Wesley C. Salmon, Logic (1984).
|a]||All diamonds are hard.||True|
|Some diamonds are gems.||True|
|∴||Some gems are hard.||True|
|b]||All cats have wings.||False|
|All birds are cats.||False|
|∴||All birds have wings.||True|
|c]||All cats have wings.||False|
|All dogs are cats.||False|
|∴||All dogs have wings.||False|
In each of these arguments, if the premises were true, the conclusion would have to be true. It is impossible for a valid deductive argument to have true premises and a false conclusion.
Invalid arguments may have any combination of truth and falsity for premises and conclusion. We shall not give examples of all possible combinations, but we must re-emphasize the fact that an invalid argument may have true premises and a true conclusion. This point was illustrated by example b of section 1.
Iln order to study validity and invalidity, we classify arguments in terms of their forms. From the standpoint of deductive logic, the subject matter of an argument is unimportant; the form or structure is what counts. Validity or invalidity is determined by the form, not by what the premises and conclusion talk about. By examining the form of an argument, in abstraction from the content of the premises and conclusion, we can investigate the relation between the premises and conclusion without considering their truth or falsity.
Different arguments may share the same form; and since form determines validity, we may speak of the validity of a form as well as the validity of an argument. If we say that a form is valid, we mean that it is impossible for any argument that has that form to have true premises and a false conclusion. Any argument that has a valid form is a valid argument. We test an argument for validity by seeing whether it has a valid form.
Consider example b. This argument contains terms referring to three classes of things: birds, cats, and things that have wings. If we are dealing with the form of this argument, we do not care about the particular characteristics of these things, so we may deliberately refuse to mention them. We may substitute letters for each of these terms, using the same letter to replace the same term each time it occurs and different letters to replace different terms.
Using the letters "F," "G," and "H," respectively, we get
|d]||All G are H.|
|All F are G.|
|∴||All F are H.|
We may do the same thing with example c. This time, let "F," "G," and "H" stand for dogs, cats, and things that have wings, respectively. Again, the result is d. Examples b and c have the same form; this form is given by d. Schema d is not, itself, an argument; it is a form that becomes an argument if particular terms are substituted for the three letters. It is a valid form. Regardless of what terms are substituted for "F," "G," and "H," provided the same term is substituted for a given letter each time it occurs, the result will be a valid argument. It does not matter to what kinds of things "F," "G," and "H" refer. If it is true that all G are H, and it is true that all F are G, then it must be true that all F are H.
We can easily illustrate the fact that the validity of an argument depends only upon its form, not upon its content or the truth, or falsity of the statements occurring in it.
|e]||All flageolets are fipple-flutes.|
|All monauli are flageolets.|
|∴||All monauli are fipple-flutes.|
Argument e has the form d. You may not have any idea which statements, if any, in this argument are true; however, it is obvious that the argument is valid: If the premises are true the conclusion cannot fail to be true. We can see this by examining the form. It is not necessary to find out what the statements mean, let alone whether they are true or false. The form is valid.
An invalid form of argument is called a deductive fallacy. A common way of exposing a fallacious argument is to compare it with another argument of the same form in which the premises are true but the conclusion is false. We shall call this method of proving invalidity the method of counterexample. To say that an argument is valid means that it has a valid form. To say that a form is valid means that no argument with that form can have true premises and a false conclusion. Thus, when we say that an argument is valid, we are making a universal statement about all arguments of that form.
A universal statement can be refuted by one negative instance -- one counterexample. In order to prove that a form is fallacious, it is sufficient to find a counterexample, an argument having that form but having true premises and a false conclusion. At this point we must insert a word of caution about the method of counterexample. It is important to note that this method conclusively proves the invalidity of a form but not necessarily that of a particular argument. To say that an argument is fallacious means that there is no valid form that it exemplifies. Hence, if we merely show that an argument has a certain form, and that this form is invalid, we have not thereby proved that the argument is invalid. In order to establish conclusively the invalidity of an argument, it is necessary to show that there is no other form it possesses by virtue of which it is valid.
Any particular argument exemplifies more than one logical form as the following consideration makes obvious. As we shall see in section 10, certain kinds of arguments depend for their validity only upon the relationships among simple statements viewed as wholes: the internal structures of these statements do not matter. Arguments of this sort -- so-called truth-functional-arguments -- can best be symbolized by using a single letter to stand for a given statement. If we analyze argument e, for example, by the method of section 10, its form would be given by
where the letters "p," "q," "r" can stand for any statements whatever. Obviously, f is not a valid form. If, however, we analyze argument e by the syllogistic method of section 14, its form would be d. Thus, e may be said to have a truth-functional form f and a syllogistic form d. You might be tempted to say that d rather than f is the form of argument e, but that would not be quite correct. Both d and f are forms of argument e; however, d is a valid form whereasn f is a fallacious form. By showing that argument e has form d we show that argument e is valid. By showing that argument e exemplifies form f as well, we certainly do not show that it is an invalid argument. We may say that d is the form by virtue of which e is a valid argument, whereas form f has no such unique or important role. The point is this: If we had started by analyzing argument e truth-functionally (see section 10) and attributed form f, we would not have shown that argument e is invalid. We could only have concluded that it is not valid by virtue of form f and that we shall have to find another form it possesses if we are to show it to be a valid argument.
In practice, we can often tell which form reveals the essential logical structure of a given argument. If that form happens to be valid, our job is done, for an argument is valid if it possesses any valid form, regardless of which other forms it might exemplify. If the form we choose to represent the given argument is fallacious, we can usually conclude at least tentatively that the argument is invalid. Anyone who wishes to convince us otherwise must show us a valid form the argument possesses.
We shall use the method of counterekample to show that the various fallacies we discuss are, indeed, invalid. As a preliminary illustration of the method, let us consider example b of section 1.
|g]||All mammals are mortal.|
|All dogs are mortal.|
|∴||All dogs are mammals.|
This argument has true premises and a true conclusion; nevertheless, it has an invalid form.
|h]||All F are H.|
|All G are H.|
|∴||All G are F.|
For "F" substitute "mammals," for "G" substitute "reptiles," and for "H" substitute "mortal." This yields
|i]||All mammals are mortal.||True|
|All reptiles are mortal.||True|
|∴||All reptiles are mammals.||False|
The resulting argument i has the same form as g, but its premises are obviously true, whereas its conclusion is patently false. Hence, i is a counterexample, proving that h is an invalid form. Since h seems to exhibit the basic logical structure of argument g, we conclude that it is an invalid argument.
