Wesley C. Salmon, Logic (1984).
Logic and Language
|d]|| 9/12 has a nine in its numerator.
3/4 = 9/12.
|∴||3/4 has a nine in its numerator.|
This argument might appear valid, for the conclusion results from the first premise by substituting equals for equals. Yet the premises seem to be true and the conclusion false.
In order to clarify this example, let us begin by noting that numbers are abstract objects of some sort, and numerals are the names we give to them. Twelve, for example, is the number of apostles; this number is designated by a variety of numerals -- for example, "12," "XII," "a dozen," "36/3," etc. The first statement in argument d can only be construed as a statement about a numeral; as such, it is not correctly written. There is a certain rational number that is denoted by various numerals -- for example, "3/4," "6/8," "9/12," "0.75." The last of these numerals is not written in fractional form, so it does not even have a numerator, and those that do have numerators have different ones. The conclusion is likewise a statement about a numeral. If we want to talk about numerals, we must make names for them, and we can do so by the method just described. When we rewrite the argument properly,
|e]||"9/12" has a nine in its numerator.
3/4 = 9/l2.
|∴||"W has a nine in its numerator.|
we see that it loses any appearance of validity. Of course, we could make it into a valid argument by replacing the second premise,
|f]|| ''9/12" has a nine in its numerator.
"3/4" = "9/l2."
|∴||"3/4" has a nine in its numerator.|
but the second premise is now patently false, for it says that two numerals that are not at all alike are identical.1
The confusion of numbers with numerals seems nowhere so common as in talk of numbers to the base ten or binary numbers. Actually, there are no such things as binary numbers, any more than there are Roman numbers and Arabic numbers. Roman numerals and Arabic numerals are simply different ways of naming the very same numbers; decimal notation and binary notation are, likewise, two different systems of numerals for referring to the same numbers. Thus, for example, the statement
|g]||The number 100 is a perfect square in the system of binary numbers as well as in the system of numbers to the base 10, for 1002 = 4 = 22 and 10010 = 102|
involves a horrible confusion. The correct statement is
|h]||The symbol "100" is a numeral in both the base 2 notation and also the base 10 notation; in these two different systems of notation this numeral refers to two different numbers, but both of these numbers are perfect squares.|
If the "new math," with its emphasis upon understanding mathematical concepts, should do anything, it ought to help us distinguish clearly between numbers and their names, but I fear that it has, with its extensive reference to different systems of numerals, actually engendered confusions like g.
Proper differentiation between the use and mention of symbols leads to the recognition of a hierarchy of languages. When we want to talk about one language -- for example, French -- we often do so in a different language -- for example, English. The language being mentioned is the object language (it is the object of discussion), whereas the language being used to talk about it is the metalanguage. In a beginning French textbook written for English-speaking students, for instance, French would be the object language and English would be the metalanguage. If, however, the textbook is an English grammar written for English-speaking students, English constitutes both the object language and the metalanguage. In this case it is especially important to distinguish the instances in which English is the object language from those in which it is the metalanguage. The use of quotation marks to make names of linguistic expressions is extremely helpful in this regard. Consider, for example, statements a and b; a is in the object language, whereas b is in the metalanguage, for it contains an expression in quotes that is the name of a word in the object language. Likewise, the statement
i] Man's mortality fills me with despair
is in the object language, and
j] "All men are mortal" is a trite example of an A statement
is in the metalanguage.2
To emphasize the importance of observing the distinction between object language and metalanguage, consider an example famous from antiquity:
k] Epimenides, the Cretan, says that all Cretans are liars.
Is his statement true or false? Neither answer seems acceptable. This is the liar paradox; it can be put in more modern dress and expressed more precisely in terms of the statement
l] This statement is false.
Is statement l true? If so, it is false. Is statement l false? If so, it is true. We seem to have a statement that is false if and only if it is true. Such a statement is a plain contradiction. The source of the difficulty is that statement l violates the distinction between object language and metalanguage -- that is to say, the distinction between use and mention -- for it is being used to mention itself. We must rule it out as a nonsensical sentence that can be neither true nor false. We now realize that failure to respect these distinctions may lead to outright logica inconsistency.
