characteristics, can rightly be termed definitions of the thing. This distinction of Real and Nominal has largely fallen into disuse for reasons to be mentioned presently. But it should be carefully noticed, for the principle it embodies is one of importance. 

(ii) Essential Definitions. These are the definitions which are formed by genus and differentia In the stricter sense. Such for instance is our definition of man as ‘a rational animal.’ Such too are our definitions of mathematical figures. The limits by which a plane figure is bounded, constitute its specific differentia. ‘A plane figure contained by three straight lines,’ is the essential definition of a rectilinear triangle.

In the case, however, of every other natural type except man, it is impossible to obtain an Essential definition. The specific differentia, from which its peculiar properties flow, is unknown to us. While we recognize that the substantial principle, which determines the distinctive characteristics of, e.g. a lion, must needs be totally different from that of a horse, we can never hope to penetrate to any knowledge of the two principles, except in so far as they are manifested by their properties. We must then be content with definition by properties, or as it is often called:―

(iii) Distinctive Definition. It is at definitions of this kind that the student of natural history or of physical science, aims. He seeks to state the most characteristic properties of the type with which he is dealing. More over he recognizes that he can scarcely hope to attain finality in his quest. For the number of distinctive properties in every natural type is vast, and it is always possible that he may discover some property of primary moment hitherto overlooked.

These definitions are of the highest importance in science. For it is on them that is based the scientific classification of natural types. This application of definition will form the subject of a later chapter.

It is because these Distinctive definitions are the only definitions attainable in the case of natural classes, that Aristotle’s division of Real and Nominal definitions has been discarded. It would be manifestly unsatisfactory if a class of definitions, which, within their own sphere, are the highest result of science, were ranked as merely Nominal, because they fail to reveal the specific essence of the nature. Yet between the Distinctive definition which states properties only, and the Essential definition which unfolds the essence, there is a great difference. For wherever we possess the Essential definition we can reason from the essence to the properties. But we cannot from the properties given in the Distinctive definition, conclude to the essence. Moreover, the same substance under different circumstances, manifests itself by different properties. Thus the properties of phosphorus are totally different from those of what is termed amorphous phosphorus: while it is needless to point out how much those of carbon, graphite and diamond differ from each other. Yet in each of these cases, we are supposed to be dealing with the same substance, manifesting itself by different properties when affected by different conditions.

(iv) Genetic Definitions give us neither the essential nature of the thing nor its properties, but the elements which, taken in conjunction, result in its production. The term is most frequently used in reference to certain mathematical definitions, which express the nature of a figure by a statement of the manner in which it may be constructed. Thus a circle may be defined, as a figure formed by the revolution of a line in a plane round one of its extremities. Here the several segments gradually traced by the line are not themselves a circle. Yet when the process is complete, and all the parts are seen in conjunction, they constitute a circle. But the employment of Genetic definitions is by no means limited to this special case. The definitions employed in chemistry are of this type, as, e.g. the definition of water as ‘two atomic weights of hydrogen chemically combined with one of oxygen.’ For these definitions do not inform us regarding the qualities of the substance. They tell us what constituents are requisite to its production.

(v) Causal Definitions are of two kinds.

(a) One class defines by indicating the Final cause or purpose of the object (Arist., Met., VII., c. 2, §§ 7,8). This form of definition is the one ordinarily used in the case of the works of human ingenuity. We define a clock by saying that it is ‘a mechanism destined to indicate the hours of the day,’ a stirrup by terming it ‘a metal hoop for the purpose of supporting the foot when riding.’

(b) The other class of Causal definitions defines by stating the Efficient cause . It may often happen that the most satisfactory explanation of the nature of the thing, is given in this manner. Thus since the days of Koch, who first discovered that certain diseases were due to the presence of specific microbes, it is a true definition of anthrax (wool-sorter’s disease) to say that it is ‘an illness caused by the introduction into the body of the bacillus anthracis.’ Some things, e.g. natural products, can hardly be defined in any other way. We naturally define the double cocoa-nut as the fruit of the tree Lodoicea Seychellarum.’

(vi) Accidental Definitions. These are employed to define those sub-classes, which do not constitute distinct species. Thus a negro may be defined by his colour as ‘a black man,’ even though colour be an accident and not a property of the nature. These are, however, rather to be regarded as descriptions , than as definitions.

(vii) Analytically-formed and Synthetically-formed Definition. This distinction has been employed by recent logicians: it is only applicable when definitions are regarded not as declaring the nature of the thing, but simply as stating the meaning of the word. If the definition gives the recognized intension of the term, it is said to be analytically-formed. It, however, sometimes happens that a special meaning is attached to a word, which hitherto it has not borne. Such for instance is the employment of the term ‘wave’ by physicists, as signifying ‘a change periodically recurring in time and space.’ When a new definition of this character is introduced, it is said to be synthetically-formed.

The Aristotelian doctrine of the four causes throws considerable light on this long list of definitions. Every material thing has four causes― the efficient, the final , the formal , and the material cause. The meaning of these terms may be illustrated in the case of a statue. The efficient cause of the statue is the sculptor: the final cause is the motive, be it honour or profit, that leads him to execute the work. These are known as the extrinsic causes. The formal and material causes are termed intrinsic, since they are internal constitutive principles of the thing itself. The formal cause is the determining principle which gives the thing its specific character. The shape of the statue, e.g. Apollo, Julius Caesar, Charles I., may be viewed as such. In a natural entity such as e.g. a man, or anyone of the animals, the formal cause is not, of course, the mere external shape. The unifying principle of the material object, that which makes it to be the kind of thing it is, is the vital principle, the soul. The material cause is the substratum to which  the formal principle gives its character. In the statue it is the marble.

We stated above that the true definition always gives us the cause of the thing defined. Definitions are formed from each of the four causes which we have mentioned. Sometimes a more satisfactory explanation is given by assigning one cause rather than another: sometimes only one cause is fully known to us. The definitions drawn from the efficient and final causes have been explicitly noticed in our list, and need not be further discussed. Essential definition is definition by the formal cause. We are not speaking here of the formal cause in the real order, which is, as we have seen, the soul. But we can form an abstract notion of ‘humanity,’ expressing those attributes alone, in virtue of which an individual is a man. In the conceptual order this ‘humanity’ is viewed as though it were a formal cause by which the individual is constituted man. He may have many other attributes: he may be a Greek, a philosopher, etc., etc. But it is only those attributes which belong to his ‘humanity,’ which make him a man. The definition ‘rational animal’ expresses the notes which are found in this formal cause.

Finally Genetic definition is definition by the material cause. Even when the conjunction of constitutive parts does not, as in the case of chemical combination, result in internal changes, the parts always stand to the whole in the relation of material cause. For taken separately they lack the distinctive attributes of the thing in question. It is only to the whole produced by their conjunction, that the attributes belong.

§ 3. Limits of Definition. It is manifest that the highest genera will be incapable of definition properly so called. We cannot assign any higher genus under which they may fall. Much more is this the case with concepts such as Being, Unity and the like, which transcend the limits of the highest genera, and are found in all of them. At the other end of the scale, there can be no definition of individuals. They can only be described, not defined: for definition is always of the universal. Further it is impossible to define the simple qualities which are the immediate data of sense perception, e.g. sweetness, cold, whiteness, pain, etc. The purpose of definition is to unfold the nature of the object by an analysis. Here there is no room for analysis. These are the elements from which knowledge begins. We can, of course, enumerate the properties of such qualities, as when, e.g. we say that red is the colour which manifests itself through vibrations varying from 360 billions to 500 billions per second. But this is not properly speaking a definition: for apart from this information we have a perfectly clear conception of the colour itself.

§ 4. Rules of Definition. We must now consider the traditional rules of definition. These are four in number:―

(1) The definition must be adequate to its object; erring neither by excess nor defect. It must, that is, be applicable to every member of the class defined, and to no other objects. It must thus be convertible with the class name. We constantly come across definitions which are too wide or too narrow. Thus the definition, ‘Logic is a machine for combating fallacy,’ is far too narrow. It reduces the whole science to what is, in fact, one of the least important of its properties. Mill’s definition of eloquence as ‘the power of influencing the feelings by speech or writing,’ is too wide. It is possible to influence men’s feelings by speech without eloquence. So too, to define religion as ‘the totality of man’s relations with God,’ is to err by excess.

(2) The definition must not be obscure. This rule must not be misunderstood. It does not signify that the definition must in no case appear obscure to the uninstructed. Sometimes the elements of the definition may to the uninstructed be less comprehensible than the thing defined. In a scientific definition, e.g. of lightning, or a philosophic definition, say of free-will, obscurities are unavoidable. A definition is the result of long study, and many definitions will, to those whose minds are unprepared for them, seem more obscure than the thing they profess to explain. When the mind of the learner has been prepared by the requisite instruction, he will realize that in the definition, he has received the summarized results of science. Such definitions do not really offend against this rule; for they are true analyses of the phenomenon into its simpler elements. In regard to this class of definitions, the purpose of the rule is to forbid ambiguous and metaphorical expressions.

Where, however, the definition is intended to serve as a brief explanation for those not versed in the special subject under consideration, it is requisite that it should be couched in simple terms. Those who violate this rule, are said to explain obscurum per obscurius.

(3) The definition must not be tautologous. This fault is committed when the subject of the definition reappears either explicitly or implicitly in the defining predicate. Thus to define a horse as ‘a member of the species equus,’ would convey no information whatever. This rule is violated when we have what is termed circulus in definiendo, e.g. ‘A day is a period of time consisting of twenty-four hours,’ and ‘An hour is a twenty-fourth part of a day.’ It is not, however, regarded as a circular definition when, in the case of two relative terms, each appears in the definition of the other; since their concepts are mutually dependent. We cannot define ‘antecedent’ without mention of ‘consequent,’ nor ‘consequent’ without mention of ‘antecedent. ’

(4) The definition must not be negative if it can be positive. We are not to define wisdom by saying that it consists in the ‘avoidance of folly’; nor health as ‘the absence of sickness’.

There are two cases where positive definitions cannot be given. The first of these is when we have to define objects which are incorporeal or unextended. Our cognitive faculties have direct knowledge only of what is corporeal and extended. We are therefore compelled to define the unextended by negative expressions. We define a line as ‘length without breadth’; a point as ‘that which has no parts;’ a spirit as ‘an immaterial substance.’ The second case occurs in regard to negations and privations. These consist essentially in the mere absence of a positive quality, and hence must be defined in that manner. Blindness is ‘the absence of sight in an animal usually found with that sense.’ Darkness is ‘the absence of light. ’
George Hayward Joyce; Principles of Logic; 1908; p150ff

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CHAPTER VII

DEFINITION, DIVISION, CLASSIFICATION, AND ANALYSIS

1. The Predicables

Aristotle’s genius for clear analysis, which enabled him to give to logic a terminology and form that persisted for two thousand years, is nowhere better exemplified than in his theory of the predicables. This doctrine is introduced in the Topics. It is concerned with certain types of relation that a predicate may bear to a subject, namely, the relations of being a definition, property, genus, or accident of the subject.

The Topics as a whole classifies the problems that can be raised for dialectical discussion in syllogisms. Thus the theory of the predicables really precedes the theory of the syllogism in logical order, and is a part of the Aristotelian analysis of propositions. The theory is connected in Aristotle’s mind with a metaphysics of natural kinds, or fixed species and essences. It reflects his general philosophical view that the objects of scientific knowledge constitute a hierarchy of forms. The four predicables exhibit the possible types of relation between these forms as they are expressed in propositions. Is P the genus, definition, property, or accident of S?– this becomes a logical formula into which the most important scientific problems are fitted, a formula bearing witness to the fact that the Aristotelian science is still largely in the classificatory stage.¹

“A definition is a phrase signifying a thing’s essence.”²

This statement immediately raises the question, what is a thing’s essence? First of all, Aristotle does not mean by thing an individual thing. He is dealing with relations between forms, i. e., universals, and is seeking the definition of man, motion, virtue, not of Socrates, of a particular motion, or an individual case of virtue. The particular is not an object of scientific knowledge or definition for Aristotle.

Now, the essence of anything is that which makes it what it is; it is not arbitrary, but is necessary to the thing. The essence of water, and hence its definition, would be (let us say) that “water is a chemical compound of hydrogen and oxygen in the proportions H ₂ O”; the essence of a circle, that “it is a plain figure bounded by a line whose points are equidistant from a central point.” It is characteristic of the Aristotelian theory of essence to insist that any kind of thing has one and only one essence, one and only one definition. The essence is peculiar to, and convertible with, the subject of which it is a predicate, e. g., nothing but water is a chemical combination of hydrogen and oxygen in the proportions H ₂ O, and all water is such a combination. But the essence is not the only predicable peculiar to and convertible with the subject. This is true also of a property in Aristotle’s technical sense. Thus, it would be a property of water to freeze at 0° Centigrade and to boil at 100° (if this is true only of water). But these properties would not form a part of the essence of water. Similarly, taking the essence of a triangle to be “a plane figure bounded by three straight lines,” we must reckon it as a property that the sum of the angles is equal to two right angles; this being true only of triangles, and of all triangles.

The soundness of this distinction between essence and property will be presently discussed; but it is important to observe what Aristotle says on this point in order to be clear that every kind of thing has, for him, only one essence– what the mediaevals called its substantial form. “Since, however, of what is peculiar to anything part signifies its essence, while part does not, let us divide the ‘peculiar’ (i. e., that which is convertible with the subject) into both the aforesaid parts, and call that part which indicates the essence a ‘definition,’ while of the remainder let us adopt the terminology which is generally current about these things, and speak of it as a ‘property.’”³

The essence (or definition) is constituted by two factors, the genus and differentia. Although Aristotle does not (in the Topics) rank the differentia as one of the predicables, since it would usually be predicated only in connection with the genus to make up the definition, still the differentia is entitled to a place in the list, and has always been given one, making the “Heads of the Predicables” five⁴ as follows:

“A genus is what is predicated in the category of essence of a number of things exhibiting differences in kind.”⁵

That is to say, the genus is that part of the essence (or definition) of a kind of thing, which is shared by the essences of other kinds of things. The genus of triangle, circle, square, ellipse, etc. is “plane figure” since this is shared by the essences of all these geometrical forms. The genus of water, sulphuric acid, nitrous oxide, etc. is “chemical compound.”

The differentia is that part of the essence (definition) which, being taken with the genus, distinguishes one kind of thing from other kinds within the same genus. Thus, a circle is a plane figure bounded by a line whose points are equidistant from a central point, while a triangle is a plane figure bounded by three straight lines; and so on for other plane figures. The italicized phrase states the differentia. Water is a chemical compound of hydrogen and oxygen in the proportions H ₂O, while sulphuric acid is a chemical compound of hydrogen, sulphur, and oxygen in the proportions H₂SO₄ ; and so on for other chemical compounds. Definition, for Aristotle, is thus always per genus et differentiam.

The conception of the differentia arises from Aristotle’s metaphysical doctrine of fixed kinds. Each kind within the same genus has its peculiar and appropriate differentia, and the ideal of definition is to discover the essential lines of cleavage between the kinds falling under a genus.⁶ Nature has secretly organized herself along the lines of the predicables, if we can only find it out. The notion that there is one and only one essence of a thing carries with it the notion that there is one and only one genus, and one and only one differentia. This can be seen in the following passage from the Topics:

“Again, in regard to the differentire, we must examine in like manner whether the differentire, too, that he (the proponent of a question) has stated be those of the genus. For if a man has not defined the object by the differentire peculiar to it, or has mentioned something such as is utterly incapable of being a differentia of anything, e. g., ‘animal’ or ‘substance,’ clearly he has not defined it at all.”⁷

Ordinarily we should say that one kind could be differentiated from another within the same genus in several alternative ways. Man is distinguished from other animals by the weight of his brain, by the fact that he walks erect and speaks, by the fact that he can adapt himself to a wider variety of environments, and so on. But this is not the Aristotelian view. The forms of things exhibit a rigid and necessary structure, so far as their definitions are concerned. Behind the changing and accidental features of the world lies the hierarchy of natural kinds, each one having its inalienable essence. The Aristotelian metaphysics is needed to make the doctrine of definition by genus and differentia intelligible; each kind of thing moves always to the realization of its substantial form, and each substantial form (or essence) stands inalterably fixed by a genus and differentia which belong to the ultimate nature of reality. Whether this metaphysics be true or false, it is in metaphysics, and not in logic, that the Aristotelian theory of the predicables is rooted.⁸

The Tree of Porphyry, usually given in connection with the predicables, shows how the definition of the infima species, man, is arrived at by differentiation from the summum genus, substance:

The division of each genus in the Tree of Porphyry is made by dichotomy, i.e., the genus is divided into those members marked by a certain differentia, and those not marked by this differentia. The infinite (negative) classes at the right are rejected at each step by the process of abscissio infiniti– “cutting off the infinite”– as irrelevant to the definition.

“A property is a predicate which does not indicate the essence of a thing, but yet belongs to that thing alone and is predicable convertibly of it. Thus it is a property of man to be capable of learning grammar, and if he be capable of learning grammar, he is a man. For no one calls anything a ‘property’ which may possibly belong to something else, e. g., ‘sleep’ in the case of man, even though at certain times it may happen to belong to him alone.”⁹

Aristotle means by “the predicate being convertible with the subject” that the predicate is logically equivalent to the subject; that is, “x is P” implies and is implied by “x is S.” Both essence and property are logically equivalent to the subject of which they are predicated. Thus, if S≡P1≡P2≡P2≡ … Pn, all of these predicates excepting one will be properties of S, and this one, let us say P1, will be the essence. The only warrant for such a theory is to be found in the Aristotelian metaphysics. Why not take any of these equivalent predicates as the definition of S? The others would then follow from this definition. A triangle could be defined as a three-sided plane figure and it would follow (with other principles of geometry) that it was also three-angular; or it could be defined as three-angular, and it would follow that it was three-sided. Or it might even be defined as a figure having the sum of its angles equal to a straight angle, and it would follow that it was three-sided and three-angular. The choice of the definition from among the properties of the thing defined, apart from metaphysical considerations, seems to be a matter of convenience only. Some properties chosen as definitions could be more easily handled in making deductions than others, but if all are equivalent, they must yield the same theorems. Aristotle’s answer, however, has already been given: definitions are fixed in nature, every kind has its peculiar essence, a definition is distinct from a property.¹⁰

The accident stands by itself in contrast to the other predicables: it is any non-necessary predicate, while genus, differentia, definition, and property are necessary predicates. What Aristotle means by a necessary predicate is not easy to say.¹¹ The proper interpretation probably is, that it is a predicate intensionally connected with the subject;¹² one which enters into the meaning of the subject, so that without having that predicate the subject could not be what it is. Thus man must be an animal, and must be differentiated from other animals by rationality, in the same sense in which a green thing must be colored, or a finite number must be greater by one than some other finite number. Otherwise, man could not be man, green could not be green, number could not be number; for the predicates are essential elements in the meaning of the subject terms. When we pass to the accident, however, the realm of intensional connections is left behind; we are dealing with material¹³ rather than necessary relations of forms.

“An accident,” the Topics declares, “is (1) something which, though it is none of the foregoing– i. e., neither a definition nor a property nor a genus– yet belongs to the thing: (2) something which may possibly either belong or not belong to any one and the self-same thing, as (e. g.) the ‘sitting posture’ may belong or not belong to the self-same thing. … Of the definitions of accident the second is better: for if he adopts the first, any one is bound, if he is to understand it, to know already what ‘definition’ and ‘genus’ and ‘property’ are, whereas the second is sufficient of itself to tell us the essential meaning of the term in question.”¹⁴

The first is a negative, the second a positive definition of accident; and positive definitions are always preferable to negative ones.

Aristotle seems to imply that an accident may be (1) that of an individual, as “sleeping” would be of Socrates, since Socrates might be awake, or (2) that of a class-concept, i. e., a form or universal. Thus “white-skinned” is an accident of the concept, man; it is an accident of organisms to be eaten by other organisms; an accident of polar bears to be kept in cages in circuses. The second is the more important use of the term accident, since it parallels the idea that the definition, genus, differentia, and property– and thus all the predicables– are predicable of kinds rather than individuals. They exhibit relations between forms.¹⁵ Mr. H. W. B. Joseph rejects the idea that Aristotle intends the accident to be taken as a predicate of an individual at all; he says,¹⁶ “But we cannot distinguish between property and accident, so long as the subject whose predicates we wish to refer to these heads is an individual. A property is necessary to its subject and an accident is not; but all the attributes which belong to Cetewayo (an individual) are equally necessary to him as Cetewayo (as an individual); on what grounds then are some to be called properties and the others accidents? An accident is an attribute which coincides in an individual with another general character, or universal; its accidental relation lies towards that other universal, and not towards the individual, in which its presence is, historically, necessary.” This interpretation renders the doctrine of the predicables consistent with itself on the point that each predicable indicates a type of connection between general characters. The major division of the whole doctrine is that between accidental and necessary connections.¹⁷

2. Division and Classification

From what has been said it is clear that a definition in the Aristotelian sense is arrived at by a process of division. Begin with a summum genus; divide this genus by its peculiar differentiæ, thus giving sub-genera; divide these genera further by their peculiar differentiæ, and so on, until the species to be defined is reached.