The first few valid and invalid forms of argument we shall examine contain a very important type of statement used as a premise: the conditional (or hypothetical) statement. It is a complex statement composed of two component statements joined by the connective "if. . . then . . ." For example,
a] If today is Wednesday, then tomorrow is Thursday.
b] If Newton was a physicist, then he was a scientist.
Both statements are conditional or hypothetical. In a conditional statement, the part that is introduced by "if" is called the antecedent; the part that comes immediately after "then" is called the consequent. "Today is Wednesday" is the antecedent of a; "Newton was a physicist" is the antecedent of b. "Tomorrow is Thursday" is the consequent of a; "he [Newton] was a scientist" is the consequent of b. The antecedents and consequents of conditional statements are, themselves, statements. A conditional statement has a definite form, which may be expressed
c] If p, then q
where it is understood that statements are to replace "p" and "q." The content of a conditional statement depends upon the particular statements that occur as antecedent and consequent. The form is determined by the fact that the connective "if . . . then . . ." places these two statements, whatever their content, in definite relation to each other.
In logic, it is useful to have standard forms, and we shall take c as our standard form for a conditional statement. Nevertheless, it is important to realize that a conditional statement can be equivalently reformulated in a variety of ways. We must expect to encounter the alternatives when we are dealing with arguments that we find in ordinary contexts. Examination of some of these different formulations will help us to recognize them when they occur, and it will deepen our understanding of the extremely important conditional form.
1. "If p then q" is equivalent to "If not-q, then not-p." This relation is so fundamental that it has a special name: "contraposition." "If not-q, then not-p" is the contrapositive of "If p, then q."
Application of contraposition to b gives
d] If Newton was not a scientist, then he was not a physicist.
You should satisfy yourself that b and d are equivalent to each other (see p. 49).
2. "Unless" means the same as "if not . . . ."
Thus, d can be translated directly into
e] Unless Newton was a scientist, he was not a physicist.
Again, you should satisfy yourself that e is equivalent to d and also to b.
3. "Only if" is the exact converse of "if; that is, "If p, then q" is equivalent to "Only if q, p."
Applying this equivalence to b we get
f] Only if Newton was a scientist was he a physicist.
You should satisfy yourself that f is equivalent to b. Furthermore, you should be perfectly clear that b differs in meaning from
g] Only if Newton was a physicist was he a scientist.
This statement says that Newton would not have been a scientist if, instead of being a physicist, he had been a chemist or a biologist. Statement b is certainly true, but g is surely false; g means the same as
h] If Newton was a scientist, then he was a physicist which is clearly different from b in content.
4. The word order of conditional statements can be inverted without changing the meaning, provided that the same clause is still governed by "if." The antecedent of the conditional statement need not come first; it may come after the consequent. The antecedent is the statement introduced by "if." wherever "if" happens to occur. The same thing may be said of conditional statements expressed in terms of "only if" and "unless." The statement may be inverted, provided the same clause is still governed by "only if" or "unless."
Using this equivalence, b, d, e, and f, respectively, may be expressed as follows without changing their meaning:
i] Newton was a scientist if he was a physicist.
j] Newton was not a physicist if he was not a scientist.
k] Newton was not a physicist unless he was a scientist.
l] Newton was a physicist only if he was a scientist.
This list by no means exhausts the possible ways of expressing conditional statements, but it gives a fair idea of the kinds of alternatives that exist. In the arguments we examine subsequently we shall encounter some of these alternatives.
Our examination of specific argument forms will begin with the consideration of four very simple and basic ones. Two of them are valid, and the other two are invalid. In each case there are two premises, the first premise being a conditional statement.
The first valid argument form is called affirming the antecedent (or sometimes modus ponens). Consider the following example:
|a]||If Smith fails her English examination, then she will be disqualified for the swimming meet.|
|Smith fails her English examination.|
|∴||Smith will be disqualified for the swimming meet.|
This argument is obviously valid; its form is described by the following schema:
|b]||If p, then q.|
Here is another example of the transition from an argument to its form or structure: b is not an argument but the schema of an argument. The letters "p" and "q" are not statements -- they are merely letters -- but if statements are substituted for these letters, an argument results. Of course, it is essential that the same statement be substituted for "p" in each place it occurs and that the same statement be substituted for "q" in each place it occurs. If the substitution is made in this way, the resulting argument will be valid, regardless of what statements are substituted for "p" and "q." Indeed, we may substitute statements for "p" and "q" in such a way that the premises are either doubtful or known to be false, and still we may be assured that the conclusion would be true if the premises were true. This again illustrates the fact that the validity of an argument depends only upon its form and not upon its content.
It is easy to see why form b is called affirming the antecedent. The first premise is a conditional statement, and the second premise affirms (asserts) the antecedent of this conditional. The conclusion of the argument is the consequent of the first premise. Here is another example of affirming the antecedent:
c] Is 288 divisible by 9? It is if its digits add up to a number that is divisible by 9. Since 2 + 8 + 8 = 18, which is divisible by 9, the answer is "yes."
Like most arguments we encounter, c is not given in standard logical form. We rewrite it:
|d]||If the sum of the digits of 288 is evenly divisible by 9, then 288 is evenly divisible by 9.|
|The sum of the digits of 288 is evenly divisible by 9.|
|∴||288 is evenly divisible by 9.|
This argument has form b.
Another valid form of deductive argument is denying the consequent (sometimes called modus tollens).
|e]||If there is going to be a storm tonight, then the barometer is falling.|
|The barometer is not falling.|
|∴||There is not going to be a storm tonight.|
This argument has the form
|f]||If p, then q.|
It is easy to see why this form is called denying the consequent. The first premise is a conditional statement, and the second premise is the denial or negation of the consequent of that conditional. Here is another example:
g] He would not take the crown;
Therefore, 'tis certain he was not ambitious.1
Again, we have an argument that must be translated into standard form. This time a premise is missing, but we can easily supply it.
|h]||If Caesar had been ambitious, then he would have taken the crown.|
|He did not take the crown.|
|∴||Caesar was not ambitious.|
The term "therefore" indicates the conclusion; the phrase " 'tis certain" indicates the necessity of a deductive argument. Clearly, h has the form f.