1 The word "nine" in English is ambiguous; sometimes it stands for a number and sometimes for a numeral. This fact, no doubt, tends to facilitate the confusion of use and mention in arithmetical contexts. In arguments d to f we are obviously using "nine" as a name of an Arabic numeral.
The perceptive reader might ask why we reject argument e, since its conclusion seems to result from substitution of equals for equals inside the quotation marks in the first premise. What is wrong with that? The answer is that the name of the numeral used in the first premise must be taken as a single word, and that substitutions are not permitted for parts within words. Thus, even though ten equals five plus five, we will not tolerate a translation of "Ladies and gentlemen, your attention please!" into "Ladies and gentlemen, your atfive-plus-fivetion please!"
2 Displaying a statement or other expression on a separate line is also a way of mentioning it; such display is a satisfactory substitute for quotation marks. Using quotation marks for the same purpose, we could say, "The statement, 'All men are mortal,' is a trite example of an A statement" is a statement in the metalanguage. Here, we have alternated single quotes with double quotes for clarity, and we have used a sentence in the metametalanguage in order to mention a statement in the metalanguage. Evidently, we can construct as many levels as we wish in our hierarchy of languages.
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 occurences. 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 that people discover; 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" that 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 meaning 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. 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 that 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 reafize 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 people such as Aristotle. George Boole, Gottlob Frege. Bertrand Russell, Kurt Gödel, 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 obiects to which that word correctly applies. Extension is one aspect of the meaning of the word. In the second place, there are certain propertiea that 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 ot 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 that 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 definition 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 bv 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, 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 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 that 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.
Definitions of another type are often used to introduce new terms, or new meanings for old terms, in scientific contexts. The sciences -- physical, biological, and social -- endeavor to provide precise statements that contain objective information about the world. If this aim is to be achieved, scientific terms must have exact meanings in reference to objects and events found in the world. Operational definitions are often used by scientists to provide a suitable scientific vocabulary.3
The basic idea behind an operational definition is that a term is defined by specifying some physical operation that provides a criterion to determine whether the term applies to an object. Suppose, for instance, that we want to use the concept of temperature. We will need an appropriate way to measure temperature.
|d]||To say, for example, that John Jones is running fever of 102 degrees means: If a suitably calibrated thermometer is placed in his mouth and held under his tongue long enough to come into thermal equilibrium with his body, the end of the mercury column will coincide with the mark labeled "102."|
The concept of hardness of minerals can also be operationally defined.
|e]||To say that diamonds are harder than glass means: If a sharp corner of a diamond is drawn firmly across a glass surface it will leave a scratch; if a sharp corner of a piece of glass is drawn firmly across the surface of a diamond it will not leave a scratch.|
The concept of human intelligence has also been approached operationally. Psychologists have devised a number of intelligence tests and have defined the "intelligence quotient" or IQ. Some psychologists have used such intelligence tests as operational definitions of "intelligence."
|f]||To say that Mary Smith is more intelligent than John Brown means that Mary Smith achieved a higher IQ rating than did John Brown, based upon their scores on some particular intelligence test (e.g., the Stanford-Binet).|
It should be explicitly recognized that the passing out of test papers, the marking of such papers by individuals taking the test, and the scoring of the answer sheet by an optical scanning device constitute a series of physical operations even though the term being defined refers to a psychological characteristic of human beings.
In each of the foregoing three examples it should be noted that we begin with an imprecise commonsense concept -- temperature, hardness, intelligence. Everyone has a vague intuitive notion of what is meant by saying that today is a hot day or that someone is running a fever. We all have the same kind of intuitive idea of the meaning of the statement that granite is hard but chalk is not very hard. All of us have a similar sort of understanding of the claim that Einstein's intelligence was far above average. The aim of an operational definition is to render a term of that sort precise by specifying a definite operational procedure -- placing a thermometer in the patient's mouth, drawing a sharp corner on one mineral across the surface of another, administering an IQ test. The operational definition may involve logical or mathematical manipulations, such as calculating an IQ from a raw test score, but it must involve some physical operation of the previously mentioned sort. Operational definitions share a fundamental property with ostensive definitions: Each provides some direct connection between the words in our language and the nonlinguistic physical facts that we want to be able to talk about. Operational definitions are extremely important, but they involve certain dangers and limitations, which will be discussed later.