This process of division, which proceeds from the more general to the less general, by breaking up wider concepts (genera) into narrower ones (species), has as its converse classification. Classification begins with individuals; these are grouped into classes (infima species) through analogies of structure; these classes are joined into wider classes (genera); and so on, till a summum genus is reached. Thus classification proceeds from the less general to the more general; it retraces in the converse direction the structural lines of genera and species revealed by division. Both division and classification may– and usually do– go on at once, lending support to one another; and classification is equally relevant, with division, to an Aristotelian definition.

Modern logic, leaving aside the metaphysics of a hierarchy of natural kinds, views definition in a much more formal and strictly logical way, which dissociates it from classification and division. As processes useful in science, classification and division belong to the more elementary stages of development, the pre-theoretical stages.

The rules for division, applicable also to classification, are three:
(i) The division must be exhaustive; that is, every member of the genus must be provided for in some one of the differentiated species.
(ii) The species into which the genus is divided must exclude one another.
(iii) The division must employ a single principle of differentiation, i. e., a single fundamentum divisionis, for each genus and, so far as possible, must continue with the same fundamentum divisionis at all stages.

Unfortunately, the first rule is a counsel of perfection, fulfilled only in the most abstract sciences. There is no way of knowing– unless, as in mathematics, we can lay it down as an a priori condition for the subject-matter in question– that a genus is exhausted by a division. Take for instance the division of vertebrates into mammals, birds, reptiles, amphibians, and fishes. What assurance is there that these species comprise all possible animals with backbones? In the future course of evolution vertebrates that could not be fitted into any of these classes might turn up; and only if we could believe with Linnæus that “there are just so many species as in the beginning the Infinite Being created,” could we maintain the contrary. On the other hand, we can be certain that, if space conforms to the axioms of geometry, the genus of rectilinear plane figures must be divided into the species, three-sided, four-sided, five-sided, etc. to n-sided, and that this division is exhaustive.

The second rule, that the species must exclude one another, is violated (1) by including among the species, as co-ordinate with them, a subordinate or super-ordinate class; e. g., “fish, flesh, fowl, and good red herring,”¹⁸ or “animal, vegetable, mineral, and organic kingdoms;” (2) by a cross-division, which technically arises through the violation of the third rule, that a single fundamentum divisionis must be employed throughout the division. (It should also be noted that if a division is not exhaustive, we cannot be sure that it is exclusive. If there could be other vertebrates than mammals, birds, reptiles, amphibians, and fishes, they might possibly exhibit the distinguishing features of two or three of these species at once.)

The fundamentum divisionis is some principle according to which the differentiæ of the species separated out from the genus are selected. Thus, in the division of rectilinear figures into three, four, etc. to n-sided ones, the fundamentum divisionis is the number of sides; in Cuvier’s division of races into Caucasian (white-skinned), Mongol (yellow-skinned), and Negro (black-skinned), the fundamentum divisionis is the color of the skin. To classify races as white, black, and round-headed would be a departure from the rule; and would in this case result in an overlapping of the species,– for there are round-headed races of both whites and blacks. In the same way, animals might be divided for economic purposes into domestic animals, game animals, fur-bearing animals, gnawing animals, predacious animals.¹⁹ There is no single fundamentum divisionis here: the classes are not exclusive.

The second part of the rule demands the use, so far as possible, of the same fundamentum divisionis at all stages of the division. This embodies Aristotle’s idea that each differentia should be a differentiation of the previous differentia. Thus, if we classified the inhabitants of the United States into genera according to the state in which they reside, into species according to the county, and into sub-species according to the township or smaller unit of residence, we should be using the same fundamentum divisionis throughout, namely, the region of residence. Or, if we further subdivided white, black, and yellow-skinned races by the color of their skin, we should be following out this principle. The application is plainly difficult, and would often serve no useful purpose, for the important subdivisions of the species may be those that proceed on a new fundamentum divisionis, e. g., the important subdivisions of white-skinned, black-skinned, or yellow-skinned races may not be at all those determined by complexion. Hence, this part of the rule is not always pressed by logicians.

The failure to adhere to a single fundamentum divisionis does not necessarily lead to an overlapping of the species. The rule prevents overlapping, but its violation does not always produce this result. Here again the logician seems to be giving to the scientist a counsel of perfection. Actual classifications and divisions proceed in a much less formal way. The eye of the scientist is fixed on complex similarities and differences; the web of interlacing characters is too tangled to exhibit any single principle of differentiation; and though the scientist wishes his divisions to be exclusive, he cannot secure this by adherence to the logical rule. The facts are not so simple.

Consider, for example, the following zoological division of the animal kingdom into genera:²⁰
(a) Vertebrates: animals with a backbone;
(b) Arthropoda; animals with jointed appendages and segmented bodies, e. g., crayfishes, centipedes, insects, etc.;
(c) Molluscs; animals with a ventral muscle, called the foot, usually serving as an organ of locomotion; often with a heavy shell;
(d) Annelida: segmented worms;
(e) Echinodermata: spiny-skinned sea animals, e. g., the starfish;
(f) Nemathelminthes: unsegmented round or thread worms;
(g) Platyhelminthes: flat unsegmented worms;
(h) Coelenterata: animals with simple sac-like bodies, e. g., the jellyfish;
(i) Porifera: sponges;
(j) Protozoa: the simplest animals, only visible to the microscope.

No attempt to employ a single fund amen tum divisionis is in evidence here, unless it be degrees of simplicity and complexity of structure; and this is an extremely elastic principle. Again, if we consider the species within one of the above genera of animals, namely, the species of vertebrates, we can find no one principle of division. Vertebrates fall into,
(a) Mammals: vertebrates which possess hair, and, with few exceptions, nourish their young with milk secreted from mammary glands; they breathe air by means of lungs and are said to be warm-blooded;
(b) Birds: vertebrates characterized by the presence of feathers;
(c) Reptiles: vertebrates having lungs, and in most cases covered with armor of scales or bony plates; called cold-blooded;
(d) Amphibians: vertebrates that resemble reptiles, but do not possess scales and are anatomically different; in early life they breathe through gills, but later become air-breathers;
(e) Fishes: vertebrates with scales, who spend their entire existence in water; breathe through gills, and swim by means of fins.

The whole conception of a rigid differentia, and a single principle for its selection, seems to be abandoned in these actual cases of classification; and the possibility of intermediate forms– of animals that fall, e. g., between birds and reptiles, amphibians and fishes,– is never excluded. At the same time, the naturalist is attempting to separate out distinct species, and to avoid as far as possible overlapping; but he does not achieve this result in the simple manner prescribed by the Aristotelian logic, i. e., by following the rule of the single fundamentum divisionis. Departure from this rule, whether the species do or do not overlap in fact, is technically cross-division, since there might be forms (though there may be none in fact) that fall into more than one species. When the Darwinian theory of the evolution of species became a factor in natural classifications, the possibility of intermediate forms between all species, taken in historical order, was admitted. The ideal of a classification or division that formally avoids cross-division became unworkable, unempirical, and in fact positively false.

In dichotomous division (or dichotomy), a genus is divided by a differentia and its negative, e. g., the genus animals could be divided into vertebrates and non-vertebrates; the genus man into white and non-white, and so on. The division of vertebrates would continue as shown in the table on the opposite page.

This type of division possesses the merely formal merit of being exclusive and exhaustive. There can be no question that all members of a genus are exhausted by the classes A

and not-A, if A is some character belonging to certain members; nor can there be a question that the species A excludes the species not-A. But this formal advantage is gained at the cost of (1) a cumbersome multiplication of “infinite classes” (i. e., negative classes) which (2) obscures the simpler relationships of genera and species that appear where only positive divisions are made.

The simpler classification of animals (corresponding to the dichotomous division above) by positive characters alone would be schematized thus:

The negative classes of the dichotomous division require to be further divided if the division is to go forward at all. Thus, nothing whatsoever is gained by including these negative classes, excepting an a priori guarantee of exclusiveness and exhaustiveness; a guarantee that has no empirical connection with the subject-matter. The saner method is to proceed at once to the positive differentia, omitting the negative classes and thus exhibiting clearly the co-ordination of the various species and genera, as in the second table above. The chief argument against dichotomous division as a scientific method is thus pragmatic; it continually introduces, for no good scientific reason, negative classes.²¹

The two types of division we have described– the positive differentiation of genera into species and sub-species; the dichotomous differentiation, which employs negative concepts at every stage– are usually called logical division, to distinguish them from physical and metaphysical division. The division of an individual thing into its separate parts is physical division (or partition). Thus, if I divide a watch into its case, hands, face, and works, I have performed a process of physical division. Metaphysical division separates out (conceptually) the various qualities, rather than the physical parts, of a thing. When I enumerate the qualities of the watch– its size, accuracy, color, etc.– I am dividing it metaphysically.

These seemingly trivial distinctions touch an important logical point, namely, the difference between logical division and analysis. The use of the term “division” for both obscures the issue; physical and metaphysical division are more clearly described as two types of analysis.

Logical division traces the route from less specific to more specific concepts, from wider to narrower classes,– from what Mr. W. E. Johnson calls the “determinable” to its “determinates.”²² This is scarcely a process of analysis; or at least it does not seem wise to speak of it as analysis, if we wish to give precision to this term. We should not say that we had analyzed the idea of color when we exhibit red, green, orange, etc. as colors; or that we had analyzed man when we name white, black, and yellow-skinned races as types of men. This is a specification of a more general concept in its less general forms. On the other hand, we have analyzed the watch when we describe its parts, its balance-wheel, jewels, mainspring, etc., together with their functioning in the whole, which causes the watch to keep time. And we have also analyzed the watch in a second– and different–sense when we assign its qualities; i.e., it is round, flat, made of gold, it belongs to Mr. X, etc. Both of these processes are different from the exhibition of a less specific concept in its more specific forms. The first can be spoken of as partitive analysis (following the term partition which is used for physical division), and the second as qualitative analysis.

Analysis and synthesis are inseparable from one another, and both are connected with definition.²³ To analyze is to display the components– we should probably be tempted to say, the simple components– that constitute some whole; further, it is to display these components as constituting this whole. We analyze water when we resolve it into the simpler components, hydrogen and oxygen; we analyze our perception of space when we resolve it into visual, tactile, and kinæsthetic sensations. But it is not enough merely to decompose water into hydrogen and oxygen; we must show that, in the proportions H₂O under certain conditions of temperature and pressure, they actually do constitute water. We must exhibit water as a synthesis of these elements in order that the analysis may have meaning, that is, may be the analysis of something. In the same way, we must show how visual, tactile, and kinæsthetic sensations fuse to give the perception of space; otherwise, the components are components of nothing. They are disparate and scattered simples. The recognition that what is analyzed already constitutes a whole (a synthesis) is implied in the meaning of the term “analysis”: just as it is implied in the meaning of the term “synthesis” that something analyzable, namely, a whole of constituents, is presented to us.²⁴

Mr. Johnson puts this point as follows:
“… since the important process is– not the mere revelation of the parts contained– but rather the indication of their mode of combination within the whole, analysis is better defined as the exhibition of a given object in the form of a synthesis of parts into a whole. In this way we can say that any process of analysis can also be described as a process of synthesis; but this does not amount to saying that analysis means the same as synthesis, any more than that the relation ‘grand-father’ is the same as the relation ‘grandson,’ although the fact that A is the grandfather of B is the same fact as that B is the grandson of A. In short, analysis is the inverse of synthesis; i. e., when the whole X is analyzed into several components a, b, c, d; then a, b, c, d have to X the inverse relation which X has to a, b, c, and d. In this way it is clear that to analyze X simply means the same as to exhibit X as a synthesis.”²⁵

Partitive and qualitative analysis are both ways of exhibiting elements as constituting a synthesis, though the type of whole– and the meaning of the term “part”– is different in each case. The elements analyzed out in a partitive analysis are always related spatially or temporally (or both). They fall spatially or temporally within the whole; hence the name “physical division,” since the analysis of such wholes is relevant only to the physical world. The springs, wheels, etc. are spatio-temporal parts of the watch. Again, our perception of space is composed of visual, tactile, and kinæsthetic sensations forming at least a temporal whole. (And if we admit, from a naively realistic point of view, that spatial relations between sensations are presupposed in the perception of space, we could say that our visual, tactile, and kinæsthetic sensations are spatially related to one another in yielding space-perception.)

Partitive analysis can be divided into the two kinds, homogeneous and heterogeneous. If the spatial or temporal parts are of the same genus as the whole (and hence as the other parts), the analysis is homogeneous; for example, the division of a cube into other cubes, a surface into surfaces, a line into lines, a quantity of heat into quantities of heat. But if the parts are of a different genus from the whole (and, as a rule, from one another), the partition is heterogeneous. The partition of the watch is of this type: the analytical elements, the springs, wheels, etc. are not themselves watches.

Homogeneous partitive analysis is required for measurement. This is what is ordinarily meant by divisibility. Thus, to measure a volume, we must analyze it into volumes; to measure a duration, we must analyze it into durations; and so on. The measurement consists in assigning a number of equal parts, in this sense of “part,” to the whole analyzed. Heterogeneous partition, on the other hand; is not the type of division which is relevant to measurement. When we exhibit the mechanisms within mechanisms which constitute the watch, we are not dividing the whole into measurable units. The parts can, of course, be enumerated; let us say that there are nine hundred and ninety-nine parts in a watch; but this tells us nothing about the size of the watch.²⁶

Qualitative analysis considers– not the spatial or temporal relatedness of elements with a whole– but the qualities and relations beyond itself which belong to this whole as a whole. Such analysis need not be restricted to the physical world; this gives a justification for the name “metaphysical division.” Clearly, the qualities and relations of an object are not parts of it in the spatio-temporal meaning of this term. The blueness, foaminess, saltiness, etc. of the sea are not in the sea, as are the individual waves and the individual molecules of salt. Nor are we analyzing water from the same angle when we describe it as colorless, odorless, and tasteless, as when we decompose it into H₂O. We can define qualitative analysis as follows: all the conjoined qualities and relations which can be predicated of a given subject S constitute the qualitative analysis of S. This is by far the most important sense of the term “analysis,” and the one most purely logical in meaning.²⁷ Obviously, partitive analysis can be brought under it, though the two are not equivalent; i. e., the fact that S can be partitioned into certain spatio-temporal parts is a quality of S, and can be predicated of S as a whole. It is a quality of water– in fact, the most important one,– to be analyzable by partition into H₂O, just as it is a quality of water to be colorless and odorless.

Since the world is so full of a number of things– both the world of the actual and the possible– the complete qualitative analysis of anything would lead to infinite complexity. Whatever can be truly said about a thing is a part of its qualitative analysis. Thus, for purposes of definition, it becomes essential to select: we might characterize (real) definition in general as selective qualitative analysis.

Whatever the type of analysis, the question arises– Are the elements yielded by any analysis absolutely simple?

To say that there are absolute simples seems a dangerous and unverifiable assumption. For if we can make any statement at all about anything we are, in one of the senses described above, analyzing that thing; and it would be rash to maintain that there is something about which nothing at all can be said. In so far as analysis is partition, the physical division of spatial or temporal objects into parts, the question becomes– Is there anything absolutely simple in the physical world?– a simple, indivisible substance?– a simple physical individual? The question cannot, in any case, be answered by logic, and the answer seems doubtful from a metaphysical point of view. The sanest solution is, that what appears to be unanalyzable is probably merely unanalyzed; what is taken in analysis as simple lies at the limit of the process, but is not– in its nature– incapable of further characterization or partition.

Mr. Bertrand Russell, holding the view that there must be simples, declares:²⁸
“When I speak of ‘simples’ I ought to explain that I am speaking of something not experienced as such, but known only inferentially as the limit of analysis. It is quite possible that, by a greater logical skill, the need for assuming them could be avoided. A logical language will not lead to error if its simple symbols (i. e., those not having any parts that are symbols, or any significant structure) all stand for objects of some one type,²⁹ even if these objects are not simple. The only drawback to such a language is that it is incapable of dealing with anything simpler than the objects which it represents by simple symbols. But I confess it seems obvious to me (as it did to Leibniz) that what is complex must be composed of simples, though the number of constituents may be infinite. It is also obvious that the logical uses of the old notion of substance (i. e., those uses which do not imply temporal duration) can only be applied, if at all, to simples; objects of other types do not have that kind of being which one associates with substances. The essence of a substance, from the symbolic point of view, is that it can only be named– in old-fashioned language, it never occurs in a proposition except as the subject or as one of the terms of a relation. If what we take to be simple is really complex, we may get into trouble by naming it, when what we ought to do is to assert it. For example, if Plato loves Socrates, there is not an entity ‘Plato’s love for Socrates,’ but only the fact that Plato loves Socrates. And in speaking of this as ‘a fact,’ we are already making it more substantial and more of a unity than we have any right to do.”

If we reject Mr. Russell’s notion that there are absolute simples, there is no reason why we should get into trouble by treating a complex– e. g., a fact– as if it were simple. A proposition analyzes a fact; it exhibits terms in relation, or qualities as characterizing a subject. But, so far as logic is concerned, we can substitute an unanalyzed x for any fact, and begin our analysis at a different level. The corollary of the view that there are– for logic at least– no absolute simples, would be that any complex could be treated as if it were simple.

A tangle of ambiguities clusters about the idea of definition. We find Aristotle holding, on the one hand, that the primary truths of science are indemonstrable definitions stating what the essences of certain things are;³⁰ and, on the other, we read in the works of some modern logicians that “a definition is, strictly speaking, no part of the subject in which it occurs” and “is not true or false.”³¹ This wide divergence of opinion springs from two utterly different interpretations of the term “definition.” Aristotle is speaking always of real definitions, while recent logic tends to treat all definitions as verbal, or nominal.

A verbal or nominal definition is a declaration of intention to use a certain word or phrase as a substitute for another word or phrase. The original word or phrase is the definiens; the substituted one, the definiendum. Such definitions have value, usually, as linguistic or symbolic conveniences. If an expression becomes too long to be easily handled, we can– by stating our intention to do so in a nominal definition– replace this expression by a shorter one. In writing on logic, for example, where reference is frequently made to “the type of relation such that if a R b and b R c, then a R c,” I can shorten my exposition by substituting for this expression the term “a transitive relation.” Or, in Euclidean geometry, I can replace the phrase, “lines that do not intersect in a plane” by the words “parallel lines.”

What is important about a verbal definition is this: (1) the meaning of the definiendum is not independent of that of the definiens, i. e., the expression defined has literally no other meaning, in the discussion, than that given arbitrarily to it; (2) a verbal definition is neither true nor false, and therefore (3) cannot serve as a premise for deductions.

It is often said that definitions cannot be questioned. This refers to nominal definitions. Thus, if I take the word “good” to mean “any object of desire,” it would be vain– and meaningless– to question my statement that “any object of desire is good,” for I am not asserting a truth about objects of desire or goods. All I am saying is that this is how I intend to use the word “good,” and one may use words in any way he chooses. However, I could not conclude from this definition that “no objects of desire are evil.” The only legitimate process which the definition permits is verbal substitution; and all that this statement can mean is that “no objects of desire are non-good, i. e., are not objects of desire.”

In adopting this notion of definition, the authors of Principia Mathematica give the following explanation:³²

“A definition is a declaration that a certain newly-introduced symbol or combination of symbols is to mean the same as a certain other combination of symbols of which the meaning is already known. … We will give the names of definiendum and definiens respectively to what is defined and to that which it is defined as meaning. We express a definition by putting the definiendum to the left and the definiens to the right, with the sign ‘=’ between, and the letters ‘Df’ to the right of the definiens. … An example of definition is, p ⊃ q = ~ p ⋁ q Df.

“It is to be observed that a definition is, strictly speaking, no part of the subject in which it occurs. For a definition is concerned wholly with the symbols, not with what they symbolize. Moreover it is not true or false, being the expression of a volition, not of a proposition. (For this reason, definitions are not preceded by the assertion-sign.) Theoretically, it is unnecessary ever to give a definition: we might always use the definiens instead, and thus wholly dispense with the definiendum. Thus although we employ definitions and do not define ‘definition,’ yet ‘definition’ does not appear among our primitive ideas, because the definitions are no part of our subject, but are, strictly speaking, mere typographical conveniences. Practically, of course, if we introduced no definitions, our formulæ would very soon become so lengthy as to be unmanageable; but theoretically, all definitions are superfluous.”