Denying the consequent often takes a slightly different form. For example,
|i]||Brutus:||Ay, Casca; tell us what hath chanced today, That Caesar looks so sad.|
|Casca:||Why, you were with him, were you not?|
|Brutus:||I should not then ask Casca what had chanced.2|
This argument can be reconstructed as follows:
|j]||If I had been with Caesar, then I would not have asked what happened.|
|I have asked what happened.|
|∴||I was not with Caesar.|
The consequent of the first premise is a negative statement, so the second premise, which is its denial, is affirmative. This gives rise to a slight variant of f, namely,
|k]||If p, then not-q.|
This form is also called denying the consequent.
Further examples of these two forms of argument lie at the heart of the philosophical problem of free will. We cite a classical source:
l] Lucretius, a Roman poet of the first century B.C., in his famous work, De Rerum Natura, argued that everything consists of atoms. Furthermore, he held that these atoms are subject to spontaneous and indeterminate swervings; for if each atomic motion were rigidly determined by prior motions, where could we find a source for free will? To Lucretius, it was clear that living things do have free will, so he concluded that determinism cannot hold.
The central core of the argument may be put as follows:
|m] If determinism holds, then people do not have free will.|
|People have free will.|
|∴||Determinism does not hold.|
This argument is valid; it is an instance of denying the consequent. The only grounds upon which it can be attacked are by attacking the truth of the premises. Some people, however, taking it as more obvious that determinism holds than that man has free will, have constructed a different argument.
|n]||If determinism holds, then people do not have free will.|
|∴||People do not have free will.|
This argument is also valid; it is an instance of affirming the antecedent (schema b). In order to accept the conclusion of this argument, however one must deny the truth of the second premise of m. The controversy between those who accept m and those who accept n does not lie in th validity of the arguments; they are both valid. It lies in the question of the truth of the premises. Between the two arguments we have three premises;. they cannot all be true, for they are mutually incompatible. The philosophical controversy hinges upon the question of which premises are false.
There are two invalid forms of argument that are deceptively similar to the two valid forms we have just discussed. The first of these is called the fallacy of affirming the consequent. For example,
o] Men, we will win this game unless we go soft in the second half. But I know we're going to win, so we won't go soft in the second half.
In standard form this argument becomes
|p]||If we do not go soft in the second half, then we will win this game.|
|We will win this game.|
|∴||We will not go soft in the second half.|
This argument has the form
|q] If p, then q.|
This form bears some resemblance to the valid form of affirming the antecedent (schema b), but there are crucial differences. In affirming the antecedent, the second premise asserts the antecedent of the first premise, and the conclusion is the consequent of the first premise. In the fallacy of affirming the consequent, the second premise asserts the consequent of the first premise and the conclusion is the antecedent of the first premise.
The invalidity of affirming the consequent can easily be shown by the method of counterexample (section 5). We construct an argument of this form that has true premises and a false conclusion.
|r]||If Harvard University is in Vermont, then it is in New England.|
|Harvard University is in New England.|
|∴||Harvard University is in Vermont.|
The second invalid form of argument is called the fallacy of denying the antecedent. This one bears some resemblance to the valid form of denying the consequent. Consider the following argument:
|s]||If Richard Roe is willing to testify, then he is innocent.|
|Richard Roe is not willing to testify.|
|∴||Richard Roe is not innocent.|
This argument has the form
|t]||If p, then q.|
The following fictitious bit of campaign oratory provides another instance of this fallacy:
u] So I say to you, ladies and gentlemen, that you should vote for my opponent if you want to pay higher taxes and get less for your money -- if you feel that clean and honest government is not worth having. But I know that you are decent and intelligent people, so I ask for your support on election day.
This argument may be analyzed as follows:
|v]||If you want to pay higher taxes and get less for your money, and you feel that clean and honest government is not worth having, then you should vote for my opponent.|
|It is not true that you want to pay higher taxes and get less for your money or that you feel that clean and honest government is not worth having.|
|∴||You should not vote for my opponent.|
Actually, people often say "if" when they mean "if and only if"; if the first premise is construed in that way, the argument, of course, becomes valid, though it loses some of its rhetorical force.
It is easy to show that denying the antecedent is invalid by the method of counterexample.
|w]||If Columbia University is in California, then it is in the United States.|
|Columbia University is not in California.|
|∴||Columbia University is not in the United States.|
Both the fallacy of affirming the consequent and the fallacy of denying the antecedent have special instances that deserve explicit mention. As we have explained, in a valid deductive argument, if the premises are true the conclusion must be true. Suppose we have an argument that is known to be valid and that is known to have a true conclusion. What can we say about the premises? It might be tempting to say that the premises of this argument are true. To do so would be to commit the fallacy of affirming the consequent.
|x]||If the premises of this argument are true, then the conclusion of this argument is true (i.e., the argument is valid).|
|The conclusion of this argument is true.|
|∴||The premises of this argument are true.|
It is a logical error to infer the truth of the premises from the truth of the conclusion. Similarly, if we have a valid argument with false premises it might be tempting to say that the conclusion is false. This would be the fallacy of denying the antecedent.
|y]||If the premises of this argument are true, then the conclusion of this argument is true (i.e., the argument is valid).|
|The premises of this argument are not true.|
|∴||The conclusion of this argument is not true.|
When an argument is given with missing premises, it is sometimes impossible to tell what premises the person had in mind. It is up to us to choose, and more than one choice may be available. Consider the following imaginary conversation:
z] It was Monday morning. Neither John nor Harvey felt much like working, so they killed a bit of time at the water cooler, gossiping about their fellow workers.
"Have you noticed," John asked, "that Henry never seems to take a drink? After work last Friday, we all stopped at that little lounge over on Elm Street -- sorry you couldn't make it, Harv -- and Henry had nothing but coffee. And at the company picnic last spring -- man, the beer was really flowing that time, wasn't it? -- he was drinking iced tea. What is it with him?"
