Theory of Definition Arthur Pap Philosophy of Science, Vol. 31, No. 1. (Jan., 1964), pp. 49-54. Stable URL: http://links.jstor.org/sici?sici=0031-8248%28196401%2931%3A1%3C49%3ATOD%3E2.0.CO%3B2-U Philosophy of Science is currently published by The University of Chicago Press.
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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 classiJication. 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 ec 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,l we call a definition which is an empirical proposition about verbal usage lexical. But we split Copi's category of "theoretical" definitions into tlzeoretical
* 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 cpinion, merits wider circulation. T h e 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" and of the wording in Exercise 1 at the end; it was prepared for publication by John T. Wilcox, Assistant Professor of Philosophy, Emory University. T h e reference is to I. M. COPI,Infroduction t o Logic, 1st ed. (New York, Macmillan, 1953). -JTW.
50
ARTHUR PAP
in the sense of empirical propositions about scientific objects, and analyses of concepts. T o see the difference, compare "Water is a substance composed of molecules consisting of two hydrogen atoms and one oxygen atom (H,O)" 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. I t is easy to confuse a lexical definition with an analysis because one tends to confuse the use of a word with its mention. When 1 say, "John is a tall boy," I use the name et 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 langdage 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 nnuld 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 Lin oncle est un hornme qui a les &me parents que qz~elqueautre personne qui est un parent" malies (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 may 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 altvays perfectly sharp. What Copi calls a precising dejkition 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 J15,000), 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,
THEORY OF DEFINITION
51
prior to the precising definition, vague just means that there are borderline cases with respect to it, i.e., persons who wouid not uniformly be called "wealthy" and would not uniformly be called "not wealthy" either. T h e 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 dejinition 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." T o 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). ~
EXERCISE: Identify the analytic statements in the following group, and prove that they are analytic by a) formulating the correct definitions by which they are true; and b) transforming them, with the help of those definitions, into logically true statements: 1 ) all squares are equilateral; 2) no spinsters are married; 3) a parent is either a mother or a father; 4) if John is smarter than B111, then he is smart; 5) either John is smarter than Bill or Bill is smarter than John; 6) if John is taller than Bill, then Bill is shorter than John; 7) every hour contains sixty minutes; 8) if one box is inside a second, and the second box is inside a thsrd box, then the first box is inside the third box.
Formal classiJication. 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 dejiniendum with the dejiniens 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 sentenc:: "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." I t 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"!
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ARTHUR PAP
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"Vn 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. I n 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. T o 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 9 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). T h e following kinds of explicit definition should be distinguished: genus-difference, disjunctive, and quantitative. T h e word "sibling" may be disjunctively defined as "brother or sister" (provided you don't define "brother" as "male sibling" and e< sister" as "female siblingm!),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. I t 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. Don't confuse the property-tern1 "brother" with the relation-term " b r ~ t ~ of." e r TI-~e for~ncr 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 tcrms of "brother of": a brother is a person who is brother of some other person. I t is true that in a dictionary you are lilwly to find "sibling" dcfined in terms of "brother" and "sister," and also the latter words in terms of the former. When such ci~czllardefinitions 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. Rut the dictionary maker cannot easily predict which are the words already understood and which the words that prospective uscrs of the dictionary will "look up." T o 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".
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THEORY OF DEFINITION
Momentum is not a species of velocity, the way lions are a species of animals. Similarly, the definitions "x3 = x x . x , < t2 = 1 I", "i = d--jy'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 dejiniens 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. T o 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 o) = x. Here 'j"' 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 2' = (2 2)' = (2 1')' = (2 1)" = (2 -k 0')" proceeds as follows' 2 = (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).
+
9,
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+ +
+
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EXERCISES: 1) Construct contextual definitions of the relation-terms "uncle of," "grandfather of," and "nephew of," and of the property-terms "sister-in-law" and "cousin," on the basis of the following terms: "male," "female," "parent of," and "married." 2) Construct operational definitions of the following terms: "aggressive," "elastic," "inflammable," "stubborn." 3) Classify the following definitions with respect to the enumerated forms: a) a physically homogeneous substance is a substance all samples of which are equally dense and melt or solidify or vaporize (under standard pressurc) at the same temperature b) a parent is a mother or a father c) acceleration is rate of change of velocity d) to say that a man has freedom of speech is to say that he is legally permitted to say whatever he wants to say, at whatever time and place e) two bodies have equal mass if they stretch a spring the same amount f) n! = n((n - I)!), and O! = 1. 4) Compute 5 ! on the bassis of definition f) above. 5) Defiine contextually the inclusive sense of "or" in terms of its exclusive sense, "and," and "not."
ARTHUR PAP
Epistemological Classi$cation stipulative definition
/'
abbreviatory (wholly arbitrary)
propositional definition
\ precising
\
/
(partly arbitrary)
empirical proposition
/ \
analysis of a concept (of a property or relation)
lexical theoretical (about (about linguistic objects) (usage)
Formal Classification definition by examples
)denotative)
ostensive
\
not ostensive (inductive) operational
disjunctive (enumeration of species)
by genus and difference
by quantitative
simple
synonym