A real definition does not merely provide verbal substitutes for expressions already introduced: (1) it states that two expressions, each of which has an independent meaning, are equivalent to one another; (2) it is either true or false, and (3) can therefore serve as a premise in deductions.

Such definitions– which would not be definitions at all, but statements of equivalence, for Principia Mathematica, and which would be the only kind of definitions worth taking seriously for Aristotle– could be questioned and even disproved. If the term “good,” for example, has a meaning of its own, and the phrase “object of desire” has also a meaning of its own, the real definition– “the good is any object of desire”– might be false. We can ask, is it true that what is meant (independently) by “good” is the same as what is meant by “any object of desire”? And, if evil, as most religions maintain, can be desired, this definition cannot stand as “a primary and indemonstrable truth.”

In the same way, the definition of implication given in Principia Mathematica, that “p implies q” is a verbal substitute for “either p is false or q is true,” can be questioned if we treat it (though it is not intended to be taken in this way) as a real definition. Is this really what implication means?– is it a proper analysis of this idea, considered as having an independent meaning? We can then bring forward the paradoxes that a false proposition implies any proposition, and a true proposition is implied by any proposition; “today is Wednesday,” whether true or false, implies that “Columbus discovered America in 1492.” Does the proper meaning of implies permit these paradoxes?³³ But these objections are precluded by the theory of definition itself which Principia Mathematica advances. “Implies” is simply shorthand for the longer expression, “either p is false or q is true.”

This divergence in the use of the term “definition” leads to the question– What is the purpose of a definition?

We can differentiate its logical from its psychological purpose. Psychologically, a definition serves to make the meaning of a concept, or group of concepts, clear to the mind. Logically, its purpose is to aid in the adequate exposition of the subject-matter in question. That is, a good definition from a logical point of view would be one which, taken in conjunction with certain premises, leads to true conclusions (or theorems) completely covering, so far as possible, the field under investigation. Thus if, in the field of economics, money were defined as “any imperishable medium of exchange,” paper currency would be left out; we could not, by this definition, cover most of the intangible transactions of business, and the definition would be logically bad because of its inadequacy. In the same way, if we defined number so that infinite numbers were excluded, the definition would be inadequate. The logical test of a good definition is not, then, its clarity to the mind, but its ability to give us what we want in our conclusions.

Now, obviously the process of definition cannot be led back ad infinitum. It must terminate in the undefined, or indefinable. Any discussion must begin with certain primitive concepts from which it proceeds in an orderly way. Psychologically, it would always be best to take as undefined those concepts which are most clearly understood; but logically, such a procedure might be very limiting indeed, being confined by the circumference of the understanding to which we appeal.³⁴ What is best taken as primitive (or undefined) may therefore be some extremely complex idea, so far as understanding it is concerned. A glance at any mathematical system easily convinces us that this is the case. Mr. H. M. Sheffer,³⁵ for example, uses as primitive in logic (and defines all other propositional relationships through it) the idea of rejection, i. e., “neither p nor q,”– a notion much less easily grasped by most people than that of “p or q,” “p and q,” etc.

How, then, is the undefined understood, if not through definition? The only answer is, by apprehending it, or “demonstrating” it, not in the sense of proving but of showing what it is. If I define the parallelism of lines as non-intersection, and define the intersection of lines as “having a point in common,” I cannot (perhaps) go further and define what is meant by having a point in common, but must show or exhibit to you what I mean. The undefined, therefore, is not that which is without meaning, but that which has its meaning by an external reference to the realm of objects thought about.

Returning to the distinction between nominal and real definitions, we see that nominal definitions can fulfill only a psychological purpose: a (more or less) clear significance is assigned as a matter of convenience to an expression which is otherwise taken as meaningless. Nominal definitions do not analyze the ideas they define, in the sense of stating that these ideas exhibit such and such components; for they are not statements of truths about these ideas, as are real definitions. Nor could the question as to whether they do or do not adequately cover the subject-matter be raised.

Having introduced definitions as purely nominal, many writers tend later to treat them as if they were real– as if they conveyed some information about the concept defined, and so, analyzed this concept. Ethical philosophers who nominally define “the good” as “any object of desire” often end by arguing that this is the only meaning “good” can have, since everything that is good is an object of desire, and there is no object of desire that is not good. Tacitly they assign an independent meaning to the term “good”; and their erstwhile nominal definition becomes an important truth in their minds.

The same tendency is illustrated in logic by the further remarks on definition made in the Introduction to Principia Mathematica³⁶

“In spite of the fact that definitions are theoretically superfluous, it is nevertheless true that they often convey more important information than is contained in the propositions in which they are used. This arises from two causes. First, a definition usually implies that the definiens is worthy of careful consideration. Hence the collection of definitions embodies our choice of subjects and our judgment as to what is most important. Secondly, when what is defined is (as often occurs) something already familiar, such as cardinal or ordinal numbers, the definition contains an analysis of a common idea, and may therefore express a notable advance.”

A nominal definition can never be an analysis of an idea; a real definition is always an analysis of an idea.³⁷ For, a real definition states an equivalence between expressions of independent meaning, each of which is undefined and can be merely exhibited as having that meaning.

By what warrant can such statements of equivalence be called definitions if both sides of the equation are undefined?– if one side does not assign a meaning to the other? The warrant is this: (1) they are primary, i. e., they come first in the discussion or proof, and any other statements of equivalence that occur follow from them; (2) though both sides of the equation are undefined, one meaning elucidates the other; i. e., any idea is more clearly understood if it is analyzed or expressed in different terms; thus, they serve the psychological purpose of definitions. They are in Aristotle’s sense “primary and indemonstrable truths.” Recent logic, however, would speak of such definitions as “axioms,” “primitive propositions” or “postulates,” reserving the term “definition” for the assignment of a meaning, in terms already introduced, to an expression that otherwise has no meaning.

The usual rules of definition (as given by Mr. Joseph)³⁸ are as follows:
(i) A definition must give the essence of that which is to be defined.
(ii) A definition must be per genus et differentiam.
(iii) A definition should not be expressed in obscure or figurative language; and it is sometimes added that the definition should be clearer and simpler than the thing to be defined.
(iv) A definition must not be in negative where it can be in positive terms.
(v) A definition must be commensurate with that which is to be defined.
(vi) A definition must not, directly or indirectly, define the thing by itself.

The first two rules, that the definition must state the essence of what is defined and must be per genus et differentiam, are relevant only to definition in the Aristotelian sense. If we reject the notion of a fixed essence, peculiar to a thing and different from its properties– which would also entail the rejection of a peculiar genus and a rigid differentia,– these two rules will not be applicable. Any concept equivalent to the one defined could be used as a definition, and the choice would depend on the adequacy of the definition to cover the subject matter we wish to include in the discussion. The choice, in other words, would be pragmatic. For the purposes of a lawyer it would be sufficient, perhaps, to define an argument as “a discussion in which various sides of an issue are put forward;” but such a definition would never do for a logician.

The third rule turns upon the psychological purposes of definition. It must always be recognized that the obscurity or clarity of the definiens is a relative matter; what is obscure or clear to one person may not be so to another. Figurative definitions ought not to be completely excluded; e. g., “money is the root of all evil.” They are illuminating for some purposes, for those of the poet or stylist, though scarcely for the scientist or philosopher.

Rule iv has its basis in the same considerations as the previous rule; namely, in the ambiguity and obscurity of negative concepts. Strictly speaking, a negative concept, construed as infinite or purely negative, includes, as we have seen, everything excluded by the positive. Thus, if I were to define a chair as a “non-table,” this might refer to a bed, a book, or an infinity of other things. In the same way, if parallel lines are defined as lines not-intersecting in a plane, a series of concentric circles would conform to the definition-and the question arises whether this is the meaning of parallel lines.³⁹

The fifth and sixth rules, like the third and fourth, apply to definition in any sense, not only to the Aristotelian type of definition. Rule v states what we have considered in the previous section on nominal and real definition: that the definiens must be coextensive with the definiendum, either in the sense of equivalence, i. e., of implying and being implied by it, where the definition is real; or in the sense of being a verbal substitute for it, where the definition is nominal. No definition could violate this rule.

The sixth rule, that the definition must not be circular, is equally important with the fifth. Whether we view definition as resting on real analysis, or as a provision for verbal substitution, no definition should use the concept to be defined as the defining notion. Take the following flagrant example: “Justice is the doing of just acts.” Considered as an analysis of “justice,” this is faulty, since the unanalyzed idea “justice” is represented as the chief element in the analysis. We are still faced with the same unanalyzed notion. Considered as a verbal substitute for the word “justice,” which is otherwise meaningless, the definition assigns no meaning to this word. As an example of circular definition, Aristotle gives the following: “supposing anyone had defined the sun as a ‘star that appears by day,’ … in bringing in ‘day,’ he brings in the sun.”⁴⁰

The fault of defining a thing by itself is said to be committed (1) whenever the term to be defined, or any other term synonymous with it, is introduced in the definition (as illustrated above); or (2) when we define relative terms by their correlatives, or counter-alternative terms by one another.⁴¹

Relative terms are those into whose meaning a relation to some other term enters. Thus, parent, wife, successor, etc. are relative terms; the correlatives being, respectively, offspring, husband, predecessor. If a wife is defined as “a woman who has a husband,” the definition is circular, since “having a wife” enters into the meaning of “being a husband.” The proper way to treat such terms is to define the relation between them; in defining the relation, we also define its converse and hence, both correlative terms at once. “A is the wife of B” means “A is a woman who has entered into a marriage contract with a man B.” The converse of this relation is, “B is a man who has entered into a marriage contract with a woman A;” and this defines “B is the husband of A.”

The notion that no part of the expression to be defined can appear in the definiens has often been construed too rigidly, so that definitions which are not really circular are excluded on the grounds of circularity. I could correctly define “the first element in a series” as “any element of that series such that all the other elements follow it;” e. g., Adam was the first man because all other men were his descendants. Here the definiendum is “the first element in a series,” and the words “element” and “series” are repeated in the definiens. Is the definition circular? If the rule is literally construed to mean that no word can appear on both sides of the defining equation, the definition is circular. But, plainly, we are not defining element or series, though these terms enter into the definiendum. We are defining one part of the whole expression, “the first element in a series,” namely, first.

Mr. W. E. Johnson remarks on the rule forbidding circularity in definition:⁴²
“In this connection it is worth noting that, when what has to be defined is a verbal phrase rather than a single word, we may italicize– so to speak– that part of the phrase for which an explanation is asked. In such cases the remaining components of the phrase may be, and generally ought to be, repeated in the phrase constituting the definition. … this mode of definition, so far from being a ground of condemnation, exactly answers in the most adequate sense the requirements. The more exactly we repeat in our definition the actual words and their form of combination, used in the phrase to be explained, the more precisely do we meet the demands for an explanation.”

Thus, it would be less accurate to define “a regular student in Harvard College” as “a young man who enters Harvard College by examination,” than to repeat verbatim, with the exception of the italicized word, the other words in the definiendum: i. e., “a regular student in Harvard College” is “a student in Harvard College who enters by examination.” For, there are old men as well as young ones who enter in this way.

NOTES
1 The Aristotelian doctrine of the categories, as well as that of the predicables is usually given a place in treatises on logic. But the bearing of this doctrine on purely logical issues is very remote indeed; and for that reason it is not treated at length here. The theory of the predicables, though its background is metaphysical, does constitute in Aristotle’s mind a part of the analysis of propositions. It is an inquiry into the formal types of relationship which the predicate bears to the subject in any proposition; i. e., is the relationship necessary or accidental?– and if it is necessary, is it that of definition, genus, differentia, or property? The doctrine of the categories, on the other hand, is not a part of the analysis of propositions, or of the formal types of relationship which a predicate may bear to the subject in a proposition. It examines the meaning of “words uncombined,” and lists “the widest predicates which are predicable essentially of the various namable entities, i. e., which tell us what kinds of entity at bottom they are.” In the Metaphysics, Aristotle speaks of the categories as a catalogue of the “various meanings of being.” (The quotations are from W. D. Ross, Aristotle, pp. 21 ff.) In other words, the categories form a list of the fundamental kinds of realities with which the metaphysician must deal; the doctrine belongs to general philosophy, and not specifically to logic. The ten categories as given by Mr. Ross (op. cit., p. 21) are: (1) substance, e. g., man; (2) quantity, e. g., two cubits long; (3) quality, e. g., white; (4) relation, e. g., double; (5) place, e. g., in the Lyceum; (6) date, e. g., yesterday; (7) position, e. g., sits; (8) state, e. g., is shod; (9) action, e. g., cuts; (10) passivity, e. g., is cut. Substance is divided into (a) primary substance, i. e., the individual, which is always a subject and never a predicate, and (b) secondary substance, i. e., the species and genera in which the primary substances are included.
2 Aristotle, Topica, 101b-39, W. D. Ross trans., Oxford Press, 1928.
3 Aristotle, Topica, 101b-19.
4 Porphyry (A. D. 233-304) in his Isagoge (An Introduction to the Categories of Aristotle) “hopelessly muddled” the doctrine of the predicables, according to W. D. Ross, by adding the idea of species in place of definition. Porphyry’s list, which was more current than Aristotle’s almost up to the present time, was: (1) species, (2) genus, (3) differentia, (4) property, (5) accident.
5 Aristotle, Topica, 102a-31.
6 The idea of the differentia, however, went through an evolution in Aristotle’s mind, which is traced by W. D. Ross, Aristotle, Scribners, 1924, p. 57, as follows: “In the present passage (the Topics) the distinction between genus and differentia is slurred over. Differentia, like genus, is treated as being wider than that whose differentia it is. The implied doctrine is one which we find also in the Posterior Analytics (96a-24-b14), that a definition is made by collecting attributes each wider than the term to be defined but collectively coextensive with it. In the Metaphysics (Z. 12), on the other hand, Aristotle lays it down that each differentia stated should be a differentiation of the previous differentia, and that the last differentia should be coextensive with the definiendum.”
7 Aristotle, Topica, 143a-29.
8 The distinctions between the specific difference and the generic difference, the summum genus and the proximate genus should be noted. The proximate genus is the genus immediately above the kind defined; e. g., “animal” is the proximate genus of “ox”, but “organism” is the proximate genus of “animal,” since both animals and plants are organisms, A summum genus is one above which no further genus stands, e. g” “substance,” “thing,” “being,” A specific differentia is one which marks off kinds within the proximate genus; e. g., “having three sides” distinguishes triangles within the proximate genus “plane figure” and is a specific difference; whereas “plane” is, with reference to “triangle,” a generic difference, since it distinguishes the genus “plane figure” from other geometrical figures, namely, linear and solid, within the higher genus “geometrical figure.” An infima species is one which is not differentiated further into species; only individuals, and not species, stand under it. Thus, “rat” is an infima species of the genus vertebrate.
But it is obviously a question whether there are any infima species at all; and for a logic like that of Leibnitz, which takes the individual to be an infinitely complex set of predicates, individuals alone could be ranked as infima species. There seems to be no reason, except the doctrine of fixed species, why there should not be species within species ad infinitum.
9 Aristotle, Topica, 102a-17.
10 Porphyry elaborates the Aristotelian theory of the property by departing from the idea that the property must be equivalent to the subject of which it is predicated; he was followed by the mediaeval logicians. He distinguished (1) properties that belong to the species alone, though not to all its members; e. g., only human beings are mathematicians or philosophers, though not all human beings have these properties; (2) properties that belong to all the members of a species, but not to them alone; e. g., all men have five toes, but so also do all monkeys, and some other animals; (3) properties that belong to a certain species only and to all members of this species but not always, e. g., “white-haired” as connected with the species of aged-people; (4) properties that belong always to all members of a certain species, and to it alone. The latter is the same as the Aristotelian notion of property.– These trivial distinctions illustrate the sterility of much of the post-Aristotelian logic
11 W. D. Ross, Aristotle, p. 45, characterizes this necessity of the first four predicables in relation to their subject as follows: “… they are (1) true of every instance of their subject. But (2) the relation which they state between subject and predicate must be a per se or essential relation.”
12 This is Mr. H. W. B. Joseph’s interpretation (op. cit., ch. IV). Mr. Joseph’s discussion of the predicables is admirable, and should be referred to by the student.
13 See above, pp. 227-230.
14 Aristotle, Topica, 102b-4.
15 Aristotle discusses the relation between accidents and properties as follows (Topica, 102b-21): “It is clear on the face of it that there is nothing to prevent an accident from becoming a temporary or a relative property. Thus the sitting posture is an accident, but will be a temporary property, whenever a man is the only person sitting, while if he be not the only one sitting, it is still a property relatively to those who are not sitting. So then, there is nothing to prevent an accident from becoming both a relative and a temporary property; but a property absolutely it will never be.” The notion that properties may be temporary and relative is an abandonment of the idea of a strictly necessary connection between the subject and its properties; these are really accidents; only absolute properties are necessary to the subject, and are properties in the strict sense.
16 Op. cit., p. 94.
17 Porphyry introduced a distinction between separable and inseparable accidents. The separable accident would be the accident as defined by Aristotle in the passage quoted above from the Topics– that which is sometimes present and sometimes absent in the case of an individual or a class. The inseparable accident would belong to all members of a class (or invariably to an individual), but would not be necessarily connected with the class-concept; i. e., it would be a part of the comprehension rather than of the connotation (see above, p. 244 ff.) of the class-concept. Thus, blackness is sometimes said to be an inseparable accident of crows, since all crows are as a matter of fact black, but this does not seem to be necessary to them. The idea at the back of this notion of a separable accident seems to be that something which is “always true” of a subject need not be necessarily (intensionally) connected with this subject; “always true” merely indicates a conjunction in fact, and not a necessity.
18 Mr. H. W. B. Joseph’s illustration.
19 Cf. R. W. Henger, College Zoology, Macmillan, 1920, p. 689.
20 Such classifications vary. The one above is taken from R. W. Henger (op. cit.), pp. 2 ff. It is written for elementary students.
21 Mr. H. W. B. Joseph (op. cit., p. 109) makes several interesting objections to this type of division. “A negative conception affords no basis for further subdivision, and a division which attempts to classify by dichotomy is forever subdividing negative conceptions.” This objection would hold if, as Mr. Joseph maintains, a division must always “exhibit our various species as alternative developments of a common notion.” Certainly it is more useful to exhibit alternative positive developments within the genus; but there seems to be no formal reason why a negative concept cannot be further subdivided, unless we accept the idea that infinite terms are meaningless. Strictly speaking, of course, the infinite term non-vertebrates includes, not only crayfish, spiders, etc., but also tables, chairs, moons, and suns. All this irrelevant lumber can be eliminated if we take this negative term to mean what is obviously intended in the classification, non-vertebrate animals. But in any case a negative class is logically divisible.– Mr. Joseph also points out that the negative classes overlap some of the positive classes in dichotomous division; he could have added that they also overlap other negative classes. Thus, non-mammal, taken to include everything that is not a mammal, i. e., as an infinite class, would overlap molluscs, arthropoda, insects, etc. in the first table above; and this infinite class would also overlap non-vertebrates, non-molluscs, etc. This is avoided, however, if each negative class is construed (as is obviously intended) as being conjoined with the positive and negative classes standing above it. For example, the class non-reptiles, in the dichotomous division above, means non-reptiles that are non-mammals and also vertebrates, e. g., birds, fishes. With this construction, it does not overlap insects, non-insects, etc.– In general we can say that where the negative classes are taken as infinite (as including everything excluded by the co-ordinate positive classes) there must be overlapping in a dichotomous division; but this is not the meaning usually given to the negative classes.
22 W. E. Johnson, op. cit., part I, p. 174. “I propose to call such terms as colour and shape determinables in relation to such terms as red and circular which will be called determinates; and in introducing this new terminology, to examine the distinction between the relation of red to colour and the relation of Plato to man.” In Aristotelian terminology, the determinable would be the genus of which the determinate is a species.
23 Mr. W. E. Johnson, op. cit., part I, 106 ff., discusses briefly and brilliantly the notions of analysis and synthesis, together with their relation to definition. What is said in the text largely parallels his discussion.
24 Two extreme metaphysical positions can be taken here: (1) that nothing but the components are real, e. g., Hume, who says (Appendix to the Treatise of Human Nature): “all our distinct perceptions (sense-data) are distinct existences, and the mind never perceives any real connection between distinct existences;” (2) that nothing but the whole is real, and that the components are falsified and destroyed when considered apart from the whole, e. g., Absolutism.– Kant, adopting Hume’s alternative, that the manifold of disparate sensations is the originally given element in knowledge, makes all synthetic wholes creatures of the mind’s activity. He has fastened on philosophy the idea that a synthetic whole requires an act of synthesis in order to be in any sense. From a logical point of view, the synthetic-analytic character of wholes must be considered apart from all these metaphysical and epistemological theories.
25 Op. cit., p. 107
26 What we have termed “homogeneous partitive analysis” is called by Mr. Johnson simply “partition.” “Heterogeneous partitive analysis” corresponds roughly to what he calls “resolution.” “… resolution means the exhibition of what is presented as simple in the form of a composite of which the components are assigned.” (Op. cit., p. 111.)
27 Mr. Johnson, loc. cit., limits the term” analysis” to this meaning: “… I should restrict the word ‘analysis’ to a process which is distinctively logical, and which assumes its simplest form when we combine various adjectives as predicable of one and the same substantive, by means of the conjunction ‘and.’’’
28 B. Russell in the article, Logical Atomism, in Contemporary British Philosophy, ed. by J. H. Muirhead, Macmillan, 1924, p. 375.
29 Mr. Russell is referring to “types” as defined in the “theory of types;” cf. pp. 452 ff.
30 Cf. Analytica Posteriora, 90b, 23-33.
31 Whitehead and Russell, Principia Mathematica, 1st ed., p. 11.
32 1st ed., p. II.
33 Cf. above, pp. 228-229.
34 Mr. W. E. Johnson, viewing the indefinable largely from the psychological angle, remarks (op. cit., part I, p. 106): “A certain misunderstanding as to what in logic is meant by the indefinable must here be removed; for it has been frequently supposed that the indefinable means that which is admittedly not understood. But so far from meaning the ‘not-understood,’ the indefinable means that which is understood; and philosophy or logic may ultimately adopt a term as indefinable only where, because it is understood, it does not require a further process of definition.”
35 A Set of Independent Postulates for Boolean Algebras, Trans. Amer. Math. Soc., vol. XIV, No.4, pp. 481-488.
36 1st ed., p. 12.
37 With reference to the “nominal definitions” of Principia Mathematica, the present writer believes them not to be nominal at all, as they seem. (The point is difficult and may be passed over by the elementary student.) From among the equivalences which hold for truth-functions of propositions, e. g., ~p ⋁ q ≡ p ⊃ q ≡ ~(p ~q), etc., certain ones are selected and treated as nominal definitions, with the astonishing result that what is originally introduced as a nominal definition, p ⊃ q = ~p ⋁ q Df, comes out *4.6 as a true statement of equivalence, p ⊃ q. ≡ ~p ⋁ q– The present writer holds that, where any phrase has a structure, it cannot be nominally defined by a phrase with a structure. What the so-called nominal definitions of Principia Mathematica, such as that of “implication” given above, really mean is that a propositional structure like that on the left side is equivalent to the different structure on the right side. Only a single word (or symbol) could be given a nominal definition in the strict sense, for it has no structure. The existence of a structure in a propositional expression is already the existence of an independent framework of meaning; that which has an independent meaning can be analyzed, or shown to be equivalent to, some other independent meaning, but it cannot be nominally defined.
38 Op. cit., 1st ed., p. 97ff. The order is altered here.
39 Some concepts of negative form, those signified by words with the prefixes in, un, etc., have through use a positive significance, e. g., injustice, instability. Such terms are sometimes wrongly taken to be negative, and are said to be susceptible of a negative definition. Mr. Joseph quotes Hobbes’s definition of injustice as permissible: “Injustice is the not keeping of covenant.” But here the same objection to negative definitions holds: the not-keeping of covenants could, logically, be almost anything. A negative definition is acceptable only where there is a tacit limitation of the meaning of the negative to something specific and positive. Hobbes’s definition of injustice is really understood to mean, “injustice is the violation of covenant.” Violation is a positive idea, as positive as keeping a covenant. “Privative” concepts are those (not negative in form) such as “blindness,” “baldness,” etc., which mean “the privation or absence of some quality.” It is generally said that privation or absence is a negative notion, and that such concepts must be negatively defined– since negation is a part of their meaning. “Blindness is not seeing,” “baldness is not having hair.” But there is a question whether “privation” is essentially negative; the loss of one’s sight or one’s hair seems a positive phenomenon. If this is so, to define blindness as “the loss of sight” is not to give it a negative definition. In general, negative definitions should be avoided.
40 Topica, 142a-35.
41 Counter-alternatives are two mutually exclusive terms such that everything to which they are applicable is an example of either one or the other; e. g., odd and even as predicated of numbers, male and female as predicated of animals, straight and curved as predicated of lines, etc. Now, if we use such counter-alternatives to define one another, as is only natural, our definition will be circular. We might say “an odd number is one that immediately follows an even number (or the number zero) in the series of positive integers,” and “an even number is one that immediately follows an odd number.” To avoid this circularity, one of the counter-alternatives must be independently defined. Thus, if we defined an even number as “any number divisible by two,” the previous definition of an odd number could stand. (On the surface, at least, there is no circularity here, though further analysis might reveal a circle.) Such counter-alternatives offer traps in definition, the escape from which often requires much ingenuity, and it may sometimes be necessary to take both alternatives as undefined.
42 Op. cit., Part I, p. 104.
 