"Well, as you know," Harvey answered, "I've known old Hank for many years, and I've never seen him touch a drop."
"You mean he really is a teetotaler?" John asked in some amazement. "That's funny, he never struck me as the puritanical sort."
This conversation involves arguments, but as usual they need to be sorted out. In the first place, an inductive argument is given to support the conclusion that Henry is a total abstainer from alcoholic beverages. Using that conclusion as a premise, John goes on to inter that Henry is puritanical, that is, that he has moral principles which prevent him from drinking. Clearly, a premise is missing from the argument, so we must supply it. We might reconstruct it as follows:
|aa]||If Henry never drinks, then he has moral scruples against drinking.|
|Henry never drinks.|
|∴||Henry has moral scruples against drinking.|
This argument is an instance of affirming the antecedent and is, therefore, valid. The trouble is that there is not much reason to believe that the premise we have supplied is true. Henry might, for all we know, abstain for reasons of health or because he doesn't like the taste of alcoholic beverages. We might try a different premise.
|ab]||If Henry has moral scruples against drinking, then Henry never drinks.|
|Henry never drinks.|
|∴||Henry has moral scruples against drinking.|
The premise we have introduced this time is far more plausible than the one we used previously, but now the argument is invalid, for it is an instance of the fallacy of affirming the consequent, and it is hard to see any alternative form it could have that would make it valid. Following the advice offered in section 1, we have done our best to reconstruct the argument as one that has a valid form and true -- or, at least, plausible -- premises. In this case, there seems to be no way in which this can be done.
2 Ibid., Act I, Scene III.
The reductio ad absurdum is a valid argument form that is widely used and highly effective. Sometimes it is used to establish a positive conclusion; often it is used to refute the thesis of an opponent. The idea of this form of argument is quite simple. Suppose we wish to prove that a statement p is true. We begin by assuming that p is false; that is, we assume not-p. On the basis of this assumption, we deduce a conclusion that is known to be false. Since a false conclusion follows from our assumption of not-p by a valid deductive argument, the assumption must have been false. If not-p is false, p must be true, and p was the statement we set out to prove in the first place.
Let us call the argument by which we deduce a false statement from the assumption not-p the sub-deduction. It may have any form whatever, as long as it is valid. The validity of any particular reductio ad absurdum depends upon the validity of its sub-deduction. A particular reductio ad absurdum may be attacked by showing that the sub-deduction is invalid, but the general form of reductio ad absurdum (which requires that the sub-deduction be valid) is not open to attack, for it is a valid form. Furthermore, the conclusiveness of the proof of p by reductio ad absurdum depends upon the falsity of the conclusion of the sub-deduction. The conclusion of the sub-deduction may be either some statement we are simply willing to accept as false, or it may be an actual self-contradiction (see section 33). Often, the conclusion of the sub-deduction is p itself. This is a special case of a self-contradiction. If, on the assumption of not-p, we can deduce p, then, on the assumption of not-p, we have both p and not-p, which is a self-contradiction.
The reductio ad absurdum may be schematized as follows:
|Deduce:||A false statement; either
p (which contradicts the assumption not-p), or
q and not-q (a self-contradiction), or
some other statement r, which is known to be false.
|Conclude:||∴ Not-p is false; therefore, p.|
Reductio ad absurdum is closely related to denying the consequent. This relation is shown by the following argument:
|b]|| If the premise (assumption) of the sub-deduction is true, then the conclusion of the sub-deduction is true (i.e., the sub-deduction is valid).
The conclusion of the sub-deduction is not true.
|∴||The premise of the sub-deduction is not true.|
Reductio ad absurdum is frequently used in mathematics, where it is often called indirect proof. Here is a classical mathematical example that is famous for its simplicity and elegance:
|c]|| A rational number is one that can be expressed as a simple fraction, that is, as the ratio, of two integers (whole numbers). The Greek philosopher and mathematician PytKagoras (sixth century B.C.) is credited with the discovery that there is no rational number whose square equals two -- in other words, that the square root of two is an irrational number. This conclusion is easily proved by reductio ad absurdum.
Assume that there is some rational number whose square is equal to two. Let this number be expressed in lowest terms; that is, if the numerator and denominator have a common factor greater than one, remove it. Thus we have
where a and b have no common factor greater than one. a2 is an even number because it is twice b2; hence, a is an even number because the square of any odd number is odd. Since a is even, it can be written as 2c; a2 equals 4c2. Then
It follows that b2 is even, and so is b. We have shown that a and b are both even. This contradicts the assumption that a/b is a rational number written in lowest terms. Therefore, there is no rational number whose square equals two.
Another excellent source of reductio ad absurdum arguments is the dialogues of Plato. Typically, in these dialogues, Socrates asks a question and proceeds to refute the answers given by showing that they lead to unacceptable consequences. Here is a brief and simple example:
|d]|| Well said, Cephalus, I replied; but as concerning justice, what is it? -- to speak the truth and pay your debts -- no more than this? And even to this are there not exceptions? Suppose that a friend when in his right mind has deposited arms with me and he asks for them when he is not in his right mind, ought I to give them back to him? No one would say that I ought or that I should be right in doing so, any more than they would say that I ought always to speak the truth to one who is in his condition.
You are quite right, he replied.
But then, I said, speaking the truth and paying your debts is not a correct definition of justice.3
The structure of the argument is quite clear.
|e]||To prove:||Speaking the truth and paying debts is not a correct definition of justice.|
|Assume:||Speaking the truth and paying debts is a correct definition of justice.|
|Deduce:||It is just to give weapons to a madman. But this is absurd.|
|Conclude:||∴ Speaking the truth and paying debts is not a correct definition of justice.|
For a final example of reductio ad absurdum we go to Kant's antinomies. Each of the four antinomies involves the proof of a thesis and the proof of an antithesis. Each of these eight proofs is by reductio ad absurdum. We illustrate with the proof of part of the thesis of the first antinomy.
|f]||Thesis:||The world has a beginning in time. . . .|
|Proof:||Granted, that the world has no beginning in time; up to every given moment of time, an eternity must have elapsed, and therewith passed away an infinite series of successive conditions or states of the things in the world. Now the infinity of a series consists in the fact that it never can be completed by means of a successive synthesis. It follows that an infinite series already elapsed is impossible, and that consequently a beginning of the world is a necessary condition of its existence. And this was the first thing to be proved.4|
In the examples we have cited, we are not particularly concerned with the validity of the sub-deduction; rather, we wish to show the form of the reductio ad absurdum. We might remark, however, that the sub-deduction in example c is correct, the sub-deduction in example f is almost certainly incorrect, and the sub-deduction in d is at best somewhat dubious.