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 of 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 of the form "Only G are F." This is a contextual definition of 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, we may replace the defined word by its definiens without changing the meaning of the statement
|g]||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.|
|h]||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 doespermit 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 that 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 people have, we cannot say whether they are 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 one million dollars."
4. Sometimes we seek an intensional definition for a term whose extension is quite well known. For instance, most of the time we have very little trouble in applying the term "human being"; when we encounter an object we can almost always say definitely whether or not it is a human being. Still, we might have considerable difficulty in saying what properties distinguish human beings from nonhumans. The problem is to find an intensional definition that will provide the extension we already accept for the term.
Although we use the term "human being" quite adequately in most contexts, its extension is not precisely determined. The case is typical. There are many objects that definitely belong to the extension of the term; there are many other objects that definitely fall outside its extension; and there are some objects that are borderline cases -- neither definitely within the extension nor definitely outside the extension. To find an adequate intensional definition of "human being" requires that we find a set of properties shared by all the objects definitely within the extension of the term 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 that are definitely outside the extension. The defnition will be too narrow if it excludes from the extension objects that are definitely within the extension. Notice that a definition could be both too broad and too narrow. For example, the definition " 'human being' 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, severely retarded individuals, and insane persons as human beings. However, it is doubtful that such beings are rational, so the proposed definition would seem to exclude them from the extension of "human being." 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 being" as we understand the term, 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 that 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 a human being. Does a person first qualify as a human being at the moment of birth? Is an unborn baby a human being? Does a fetus become a human being when the mother first "feels life"? Is the fertilized egg a human being from the moment of fertilization? These are not purely academic questions; legislation stipulating that human life begins at conception has been seriously proposed. The issue is hotly debated. 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 a human being.)
Even if we have succeeded in framing an intensional definition that 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 being."
Until the term "human being" is clearly defined, it does not make sense to ask whether such visitors from space are really human beings, for the answer depends upon the definition of "human being." 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 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 that 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.
Operational definitions, as we noted, are often used to introduce precise scientific terminology. It is important to say something about their limitations. Take the concept of temperature. If we are considering a particular operational definition, we must insist that it not depart too radically from our subjective judgments. The temperature of a steaming cup of coffee must turn out to be higher than that of a glass of icewater; the temperature on a typical summer day in the Sahara Desert must turn out to be higher than that of a typical winter day at the North Pole. A definition in terms of the readings on mercury thermometers does yield such results. Some of our subjective temperature judgments will, however, be rejected. Our sense of coldness will be different on a windy winter day than on a calm day when the thermometer gives the same reading. We realize, however, that our subjective judgments are sometimes unreliable, and we are willing to accept the thermometer as an objective standard.
A satisfactory operational definition of temperature will also have to fit reasonably well with more sophisticated scientific theories. One important factor is that our operational definition applies only over a limited range; we cannot use a mercury thermometer to measure the temperature of the surface of the sun or the temperature of liquid helium. However, we must require our operational definition to fit consistently with operational definitions that apply to temperatures above and below the range of the mercury thermometer. Another crucial consideiation arises from the fact that the temperature of a gas is also defined theoretically in terms of the average kinetic energy of its molecules. Since molecules, their masses, and their velocities cannot be individually and directly manipulated in the laboratory, this definition is not operational. Nevertheless, the operational definition in terms of the mercury thermometer fits reasonably well with this theoretical definition. The operational definition actually helps to bridge the gap between our direct sensations of temperature and the concept of temperature that appears in more advanced physical theories.
Not all operational definitions are as satisfactory as the foregoing definition of temperature. Let us return to the concept of intelligence. Many intelligence tests, it has been charged, have a bias in favor of white middle- to upper-class Americans. If native American Indians, or members of some other ethnic minority, get lower scores on these tests on the average, it may not mean that they are less intelligent than white middle- to upper-class Americans. For instance, one test included the question "Who discovered America?" The answer "Columbus" was good for two points; the answer "Leif Ericcson" was awarded one point. An American Indian child who answered "We did" received no points. Because the question of the relative intelligence of different races has enormous social significance, it is important to be sure that our tests measure intelligence rather than socio-economic status. To say that intelligence is, by definition, what is measured by intelligence tests surely begs the basic issue. A fundamental question concerning operational definitions is whether the operation stipulated actually measures the property we intend. Inappropriate operational definitions are likely to yield concepts that lack theoretical significance.