Ralph M. Eaton; General Logic; 1931; ch7

AVOIDING AMBIGUITY; DEFINITIONS

When we encounter words that cause confusion because their meanings are ambiguous, it is often helpful to define them. A traditional way of characterizing the definition of a word is to say that the definition is a verbal formulation of its meaning. However, the word “meaning” itself is ambiguous. Thus a general term may be said to mean each individual thing to which it applies (for example, the general term “man” means Socrates, Caesar, and each other man). This is called extensional meaning, and the totality of things to which the general term applies is called the extension of the term. But also a general term may be said to mean those characteristics which anything must possess in order that the term correctly apply to it (for example, the term “bachelor” means being a man and being unmarried). This is called intensional meaning, and the totality of characteristics which anything would have to possess in order that the term apply to it is called the intension of the term. A definition of a general term tries to specify the intension; the definition does not tell us what the extension is.

From another point of view, however, we can characterize definitions without employing the term “meaning.” We may say that a definition of a word is a recipe for eliminating the word by paraphrasing, that is, for transforming sentences containing the word into equivalent sentences that contain other expressions instead. Recipes of this kind are of especial practical value when they tell us how to eliminate ambiguous, confusing, or unfamiliar words by paraphrasing— replacing them with clearer or more familiar words.

The most fundamental way of explaining a word is to give examples. Sometimes we do this by pointing to visible examples. When a child asks ‘‘What’s a dog?” we respond by pointing to Fido, Rover, and Bruno. Some philosophers have called this procedure ‘‘ostensive definition,” but it is better to call it merely ostensive teaching of words. This ostensive procedure differs from definition in that it gives no recipe for paraphrasing the word. Although explaining a word by giving examples often can be indispensably valuable, it is not the same as giving a definition. Sometimes a definition is much more helpful than a list of examples.

In ordinary discourse we often express definitions in ways that do not clearly show that they are definitions. Wishing to define the word “dormouse,” a speaker may say, “A dormouse is a small hibernating European rodent resembling a squirrel.” The hearer is then expected to realize that the speaker is intending to define the word “dormouse,” rather than intending to make an ordinary statement about dormice (as he would be doing if he said, “Dormice are rather prolific animals”). A careful speaker can make his intention clearer by stating his definition in such a way as to leave no doubt that it is a definition. If he says ‘‘The word ‘dormouse’ means ‘small hibernating European rodent resembling a squirrel,’” then he has made it perfectly clear that he is defining the word. Moreover, here he has given what is called an explicit definition, that is, a definition in which the definiendum (the expression being defined) is declared to be replaceable by another explicitly given expression, the definiens (that which does the defining.)

Definitions that are useful in preventing ambiguity may be subdivided into two types. Some of them serve the purpose of describing the meaning that a word already has in language. We might call these analytical definitions. In giving this kind of definition of a word, the speaker does not aim to change its meaning; he aims only to characterize the meaning it already has. Dictionary definitions are of this type. When a definition has this purpose, we can properly ask whether the definition is correct or incorrect.

In order to be correct in its description of the meaning of a word, an analytical definition must not be too broad; that is, it must not embrace things that do not really belong. (To define “pneumonia” as “disease of the lungs” would be too broad, for there are many lung diseases besides pneumonia.) Also, in order to be correct in its description of the meaning of a word, an analytical definition must not be too narrow; that is, it must not exclude things that really belong. (To define “psychosis” as “schizophrenia” would be too narrow, for there are other kinds of psychoses.) Sometimes an incorrect definition errs by being too broad in one respect and also too narrow in some other respect (for instance, defining “liberalism” as “the view that the power of the government should be increased”).

Furthermore, analytical definitions should be clear enough to be understood by those for whom they are intended; otherwise they are of little use. When in his dictionary Dr. Johnson defined a net as “anything made with interstitial vacuities,” his readers would not have understood the definiens as well as they already understood the definiendum; the definition uses murky works to explain a relatively clear one and so is not helpful.

Finally, a definition cannot serve much useful purpose if it is circular, that is, if the definiendum occurs within the definiens in such a way that no one could understand the definiens who did not already understand the definiendum. For example, to define “straight line” as “the line along which a ray of light travels when it goes straight” is circular and uninformative.

Traditional logic used to prescribe additional rules for definitions, including the rule that definitions should be given by genus and species and the rule that a definition ought not to be negative. However, these rules need not always be obeyed. Granted, in giving a definition it often is helpful to proceed by genus and species, that is, first saying what general kind of thing the word means and then saying what the specific form is. But not all legitimate definitions follow this pattern. Also, it is often wise to avoid definitions couched in negative terms (“A lion is a big cat; not a tiger, not a leopard, not an ocelot”), for such definitions are likely to be too broad. But some negative definitions are perfectly legitimate.

A second type of definition useful in preventing ambiguity is the stipulative definition, whose purpose is to declare how a speaker intends that a certain word, phrase, or symbol shall be understood (“Let ‘S’ mean ‘Samoans’”; “Let ‘heavy truck’ mean ‘truck that can carry a load of 5 tons or more’”; etc.). Perhaps the expression being defined is one that previously had no meaning, or perhaps it had a different or a vaguer meaning. At any rate, the point of the stipulative definition is that the expression now is deliberately endowed with a particular meaning. Obviously, a stipulative definition cannot be of much use if it is unclear or circular. However, we do not have to worry about whether it is too broad or too narrow, for that sort of correctness cannot pertain to stipulative definitions. A stipulative definition is arbitrary, in that it expresses only the speaker’s intention to use the word in the stipulated manner, and the speaker is, after all, entitled to use it in any desired way, so long as it does not cause confusion.

In order to avoid causing confusion, however, a stipulative definition should not assign to a word that already has an established meaning some new meaning that is likely to be confused with it. Consider the following dialogue:

Smith: General Green is insane, you know. He ought to be dismissed.
Jones: He is? I agree that we should not have insane persons serving in the Army. But how do you know he’s insane?
Smith: It’s obvious. He says he believes in extrasensory perception, and according to my definition— surely I’m entitled to use words as I please— anyone who does that is insane.

Here the stipulative definition is used to promote ambiguity rather than to prevent it. In the ordinary sense of the term “insane,” Jones agrees with Smith that insane persons ought not to be generals. But Smith offers no evidence that General Green is insane in this sense. All that Smith shows is that the general is ‘insane’ in a special, idiosyncratic sense of the word. From that, nothing follows about whether he ought to be dismissed. Smith is causing confusion by failing to keep distinct these two very different senses of the word; this happens because he fails to recognize the difference here between a stipulative and an analytical definition.

Confusion can be caused in another way by a stipulative definition if a word or symbol that purports to name some individual thing (such a word or symbol is a singular term) is introduced even though it is not known that there is any such thing. Suppose I say “let ‘n’ stand for the largest whole number.” And then I go on to use this symbol “n” in making supposed assertions about this largest whole number. Here I am guilty of constructing a confused definition, for there is no largest whole number; hence, I have no right to introduce and use a singular term for this nonentity. I may become badly confused if I assume that this definition is enough to entitle me to start talking about this largest whole number as if it existed. There is no such number, and a mere definition cannot create a number or any other object.

The two kinds of definitions mentioned so far both aim to inform us about verbal usage. The stipulative definition expresses a speaker’s intention henceforth to use his definiendum in a certain way, and the analytical definition describes the way in which the definiendum already is used in language. These two kinds of definitions are valuable in helping to prevent ambiguity.

It would be a mistake, however, to suppose that everything called a definition belongs to one of these two kinds. In fact, the profoundest and most valuable definitions usually do not fit tidily into either kind. When Newton defined force as the product of mass times acceleration, when Einstein defined simultaneity of distant events in terms of the transmission of light rays, and when Whitehead and Russell defined zero as the class of all empty classes, these important definitions expressed stipulations about how Newton, Einstein, and Whitehead and Russell proposed to use their terms. But these definitions did not merely do this; they also reflected previously established usage. What these definitions did was to propose new verbal usages growing out of the previously established usages. It was felt that these new usages perfected tendencies of thought implicit in the old usages and offered more insight into the subject matter being treated.

We might give the name revelatory definitions to definitions like these, which do not fit into either of the two categories of stipulative and analytical. Revelatory definitions constitute a third category. Further examples of revelatory definitions can be found in other, diverse fields. For example, when a nineteenth-century writer defined architecture as frozen music, he was not trying to describe how the word “architecture” is used in our language. (He took it for granted that his readers would know what kinds of constructions are considered architecture.) Nor was he proposing some arbitrary new usage. We should not censure his definition on the ground that it is unhelpful for the purpose of preventing ambiguity; that is not the purpose of this kind of definition. This definition is a metaphor, and it suggests a new way of looking at architecture, comparing the structural organization of the parts of a building with the structural organization of the parts of a musical composition. In trying to decide whether the definition is a good one or not, we must reflect about the extent and validity of this comparison between music and buildings; the definition is a good one if and only if the comparison is revealing.

Or again, when a writer on psychoanalysis says that man is to be defined as the neurotic animal, this definition does not have the purpose of explaining the meaning of the word “man” to someone unfamiliar with it. Instead, its purpose is to call attention to something about human beings that the writer thinks is of fundamental importance in making humans what they are and in explaining the differences between the life of humans and the life of animals. The definition is a good one if it achieves this. These revelatory definitions have no relation to the elimination of ambiguity; they are mentioned merely to indicate that analytical and stipulative definitions are not the only kinds of definitions.

How frequently are definitions needed? People sometimes think that one always should define one’s terms at the beginning of any discussion. But this idea becomes absurd if carried too far. Suppose that we as speakers did undertake to define all our terms in noncircular ways. However far we proceeded, we would always still have at least one definiens containing as yet undefined terms; therefore this task is an impossible one to complete. Moreover, we do have a fairly adequate understanding of the meanings of many words that we have never bothered to define and also of many words that we would not know how to define satisfactorily even if we tried. Thus, it would be foolish to try indiscriminately to define all or even most of our terms before proceeding with our thinking. What we should do at the beginning of a discussion is seek definitions of those particular words which are especially likely to make trouble in the discussion because they are harmfully ambiguous, obscure, or vague.

This is especially true with regard to discussions in which confusion is caused by failure to notice the different meanings of a term. A verbal dispute is a dispute arising solely from the fact that some word is being used with different meanings; this kind of dispute can be settled merely by giving the definitions that clarify the situation (though to say this is not to say that such disputes always are easy to settle).

The American philosopher William James gives a classic example of such a verbal dispute (Pragmatism, Lecture II). Suppose there is a squirrel on the trunk of a tree, and a man walks around the tree. The squirrel moves around the tree trunk so as to stay out of sight, always facing the man but keeping the tree between them. Has the man gone around the squirrel or not? Some of James’s friends disputed hotly for a long time about this question. Here is a purely verbal dispute; it can be settled by pointing out that in one sense the man has gone ‘around’ the squirrel, for he has moved from the north to the west and then to the south and east of the squirrel’s location, but in another sense the man has not gone ‘around’ the squirrel, for the squirrel has always been facing him. Once we have pointed out these two different senses of the word, we have done all that can reasonably be done; there is nothing more worth discussing (though this does not ensure that discussion will cease). With a verbal dispute like this, giving definitions is the way to resolve the dispute. But it would be utterly wrong to assume that all disputes are verbal in this way. There are many serious problems for the settling of which definitions are not needed, and there are many other problems where if definitions help, they mark only the beginning of the thinking needed to resolve the issue.
Stephen F. Barker; The Elements of Logic; 1989; p172ff

CHAPTER XII
CLASSIFICATION AND DEFINITION

§ 1. THE SIGNIFICANCE OF CLASSIFICATION
We have been calling the reader’s attention to the fact that the process of classifying things really involves, or is a part of, the formation of hypotheses as to the nature of things. It is well to consider this in some detail.
There is a general feeling, shared by many philosophers, that things belong to “natural” classes, that it is by the nature of things that fishes, for instance, belong to the class of vertebrates, just as vertebrates “naturally” belong to the class of animals. Those who hold this view sometimes regard other classifications as “artificial.” Thus a division of animals into those that live in the air, on land, and in water would be regarded as artificial. This distinction involves a truth which is confusedly apprehended. Strictly speaking, the last division, or any division of animals according to some actual trait arbitrarily chosen, is perfectly natural. For in every classification, we pick out some one trait which all the members of the class in fact possess, and therefore we may call it natural. All classification, however, may also be said to be artificial, in the sense that we select the traits upon the basis of which the classification is performed. For this reason controversies as to what is the proper classification of the various sciences are interminable, since the various sciences may be classified in different ways, according to the objectives of such classification.

Various classifications, however, may differ greatly in their logical or scientific utility, in the sense that the various traits selected as a basis of classification differ widely in their fruitfulness as principles of organizing our knowledge. Thus the old classification of living things into animals that live on land, birds that live in the air, and fish that live in water gives us very little basis for systematizing all that we know and can find out about these creatures. The habits and the structure of the porpoise or the whale have many more significant features in common with the hippopotamus or the horse than with the mackerel or the pickerel. The fact that the first two animals named have mammary glands and suckle their young, while all species of fish deposit their eggs to be fertilized, makes a difference which is fundamental for the understanding of the whole life cycle. In the same way, the fact that some animals have a vertebral column, or, to be more exact, a central nervous cord, is the key which enables us to see the significance of the various structures and enables us to understand the plan of their organization and functioning. Some traits, then, have a higher logical value than others in enabling us to attain systematic knowledge or science.

When, therefore, it is said that the business of science is first to gather the facts and then to classify them, we do not have a clear or adequate account of the situation. Some classification is involved in determining what facts we should gather; but this is not all. The most important thing is to pick out that trait in the objects studied which will be the most significant clue to their nature.

Obviously, there can be no a priori rules as to how we may hit upon such significant traits. Generally it depends upon genius, except that, other things being equal, we can say that he who has more knowledge is more likely to reject irrelevant or insignificant traits.

Formal logic, however, may aid us by defining the objects or traits considered so that our reasonings about them may be accurate, and may permit of being put into systematic deductive form.

§ 2. THE PURPOSE AND THE NATURE OF DEFINITION
The language of everyday conversation is notoriously vague, and the language of even technical treatises is not always very much better. Everyone is familiar with the difficulty of deciding whether certain micro-organisms are “plants” or “animals,” whether certain books are or are not “obscene,” whether a certain symphony is or is not the work of a “genius,” whether a given society is or not a “democracy,” whether we do or do not have certain “rights.” Such words are vague, because their denotation shades off imperceptibly into the denotation of other words. Many of the fatuities of actual thinking take place because the inescapable vagueness of most words makes a careful check upon one’s thoughts well-nigh impossible. The vagueness of ordinary words is one of the principal reasons why technical vocabularies must be constructed in the special sciences.

To the vagueness of words their ambiguity must be added as a serious danger to accurate thinking. Serious blunders in reflective thinking occur because the meaning that a word has in some context is replaced, without the fact being noticed, by an allied but different meaning. A famous instance of how the ambiguity of words may invalidate a reasoned discourse, is found in Mill’s Utilitarianism. Mill is trying to prove “that happiness is desirable, and the only thing desirable, as an end.” He argues as follows: “What ought to be required of this doctrine— what conditions is it requisite that the doctrine should fulfill— to make good its claim to be believed? The only proof capable of being given that an object is visible, is that people actually see it. The only proof that a sound is audible, is that people hear it: and so of the other sources of our experience. In like manner, I apprehend, the sole evidence it is possible to produce that anything is desirable, is that people do actually desire it.” Now to say that a thing is “desirable” may mean either that it should be the object of desire, or that it is in fact the object of desire. These two meanings are different. But in order that Mill may prove his thesis that happiness is the only end, “desirable” must be taken in the first sense; all his argument shows, however, is that happiness is “desirable” in the second sense.

Ambiguity arising from the grammatical structure of sentences, rather than from the ambiguity of its constituent words, was a common feature of the deliverances of the ancient oracles. Thus a celebrated response of an oracle was, “Pyrrhus the Romans shall, I say, subdue.”