4 Immanuel Kant, Critique of Pure Reason, trans. J. M. D. Meiklejohn (New York: The Colonial Press, 1900), p. 241.
In everyday language, we say that one faces a dilemma if one is forced to make a choice between two undesirable alternatives. For example,
|a]|| Brown has been compelled to appear in court because she has been charged with a minor traffic violation of which she is innocent. The judge asks whether she pleads guilty or not guilty. This is Brown's dilemma:
Either I plead guilty or I plead not guilty.
If I plead guilty, then I must pay a fine of twenty-five dollars for an offense I did not commit.
If I plead not guilty, then I must spend another whole day in court.
|∴ Either I must pay a fine of twenty-five dollars for an offense I did not commit or I must spend another whole day in court.|
In some cases, however, the only undesirable aspect of the situation is the fact that a choice is forced when one would rather not give up either alternative.
|b]||Johnson is coming to the end of a delicious meal in a good restaurant. The waiter asks what he would like for dessert. "This is a terrible dilemma," he says with feigned distress, "because I can't decide between blueberry pie and chocolate cake." On the assumption that he cannot have both, the dilemma results from the fact that he must forego the cake if he chooses the pie and he must forego the pie if he chooses the cake. Either way, he must forego one or the other of his favorite desserts.|
These arguments are valid; they have the following form:
|c]||Either p or q.|
|If p, then r.|
|If q, then s.|
|∴||Either r or s.|
In the terminology of logic, any argument which has form c is called a dilemma, whether or not the conclusion is undesirable. The dilemma is an extremely effective type of argument in controversy or debate, where it is used to try to force an opponent to accept an unwanted conclusion.
A famous example, which has a more specialized form, illustrates this use.
|d]|| An ancient teacher of argumentation made a contract with one of his pupils. The pupil would not have to pay for the lessons if (and only if) he did not win his first case. After the lessons were completed, the student did not take any cases. In order to receive payment, the teacher sued. The pupil defended himself with the following argument:
Either I will win this case, or I will lose it.
If I win this case, I will not have to pay my teacher (because he will have lost his suit for payment).
If I lose this case, I will not have to pay my teacher (because of the terms of our agreement).
|∴||I do not have to pay.|
The teacher, however, presented this argument.
|Either I will win this case, or I will lose it.
If I win the case, the pupil must pay me (because I will have won my suit for payment).
If I lose the case, the pupil must pay me (because he will have won his first case).
|∴||The pupil must pay me.|
It is not known how the case was decided, but these two dilemmas show that the original contract contained a hidden self-contradiction (see section 33). Both of the dilemmas in d have this form:
|e]|| Either p or not-p.
If p, then r.
If not-p, then r.
Clearly, e is a special case of c.
The ancient theological problem of evil can be posed as a dilemma. The argument runs as follows:
|f]|| There is evil in the world. This means that God cannot prevent evil, or else He does not want to prevent evil. If God cannot prevent evil, He is not omnipotent. If God does not want to prevent evil, He is not benevolent. |
Therefore, either God is not omnipotent, or else He is not benevolent.
This argument is valid; it has the form c. Theological controversies being what they are, we hasten to remark once more that validity has nothing to do with the truth of the premises. Some theologians reject the premise that there is evil in the world. Some theologians reject the premise that God is not benevolent unless He wants to prevent evil. The validity of the argument is not, however, open to question.
In section 7 we made reference, by way of example, to the problem of free will and determinism. This problem gives rise to another dilemma.
|g]|| Many philosophers have argued that people could not have free will if all events, including willing and acting, are completely determined by previous causes. Other philosophers, however, have said that free will would be equally impossible if some events are not entirely determined by previous causes. To whatever extent events are due to chance, they have nothing whatever to do with our will. Free will is just as incompatible with chance as it is with rigid causation. These doctrines, taken together, give rise to this dilemma.
If determinism holds, people do not have free will.
|∴||People do not have free will.|
This argument has the special form e; hence, it too is valid.
The fact that the dilemma (form c or form e) is a valid argument has an important consequence for examples like f and g. Many people would prefer to reject the conclusions of either or both of these arguments. They may believe in a God who is perfect; they do not want to admit that He is not both omnipotent and benevolent. They may place a high value on human freedom of choice and action. Since no valid argument has true premises and a false conclusion, the only way to avoid either of these conclusions is by rejecting at least one of the premises from which it is deduced.
Because of the rhetorical importance of the dilemma, a special terminology has traditionally been used. A person who cannot successfully challenge any of the premises from which an unwanted conclusion is deduced is said to be "impaled on the horns of a dilemma." A disputant who denies the first premise -- the disjunctive premise containing "or" -- is said to "go between the horns." The person who denies that there is really any evil in the world -- that everything is for the best in the total scheme of things -- is maintaining that it is false to say that God either is not able to prevent evil or does not want to do so. Going between the horns in this manner is one standard way of attacking argument f. As we shall see (example e of section 34), this kind of maneuver is not feasible for arguments that, like g are instances of form e. If dilemmas of this sort are to be challenged successfully, it must be by rejecting at least one of the conditional premises. This strategy is called "taking the dilemma by the horns." The dilemma of free will has often been attacked by those who deny that free will is incompatible with indeterminism and by those who deny that free will is incompatible with determinism. One strategy that holds no promise whatever for dealing with dilemmas is to accept the premises and deny the conclusion, for the validity of the argument forms can easily be demonstrated (section 10).
In sections 7 to 9 we discussed several deductive argument forms. Arguments like these are built out of constituent statements as parts, and the parts do not have to be further broken down to determine the validity or invalidity of the form. For example, affirming the antecedent,
|a]||If p, then q.
is valid because its first premise is a conditional statement, its second premise is the antecedent of that conditional, and the conclusion is the consequent of the same conditional. Arguments of this type are valid regardless of what statements the letters "p" and "q" stand for. The validity of the argument does not depend in any way upon the internal structure of the constituent statements "p" and "q." Similar remarks apply to denying the consequent, reductio ad absurdum, and the dilemma. All such arguments can be handled by an extremely efficient device known as a truth table (see below).