In philosophy, too, we seek definitions that 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 that 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. For many people "thorough" and "scholarly" are terms ot praise: for almost everyone "tiresome" and "pedantic" carry strong negative connotations.
Definitions are often designed to transfer emotive force. This can be done in either of two ways. First, we may take a word that 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 "socialist" 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 sordidriess 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.4
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 being," we were concerned with characterizing customary usage to some extent (purpose 1), making a somewhat vague term more precise (purpose 3), providing an intensional definition for a term whose extension is fairly clear (purpose 4), and providing a term 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.
3 The term "operational definition," was invented by the physicist Percy W. Bridgman, who also advocated a philosophical view called operationism or operationalism. According to this doctrine, every legitimate scientific term must be operationally defined. In discussing operational definitions, I do not intend to advocate the operationist philosophy of science. We can readily admit that operational definitions are often useful without insisting that every, scientific term must have an operational definition. I seriously doubt that the term "electron" can be defined operationally.
4 The term "persuasive definition" was originated by Charles Stevenson; see his Facts and Values (New Haven, Conn.: Yale University Press, 1963), Essay III. Stevenson provides many excellent examples that illustrate both the utility and the dangers of persuasive definitions.
In section 12 we saw that there are certain statements, known as tautologies, whose truth value is T under all circumstances: we observed that these statements constitute a small portion of all statements whose truth can be certified by logical means alone. Now that we have discussed definitions, it is time to consider the broader class. Let us begin with the tautological form
a] p ⊃ p
which has an instance,
b] If x is a bachelor, then x is a bachelor.
This obvious truism holds generally; hence,
c] All bachelors are bachelors.
is a logical truth. If we now invoke the definition
d] "Bachelor" means "unmarried adult male,"
we can substitute in c and assert
e] All bachelors are unmarried adult males,
from which it follows that
f] All bachelors are unmarried.
Logical truths like c are statements whose truth follows from their logical form -- that is, the meanings of the logical terms -- alone. The class of analytic statements contains all such logical truths, as well as all statements like e and f whose truth follows from logical truths by virtue of definitions. Statement f is not, itself, a definition, but no information other than a knowledge of the meanings of its logical and nonlogical terms is needed to establish its truth. It is not necessary to observe a large number of bachelors to ascertain the truth of f. Since a bachelor is, by definition, an unmarried adult male, the word "bachelor" cannot correctly be applied to any person who is not unmarried. Statements of this type, whose truth follows from the definitions of the words (logical and nonlogical) that occur in them, are known as analytic statements.5
g] Some bachelors are married.
is like statement f, except that the meanings of the words occurring in g make it false rather than true. Statements of this type are contradictory statements (or self-contradictions, or merely, contradictions). A contradictory statement is one whose falsity follows from the meanings of the words that occur in it. In a broad sense, then, analytic statements and contradictory statements may be considered logical truths and logical falsehoods respectively.
h] Some bachelors do not own cars.
i] All bachelors are blind.
are statements whose truth or falsity is not determined solely by the meanings of the words they contain. They are called synthetic statements. To be sure, you have to know the meanings of the words that occur in them before you can find out whether they are true or false, but it would be possible to understand the meanings of the words perfectly without knowing whether either one is true or false. Statement h is obviously true, but there is nothing in the definition of the word "bachelor" that implies that some bachelors do not own cars. It is a matter of fact, over and above the meaning of the word, that some bachelors do not own cars. Similarly, statement i is obviously a false sentence, but again this is not a consequence of the definitions of the words. The only way to find out that h is true and i is false is to investigate bachelors. It is not sufficient merely to investigate the word "bachelor" (and the other words in these statements). Synthetic statements are not logical truths or falsehoods; they are factual statements.
There are important consequences of these distinctions. The reason that the truth of an analytic statement can be ascertained without recourse to anything beyond the meanings of words is that analytic statements contain no information about matters of fact outside of language. Whatever information is conveyed by an analytic statement is information about language itself and not about the facts that language refers to. It is not that analytic statements refer explicitly to language; rather, they express relationships among definitions. Definitions are, as we have said, conventions. These conventions are neither true nor false. The conventions do, however, have consequences, and analytic statements are among the consequences of our definitions.