Much of the best effort of human thought must go, therefore, to delimit the vagueness of words and eliminate their ambiguity. Vagueness can be reduced, but never completely eliminated. Ambiguity also can with care be successfully overcome. Thus the specific meaning of an ambiguous word may be determined from the context in which it is found on a specific occasion. For example, as we have noted before, when Christ declares, “Blessed are they that mourn: for they shall be comforted,” it is clear from the context that the “mourners” meant are those who “hunger and thirst after righteousness.”

But such a method of clarifying the meaning of a word is not always possible or even desirable. A much more deliberately devised process must be employed, and a standard or formal rule for defining symbols must be adopted. Let us examine it.

The reader is no doubt familiar with the famous scene in Moliere’s Le Bourgeois Gentilhomme between Monsieur Jourdain and the Teacher of Philosophy. We reproduce it somewhat abridged:

Teacher. What do you wish to learn?
M. Jour. Everything I can, for I am intensely anxious to be learned; it troubles me that my father and mother did not see to it that I was thoroughly grounded in all knowledge when I was young.
Teacher. An admirable sentiment: Nam sine doctrina vita est quasi mortis imago. Doubtless you know Latin and understand that?
M. Jour. Yes, but proceed as though I did not know it: explain to me what it means.
Teacher. It means, “Without knowledge, life is little more than the reflection of death.”
M. Jour. That Latin is right. … I must tell you something. I am in love with a person of high estate, and I would like you to help me to write something to her in a billet-doux, which I propose to let fall at her feet.
Teacher. Very good.
M. Jour. Something very gallant.
Teacher. Certainly. Do you wish to write in verse?
M. Jour. No, no, no verses.
Teacher. You only want prose?
M. Jour. No. I do not want either prose or verse.
Teacher. It must really be either one or the other.
M. Jour. Why?
Teacher. Because, monsieur, one can only express oneself in prose or verse.
M. Jour. Is there nothing but prose or verse?
Teacher. No, monsieur: all that is not prose is verse; and all that is not verse is prose.
M. Jour. And what is it when one speaks?
Teacher. Prose.
M. Jour. What? when I say, “Nicole, bring me my slippers, and give me my nightcap,” is that prose?
Teacher. Yes, monsieur.
M. Jour. Upon my word! I have spoken prose for more than forty years without knowing anything about it; I am infinitely obliged to you for having taught me this.

We shall compare the above “lesson” with the following (also abridged) scene from Plato’s dialogue Euthyphro. Socrates meets Euthyphro, who is on his way to the Athenian court in order to accuse his father of murder. Socrates is surprised, and asks Euthyphro whether it is pious to behave thus to one’s father. Euthyphro thereupon claims adequate knowledge about the nature of piety.

Soc. … What is piety, and what is impiety?
Euth. Piety is doing as I am doing; that is to say, prosecuting anyone who is guilty of murder, sacrilege, or of any other similar crime and not to prosecute them is impiety.
Soc. But … I would rather hear from you a more precise answer, which you have not as yet given, my friend, to the question, What is ‘piety’? When asked, you only replied, Doing as you, charging your father with murder.
Euth. And what I said was true, Socrates.
Soc. No doubt, Euthyphro; but you would admit that there are many other pious acts?
Euth. There are.
Soc. Remember that I did not ask you to give me two or three examples of piety, but to explain the general idea which makes all pious things to be pious. Do you not recollect that there was one idea which made the impious impious, and the pious pious? Tell me what is the nature of this idea, and then I shall have a standard to which I may look.
Euth. I will tell you, if you like.
Soc. I should very much like.
Euth.Piety, then, is that which is dear to the gods, and impiety is that which is not dear to them.
Soc. Very good, Euthyphro; you have now given me just the sort of answer which I wanted. But whether what you say is true or not I cannot as yet tell, although I make no doubt that you will prove the truth of your words.

Nominal Definition
We have before us now several attempts at the definition of verbal symbols. There are important differences between some of them, which we must note. To M. Jourdain, who knew no Latin, the explanation of the Latin sentence consisted in a translation. He was informed of the meaning of a set of symbols with which he had previously been totally unfamiliar by being told that they were equivalent to a set of symbols with which he had been familiar. Ordinarily, we regard translations as true or false. Thus if the words “sine pecunia” were used for “without knowledge” those who know Latin would call it a false translation. If, however, there were no reference to the fact that the new words were part of the language historically called Latin, the question of truth or falsity would not be involved. There would simply be a substitution of a new set of words or symbols for old familiar ones, as is the case’ in the creation of cryptograms, private codes, and artificial languages, as well as in the invention of technical terms in the various sciences. Thus the word “sociology” was invented by Auguste Comte as a name for the study of human relations in organized group life, and other writers have chosen to follow him. But the word might have;! been introduced to denote the study of legal or business partnerships, the phenomena of clubbing together, or the way things in general are associated together. That, unlike many other proposed new terms, this one has been generally adopted, and that its denotation has been confined to human relations but not restricted to any special form of them, are results of choice, to which we may agree or not as we please without thereby asserting anything true or false. This is also the case when in mathematics we introduce symbols like + for “plus” after the latter had become used as equivalent to “added to.” Careful writers since Aristotle have been aware of this and have often used the imperative form to define a new word, for example, Let the process of grasping meanings be called “apperception.”

A nominal definition, then, is an agreement or resolution concerning the use of verbal symbols. A new symbol called the definiendum is to be used for an already known group of words or symbols (the definiens). The definiendum is thus to have no meaning other than the definiens. In the Principia Mathematica by Whitehead and Russell a definition of this type is written by putting the definiendum to the left and the definiens to the right with the sign of equality between them and the letters “Df.” to the right of the definiens. Thus implication, symbolized by ⊃, is defined thus: p ⊃ q = p´ ⋁ q. Df. Or, in words, “p implies q” is equivalent by definition to “not p or q.” In algebra the same procedure could be followed. Exponents could be introduced as follows: a ² = a • a. Df.

A nominal definition, then, is a resolution and not anything true or false― though of course the assertion that anyone has or has not consistently lived up to his resolution may be true or false. And since that which is neither true nor false cannot be a proposition, nominal definitions cannot be real premises of any argument. There are no implications of truth or falsity in words themselves.

But while nominal definitions do not extend our real knowledge, they aid in scientific inquiry in the following ways:
1. In the first place, we economize space, time, and attention or mental energy if we use a new and simple symbol for a group of old familiar ones. Thus if we continued to use ordinary words and did not introduce such technical terms of higher mathematics and physics as “differential coefficient,” “energy,” “entropy,” and the like, our expressions would become so long and involved that we could not readily grasp the complex relations indicated by these terms. Thus, it is easier to read Newton’s Principia translated into the technical language of the modern calculus than in the more familiar language of geometry in which Newton wrote.
2. The translation of the familiar into unfamiliar terms tends to clarify our ideas by depriving our symbols of accidental or irrelevant associations. Familiar or ordinary words have strong emotional associations and carry penumbras of suggested meanings which obstruct the process of rigorous deduction.

Definition by Denotation
Another way in which the meaning of words is clarified is by exhibiting a part of their denotation. Thus the word “prose” was explained to M. Jourdain by giving him examples to which it can be correctly applied. For psychological reasons this method may have something to recommend it. Such a method, however, does not yield a “definition” in any usual sense of that word. We may understand what a word means when we know what it symbolizes, that is, to what it may be applied; but we do not thereby define its meaning.

Euthyphro’s attempt to define “piety” in this way was naturally unsatisfactory to Socrates. That which is offered as an instance of piety may also be an instance of something else. Unless we have some sense of the connotation of the term, how can we be at all sure that we can recognize in the example what it is an instance of? It is partly for this reason that Socrates rejected Euthyphro’s first attempt.

Real Definitions
Euthyphro grasped the nature of a satisfactory definition in his second trial. We must examine his attempted definition of “piety,” for it introduces us to real, as contrasted with verbal definition.

Both Socrates and his friend knew, in a rough way, what “piety” was. They understood, that is, to what sort of acts the term could be applied correctly. But in seeking for a definition of “piety,” Socrates was searching for an analysis of that which the term represented. Consequently, he was pleased with the sort of answer Euthyphro gave, although, as the dialogue shows, he rejected it as false. Euthyphro’s definition may be put in the form:

Piety = that which is dear to the gods. Df.

Like a nominal definition, this real definition defines the word “piety” by means of an equivalent group of words. But, and this is the important point, the definiens is an analysis of the idea, form, type, or universal symbolized by “piety.” Both the definiens and the definiendum refer to the same thing or character. They each possess a meaning independently of the process of definition which equates them. The definiens, however, indicates the structure of that to which both refer.

A real definition, therefore, is a genuine proposition, which may be either true or false. Since the definiendum and the definiens must symbolize the same universal, and since the definiens must express the structure of that universal, a real definition can be true only if the two sides of the definition are equivalent in meaning and the right-hand side represents a correct analysis of it.

We may give another illustration of real definitions. Everyone may be supposed to be familiar with the meaning of “similar figures.” Such figures resemble one another in a way most people untutored in geometry would find it hard to state, but which they can identify in a crude way. The following is a real definition of similarity:

Figure A is similar to figure A´. = .
The ratio of the distance between any two points P, Q, on A and the distance between the corresponding points P´, Q´, on A´, is constant. Df.

This is a true definition of what is ordinarily meant by similar figures, because the right-hand side means precisely what the left-hand side does, and at the same time the right-hand side offers an analysis of the structure of that which both sides symbolize.

We may now survey some of the purposes of definitions.

Psychological Motives for Definitions
There is, in the first place, the desire to learn the meaning of new words. This may be satisfied by expressing that meaning in more familiar words. In the second place, there is a desire to find a conveniently short expression for one that is long and cumbersome. Thus instead of using the phrase “the son of my mother’s sister” we introduce the shorter one “my cousin.” In the third place, we wish to make the meaning of a word better known to us by resolving that meaning into its constituent elements. This requires a real definition. All these motives are psychological.

The reader may have noted that the definiens is generally a longer expression than the definiendum, not only in nominal definitions, where it is to be expected, but in real definitions also. This fact is intimately connected with the psychological purpose of definitions. Since the definiens contains a larger number of symbols than the definiendum, it brings to the mind a larger number of ideas also. These ideas, however, are structurally related, so that they limit one another and at the same time are equivalent as a whole to the meaning of the definiendum. Thus in the definition of similarity above, the right-hand side contains the symbols “ratio,” “distance,” “corresponding points,” “constant”; the notions which they represent are familiar, and they are so organized that the fringe of vagueness each one may have does not affect the sense of the complex whole.

This same psychological phenomenon is more clearly observable if, as is sometimes done, the meaning of a word is clarified by means of a series of synonyms. Thus “to be honest” means “to be candid, equitable, frank, genuine, ingenuous, straightforward, trustworthy, upright.” No one of the so-called synonyms has precisely the same meaning as “honest.” But the intensions of the synonyms overlap, so that they mutually delimit each other. The part of the intensions which is common to all may then convey, more or less precisely, the meaning of the word required.

Logical Purpose of Definitions
But these psychological reasons for making definitions must not be confused with the logical function that definitions have. Logically, definitions aim to lay bare the principal features or structure of a concept, partly in order to make it definite, to delimit it from other concepts, and partly in order to make possible a systematic exploration of the subject matter with which it deals. A real definition may always serve as the premise, or part of the premise, of a logical inquiry concerning a subject matter. Thus from the definition of similar figures, together with other premises, we can deduce the theorem that the volumes of any two similar figures are to each other as the cubes of any two corresponding distances. Aristotle saw this clearly when he declared that “the basic premises of demonstrations are definitions.”

Unfortunately the terminology concerning these matters has undergone much change, so that any attempt to bring together traditional and modern opinions must seem confusing. In the technique of modern mathematics, as we have already seen, all real definitions are implicit. No explicit definitions except nominal ones are required. However, what Aristotle called “undemonstrable definitions” which reveal the essence of a subject matter, appear in modern logical techniques as axioms or primitive propositions. Such axioms define the subject matter implicitly, as one which satisfies or verifies the axioms. For example, the nature of electricity is defined by Maxwell’s equations, the nature of gravitation by Newton’s laws. It is perhaps unnecessary to remind the reader, however, that while in a given system the real definitions or axioms may be logically prior to all the theorems, these axioms are not first in the order of the development of our knowledge, nor are they more evident or certain than any of the theorems which they imply.

We have drawn a sharp distinction between verbal and real definitions. In practice, however, the distinction is never so sharp, and even in definitions which seem altogether verbal there is generally some reference to the analysis of what the words stand for. Words are so fundamentally symbolic that it would be strange if it were otherwise. Moreover, the emotional associations and overtones of words may often prevent a clear apprehension of the issues at stake. This is particularly true in the social sciences. Words like “democracy,” “liberty,” “duty,” have a powerful emotive function; they are frequently used as battle cries, as appeals to emotions, and as substitutes for thought. Many of the disputes about the true nature of property, of religion, of law, which undoubtedly arise from a conflict of emotional attitudes, would assuredly disappear if the precisely defined equivalents were substituted for these words.

However, issues other than emotional ones may also be involved. Religion, for example, has sometimes been defined in terms of some dogma, sometimes in terms of a social organization and ritual, and sometimes in terms of emotional experiences. The resulting conflicts over the meaning or essence of religion have been regarded, perhaps not without some justice, as conflicts over words. But this is only a half-truth. For the disputants frequently have their eye on a concrete phenomenon which presents all these aspects. The quarrels over the right definition of religion are attempts to locate the fundamental features of a social phenomenon. For if those features are taken as the definition of religion, it is possible to deduce many important consequences from it. Thus if belief in some doctrine is the essence of religion, other things follow than if some type of emotional experience is taken as defining religion: in the one case there is an emphasis upon intellectual discipline and conformity, in the other, an emphasis upon esthetic elements and a neglect of theology.

The age-long dispute about the nature of law involves similar issues. Is “law” to be construed as a command, as a principle certified by reason, or as an agreement? The controversy is not simply about words. It is concerned with making one rather than another aspect of law central, so that the appropriate consequences may be drawn from it. A schoolroom illustration is the question, “Is a bat a bird?” The two parties to the dispute concerning the answer may agree that a bird is a warm-blooded vertebrate having its fore limbs modified as wings, and yet not agree as to whether a bat is a bird. Why? Because one party to the dispute may believe there is a closer affinity of the bat to rodents than to birds, and may wish to regard those common features of rodents as central in the bat.

We may now summarize our discussion of real definitions. A real definition involves two sets of expressions, each with a meaning of its own and these meanings are equivalent if the definition is true. In a true definition, the definiens may be substituted for the definiendum without any alteration of sense. The definiens must be easier to understand, even though a longer or more complicated expression than the definiendum, if the psychological as well as the logical function of the definition is to be fulfilled.

Are there any general rules which are of help in the formulation of definitions? We shall reserve the reply until after we have examine the traditional discussion of definition.

§ 3. THE PREDICABLES
Aristotle’s discussion of definition is central to his entire theory of science, and is itself based upon his analysis of the possible ways in which a predicate may be related to the subject. His inquiry grew out of his reflections upon the method and results of the speculations of Socrates and Plato. His writings upon the syllogism cannot really be understood without reference to his analysis of the possible kinds of propositions, each kind depending upon the nature of the relation between subject and predicate. This analysis, called the theory of the predicables, was in turn closely connected with fundamental metaphysical doctrines, especially with the doctrine of fixed natural kinds or types. Into these important matters we cannot go except for a brief discussion of the predicables.

Aristotle obtains an exhaustive enumeration of the possible relations between predicate and subject, in the following way: Every predicate must be either convertible with its subject or not; that is, if A is B, then either B is so related to A that if anything is B it is A, or this is not the case. If it is convertible (Aristotle also calls it commensurable), it either signifies its essence, in which case it is the definition; or it is a property. If the predicate is not convertible with the subject, either it is contained in the definition of the subject, in which case it is the genus or the differentia, or it is not contained in the definition, in which case it is an accident. A predicate must therefore stand to the subject in some one of these five possible relations: it must be either definition, property, genus, differentia, or accident. We must now explain the significance of each of these distinctions. But the reader must understand that the subject term is taken by Aristotle to represent a form, type, or universal, and not a singular, concrete thing. The predicables indicate the possible ways in which universals are related to one another. The concrete individual as such is not a subject matter for science, according to Aristotle; only in so far as the individual embodies a type or form is a science of individuals possible. It is never of Socrates as an individual, but only of Socrates as “man” that we may have scientific (or systematic) knowledge. Aristotle’s discussion of the predicables, therefore, stressed the intensional aspect of terms. But it is possible to give an extensional interpretation of them also, and traditionally this has usually been done.

Definition
“A ‘definition,’ according to Aristotle, “is a phrase signifying a thing’s essence.” By the essence of a thing he understood the set of fundamental attributes which are the necessary and sufficient conditions for any concrete thing to be a thing of that type. It approximates to what we have called the conventional intension of a term. Thus the essence or definition of a circle is that it is a plane figure every point of which is equidistant from a fixed point. The predicate (a plane figure every point of which is equidistant from a fixed point) is convertible or commensurate with the subject: it may be predicate of everything that is a circle, and everything to which it can be applied is a circle. The predicate is the essence, because it tells what a circle is, so that all the “peculiarities” of the circle necessarily follow from it.

Genus
The definition contains two terms as components, the genus and the differentia. “A ‘genus’ is what is predicated in the category of •essence of a number of things exhibiting differences in kind.” Thus the genus of “circle” is “plane figure.” The circle, on the other hand, is a species of plane figure. But “plane figure” is also the genus of “triangle,” “ellipse,” “hyperbola,” and so on. These different species exhibit differences in kind, but they all belong to the same genus.

Differentia
The differentia is that part of the essence which distinguishes the species from the other species in the same genus. The differentia of “circle” is “having all its points equidistant from a fixed point”; the differentia of “triangle” is “being bounded by three straight lines.”

The distinction between genus and differentia was absolute for Aristotle, and was connected with his metaphysical views. But from a purely logical or formal point of view, the distinction is absolute only within a specific context. For consider the definition, “Man is a rational animal.” According to Aristotle, the genus is “animal,” the differentia is “rational.” But formally we may regard with equal right “rational” as the genus and “animal” as the differentia. This will be clear if we express the definition explicitly as a logical conjunction of two attributes. Thus, X is a man: =: X is rational and X is an animal. It doesn’t make any logical difference which conjunctive is regarded as the more important. The logical function of the differentia is to limit or qualify the genus. And this function is performed by either term in the definition with respect to the other. A definition may, therefore, be regarded as the logical product of two terms. This interpretation is particularly adapted to an extensional emphasis upon the predicables.

The relation of a genus to its species is clearly illustrated by the device known as the Tree of Porphyry. The following is the traditional illustration, and has evoked from Bentham the characterization of “the matchless beauty of the Tree of Porphyry.”

The reader will note, however, that the relation between the genus “animal,” say, to its species “man,” is different from the relation of the species man to its individual members. The first is a relation between a class and its subclass, the second a relation between a class and its members. Porphyry, who considerably modified Aristotle’s theory of the predicables, also confused it irreparably.

Property
“A ‘property’ is a predicate which does not indicate the essence of a thing, but yet belongs to that thing alone, and is predicated convertibly of it. Thus it is a property of man to be capable of learning grammar: for if A be a man, then he is capable of learning grammar, and if he be capable of learning grammar, he is a man.” Thus, a property of a circle is that it has the maximum area with a given perimeter; another property is that the product of the segments of the chords passing through a fixed point is constant. The property is an attribute which follows necessarily from the definition.

The distinction between essence and property was regarded by Aristotle as absolute, for a subject has, according to him, only one essence. From a purely logical point of view, however, the distinction is absolute only relatively to a given system. Thus if we define a circle as the locus of points equidistant from a fixed point, we can formally deduce the property that its area is maximum with a given perimeter. On the other hand, if the circle is defined as the plane figure having a maximum area with a given circumference, it follows necessarily that all its points are equidistant from a fixed point. The roles of definition and property are therefore interchangeable. Which character of a subject is taken as the definition turns upon extralogical considerations. Hence, while the distinction between essence and property is perfectly sound, it is absolute only within a given system. We have already seen, in connection with the discussion of the nature of mathematics, that there are no intrinsically undemonstrable propositions or intrinsically undefinable terms. The points we made there are relevant here. We have also suggested above that the “undemonstrable definitions” of Aristotle are the axioms of modern mathematical technique. The reader will therefore have no difficulty in interpreting the “properties” which flow from the definition as none other than the theorems of a system which are implied by the axioms. Unfortunately, in the example above we have quoted from him, Aristotle does not show how the property of being capable of learning grammar follows from the definition of man.