Many English sentences -- for example, imperatives ("Do not smoke.") and ejaculations ("Ouch!") -- are neither true nor false. In the present context we are not concerned with sentences of that type. Most declarative sentences -- sentences whose main function is to convey information of some sort -- are either true or false, though in many cases we do not know which. These are the kinds of sentences we have in mind when we speak of statements. Truth and falsity are known as truth values of statements; every statement has one and only one of these truth values. In analyzing deductive arguments, we are obviously concerned with the types of sentences that are capable of assuming truth values, for we have characterized deductive validity in terms of truth values -- that is, the conclusion must be true if the premises are true.
The basic idea behind the construction of truth tables is that there are certain ways of making compound statements out of simple parts so that the truth value of the compound statement is completely determined by the truth values of the component parts. The easiest example is the negation of a statement. The statement "Jefferson was the first president of the United States" can be negated by forming the statement "Jefferson was not the first president of the United States." If we begin with any true statement, its negation will be false; if we negate a false statement, the resulting statement will be true. The negation merely reverses the truth value of the statement to which it is applied.
Negation is applied to a single statement to form another statement. It is also possible to combine two statements in such ways that the truth value of the compound statement is completely determined by the truth value of the constituent parts. For example, the sentence "Abraham Lincoln was assassinated in Washington and John F. Kennedy was assassinated in Dallas" is built out of two simple statements joined together by the connective "and." The compound statement is true because it is a conjunction of two true components. The internal structure of the component statements is irrelevant to the truth value of the compound statement. Any compound statement of the form "p and q" will be true if the constituent statements "p" and "q" are both true; otherwise, if either or both component statements are false, the compound statement will also be false.
Other statement connectives, such as "or," "if . . . then . . .," and "if and only if" can be handled similarly. All of these connectives are known as truth-functional connectives, for the truth value of any compound statement constructed by means of these connectives is completely determined by (is a function of) the truth values of the parts. To see how this works, let us begin by introducing some standard symbols:
|Compound statement form||Symbol||Name of form|
p and q
p or q
if p then q
p if and only if q
p · q
p ∨ q
p ⊃ q
p ≡ q
We can now exhibit the meaning of each of these connectives in a convenient tabular form:
|p||q||p · q||p ∨ q||p ⊃ q||p ≡ q|
These are truth tables, and they serve to define the truth-functional connectives. They can be construed in the following way. In Table I we are dealing with a connective that applies to a single statement. (We call negation a singulary "connective" by courtesy, even though it operates on only one statement.) A single statement can be either true or false; those are the only possible cases. The table tells us, for each of these cases, the result of forming the negation: A true statement, when negated, is transformed into a false statement; a false statement, when negated, is transformed into a true statement.
Table II defines the so-called binary connectives -- connectives that join two statements to make a third statement. There are four possible ways in which the two truth values can be assigned to two statements; these four possibilities are displayed in the first two columns of Table II. To define the binary connectives, we need merely state the truth value of the compound statement for each possible combination of truth values of the two constituent statements. Thus, column 3 summarizes what we said about conjunctions; the conjunction is true if both components are true, but it is false in all other cases.
We noted in section 6 that conditional statements can be expressed in a variety of ways; similarly, there are words other than "and" that can be used to form conjunctions. The leading example is "but." If one says, "Abraham Lincoln was killed in Washington, but John F. Kennedy was killed in Dallas," the logical import is the same as if the word "and" had been used -- both conjuncts must be true if the compound statement is to be true. "But" reveals some astonishment at the conjunction, but such surprise is an expression of a psychological attitude, not part of the informational content of the statement. Similarly, words like "although," "nevertheless," and "however" often serve, as "but" does, to state a conjunction and to heighten the contrast between the conjuncts.
We must also note emphatically that the truth-functional conjunction, which we symbolize with a dot and define in column 3 of Table II, does not correspond exactly to the English word "and." In ordinary conversation, the statement "Jane became pregnant and got married" would have a significance radically different from that of the statement "Jane got married and became pregnant." The difference arises from the fact that the English word "and" sometimes has the force of "and then." Let us suppose, without making any temporal assumptions, that both component statements, "Jane became pregnant" and "Jane got married," are true. Then, according to the truth table definition of conjunction, the conjunction of those two statements is true regardless of the order in which they are conjoined. Thus, there is a serious discrepancy between the word "and" as it is often ordinarily used and the truth-functional connective. One very important aspect of this difference is the fact that the temporally loaded "and then" sense of "and" is non-truth-functional, for the truth value of the compound statement is not fully determined by the truth values of the constituent parts. We use the truth-functional conjunction, as defined in the truth table, because it works neatly from a logical standpoint, but we must not forget the differences between our dot and the English word "and."
The word "or" has two distinct meanings in English: In one sense (known as the exclusive sense) it means "one or the other but not both." This is the meaning it has on a menu in the phrase "soup or salad," used to tell what comes with the entree. The other sense (known as the inclusive sense) is often rendered by the expression "and/or," which frequently appears in documents like insurance policies or wills. If, for example, a degree in law or a degree in accounting were stipulated as qualification for a position, we would not expect people who had both degrees to be automatically disqualified, as they would if "or" were meant in the exclusive sense. A glance at column 4 of Table II reveals that we have chosen the inclusive disjunction to be symbolized by the wedge, for the compound statement is true if either or both constituent statements is true, and false only if both components are false. The choice here is one of convenience. The inclusive disjunction is a useful logical operation, and the exclusive disjunction is easily symbolized in terms of others we are adopting, for example, as the negation of a biconditional
The horseshoe-shaped symbol defined in column 5 is known as the sign of material implication; it is used to symbolize conditional statements, and it can be read, roughly, "if . . . then . . ." This connective is somewhat problematic, for its import differs markedly from the phrase "if . . . then . . ." as it is used in most conditional statements of ordinary discourse. In defining this connective truth-functionally, we find ourselves committed to treating as true any conditional that has a true antecedent and a true consequent -- for example, "If Mars is a planet, then diamonds are composed of carbon." Such a statement would not normally be regarded as a reasonable conditional, for there seems to be no connection between the truth of the antecedent and the truth of the consequent. In this respect it contrasts sharply with our example of section 6, "If Newton was a physicist then he was a scientist," which exhibits a clear relation between antecedent and consequent Moreover, according to the meaning given in the truth table, we must regard both the statements "If Mars is not a planet, then coal is black" and "If Mars is not a planet, then coal is white" as true material conditionals, simply by virtue of the fact that both have a false-antecedent. We see from lines 3 and 4 of Table II that any material conditional with a false antecedent is true, regardless of its consequent. Similarly, examination of lines 1 and 3 reveals that any material conditional with a true consequent is true, regardless of the antecedent. Thus, "If cats bark, then two plus two equals four" and "If Beethoven was a composer, then two plus two equals four" are true material conditionals because of their true consequents. The fact that a material conditional with a false antecedent is automatically true, and the fact that a material conditional with a true consequent is likewise automatically true, are sometimes called paradoxes of material implication.