We may explain in the following way why analytic statements have no factual content. Suppose there is a room whose contents we know nothing about. We can imagine all sorts of possibilities concerning what is in that room, but we have no way of knowing which of these imaginable possibilities is actually true. Now, suppose we learn that the synthetic statement "There is a book in the room" is true. We can immediately exclude as not actual any of the imaginable possibilities about the contents of the room that did not include at least one book among the contents. For example, we now know that the room is not completely bare; we know that it is not completely filled with corn flakes to the exclusion of everything else; we know that it does not have just a bed, chair, table, lamp, and rug, and so on. Now we have some information about the contents of the room, but not very much. Next, suppose we learn that the statement "There is exactly one book in the room, and this book is green, has hard covers, has 348 pages, is a mystery novel, and rests upon a desk" is true. Now we have much more information about the contents of the room, for the truth of this latter statement is incompatible with many more imaginable possibilities than the truth of the former is. For example, we now know that the room does not have more than one book in it; we know that it is not without a desk; we know that it does not contain a desk with a paperback philosophy book; and so on. All of these possibilities that have been excluded were compatible with the truth of the former statement. In general, the amount of factual information in a statement is a function of the number of possibilities its truth excludes. The more information a statement contains, the fewer possibilities it leaves open.
Let us consider analytic statements from this standpoint. How many possibilities does the truth of an analytic statement exclude? The answer is none. Since an analytic statement is necessarily true, it will be true regardless of what circumstances hold. Its truth does not exclude any imaginable possibilities, and knowing it to be true does not enable us to draw any specific conclusion about what possibilities are actualized. Suppose, referring back to the previous example, we were to learn that the statement "All green books in the room are green" is true. This statement is analytic and it is completely uninformative. It does not tell us whether there is a book in the room (see section 13), and it does not tell us anything about the color of any book that might be in the room. It is true regardless of the actual contents of the room. It is a statement that cannot be false, and for this reason it does not convey information.
One important feature of analytic statements is that observational evidence is not relevant to their truth. Consequently, it would be pointless to try to refute them by reference to observational data. Such attempted refutation would be irrelevant. For example, many (if not all) truths of mathematics are analytic. Consider the statement "Two plus two equals four." We would establish the truth of such a statement by reference to the definitions of "two," "four," "plus," and "equals." Small children sometimes learn their sums by counting fingers, blocks, or apples, but the truths of arithmetic do not rest upon observational evidence of that sort. If they did, we would have to consider negative observational evidence, as well. For instance, it is a fact that two quarts of alcohol mixed with two quarts of water yield less than four quarts of solution. If observational evidence were relevant, we would have to conclude that two plus two sometimes equals four but sometimes does not. We do not draw any such conclusion, however, because the experimental facts cited are not relevant. We say instead that if we put two quarts of alcohol into a container and two quarts of water into the same container, then we put four quarts of liquid into the container. The fact that the resulting solution occupies less than four quarts volume is an interesting physical fact that has no bearing upon any truth of arithmetic.