Accident
Finally, “an ‘accident’ is (1) something which, though it is none of the foregoing, i.e., neither a definition nor a property nor a genus-yet belongs to the thing: (2) something which may possibly either belong or not belong to anyone and the self-same thing, as (e.g.) the ‘sitting posture’ may belong or not belong to the self-same thing.” To have a triangle inscribed in it is, therefore, an accident of the circle. From a purely logical point of view, an accident is a proposition not formally derivable from the definition. So stated, it is perhaps unnecessary to warn the reader once more that an accidental predicate is not to be predicated of a concrete individual, but only of an individual as representing a kind. Thus, snub-nosedness is an accident not of Socrates as an individual; but of Socrates as a man. Man, the type, need not be, although it may be, conjoined with snub-nosedness. Snub-nosedness is an accident, because it is not a necessary consequence of being a man.

Such, in brief, is the Aristotelian theory of predicables. In terms of the doctrine, therefore, the condition which satisfactory definitions must satisfy is that they be stated in terms of genus and differentia.

§ 4. RULES FOR DEFINITIONS
It is convenient, however, to discuss the rules for satisfactory definitions without restricting ourselves to the Aristotelian analysis. The following rules are the substance of those usually given:
1. A definition must give the essence of that which is to be defined. The definiens must be equivalent to the definiendum― it must be applicable to everything of which the definiendum can be predicated, and applicable to nothing else.
2. A definition must not be circular; it must not, directly or indirectly, contain the subject to be defined.
3. A definition must not be in the negative where it can be in positive terms.
4. A definition should not be expressed in obscure or figurative language.

We shall comment briefly upon each of these precepts.
1. The first rule expresses in different words the substance of our discussion in the previous sections of this chapter. When the traditional doctrine of the predicables is made the basis for discussion, this rule may be replaced by the injunction that a definition must be per genus et differentiam. Real definitions are definitions of words, and at the same time are analyses of the universal symbolized by both the definiens and the definiendum.

We have already called the reader’s attention to the fact that in modern treatments of mathematics real definitions are implicit, the subject being defined in terms of the axioms which it must satisfy. It frequently happens, therefore, that several undefined terms must be defined on the basis of their relations to one another, and not in isolation from one another. Thus in Hilbert’s study of the foundations of geometry, points, lines, and planes are taken as the “undefined” elements. But they are implicitly defined by the axioms. These axioms state the relations which must hold between points taken by themselves, lines taken by themselves, planes taken by themselves, and also the relations between points and lines, points and planes, and so on. But whether explicit or implicit, the definition should be so selected that the attributes known to belong to the things defined must be formally derivable from the definition.

Since, therefore, the logical aim of definitions is to state those features of a thing from which its other features follow, it is not always possible to satisfy the psychological motives behind the need for definitions. When the psychological objectives of definitions are emphasized, it is often said that the definiens should contain more familiar ideas than the definiendum. But if the logical goal of definitions is in the foreground, it may be advantageous to neglect this advice and to use less familiar notions in the definiens than in the definiendum. The undefined terms (and of necessity there must be undefined terms in every system) should be so selected as to give scope for a deductive treatment of the subject matter. Such undefined terms cannot be made meaningful by further definition, but only by some carefully selected process of exhibiting that which they denote. In some instances the undefined terms may be invested with significance by a direct process of exhibition. In others, however, the denotation of such terms cannot be exhibited. This is generally true of terms defined implicitly through the axioms. Such terms, although they function as undefined elements in the system, are virtually defined by the system itself. Thus in electrical theory a hypothetical electrical fluid may be an undefined term. Its meaning becomes known to us, however, in virtue of the fact that many of the properties of such a fluid with which the theory endows it can be directly exhibited.

2. If the term to be defined, or some synonym, appears in the definiens, no logical advance has been made in the analysis of the concept for which it stands, although it may be that the psychological purpose of the definition is satisfied. Thus if “courage” is defined through its synonym “bravery,” the meaning of “courage” may have become clearer to us because we are more familiar with the meaning of “bravery.” But the net effect of the definition is verbal, and the structure of “courage” (what it signifies, not the word) has not been analyzed. Such tautological definitions sometimes escape detection. The prese nt rule is violated if the sun is defined as “the star which shines by day”: for “day” itself is defined in terms of the shining of the sun.

A definition may seem to violate this rule when in fact it does not do so. A famous example is Russell’s definition of “number.” According to him, “A number is anything which is the number of, some class.” Here “number” is defined in terms of “the number of some class.” The definition does not violate the present rule, for the definiendum is “number,” or “number in general,” while the definiens contain the term “number of some class.” Definitions of this type are frequent in mathematics. Thus the series u0 +u1 + u2 + … un + … is defined to be convergent, if the sequence of successive terms S0 = u0, S1 = u0 +u1, S2 = u0 +u1 +u2 + …,
Sn = u0 +u1 + … +un + … is convergent.

3. It is obviously preferable to define a thing in terms of what it is, rather than in terms of what it is not. For in general, to state what a thing is not does not sufficiently delimit it from other things. Thus to define a watch as a timepiece which is not a clock will be unsatisfactory if there are other timepieces besides watches and clocks. However, it is easy to overemphasize this rule, for in some cases an adequate definition can be given this way. Thus to define a scalene triangle as one which is neither equilateral nor isosceles delimits perfectly scalene triangles from all others, provided it is stated in what system of geometry the triangle is to be included. In some cases negative definitions are inescapable. Thus the definition of an orphan as a child who has not parents must of necessity be in negative terms, for the state of orphanhood is a denial of the state of having parents. Other instances, like “independence,” “parallel,” “bankrupt,” or “insolvent” will readily occur to the reader. Moreover, whether a definition is considered negative or positive often depends upon linguistic conventions. Some languages may possess a positive term for an idea which must be expressed negatively in another language. Finally, a definition may have the appearance of being negative simply because one of the terms in it is negative in form. Thus to define a drunkard as a man who is intemperate in drink is not to violate the present rule: intemperance itself is defined in terms of an excessive imbibing of alcoholic liquids.

4. The chief danger from definitions expressed in figurative language is that the metaphors which are employed may suggest meanings that they are not intended to convey. Thus to define a king as the “captain of the ship of state” may be misleading because it may suggest that a king can guide the destinies of a nation by following a charted path. The injunction that the definiens should not be obscure expresses the psychological motives for definitions. Samuel Johnson’s definition of a net as a “reticulated fabric, decussated at regular intervals, with interstices at the intersections” is a classic example of a definition which violates this psychological requirement.

However, the occurrence in the definiens of terms unfamiliar to most readers does not make the definition obscure. In physics, the definition of “the action of a system of particles,” is given as “the sum for all the particles of the mean momentum for equal distances multiplied by the distance traversed by each particle.” This definition is by no means obscure to the competent student of analytical dynamics, whatever it may appear to be to the untrained.

§ 5. DIVISION AND CLASSIFICATION
According to the traditional account, definition consists in the analysis of a species in terms of its genus and differentia. But a genus may be differentiated into other species as well. Thus the genus “plane figure” may be differentiated not only into the species “triangle,” but into the species “quadrilateral,” “conic section,” and so on. The exhibition of the various species in the same genus is called logical division, or more simply, division. The genus with which the process of division starts is called the summum genus. Now the species obtained by a division may be capable of further division. The species with which a division ends is called the infirma species, while the species intermediary between the summum genus and the infirma species are called the subaltern genera.

The process of division, from an extensional point of view, is the breaking-up of a class into its constituent subclasses. Division is therefore related to definition, because it marks off the limits of the extension of a class denoted by a term. If, however, division is looked at from the point of view not of its constituent species, but of its individual members, the process is allied to classification. While division breaks up a genus into species, classification is the grouping of individuals into classes, and these classes into wider ones.

A number of rules have been stated for satisfactory logical division. They are also applicable to classification. They are:
1. A division must be exhaustive.
2. The constituent species of the genus must exclude one another.
3. A division must proceed at every stage upon one principle, the fundamentum divisionis.
Thus if we divide rational numbers into odd integers and even integers, the first rule is violated, for we have omitted the fractions. The purpose of the first rule is to take account of every species in the genus. We violate the second rule when we divide the genus “quadrilateral” into “rhomboids,” “parallelograms,” “rectangles,” since if anything is a rectangle it is also a parallelogram. The principle upon which a division is made is called the fundamentum divisionis. In dividing the genus “professor” into “mathematicians,” “physicists,” and so on, the fundamentum divisionis is the subject matter which they profess; if we divide it into “dull lecturers,” “brilliant lecturers,” and so on, the principle is their rhetorical ability. A division which conforms to the third rule will necessarily conform to the second. But the converse is not true. Thus, the division of the genus “number” into “odd,” “even,” “fractional,” yields exclusive species, although the principle of the division is not single.

However, these rules, although unexceptional from a formal point of view, are of little help in practice. They express an ideal rather than state a method. Moreover, the ideal is inadequate for a well-developed science; it is more suitable to sciences in their infancy.

Until we have explored a subject matter thoroughly we cannot achieve either a satisfactory definition, a satisfactory division, or a satisfactory classification. In the first place, we can never be sure, in any existential subject matter, that the division or classification is exhaustive. A hitherto unknown and unpredictable aspect of the subject matter may suddenly turn up and undo, or at least call for a serious revision of, our efforts at system. Nor can we be certain that the subaltern genera are in fact exclusive. Indeed, this warning is a corollary from the proposition that the division cannot be known for certain to be exhaustive, for a hitherto unfamiliar subclass may turn up which possesses the common characters of several of the already recognized species.

In the second place, the process of scientific classification is much more groping and less formal than the rules would suggest. Even before science was deliberately pursued, everyday experiences compelled the recognition of kinds of things in which certain groupings of qualities occurred more or less invariably. Thus unreflective experience takes cognizance of trees, earth, animals, and so on, on the basis of obvious similarities between instances of these types. With growth of knowledge, however, features that are less obvious may be taken as the basis for classification or division. Thus although the porpoise is like a fish in many ways, it is classified in modern biology as a mammal because it suckles its young. The basis of classification depends on the discovery of some significant traits, significant in the sense that on the basis of the traits the subject matter can be organized into a system. Such traits, however, are only slowly discovered, and cannot be determined on formal grounds alone.

All sciences in their early days are classificatory, and almost any arbitrary scheme of grouping objects may be tentatively adopted in the interest of mastery of the subject matter. The classification of genera in modern biology still does not conform to the third of the rules above. Anthropology has not yet grown out of the classificatory stage, and until recently chemistry too was content to classify its subject matter in terms of elements, compounds, and reactions. Today, however, chemistry is organized on the basis of physical principles, which show more clearly than the older scheme the structure of its subject matter and the interrelation of chemistry and other sciences.

An exhaustive and exclusive division can always be obtained by dividing a genus in terms of a differentia and its negative. Aristotle obtained an exhaustive set of possible relations between a subject and predicate by this method. It is called dichotomous division. It can be represented as follows:

Nevertheless, although dichotomy insures exhaustiveness and exclusiveness of the species, it is not much of an advance over ordinary division. The practical difficulty of finding significant principles of division still remains. And in dichotomous division we cannot be sure that all the subclasses have members. Moreover, the method is somewhat clumsy, and modern symbolic logic has shown how dichotomous division can be effected in an almost mechanical manner. Thus suppose we wish to classify the population of the United States on the bases of sex, of being over thirty years of age, and of being in good or exceptional health. Let 1 represent, as usual, the universe of discourse; a those of male sex, a´ those of female sex; b those over thirty, b´ those thirty or under thirty; c those in good or exceptional health, c´ those in poor health. Then the population of the United States is divided into eight groups as follows:

1 = (a +a´) = (a +a´) (b + b´) = (a +a´) (b + b´) (c + c´)
    = abc +abc´ + ab´c + ab´c´ + a´bc + a´bc´ + a´b´c + a´b´c´

The symbol abc will then represent the males over thirty and in good health; a´bc´ will represent the females over thirty who are in poor health, and so on.

Morris R. Cohen; Ernest Nagel; An Introduction to Logic and Scientific Method; 1939; ch12

Summary of Chapter 3
Introduction to Logic

To explain the meaning of a term is to give the definition of it. In this chapter we have discussed the several kinds of definitions and their uses, and techniques for constructing definitions, with rules for applying these techniques.

In section 3.1, we have explained three kinds of disputes: [See graphic above.]
1. Obviously genuine disputes, in which there is no ambiguity present and the disputers do disagree, either in attitude or in belief.
2. Merely verbal disputes, in which there is ambiguity present but there is no genuine disagreement at all.
3. Apparently verbal disputes that are really genuine, in which there is ambiguity present and the disputers disagree, either in attitude or in belief.

In section 3.2, we first explained that definitions are always of symbols, and we introduced the terms definiendum (the symbol that is defined) and definiens (the symbols used to explain the meaning of the definiendum.)

We also distinguished among five kinds of definition and their principal uses:
1. Stipulative definitions, in which a meaning is assigned to some symbol. A stipulative definition is not a report and cannot be true or false; it is a proposal, resolution, request, or instruction to use the definiendum to mean what is meant by the definiens.
2. Lexical definitions, which report the meaning that the definiendum already has and which therefore can be correct or incorrect.
3. Precising definitions, which go beyond ordinary usage in such a way as to eliminate troublesome uncertainty regarding borderline cases. Its definiendum has an existing meaning, but that meaning is vague; what is added to achieve precision is partly a matter of stipulation.
4. Theoretical definitions, which seek to formulate a theoretically adequate or scientifically useful description of the objects to which the term applies.
5. Persuasive definitions, which seek to influence attitudes or stir the emotions, using language expressively rather than informatively.

Of these five kinds of definition the first two (stipulative and lexical) are used chiefly to eliminate ambiguity; the third (precising) is used chiefly to reduce vagueness; the fourth (theoretical) is used to advance theoretical understanding; and the fifth (persuasive) is used to influence conduct.

In section 3.3, we explained that a general term denotes the several objects to which that term may be correctly applied. The collection of these objects constitutes the extension of the term. We explained that the set of attributes shared by all and only the objects within a term’s extension is the intension of the term. The extension of a term is determined by its intension, but the intension is not determined by the extension; so terms may have different intensions and yet the same extension; but terms with different extensions cannot possibly have the same intension.

In section 3.4, we explained how, using the extension of a general term, we may construct extensional definitions, of which there are several varieties, whose limitations also are noted:

1. Definitions by example, in which we list or give examples of the objects denoted by the term.
2. Ostensive definitions, in which we point or indicate by gesture the extension of the term being defined.
3. Quasi-ostensive definitions, in which the gesture or pointing is accompanied by some descriptive phrase whose meaning is taken as being known.

In section 3.5, we explained how, using the intension of a general term, we can construct intensional definitions, of which there are also several varieties, whose limitations are also noted:

1. Synonymous definitions, in which we provide another word, whose meaning is already understood, that has the same meaning as the word being defined.
2. Operational definitions, which state that the term is correctly applied to a given case if and only if the performance of specified operations in that case yields a specified result.
3. Definition by genus and difference, in which we first name the genus of which the species designated by the definiendum is a subclass, and then name the attribute (or specific difference) that distinguishes the members of that species from members of all other species in that genus.

The techniques of intensional definition may be used in constructing definitions of any one of the five kinds identified in section 3.2: stipulative, lexical, precising, theoretical, or persuasive.

In section 3.6, we formulated and explained five rules traditionally laid down for definitions by genus and difference:
1. A definition should state the essential attributes of the species.
2. A definition must not be circular.
3. A definition must be neither too broad nor too narrow.
4. A definition must not be expressed in ambiguous, obscure, or figurative language.
5. A definition should not be negative where it can be affirmative.

Irving M. Copi; Carl Cohen; Introduction to Logic; 2002; p134

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GLOSSARY

Causal - Defines a word by stating how instances to which it is applicable are produced (e.g. a sphere is a solid generated by rotating a circular disc around its diameter). The use of the term “causal definition” is sometimes objectionable since it is hard to distinguish such alleged definitions from causal statements.

Deductive argument - Argument whose conclusion follows necessarily from the premises; that is, it would be self-contradictory to affirm the premises and to deny the conclusion.

Deductive logic - Science aiming at explicit formulation of the conditions under which arguments of various deductive forms are valid. Subsidiary to this aim there are such investigations as the classification of propositions on the basis of their logical form, the analysis of logical concepts such as implication, contradiction, etc.

Definiendum- Expression to be defined.

Definiens- Expression used to define.

Definition- Explanation of the meaning of a linguistic expression, i.e. word, phrase (series of words that has meaning but is no sentence— e.g. descriptions), or sentence.

Degree of a predicate- A predicate has the degree one if it designates a property; if it designates a relation its degree is equal to the number of terms between which the relation holds; thus “between” is a predicate of the third degree.

Denotation- A word is said to denote the entities to which it is applicable by virtue of its connotation. A word may have a connotation without having a denotation, viz. if the connotation is a property which nothing has; on the other hand, there are words which have a denotation but no connotation (“this,” “here,” etc.). Again, one and the same entity may be denoted by one word and connoted by another word, e.g. the property blueness is connoted by “blue” and denoted by “color.”

Explicit- Definition of the form “A = B and C” or “A = B or C,” where A is the defined concept and B and C are properties which anything to which “A” is applicable must jointly or alternately have. The defined term stands by itself to the left of the equality sign, and is, by the definition, declared substitutable for the definiens, no matter in what context it appears.

Genetic- Causal definition.

Implicit- (postulational)- A set of postulates containing several explicitly undefined predicates is said to define the latter implicitly in the sense of delimiting the number of interpretations for the predicates which satisfy the postulates. Once, however, an interpretation for one primitive term has been fixed, the interpretation for the other primitives is likewise fixed by the conditions of significant substitution.

In use- (contextual) -Defines a word or phrase by showing how sentences containing it may be translated into synonymous sentences that do not contain the defined expression.

Logical- p and q are logically dependent if either p entails q, or q entails p, or p contradicts q. In a derivative sense, one also speaks of logical dependence as a relation between properties.

Nominal- A definition whose sole purpose is abbreviation, by introducing a simpler term as a synonym for a complex term already in use.

Ostensive- Explanation of the meaning of a word by exhibiting an example of the kind to which it is applicable; also called “denotative.”

Real- In non-metaphysical uses synonymous with “explication” (Carnap) or “analysis” (Moore): analysis of the meaning of an expression already in use, involving the discovery of a necessary and sufficient condition for the applicability of an expression or the truth of sentences of a certain form (the latter case corresponds to definitions in use).

Recursive- Indicates how the definiendum may be eliminated from expressions that contain it in a finite number of steps. There are two lines which jointly constitute the recursive definition: the first line indicates how complex expressions containing the definiendum are reducible to simpler expressions that still contain it; the second line shows how it may be eliminated from the simplest expressions. In exceptional cases, there are more than two lines.

Verbal- Non-ostensive definition. The meaning of the words making up the definiens, however, must already be understood, otherwise we know only that two expressions are synonymous without knowing the meaning of either.

Arthur Pap; Elements of Analytic Philosophy; 1949; pp485-487

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Definition
From Wikipedia, the free encyclopedia
 
A definition states the meaning of a word using other words. This is sometimes challenging. Common dictionaries contain lexical descriptive definitions but there are various types of definition - all with different purposes and focuses.

A definition  is a statement of the meaning of a term (a word, phrase, or other set of symbols).[1]  Definitions can be classified into two large categories, intensional definitions (which try to give the essence of a term) and extensional definitions  (which proceed by listing the objects that a term describes).[2]Another important category of definitions is the class of ostensive definitions, which convey the meaning of a term by pointing out examples. A term may have many different senses and multiple meanings, and thus require multiple definitions.[3][a]
In mathematics, a definition is used to give a precise meaning to a new term, instead of describing a pre-existing term. Definitions and axioms are the basis on which all of modern mathematics is constructed.[4]

Contents
• 1 Basic terminology
• 2 Intensional definitions vs. extensional definitions
o 2.1 Classes of intensional definitions
o 2.2 Classes of extensional definitions
o 2.3 Divisio and partitio
o 2.4 Nominal definitions vs. real definitions
• 3 Terms with multiple definitions
o 3.1 Homonyms
o 3.2 Polysemes
• 4 In logic and mathematics
o 4.1 Classification of mathematical definitions
o 4.2 Recursive definitions
• 5 In medicine
• 6 Issues with definitions
o 6.1 Fallacies of definition
o 6.2 Limitations of definition
• 7 Notes
• 8 References

1- Basic terminology
In modern usage, a definition  is something, typically expressed in words, that attaches a meaning to a word or group of words. The word or group of words that is to be defined is called the definiendum, and the word, group of words, or action that defines it is called the definiens. In the definition “An elephant is a large gray animal native to Asia and Africa”, the word “elephant” is the definiendum, and everything after the word “is” is the definiens.[5]
Note that the definiens  is not the meaning of the word defined, but is instead something that conveys the same meaning as that word.[5]
There are many sub-types of definitions, often specific to a given field of knowledge or study. These include, among many others,  lexical definitions, or the common dictionary definitions of words already in a language;  demonstrative definitions,  which define something by pointing to an example of it (“This,” [said while pointing to a large grey animal], “is an Asian elephant.”); and  precising definitions,  which reduce the vagueness of a word, typically in some special sense (“‘Large’, among female Asian elephants, is any individual weighing over 5,500 pounds.”).[5]

2- Intensional definitions vs. extensional definitions
An  intensional definition, also called a connotative definition, specifies the  necessary and sufficient conditions  for a thing being a member of a specific set.[2]  Any definition that attempts to set out the essence of something, such as that by  genus and differentia, is an intensional definition.
An  extensional definition, also called a denotative definition, of a concept or term specifies its  extension. It is a list naming every object that is a member of a specific set.[2]
Thus, the “seven deadly sins” can be defined intensionally as those singled out by Pope Gregory I as particularly destructive of the life of grace and charity within a person, thus creating the threat of eternal damnation. An extensional definition would be the list of wrath, greed, sloth, pride, lust, envy, and gluttony. In contrast, while an intensional definition of “Prime Minister” might be “the most senior minister of a cabinet in the executive branch of government in a parliamentary system”, an extensional definition is not possible since it is not known who future prime ministers will be.