Given the strange properties of the material implication as defined in Table II, you might wonder why we don't abandon it altogether as hopelessly inadequate to capture the sense of "if . . . then . . ." in ordinary language. The answer is that the operation symbolized by the horseshoe has one basic characteristic that is essential to the meaning of any conditional statement -- namely, the conditional is false if it has a true antecedent and a false consequent. The material implication is a truth-functional operation, and thus it cannot capture the sense of connection relating the antecedent and the consequent of more familiar conditionals. The "if . . . then . . ." of ordinary English is a stronger connective than the material implication. Nevertheless, the material conditional statement has just the feature of the ordinary conditional that insures the validity of such basic deductive argument forms as affirming the antecedent, denying the consequent, and the dilemma. We shall see in a moment how that works. We shall then see that the material conditional does, indeed, possess the basic logical core of the conditional statement.
The sign of material equivalence, the triple bar, is defined in column 6 of Table II, and it is used to form biconditional statements. It can be read, roughly, as "if and only if." The material equivalence inherits a good deal of the strangeness of the material implication, for it can be viewed simply as a combination of two material implications: "p ≡ q" has the same meaning as "(p ⊃ q)·(q ⊃ p)," as we shall see. A material biconditional statement is true provided both components have the same truth value, and false if they have different truth values. Thus any two true statements are materially equivalent to each other, as are any two false statements, regardless of any apparent connection between the meanings of the components. "Mars is a planet if and only if the oceans contain salt" is a true biconditional; so also is "The moon shines by its own light if and only if Isaac Newton was a German."
|p||q||∼p||∼q||p ⊃ q||p ∨ q||q · ∼q||p ⊃ (q · ∼q)|
Let us now see how truth tables can be used to test the validity of argument forms. Table III is constructed by referring to Tables I and II. The first two columns simply give the four possible combinations of T and F for the two statements "p" and "q." We shall make it a firm policy always to list them in the same order; otherwise, mistakes would be almost inevitable. The next two columns give the truth value patterns for the negations of "p" and "q" by reversing the patterns found in the first two columns. Columns 5 and 6 are simply copied out of Table II. Column 7 results from a conjunction of columns 2 and 4; no line has T in both of these columns, so the result is all F's. Column 8 is a conditional with column 1 as antecedent and column 7 as consequent. From Table II, column 5 (which is the same as Table III, column 5) we see that the conditional is false if it has a true antecedent and a false consequent; otherwise it is true. This fact is used to arrive at the pattern in column 8.
The premises of affirming the antecedent are "p ⊃ q" and "p"; their patterns are given in columns 5 and 1 respectively. Line 1 is the only line in which both premises have the value T, and in this line the conclusion "q" has the value T as well. Thus, affirming the antecedent is valid because there is no way that truth values can he assigned that render the premises true and the conclusion false. The validity of denying the consequent is easily shown in the same way. The premises of this form are "p ⊃ q" and "∑q"; line 4 is the only line in which both have the value T. In this line, the conclusion "∼p" also has the value T. Again, there is no way for the premises to be true and the conclusion false.
The same method can be used to expose the two fallacies mentioned in section 7 -- namely, the fallacy of affirming the consequent and the fallacy of denying the antecedent. Affirming the consequent has "p ⊃ q" and "q" as premises; both of these are true in lines 1 and 3 of Table III, but in line 3, the conclusion "p" has the value F. It is therefore possible, as we saw in section 7, to concoct an instance of affirming the consequent which has true premises and a false conclusion. Denying the antecedent has "p ⊃ q" and "∼p" as premises; both are true in lines 3 and 4 of the truth table. Again, in line 3 the conclusion "∼q" is false, which shows that this form too is invalid.
Another simple argument form, disjunctive syllogism, is illustrated by the following argument:
|b]||Either you paid your taxes or you have a guilty conscience.
You did not pay your taxes.
|∴||You have a guilty conscience.|
It has the form
|c]|| p ∨ q
whose validity can be established by Table III. The truth value patterns of the premises are found in columns 6 and 3 respectively. Line 3 is the only line in which the value of both premises is T; in that line the conclusion also has the value T. Notice that the similar form
|d]|| p ∨ q
is invalid, as can be seen by inspecting line 1 of Table III. Even if you did pay your taxes -- going back to a slight modification of b -- it does not follow that you do not have a guilty conscience.
Column 8 can be used to evaluate one form of reductio ad absurdum discussed in section 8. Here, the only premise is "p ⊃ (q·∼q)"; it has the value T in lines 3 and 4. The conclusion "∼p" also has the value T in both of these lines. Hence it is a valid form.
The argument forms so far tested by truth tables have only two distinct constituent statements, "p" and "q." To accommodate argument forms with more than two constituent statements, we must expand the truth table in a straightforward way. Consider, for example, this argument:
|e]||If you read the book you will know the plot.