Analytic statements, then, are necessarily true, and they have no factual content. It is easy to see that no synthetic conclusions can be validly deduced from analytic premises alone. As we have seen, in a valid deductive argument there can be no information in the conclusion that is not in the premises. If all of the premises are analytic, they have no factual content. Therefore, the conclusion cannot have factual content either. Arguments are often given, however, in which the premises are analytic and the conclusion has content. Such arguments must be invalid. For example,
|j]||People have sometimes argued for fatalism on the basis of the premise "Whatever will be, will be." Certainly the premise, taken literally, is analytic; it says only that the future is the future. One could hardly deny the premise by maintaining that some things that will be, won't be. The conclusion frequently drawn from the premise is that the future is completely predetermined and that human action and human choices cannot have any effect upon the future. Such an argument is invalid, and it can be seen to be invalid precisely because it proceeds from an analytic premise to a synthetic conclusion.|
|k]||In the past, many a businessperson has tried to justify unscrupulous practices by appealing to the statement "Business is business." Superficially, this statement looks analytic; but if it were analytic, it could not support the far-from-analytic conclusion that dishonesty and irresponsibility are legitimate tactics in business. If the premise is to support the conclusion, it must mean something like "Business is not, and cannot be, conducted ethically." Putting the premise in a form that looks analytic gives it an appearance of indubitability; stating the synthetic premise literally is rhetorically less effective, for it no longer seems so unquestionable. Much of the same sort of remark applies to the slightly more complicated contemporary formula, "The business of business is business." It certainly cannot constitute a logical justification for the conclusion that the business of business is to make money by any means, no matter how unethical.|
|l]||People have gloomily concluded that decency, altruism, and consideration of others are completely absent in mankind. The premise from which this conclusion is deduced is "People never act unselfishly." This premise is not obviously analytic; but when it is challenged, we find that no conceivable factual evidence could possibly refute it. When we point out that saints and heroes have sacrificed everything for others, we are told that they did so because they wanted to, so they were really acting selfishly. It appears that no act would be unselfish unless it were motivated by a motive that the agent did not have. But to be motivated by a motive that one does not have is logically impossible. Hence, the premise turns out after all to be analytic, and the argument must therefore be invalid.|
5 W. V. Quine has vigorously attacked the analytic-synthetic distinction; see the essay "Two Dogmas of Empiricism" in his book From a Logical Point of View, 2nd ed (New York., Harper Torchbooks, 1963). Quine's misgivings, which are based upon subtle difficulties about the identification of definitions in rich complex languages such as English, are not shared by all philosophers. Since we are assuming -- for purposes of the present elementary discussion that many definitions are available, we shall not abandon what seems to me an extremely useful distinction. Moreover, even if Quine's view is correct, the import ot his arguments cannot be appreciated without an understanding of the distinction presented here.
Two statements may be related in such a way that if one of them is true, the other must be false; and if one of them is false, the other must be true. We may not know which is the true one and which is the false one, but we can be certain that one of them is true and the other is false. Such statements are called contradictories of each other; the relation between them is called contradiction. The following pairs of statements are contradictories; you should satisfy yourself that in each pair it is impossible for both to be true, and it is impossible for both to be false.6
a] All horses are mammals. Some horses are not mammals.
b] No logicians are philosophers. Some logicians are philosophers.
c] It is raining here. It is not raining here.
In general, for any statement p, p and not-p are contradictories. Furthermore, any statement of the form "p or not-p" is analytic and, consequently, necessarily true; this is called the law of excluded middle or tertium non datur. Also, any statement of the form "Not both p and not-p" is analytic; this is called the law of contradiction. These logical laws express the fact that every statement is either true or false and no statement is both true and false.7 Two statements can be related to each other in such a way that it is impossible for both of them to be true, but it is possible for both to be false. In such cases the statements are called contraries and the relation between them is called contrariety. For example, the following are contraries:
d] It is cold here. It is hot here.
For any given time and place, it is impossible that it should be both cold and hot, but it is possible for the temperature to be moderate. Thus, both statements can be false, but they cannot be true
Failure to understand the difference between contraries and contradictories has led to much confusion.
|e]||In example f, section 9, we referred to the dilemma of free will. This dilemma concludes that freedom of the will does not exist because it is incompatible with both determinism and indeterminism. Some people have rejected this dilemma on the grounds that there may be some third alternative besides determinism and indeterminism. To deal with this objection we must be clear about the meanings of the words "determinism" and "indeterminism." Determinism is the doctrine that everything is causally determined. Indeterminism is the doctrine that something is not causally determined. A third doctrine, which we shall call "chaoticism,'' is the doctrine that nothing is causally determined. It is easy to see that determinism and indeterminism are contradictory doctrines; one of them must be true. Determinism and chaoticism, on the other hand, are contrary doctrines; both would be false if some things were causally determined and some were not. If the free-will dilemma used as a premise the statement "Either determinism or chaoticism holds," it would be open to the objection that there is a third alternative. Actually, the premise is "Either determinism or indeterminism holds." This premise is analytic; it is an instance of the law of excluded middle.|
|f]||There is an unfortunate tendency on the part of many people to "think in extremes"8 -- for instance, to regard everything as wholly good or wholly bad. We are all too familiar with our compatriots who believe that our nation can do no wrong, whereas other nations can do no right, except when they cooperate with our goals. Likewise, we are all too familiar with people who believe that their political party is on the right side in every political dispute, whereas the opposition party is always in the wrong. Thinking in extremes is a confusion of contraries with contradictories; it is the error of treating contrary statements as if they were contradictories. "X is wholly good" is contrary to "X is wholly bad"; thinking in extremes proceeds as if they cannot both be false.|
|g]||Some people have suggested that logic is dangerous because it leads to thinking in extremes. This results from the principle that every statement is either true or false. It would be better, so the allegation goes, to regard statements as true to some extent and false to some extent, in order to avoid this species of thinking in extremes. Strangely enough, this criticism of logic rests upon the same error as that committed in thinking in extremes, the confusion of contraries and contradictories; "p is true" and "p is false" are not contraries and should not be treated as such, there is nothing in logic that proposes the confusion of contrariety and contradiction; rather, the study of logic should end the confusion.|