2.1- Classes of intensional definitions
A genus–differentia definition is a type of intensional definition that takes a large category (the genus) and narrows it down to a smaller category by a distinguishing characteristic (i.e. the differentia).[6]
More formally, a genus-differentia definition consists of:
1. a genus (or family): An existing definition that serves as a portion of the new definition; all definitions with the same genus are considered members of that genus.
2. the differentia: The portion of the new definition that is not provided by the genus.
For example, consider the following genus-differentia definitions:
• a triangle: A plane figure that has three straight bounding sides.
• a quadrilateral: A plane figure that has four straight bounding sides.
Those definitions can be expressed as a genus (“a plane figure”) and two differentiae (“that has three straight bounding sides” and “that has four straight bounding sides”, respectively).
It is possible to have two different genus-differentia definitions that describe the same term, especially when the term describes the overlap of two large categories. For instance, both of these genus-differentia definitions of “square” are equally acceptable:
• a square: a rectangle that is a rhombus.
• a square: a rhombus that is a rectangle.
Thus, a “square” is a member of both the genus “rectangle” and the genus “rhombus.”.

2.2- Classes of extensional definitions
One important form of the extensional definition is ostensive definition. This gives the meaning of a term by pointing, in the case of an individual, to the thing itself, or in the case of a class, to examples of the right kind. So one can explain who Alice (an individual) is by pointing her out to another; or what a rabbit (a class) is by pointing at several and expecting another to understand. The process of ostensive definition itself was critically appraised by Ludwig Wittgenstein.[7]
An enumerative definition of a concept or term is an extensional definition that gives an explicit and exhaustive listing of all the objects that fall under the concept or term in question. Enumerative definitions are only possible for finite sets and only practical for relatively small sets.

2.3 - Divisio and partitio
Divisio and partitio are classical terms for definitions. A partitio is simply an intensional definition. A divisio is not an extensional definition, but an exhaustive list of subsets of a set, in the sense that every member of the “divided” set is a member of one of the subsets. An extreme form of divisio lists all sets whose only member is a member of the “divided” set. The difference between this and an extensional definition is that extensional definitions list members, and not subsets.[8]

2.4- Nominal definitions vs. real definitions
In classical thought, a definition was taken to be a statement of the essence of a thing. Aristotle had it that an object’s essential attributes form its “essential nature”, and that a definition of the object must include these essential attributes.[9]
The idea that a definition should state the essence of a thing led to the distinction between nominal  and real essence,  originating with Aristotle. In a passage from the Posterior Analytics,[10]  he says that the meaning of a made-up name can be known (he gives the example “goat stag”), without knowing what he calls the “essential nature” of the thing that the name would denote, if there were such a thing. This led medieval logicians to distinguish between what they called the quid nominis  or “whatness of the name”, and the underlying nature common to all the things it names, which they called the quid rei  or “whatness of the thing.” (Early modern philosophers like Locke used the corresponding English terms “nominal essence” and “real essence”). The name “hobbit”, for example, is perfectly meaningful. It has a quid nominis. But one could not know the real nature of hobbits, even if there were such things, and so the real nature or quid rei of hobbits cannot be known. By contrast, the name “man” denotes real things (men) that have a certain quid rei. The meaning of a name is distinct from the nature that thing must have in order that the name apply to it.
This leads to a corresponding distinction between nominal and real definitions. A nominal definition is the definition explaining what a word means, i.e. which says what the “nominal essence” is, and is definition in the classical sense as given above. A real definition, by contrast, is one expressing the real nature or quid rei of the thing.
This preoccupation with essence dissipated in much of modern philosophy. Analytic philosophy in particular is critical of attempts to elucidate the essence of a thing. Russell described essence as “a hopelessly muddle-headed notion.”[11]
More recently Kripke’s formalization of possible world semantics in modal logic led to a new approach to essentialism. Insofar as the essential properties of a thing are necessary to it, they are those things it possesses in all possible worlds. Kripke refers to names used in this way as rigid designators.

3- Terms with multiple definitions
3.1- Homonyms
A homonym is, in the strict sense, one of a group of words that share the same spelling and pronunciation but have different meanings.[12]  Thus homonyms are simultaneously homographs (words that share the same spelling, regardless of their pronunciation) and homophones (words that share the same pronunciation, regardless of their spelling). The state of being a homonym is called homonymy. Examples of homonyms are the pair stalk (part of a plant) and stalk (follow/harass a person) and the pair left (past tense of leave) and left (opposite of right). A distinction is sometimes made between “true” homonyms, which are unrelated in origin, such as skate (glide on ice) and skate (the fish), and polysemous homonyms, or polysemes, which have a shared origin, such as mouth (of a river) and mouth (of an animal).[13][14]

3.2- Polysemes
Polysemy is the capacity for a sign (such as a word, phrase, or symbol) to have multiple meanings (that is, multiple semes or sememes and thus multiple senses), usually related by contiguity of meaning within a semantic field. It is thus usually regarded as distinct from homonymy, in which the multiple meanings of a word may be unconnected or unrelated.

4- In  logic and mathematics
In mathematics, definitions are generally not used to describe existing terms, but to give meaning to a new term.[15]  The meaning of a mathematical statement changes if definitions change. The precise meaning of a term given by a mathematical definition is often different than the English definition of the word used,[16]  which can lead to confusion for students who do not pay close attention to the definitions given.

4.1- Classification of mathematical definitions
Authors have used different terms to classify definitions used in formal languages like mathematics. Norman Swartz classifies a definition as “stipulative” if it is intended to guide a specific discussion. A stipulative definition might be considered a temporary, working definition, and can only be disproved by showing a logical contradiction.[17]  In contrast, a “descriptive” definition can be shown to be “right” or “wrong” with reference to general usage.
Swartz defines a precising definition as one that extends the descriptive dictionary definition (lexical definition) for a specific purpose by including additional criteria. A precising definition narrows the set of things that meet the definition.
C. L.. Stevenson has identified persuasive definition as a form of stipulative definition which purports to state the “true” or “commonly accepted” meaning of a term, while in reality stipulating an altered use (perhaps as an argument for some specific belief). Stevenson has also noted that some definitions are “legal” or “coercive” – their object is to create or alter rights, duties, or crimes.[18]

4.2- Recursive definitions
A recursive definition, sometimes also called an inductive definition, is one that defines a word in terms of itself, so to speak, albeit in a useful way.

Normally this consists of three steps:
1. At least one thing is stated to be a member of the set being defined; this is sometimes called a “base set.”
2. All things bearing a certain relation to other members of the set are also to count as members of the set. It is this step that makes the definition recursive.
3. All other things are excluded from the set

For instance, we could define a natural number as follows (after Peano):
1. “0” is a natural number.
2. Each natural number has a unique successor, such that:
• the successor of a natural number is also a natural number;
• distinct natural numbers have distinct successors;
• no natural number is succeeded by “0.”
3. Nothing else is a natural number.

So “0” will have exactly one successor, which for convenience can be called “1.” In turn, “1” will have exactly one successor, which could be called “2”, and so on. Notice that the second condition in the definition itself refers to natural numbers, and hence involves self-reference. Although this sort of definition involves a form of circularity, it is not vicious, and the definition has been quite successful.
In the same way, we can define ancestor as follows:
1. A parent is an ancestor.
2. A parent of an ancestor is an ancestor.
3. Nothing else is an ancestor.
Or simply: an ancestor is a parent or a parent of an ancestor.

5- In  medicine
In medical dictionaries, definitions should to the greatest extent possible be:
• simple and easy to understand,[19]  preferably even by the general public;[20]
• useful clinically[20]  or in related areas where the definition will be used;[19]
• specific,[19]  that is, by reading the definition only, it should ideally not be possible to refer to any other entity than the definiendum;
• measurable;[19]
• reflecting current scientific knowledge.[19][20]

6- Issues with definitions

6.1- Fallacies of definition
 
Certain rules have traditionally been given for definitions (in particular, genus-differentia definitions). [21][22][23][24]

1. A definition must set out the essential attributes of the thing defined.

2. Definitions should avoid circularity. To define a horse as “a member of the species equus” would convey no information whatsoever. For this reason, Locking adds that a definition of a term must not consist of terms which are synonymous with it. This would be a circular definition, a circulus in definiendo. Note, however, that it is acceptable to define two relative terms in respect of each other. Clearly, we cannot define “antecedent ” without using the term “consequent”, nor conversely.

3. The definition must not be too wide or too narrow. It must be applicable to everything to which the defined term applies (i.e. not miss anything out), and to nothing else (i.e. not include any things to which the defined term would not truly apply).

4. The definition must not be obscure. The purpose of a definition is to explain the meaning of a term which may be obscure or difficult, by the use of terms that are commonly understood and whose meaning is clear. The violation of this rule is known by the Latin term obscurum per obscurius. However, sometimes scientific and philosophical terms are difficult to define without obscurity.

5. A definition should not be negative where it can be positive. We should not define “wisdom” as the absence of folly, or a healthy thing as whatever is not sick. Sometimes this is unavoidable, however. For example, it appears difficult to define blindness in positive terms rather than as “the absence of sight in a creature that is normally sighted.”

6.2- Limitations of definition
Given that a natural language such as English contains, at any given time, a finite number of words, any comprehensive list of definitions must either be circular or rely upon primitive notions. If every term of every definiens must itself be defined, “where at last should we stop?” [25][26]  A dictionary, for instance, insofar as it is a comprehensive list of lexical definitions, must resort to circularity.[27][28][29]
Many philosophers have chosen instead to leave some terms undefined. The scholastic philosophers claimed that the highest genera (the so-called ten generalissima) cannot be defined, since a higher genus cannot be assigned under which they may fall. Thus being, unity and similar concepts cannot be defined.[22]  Locke supposes in An Essay Concerning Human Understanding[30] that the names of simple concepts do not admit of any definition. More recently Bertrand Russell sought to develop a formal language based on logical atoms. Other philosophers, notably Wittgenstein, rejected the need for any undefined simples. Wittgenstein pointed out in his Philosophical Investigations that what counts as a “simple” in one circumstance might not do so in another.[31] He rejected the very idea that every explanation of the meaning of a term needed itself to be explained: “As though an explanation hung in the air unless supported by another one”,[32] claiming instead that explanation of a term is only needed to avoid misunderstanding.
Locke and Mill also argued that individuals cannot be defined. Names are learned by connecting an idea with a sound, so that speaker and hearer have the same idea when the same word is used.[33]  This is not possible when no one else is acquainted with the particular thing that has “fallen under our notice.”[34] Russell offered his theory of descriptions in part as a way of defining a proper name, the definition being given by a definite description that “picks out” exactly one individual. Saul Kripke pointed to difficulties with this approach, especially in relation to modality, in his book Naming and Necessity.
There is a presumption in the classic example of a definition that the definiens can be stated. Wittgenstein argued that for some terms this is not the case.[35]  The examples he used include game, number and family. In such cases, he argued, there is no fixed boundary that can be used to provide a definition. Rather, the items are grouped together because of a family resemblance. For terms such as these it is not possible and indeed not necessary to state a definition; rather, one simply comes to understand the use of the term.[b]

7- Notes
1. Terms with the same pronunciation and spelling but unrelated meanings are called homonyms, while terms with the same spelling and pronunciation and related meanings are called polysemes.
2. Note that one learns inductively, from ostensive definition, in the same way, as in the Ramsey–Lewis method.

8- References
1. Bickenbach, Jerome E., and Jacqueline M. Davies. Good reasons for better arguments: An introduction to the skills and values of critical thinking. Broadview Press, 1996. p. 49
2. Lyons, John. “Semantics, vol. I.” Cambridge: Cambridge (1977). p.158 and on.
3. Dooly, Melinda. Semantics and Pragmatics of English: Teaching English as a Foreign Language. Univ. Autònoma de Barcelona, 2006. p.48 and on
4. Richard J. Rossi (2011) Theorems, Corollaries, Lemmas, and Methods of Proof. John Wiley & Sons p.4
5. Hurley, Patrick J. (2006). Language: Meaning and Definition. A Concise Introduction to Logic (9 ed.). Wadsworth. pp. 86–91.
6. Bussler, Christoph, and Dieter Fensel, eds. Artificial Intelligence: Methodology, Systems and Applications: 11th International Conference, AIMSA 2004: Proceedings. Springer-Verlag, 2004. p.6
7. Philosophical investigations, Part 1 §27–34
8. Katerina Ierodiakonou, “The Stoic Division of Philosophy”, in Phronesis: A Journal for Ancient Philosophy, Volume 38, Number 1, 1993, pp. 57–74.
9. Posterior Analytics, Bk 1 c. 4
10. Posterior Analytics Bk 2 c. 7
11. A history of Western Philosophy, p. 210
12. homonym, Random House Unabridged Dictionary at dictionary.com
13. Linguistics 201: Study Sheet for Semantics. Pandora.cii.wwu.edu. Retrieved 2013-04-23.
14. Semantics: a coursebook, p. 123, James R. Hurford and Brendan Heasley, Cambridge University Press, 1983
15. David Hunter (2010) Essentials of Discrete Mathematics. Jones & Bartlett Publishers, Section 14.1
16. Kevin Houston (2009) How to Think Like a Mathematician: A Companion to Undergraduate Mathematics. Cambridge University Press, p. 104
17. Norman Swartz - Biography. sfu.ca.
18. Stevenson, C.L., Ethics and Language, Connecticut 1944
19. McPherson, M.; Arango, P.; Fox, H.; Lauver, C.; McManus, M.; Newacheck, P. W.; Perrin, J. M.; Shonkoff, J. P.; Strickland, B. (1998). A new definition of children with special health care needs. Pediatrics. 102 (1 Pt 1): 137–140. doi:10.1542/peds.102.1.137. PMID 9714637.
20. Morse, R. M.; Flavin, D. K. (1992). The Definition of Alcoholism. JAMA. 268 (8): 1012–1014. doi:10.1001/jama.1992.03490080086030. PMID 1501306.
21. Copi 1982 pp 165–169
22. Joyce, Ch. X
23. Joseph, Ch. V
24. Macagno & Walton 2014, Ch. III
25. Locke, Essay, Bk. III, Ch. iv, 5
26. This problem parallels the diallelus, but leads to scepticism about meaning rather than knowledge.
27. Generally lexicographers seek to avoid circularity wherever possible, but the definitions of words such as “the” and “a” use those words and are therefore circular. [1] [2] Lexicographer Sidney I. Landau’s essay “Sexual Intercourse in American College Dictionaries” provides other examples of circularity in dictionary definitions. (McKean, p. 73–77)
28. An exercise suggested by J. L. Austin involved taking up a dictionary and finding a selection of terms relating to the key concept, then looking up each of the words in the explanation of their meaning. Then, iterating this process until the list of words begins to repeat, closing in a “family circle” of words relating to the key concept.
(A plea for excuses in Philosophical Papers. Ed. J. O. Urmson and G. J. Warnock. Oxford: Oxford UP, 1961. 1979.)
29. In the game of Vish, players compete to find circularity in a dictionary.
30. Locke, Essay, Bk. III, Ch. iv
31. See especially Philosophical Investigations Part 1 §48
32. He continues: “Whereas an explanation may indeed rest on another one that has been given, but none stands in need of another – unless we require it to prevent a misunderstanding. One might say: an explanation serves to remove or to avert a misunderstanding – one, that is, that would occur but for the explanation; not every one I can imagine.” Philosophical Investigations, Part 1 §87, italics in original
33. This theory of meaning is one of the targets of the private language argument
34. Locke, Essay, Bk. III, Ch. iii, 3
35. Philosophical Investigations
• Copi, Irving (1982). Introduction to Logic. New York: Macmillan. ISBN 0-02-977520-5.
• Joseph, Horace William Brindley (1916). An Introduction to Logic, 2nd edition. Clarendon Press repr. Paper Tiger. ISBN 1-889439-17-7. (full text of 1st ed. (1906))
• Joyce, George Hayward (1926). Principles of logic, 3d ed., new impression. London, New York: Longmans, Green and co. (worldcat) (full text of 2nd ed. (1916))
• Locke, John (1690). An Essay Concerning Human Understanding. ISBN 0-14-043482-8. (full text: vol 1, vol 2)
• McKean, Erin (2001). Verbatim: From the bawdy to the sublime, the best writing on language for word lovers, grammar mavens, and armchair linguists. Harvest Books. ISBN 0-15-601209-X.
• Macagno, Fabrizio; Walton, Douglas (2014). Emotive Language in Argumentation. New York: Cambridge University Press.
• Robinson, Richard (1954). Definition. Oxford: At The Clarendon Press. ISBN 978-0-19-824160-7.
• Simpson, John; Edmund Weiner (1989). Oxford English Dictionary, second edition (20 volumes). Oxford University Press. ISBN 0-19-861186-2.
• Wittgenstein, Ludwig (1953). Philosophical Investigations, Blackwell Publishing. ISBN 0-631-23127-7.

Definitions
First published Thu Apr 10, 2008; substantive revision Mon Apr 20, 2015

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
o 1.1 Real and nominal definitions
o 1.2 Dictionary definitions
o 1.3 Stipulative definitions
o 1.4 Descriptive definitions
o 1.5 Explicative definitions
o 1.6 Ostensive definitions
o 1.7 A remark

• 2. The logic of definitions
o 2.1 Two criteria
o 2.2 Foundations of the traditional account
o 2.3 Conservativeness and eliminability
o 2.4 Definitions in normal form
o 2.5 Implicit definitions
o 2.6 Vicious-Circle Principle
o 2.7 Circular definitions

• Bibliography
• Acknowledgements

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 XX one needs to investigate the thing or things denoted by XX; to discover the nominal definition, one needs to investigate the meaning and use of XX. 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. For ancient views about definitions, see the essays in Charles 2010.

1.2 Dictionary 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.
Definitions sought by philosophers are not of the sort found in a dictionary. 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 dictionary entry for 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 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. They 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,YX,Y, and ZZ”). 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,YX,Y, and ZZ, 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⟩⟨x,y⟩ is defined as the set {{x},{x,y}}{{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.⟨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 xx is a member of a member of the pair ⟨x,y⟩⟨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 Ostensive definitions
Ostensive definitions typically depend on context and on experience. Suppose the conversational context renders one dog salient among several that are visible. Then one can introduce the name ‘Freddie’ through the stipulation “let Freddie be this dog.” For another example, suppose you are looking at a branch of a bush and you stipulatively introduce the name ‘Charlie’ thus: “let Charlie be the insect on that branch.” This definition can pin a referent on ‘Charlie’ even if there are many insects on the branch. If your visual experience presents you with only one of these 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 as presenting the subject with a restricted portion of the world. This portion can serve as a point of evaluation for the expressions in an ostensive definition.[2] Consequently, the definition can with the aid of experience pin a referent on the defined term when without this aid it would fail to do so. In the present example, the description ‘the insect on that branch’ fails to be denoting when it is evaluated at the world as a whole, but it is denoting when it is evaluated at that portion of it that is presented in your visual experience.
An ostensive definition can bring about an essential enrichment of a language. The ostensive definition of ‘Charlie’ enriches the language with a name of a particular insect, and it could well be that before the enrichment the language lacked resources to denote that particular insect. Unlike other familiar definitions, ostensive definitions can introduce terms that are ineliminable. (So, ostensive definitions can fail to meet the Eliminability criterion explained below; they can fail to meet also the Conservativeness criterion, also explained below.)
The capacity of ostensive definitions to introduce essentially new vocabulary has led some thinkers to view them as the source of all primitive concepts. Thus, Russell maintains in Human Knowledge that all nominal definitions, if pushed back far enough, must lead ultimately to terms having only ostensive definitions, and in the case of an empirical science the empirical terms must depend upon terms of which the ostensive definition is given in perception. (p. 242)
In “Meaning and Ostensive Definition”, C. H. Whiteley takes it as a premise that ostensive definitions are “the means whereby men learn the meanings of most, if not all, of those elementary expressions in their language in terms of which other expressions are defined.” (332) It should be noted, however, that nothing in the logic and semantics of ostensive definitions warrants a foundationalist picture of concepts or of language-learning. Such foundationalist pictures were decisively criticized by Ludwig Wittgenstein in his Philosophical Investigations. Wittgenstein’s positive views on ostensive definition remain elusive, however; for an interpretation, see Hacker 1975.
Ostensive definitions are important, but our understanding of them remains at a rudimentary level. They deserve greater attention from logicians and philosophers.