6 In section 13 we noted the relation of contradiction between A and O statements and between E and I statements.
7 In section 12 we showed that the law of excluded middle is a tautology: it is a simple matter to make a truth table that will do the same for the law of contradiction.
8 Thinking in extremes has often been called black-white thinking. This would be a good name for the error except for its unfortunate tendency to reinforce the identification of white with good and black with bad.
It is a well-known fact that a single word may have a variety of meanings. There is usually no harm in this phenomenon, for the context generally determines which of the several meanings is intended. For instance, the word "table" has one meaning in which it refers to furniture and another in which it refers to lists (as in "table of contents" or "table of logarithms"), but this multiplicity of meanings could hardly ever lead to confusion. There are cases, however, in which a word is used in such a way that we cannot tell which of its several meanings is intended. In such cases, we say that the word is being used ambiguously, for the statement in which it occurs is open to at least two distinct interpretations. In the absence of additional context, the statement "Roxanne is dusting her plants" may mean that she is cleaning the plants by removing dust from them or that she is protecting them by applying an insecticidal dust.
Multiplicity of meaning leads to logical difficulty if the same word is used in two different senses in the same argument and if the validity of the argument depends upon that word maintaining a constant meaning throughout. Such arguments commit the fallacy of equivocation. Consider a trivial example:
|a]|| Only man is rational.
No woman is a man.
|∴||All women are irrational.|
This argument would be valid if the term "man" had the same meaning each time it occurred. However, for the first premise to be true, "man" must mean "human being," whereas, for the second premise to be true, "man" must mean "human male." Thus, if the premises are to have any plausibility whatever, the term "man" must shift its meaning. We would surely be misconstruing the premises if we understood the term "man" to mean the same in both. The result is an equivocation, which destroys the validity of the argument.
Let us consider some less trivial examples:
|b]||The normal male college freshman has a lively interest in athletics and liquor and a consuming preoccupation with sex. These things engult his attention. Poetry, for instance, leaves him cold. Once in a while, however, you run across a freshman who is entirely different. He spends much of his free time reading, and perhaps even writing, poetry. He does so because he likes to. He is an abnormal case. Often he has an abnormally high IQ. He is not interested in the things that interest normal boys of his age. He is a boy set apart from other boys. When we encounter boys of this kind we ask ourselves, "What can be done for them?" Surely there must be some way to help them achieve a normal adjustment to life!|
The word "normal" in this passage shifts from one meaning to another. Toward the beginning, the word "normal" means simply "average"; at the end, it means "healthy." "Average" is purely statistical term which has no valuational content; "healthy" is, among other things, an evaluative term. There is, in this passage, an implicit argument of the following sort:
|c]|| A college freshman who likes poetry is an abnormal boy.
An abnormal boy is an unfortunate case.
|∴||A college freshman who likes poetry is an unfortunate case.|
This argument contains an equivocation on the word "normal." Arguments of this sort have been seriously proposed as arguments for mediocrity and conformity.
|d]||If you think that a brick wall is solid, you are quite wrong. Modern science has shown that things like brick walls are made up of atoms. An atom is something like the solar system; electrons revolve around the nucleus much as the planets revolve around the sun. Like the solar system, an atom is mostly empty space. What common sense regards as solid, science has shown to be anything but solid.|
The equivocation that may well be present in this passage involves the word solid." On the one hand, common sense may be right in regarding a brick all as solid -- meaning that it is strongly resistant to penetration. On the ther hand, science may also be right in regarding the same brick wall as lacking solidity -- meaning that it is not composed of tightly packed material articles.
|e]||Example l of section 33 affords a good example of equivocation. To defend the premise "People never act unselfishly," the author of the argument takes the word "selfish" to apply to any act whose motivation comes from the person who performs it. When the word "selfish" is construed in this manner, the premise becomes analytic, and hence, irrefutable. When, however, attention is turned from the defense of the premise to the deduction of a conclusion from it, the author of the argument changes the meaning of the word "selfish." Now the word "selfish" applies to acts in which the agent ignores the interests of other people. With this latter meaning of the word "seltish" the premise is no longer analytic. It is not true by definition (or even true at all) that people are never motivated to take account of the interests of other people. Saints, heroes, and ordinary people act from their own motives. Still they are sometimes decent, considerate, and altruistic, for these very motives include a concern for the welfare of others.|
Now that we have dealt with a fairly wide variety of topics in logic, it is time to reconsider the goals we have been attempting to achieve. In the very first section of the book, it will be recalled, we emphasized a basic distinction between statements that are supported by evidence and those that are not. The book has been devoted almost entirely to an elaboration of that distinction and a discussion of the ways in which statements can be supported by evidence. This has been the chief focus of our excursion into the field of logic.
We noted at the outset that statements supported by evidence are the conclusions of arguments. The evidence that is taken to support the conlusion is formulated in the premises. As we have seen, there are two kinds of support, deductive and inductive. If we are to be justified in accepting the conclusion of an argument on the basis of the evidence certain conditions must be fulfilled:
First, we must have good reasons for accepting the premises as true.
Second, the argument must possess a logically correct form.
Third, in the case of inductive arguments, we must satisfy the requirement of total evidence.
We have examined a number of basic correct forms of each type. In order to facilitate detection and avoidance of common logical mistakes, we have also discussed a number of fallacies of various sorts.
We have, for the most part, directed our attention to the logical relationship between the premises and conclusions of arguments, leaving aside the question of truth or falsity of premises. Moreover, although we have often remarked on the requirement of total evidence in connection with induction, we have not given a great deal of consideration to the question of whether it is, in fact, satisfied in particular example. This has enabled us to deal with the logical properties of arguments in abstraction from such factual questions as the truth values of the premises and whether they encompass all of the available relevant evidence. The main object of our interest throughout the book has been the nature of the relation of evidential support.
What should be the result of these studies? We should, in the first place, be able to distinguish between unsubstantiated assertions and those for which supporting evidence is offered. We should be able to distinguish between the appeal to supporting evidence and mere emotional appeals. It is an interesting -- and perhaps disillusioning -- experience to examine ordinary newspapers, magazines, radio programs, television shows, etc., in a serious attempt to find arguments. They are few and far between. For the most part, one finds unsubstantiated assertions mixed with emotional appeals. This is not to deny that appeals to our emotions have important uses; they are, I should think, quite appropriate in such contexts as fund drives for worthy charities, efforts to persuade people to engage in worthwhile activities, or attempts to influence their feelings in ways we consider desirable. The point is simply that appeals of these sorts should not be mistaken for logical arguments.
In cases in which evidence is offered, we should be able, in addition, to assess critically the quality of the arguments employed. When the evidence is inadequate we should be able to see what kind of additional evidence would be needed to make the argument compelling. As a result, we should be able to function more intelligently in our public and private roles as consumers, citizens, and individuals seeking personal satisfaction.
Third, we should also be able to apply these critical techniques to our our thinking, thereby improving our ability to evaluate the results of our own intellectual efforts. This should enable us to function more rationally in deliberating, formulating plans, making choices, and deciding what to do. Even more importantly, the study of logic may contribute in some small way to our achievement of more reliable knowledge and deeper understanding of the world in which we live. Logical skill is no substitute for factual knowledge, but it is a tool for critical evaluation of knowledge claims.
It should be re-emphasized, in conclusion, that our survey of logic has been conducted at an extremely elementary level. We have barely scratched the surface. Both deductive and inductive logic are highly developed fields of study in which profound and far-reaching results have been achieved. If would be gratifying indeed if readers of this book were motivated to learn more about the areas with which we have been concerned. Some bibliographical suggestions follow.