1.7 A remark
The kinds into which we have sorted definitions are not mutually exclusive, nor exhaustive. A stipulative definition of a term may, as it happens, be extensionally adequate to the antecedent uses of the term. A dictionary may offer ostensive definitions of some words (e.g., of color words). An ostensive definitions can also be explicative. For example, one can offer an improvement of a preexisting concept “one foot” thus: “let one foot be the present length of that rod.” In its preexisting use, the concept “one foot” may be quite vague; the ostensively introduced explication may, in contrast, be relatively precise. Moreover, 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)(X), an expression containing the defined term (…X…)(…X…), and another expression (−−−−−−−)(−−−−−−−) that is equated by the definition with this expression. Such definitions can be represented thus:
X:…X…=df−−−−−−−.(2)(2)X:…X…=df−−−−−−−.
(We are setting aside ostensive definitions, which plainly require a richer representation.) When the defined term is clear from the context, the representation may be simplified to
…X…=df−−−−−−−.…X…=df−−−−−−−.
The expression on the left-hand side of ‘=df=df’ (i.e., …X…)…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 XX; the definiendum is the unspecified expression on the left-hand side of ‘=df=df’, which may or may not be identical to XX. (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; another example is provided by definitions of logical constants in terms of introduction and elimination rules governing them. Nonetheless, 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 carry over, as we shall see, 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 LL, the ground language, is expanded through the addition of a new term XX to an expanded language L+L+, where XX is stipulatively defined by a definition DD of form (2). What logical rules govern DD? 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:

1:1 man:man The True: The True=df the successor of 0,=df rational animal,=df
everything is identical to itself.(3)(3) 1:1=df the successor of 0, man:man=dfrational 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.”[3]) 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 heterogeneous 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., ‘xx 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 GG by a homogeneous definition. These languages have no resources for forming compound predicates; hence, the definiens of a homogeneous definition of GG is bound to be atomic. In a heterogeneous definition, however, the definiens can easily be complex; for example

Gx=dfx>3&x<10.(4)(4)Gx=dfx>3&x<10.
If the language has a device for abstraction— e.g., for forming sets— we could give a different sort of heterogeneous definition of GG:
the set of Gs=dfthe set of numbers between 3 and 10.(5)(5)the set of Gs=dfthe set of numbers between 3 and 10.
Observe that a heterogeneous definition such as (4) is not a mere abbreviation. For, if it were, the expression xx in it would not be a genuine variable, and the definition would provide no guidance on the role of GG in contexts other than GxGx. 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:
Gx=dfx>y&x<10.(6)(6)Gx=dfx>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 GG a meaningful unary predicate: GG expresses, in virtue of (4), a particular concept. In contrast, under stipulation (6), GG 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
It is a plausible requirement on any answer to these questions that two criteria be respected.[4] First, a stipulative 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 XX— 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 XX. There are complications here, however. What counts as a use of XX? Are occurrences within the scope of ‘say’ and ‘know’ included? What about the occurrence of XX within quotation contexts, and those within words, for instance, ‘Xenophanes’? The last question should receive, it is clear, the answer, “No.” But the answers to the previous questions are not so clear. There is another complication: even if we can somehow separate out genuine occurrences of XX, 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 XX. And let us confine ourselves to definitions that place no restrictions on legitimate occurrences of XX. The Use criterion now dictates then that the definition should fix the use of all expressions in the expanded language in which XX 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, for 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 XX, the second idea maintains, it is necessary and sufficient to explain the use of sentential items that contain XX. (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 ZZ containing the defined term is explained by reducing ZZ 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 GG, for example, need provide no way of reducing GG, 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 GG 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 LL be one such. The proof system of LL 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 LL” and “provably equivalent in LL,” and also the notions “theorem of L+L+” and “provably equivalent in L+L+” that result when the proof system of LL is supplemented with a definition DD and the logical rules governing definitions. Now, the Conservativeness criterion can be made precise as follows.
Conservativeness criterion (syntactic formulation): Any formula of LL that is provable in L+L+ is provable in LL.
That is, any formula of LL that is provable using definition DD is also provable without using DD: the definition does not enable us to prove anything new in LL. The Eliminability criterion can be made precise thus:
Eliminability criterion (syntactic formulation): For any formula AA of L+L+, there is a formula of LL that is provably equivalent in L+L+ to AA.
(Folklore credits the Polish logician S. Leśniewski for formulating the criteria of Conservativeness and Eliminability, but this is a mistake; see Dudman 1973, Hodges 2008, Urbaniak and Hämäri 2012 for discussion and further references.)[5]
Now let us equip LL with a model-theoretic semantics. That is, we associate with LL a class of interpretations, and we make available the notions “valid in LL in the interpretation MM” (a.k.a.: “true in LL in MM”) and “semantically equivalent in LL relative to MM.” Let the notions “valid in L+L+ in MM” and “semantically equivalent in L+L+ relative to MM” result when the semantics of LL is supplemented with that of definition DD. The criteria of Conservativeness and Eliminability can now be made precise thus:
Conservativeness criterion (semantic formulation): For all formulas AA of LL and all interpretations MM, if AA is valid in L+L+ in MM then AA is also valid in LL in MM.
Eliminability criterion (semantic formulation): For any formula AA of L+L+, there is a formula BB of LL such that, relative to all interpretations M,BM,B is semantically equivalent in L+L+ to AA.
The syntactic and semantic formulations of the two criteria are plainly parallel. However, even if we suppose that strong completeness theorems hold for LL and L+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 LL to be a classical first-order language with identity. The proof system of LL may contain some non-logical axioms TT; the interpretations of LL are then the classical models of TT. As before, L+L+ is the expanded language that results when a definition DD of a non-logical constant XX is added to LL; hence, XX 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 DD meets the criteria of Conservativeness and Eliminability then DD is equivalent to a definition in normal form as specified below.[6] 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,na,n-ary predicates HH, and nn-ary function symbols ff must be, respectively, of the following forms:

a=xH(x1,…,xn)f(x1,…,xn)= y= dfψ(x),= dfϕ(x1,…,xn),=
dfχ(x1,…,xn,y), (7)(8)(9)(7) a=x= dfψ(x),(8)
H(x1,…,xn)= dfϕ(x1,…,xn),(9)f(x1,…,xn)=y= dfχ(x1,…,xn,y),

where the variables x1x1, …, xnxn, yy are all distinct, and the definiens in each case satisfies conditions that can be separated into a general and a specific part.[7] 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)ψ(x) of (7) is that it satisfy an existence and uniqueness condition: that it be provable that something satisfies ψ(x)ψ(x) and that at most one thing satisfies ψ(x)ψ(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
∃yχ(x1,…,xn,y) & ∀u∀v[χ(x1,…,xn,u) &χ(x1,…,xn,v)→
u=v]∃yχ(x1,…,xn,y) &∀u∀v[χ(x1,…,xn,u) &χ(x1,…,xn,v)→u=v]
must be provable.[9]
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 DD has a sentential definiendum. (In classical logic, all definitions can easily be transformed to meet this condition.) Let DD be
ϕ(x1,…,xn)= dfψ(x1,…,xn),(10)(10)ϕ(x1,…,xn)= dfψ(x1,…,xn),
where x1x1, …, xnxn are all the variables free in either ϕϕ or ψψ. And let ϕ(t1,…,tn)ϕ(t1,…,tn) and ψ(t1,…,tn)ψ(t1,…,tn) result by the simultaneous substitution of terms t1t1, …, tntn for x1x1, …, xnxn in, respectively, ϕ(x1,…,xn)ϕ(x1,…,xn) and ψ(x1,…,xn)ψ(x1,…,xn); changing bound variables as necessary. Then the rules of inference governing DD are simply these:
ϕ(t1,…,tn)ψ(t1,…,tn)ψ(t1,…,tn)ϕ(t1,…,tn) Definiendum Elimination Definiendum Introduction ϕ(t1,…,tn)ψ(t1,…,tn) Definiendum Elimination ψ(t1,…,tn)ϕ(t1,…,tn) Definiendum Introduction

The semantics for the extended language is also straightforward. Suppose, for instance, DD is a definition of a name aa and suppose that, when put in normal form, it is equivalent to (7). Then, each classical interpretation MM of LL expands to a unique classical interpretation M+M+ of the extended language L+L+. The denotation of aa in M+M+ is the unique object that satisfies ψ(x)ψ(x) in MM; the conditions on ψ(x)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
ϕ(x1,…,xn)↔ψ(x1,…,xn).(11)(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=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=df’. Moreover, the biconditional can be iterated—e.g., ((ϕ↔ψ)↔χ)((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 FFs is the number of GGs’. 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.
GxGx=dfx>3&x<10.=dfx>y&x<10.(4)(6)(4)Gx=dfx>3&x<10.(6)Gx=dfx>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:
GxGx=df(x>3&x<10)&y=y.=df[x=0&(G0∨G1)]∨[x=1&(∼G0&∼G1)].(12)(13)(12) Gx=df(x>3&x<10)&y=y.(13)Gx=df[x=0&(G0∨G1)]∨[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):
Gx=dfx=0.(14)(14)Gx=dfx=0.
Observe that the definiens of (13) is not logically equivalent to any GG-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:
m0mn+1=1,=mn⋅m.(15)(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 nn to define the value when the exponent is n+1n+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.[10] 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.

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

ϕ=dfThe True,ϕ=df The True,

here ϕϕ is the conjunction of the members of T*T*. (If T*T* is infinite then a stipulation of the above form will be needed for each sentence ψψ in T*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*T* admissible (for a definition of X). So, the traditional account accommodates the idea that theories can stipulatively introduce new terms, but it imposes a strong demand: the theories must be admissible.[12]
Consider, for concreteness, the special case of classical first-order languages. Let the ground language LL be one such, and let its interpretations be models of some sentences TT. Say that an interpretation M+M+ of L+L+ is an expansion of an interpretation MM of LL iff MM and M+M+ have the same domain and they assign the same semantic values to the non-logical constants in LL. Furthermore, let us say that
T*T* is an implicit semantic definition of X iff, for each interpretation MM of LL, there is a unique model M+M+ of T*T* such that M+M+ is an expansion of MM.
Then the following claim is immediate:
If T*T* is admissible then T*T* is an implicit semantic definition of XX.
That is, an admissible theory fixes the semantic value of the defined term in each interpretation of the ground language. This observation provides one natural method of showing that a theory is not admissible:
Padoa’s method. To show that T*T* is not admissible, it suffices to construct two models of T*T* that are expansions of one and the same interpretation of the ground language LL. (Padoa 1900)
Here is a simple and philosophically useful application of Padoa’s method. Suppose the proof system of LL is Peano Arithmetic and that LL is expanded by the addition of a unary predicate TrTr (for “Gödel number of a true sentence of LL”). Let HH be the theory consisting of all the sentences (the “Tarski biconditionals”) of the following form:
Tr(s)↔ψ,Tr(s)↔ψ,
where ψψ is a sentence of LL and ss is the canonical name for the Gödel number of ψψ. Padoa’s method implies that HH is not admissible for defining TrTr. For HH does not fix the interpretation of TrTr in all interpretations of LL. In particular, it does not do so in the standard model, for HH places no constraints on the behavior of TrTr on those numbers that are not Gödel numbers of sentences. (If the coding renders each natural number a Gödel number of a sentence, then a non-standard model of Peano Arithmetic provides the requisite counterexample: it has infinitely many expansions that are models of HH.) A variant of this argument shows that Tarski’s theory of truth, as formulated in L+L+, is not admissible for defining TrTr.
What about the converse of Padoa’s method? Suppose we can show that in each interpretation of the ground language, a theory T*T* fixes a unique semantic value for the defined term. Can we conclude that T*T* is admissible? This question receives a negative answer for some semantical systems, and a positive answer for others. (In contrast, Padoa’s method works so long as the semantic system is not highly contrived.) The converse fails for, e.g., classical second-order languages, but it holds for first-order ones:

Beth’s Definability Theorem. If T*T* is an implicit semantic definition of XX in a classical first-order language then T*T* is admissible.
Note that the theorem holds even if T*T* is an infinite set. For a proof of the theorem, see Boolos, Burgess, and Jeffrey 2002; see also Beth 1953.
The idea of implicit definition is not in conflict, then, with the traditional account. Where conflict arises is in the philosophical applications of the idea. The failure of strict reductionist programs of the late-nineteenth and early-twentieth century prompted philosophers to explore looser kinds of reductionism. For instance, Frege’s definition of number proved to be inconsistent, and thus incapable of sustaining the logicist thesis that the principles of arithmetic are analytic. It turns out, however, that the principles of arithmetic can be derived without Frege’s definition. All that is needed is one consequence of it, namely, Hume’s Principle:

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

Another example: The reductionist program for theoretical concepts (e.g., those of physics) aimed to solve epistemological problems that these concepts pose. The program aimed to reduce theoretical sentences to (classes of) observational sentences. However, the reductions proved difficult, if not impossible, to sustain. Thus arose the suggestion that perhaps the non-observational component of a theory can, without any claim of reduction, be regarded as an implicit definition of theoretical terms. The precise characterization of the non-observational component can vary with the specific epistemological problem at hand. But there is bound to be a violation of one or both of the two criteria, Conservativeness and Eliminability.[13]
A final example: We know by a theorem of Tarski that no theory can be an admissible definition of the truth predicate, TrTr, for the language of Peano Arithmetic considered above. Nonetheless, perhaps we can still regard theory HH as an implicit definition of TrTr. (Paul Horwich has made a closely related proposal for the ordinary notion of truth.) Here, again, pressure is put on the bounds imposed by the traditional account. HH meets the Conservativeness criterion, but not that of Eliminability.

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

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

2.6 Vicious-Circle Principle
Another departure from the traditional theory begins with the idea not that the theory is too strict, but that it is too liberal, that it permits definitions that are illegitimate. Thus, the traditional theory allows the following definitions of, respectively, “liar” and the class of natural numbers NN:
• (16)zz is a liar =df=df all propositions asserted by zz are false;
• (17)zz belongs to NN =df=df zz belongs to every inductive class, where a class is inductive when it contains 0 and is closed under the successor operation.

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

Vicious-Circle Principle. “Whatever involves all of a collection must not be one of the collection (Russell 1908, 63).”
Another formulation Russell gave of the Principle is this:
Vicious-Circle Principle (variant formulation). “If, provided a certain collection had a total, it would have members only definable in terms of that total, then the said collection has no total (Russell, 1908, 63).”
In an appended footnote, Russell explained, “When I say that a collection has no total, I mean that statements about all its members are nonsense.”
Russell’s primary motivation for the Vicious-Circle Principle were the logical and semantic paradoxes. Notions such as “truth,” “proposition,” and “class” generate, under certain unfavorable conditions, paradoxical conclusions. Thus, the claim “Cheney is a liar,” where “liar” is understood as in (16), yields paradoxical conclusions, if Cheney has asserted that he is a liar, and all other propositions asserted by him are, in fact, false. Russell took the Vicious-Circle Principle to imply that if “Cheney is a liar” expresses a proposition, it cannot be in the scope of the quantifier in the definiens of (16). More generally, Russell held that quantification over all propositions, and over all classes, violates the Vicious-Circle Principle and is thus illegitimate. Furthermore, he maintained that expressions such as ‘true’ and ‘false’ do not express a unique concept—in Russell’s terminology, a unique “propositional function”—but one of a hierarchy of propositional functions of different orders. Thus the lesson Russell drew from the paradoxes is that the domain of the meaningful is more restricted than it might ordinarily appear, that the traditional account of concepts and definitions needed to be made more restrictive in order to rule out the likes of (16) and (17).

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

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

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

2.7 Circular definitions
The paradoxes can also be used to motivate a conclusion that is the very opposite to Russell’s. Consider the following definition of a one-place predicate GG:
Gx=dfx=Socrates ∨(x=Plato&Gx)∨ (x=Aristotle&∼Gx).
(18)(18)Gx=dfx=Socrates ∨(x=Plato&Gx) ∨(x=Aristotle&∼Gx).

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

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

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

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

Revision processes help provide a semantics for circular definitions.[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 we obtain depend crucially on one aspect of revision: the number of stages before objects settle down to their regular behavior in the revision process. A definition is said to be finite iff, roughly, its revision process necessarily requires only finitely many such stages.[15] For finite definitions, there is a simple logical calculus, C0C0, 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 the language. (When added to first-order arithmetic, these definitions render all Π12Π21 sets of natural numbers definable.) Because of the expressive power, the general notion of validity for non-finite circular definitions is not axiomatizable (Kremer 1993). We can give at best a sound logical calculus, but not a complete one. The situation is analogous to that with second-order logic.

Let us observe some general features of the revision theory of definitions. (i) Under this theory, the logic and semantics of non-circular definitions— i.e., definitions in normal form— remain the same as in the traditional account. The introduction and elimination rules hold unrestrictedly, and revision stages are dispensable. The deviations from the traditional account occur only over circular definitions. (ii) Under the theory, circular definitions do not disturb the logic of the ground language. Sentences containing defined terms are subject to the same logical laws as sentences of the ground language. (iii) Conservativeness holds. No definition, no matter how vicious the circularity in it, entails anything new in the ground language. Even the utterly paradoxical definition
Gx=df∼GxGx=df∼Gx
respects the Conservativeness requirement. (iv) Eliminability fails to hold. Sentences of the expanded language are not, in general, reducible to those of the ground language. This failure has two sources. First, revision theory fixes the use, in assertion and argument, of sentences of the expanded language but without reducing the sentences to those of the ground language. The theory thus meets the Use criterion, but not the stronger one of Eliminability. Second, in this theory, a definition can add logical and expressive power to a ground language. The addition of a circular definition can result in the definability of new sets. This is another reason why Eliminability fails.

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

The principal motivation for revision theory is descriptive. It has been argued that the theory helps us to understand better our ordinary concepts such as truth, necessity, and rational choice. The ordinary as well as the perplexing behavior of these concepts, it is argued, has its roots in the circularity of the concepts. If this is correct, then there is no logical requirement on descriptive and explicative definitions that they be non-circular.

For more detailed treatments of these topics, see Gupta 1988/89, Gupta and Belnap 1993, and Chapuis and Gupta 1999. See also the entry on the revision theory of truth. For critical discussions of the revision theory, see and the papers by Vann McGee and Donald A. Martin, and the reply by Gupta, in Villanueva 1997. See also Shapiro 2006.

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Acknowledgments
The author would like to thank Ed Zalta and any anonymous editor for helpful suggestions for improving this entry. Copyright ©2015 by Anil Gupta

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MORE QUOTES

Since all terms are defined by means of other terms, it is clear that human knowledge must always be content to accept some terms as intelligible without definition, in order to have a starting point for its definitions.
Bertrand Russell

[…] in the right definition of names lies the first use of speech, which is the acquisition of science; and in wrong, or no definitions, lies the first abuse; from which proceed all false and senseless tenets.
Thomas Hobbes; (1588–1679). Of Man, Being the First Part of Leviathan; Chapter IV; Of Speech

[…] it has been frequently supposed that the indefinable means that which is admittedly not understood. But so far from meaning the ‘not-understood,’ the indefinable means that which is understood; and philosophy or logic may ultimately adopt a term as indefinable only where, because it is understood, it does not require a further process of definition.
W. E. Johnson

§6. I think, it is agreed, that a definition is nothing else, but the showing the meaning, of one word by several other not synonymous terms. The meaning of words being only the ideas they are made to stand for by him that uses them; the meaning of any term is then showed, or the word is defined when by other words, the idea it is made the sign of, and annexed to in the mind of the speaker, is as it were represented, or set before the view of another, and thus its signification ascertained: this is the only use and end of definitions; and therefore the only measure of what is, or is not a good definition.
John Locke; An Essay Concerning Human Understanding; 1689/1997; Bk 3, Ch 4

The definition of an object is the declaration of its essential characteristics. Hence, a definition is given in the form of a proposition in which the object defined stands as the subject, and the essential characteristics form the predicate. It is this predicate which is the definition properly so called. […] the definition is the concept which expresses the true nature of the thing defined. […] The true definition must do more than enable us to recognize it. It must unfold its nature.
George H. Joyce; Principles of Logic; 1908; p150,151

[…] definition is ordinarily supposed to produce clarity in thinking […]
S. I. Hayakawa; Language In Action; 1939

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