If you know the plot you will be bored by the film.
|∴||If you read the book you will be bored by the film.|
This has the form
|f]||p ⊃ q
q ⊃ r
|∴||p ⊃ r|
known as hypothetical syllogism. In order to test this argument for validity,
|p||q||r||p ⊃ q||q ⊃ r||p ⊃ r|
we need a truth table with enough lines to exhaust all possible combinations of T and F for three statements; eight is the required number. Columns 1 through 3 contain every possible assignment of T and F to "p," "q," and "r." Columns 4 and 5 contain the truth patterns for the two premises, and column 6 contains the truth pattern for the conclusion. Lines 1, 5, 7, 8 contain T for both premises; the conclusion also has T in each of these lines. Hence, the form is valid.
It is essential to be quite clear on the way the entries for columns 4 through 6 are found. Referring back to column 5 of Table II, where the conditional is defined, we see that a conditional is false if it has a true antecedent and a false consequent (line 2); otherwise it is true. In column 6 of Table IV, for example, we find that "p ⊃ r" receives the value F in lines 2 and 4; these are precisely the lines in which "p" has the value T and "r" has the value F. All other lines in column 6 contain T. Columns 4 and 5 are constructed analogously.
The method of truth tables is theoretically adequate to test any argument form whose validity depends only upon its truth-functional structure. It is fairly obvious how to extend the method to schemas that contain four or more distinct constituent statements. It is simply a matter of assigning the truth values, T and F, to all of the constituent statements in all possible ways. If a schema contains n constituent statements, the truth table needed to test its validity will require 2n lines. You can see that each time you add another letter, the size of the table doubles, so the method can rapidly become unwieldy. In schema b of section 9 we gave the general form of a dilemma; it contains four different constituent statements. You can construct the 16-line truth table needed to demonstrate its validity.
We have asserted, on a couple of occasions at least, that two statement forms are logically equivalent to one another. In section 6, for instance, we said that a conditional statement is equivalent to its contrapositive; in section 10, for a further example, we said that a material biconditional is equivalent to a conjunction of two conditional statements. Let us now use our truth tables to show how such equivalences can be demonstrated. For this purpose we construct the following table:
|p||q||∼p||∼q||p ⊃ q||q ⊃ p||(p ⊃ q)·(q ⊃ p)||p ≡ q||∼q ⊃ ∼p|
The truth value patterns in the columns have been built up on the basis of the definitions given in Tables I and II. Column 6, for example, has a T in each line except the third, in which the antecedent "q" has the value T and the consequent "p" has the value F. Column 7 represents the conjunction of columns 5 and 6; it has a T in each line in which columns 5 and 6 both have T's, and it has an F in every other line. Column 8 simply repeats the truth value pattern given in Table II as a definition of the biconditional form. We note that the truth value patterns in columns 7 and 8 are identical.
What does it mean when two statement forms have the same truth value patterns? It means that truth-functionally the two are equivalent to one another. Any substitutions for the letters "p," "q," "r," etc., which make the one into a true statement will make the other into a true statement as well, and any substitutions that make the one false will make the other false as well. This is an extremely significant fact: It means that we can interchange equivalent forms for one another anywhere in any argument and be certain that we have done nothing that would affect the validity of the argument. The reason is simple. If we have statements with equivalent forms they must have the same truth value -- there is no way of assigning truth values to the constituent parts so that one is true and the other false. Since the validity of an argument form depends solely upon the fact that none of its instances can have true premises and a false conclusion, exchanging a form for an equivalent one cannot possibly transform a valid form into an invalid one. Nor can it possibly change an invalid argument form into a valid one.
Let us illustrate this important principle by a simple example. We have shown that affirming the antecedent and denying the consequent are both valid argument forms. Utilizing the equivalence between a conditional statement and its contrapositive, we can see that the validity of denying the consequent follows immediately from the validity of affirming the antecedent. Since "p ⊃ q" is equivalent to "∼q ⊃ ∼p," the two argument forms
|a]||p ⊃ q
|b]||∼q ⊃ ∼p
must both be valid, or they must both be invalid. But form b results if we substitute "∼q" for "p" and "∼p" for "q" in
|c]||p ⊃ q
which is the form of affirming the antecedent, and which is known to be valid.
Let us be quite sure we understand what is being done here. When we say that affirming the antecedent is a valid form, we mean that there is no way of substituting statements for the letters "p" and "q" in form c (the same statement must, of course, be substituted for "p" in both places it appears, and the same statement must be substituted for "q" in both of its occurrences) that will result in an argument having true premises and a false conclusion. It does not matter how simple or how complex the statements; in particular, the statements substituted for "p" and "q" may both be negative. For example, here is a simple reformulation of argument h of section 7:
|d]|| If Caesar did not take the crown, then he was not ambitious.
He did not take the crown.
|∴||He was not ambitious.|
This exhibits the form c, but it also has the form b. Indeed, any argument that has form b will also possess form c (but not conversely -- do you see why?).
We have been doing several things with argument forms. First, we have substituted statements for the letters in valid forms to show that the resulting argument is valid. Exercising due caution -- see section 5 -- we sometimes claim that arguments are invalid when they result from substitutions in invalid forms. Second, we have substituted statement forms for the letters in argument forms, and in this way we have shown that additional argument forms are valid. To carry out this procedure, we may substitute a statement form of any degree of complexity for a given letter in the original form. That is how we showed that form b is valid -- by substituting statement forms for letters in form c. Third, we have replaced a statement form by a logically equivalent statement form -- that is how we demonstrated the validity of form a.
The validity of the disjunctive syllogism (form c, section 10) can be shown by the method we just used on denying the consequent. You can carry out the demonstration by constructing a truth table showing
e] "p ∨ q" is equivalent to "∼p ⊃ q."
When this equivalence is used to replace the first premise of disjunctive syllogism, the result is a form you can get by making a suitable substitution in affirming the antecedent (form c, this section).
Equivalence e has another interesting consequence. In section 6, we pointed out that "unless" has the same meaning as "if not." Thus
f] If you do not pass English, then you cannot graduate.
means the same as
g] Unless you pass English, you cannot graduate.
Now, as a result of equivalence e, you can see that g means the same as
h] Either you pass English or you cannot graduate.
Here we must be careful to note that the word "or" has its inclusive sense. Hence, the word "unless" has precisely the same meaning as "or" (in the inclusive sense). Not many native speakers of English are aware of that fact.
Statements like f and g are often used to assert necessary conditions -- in this case, passing English is said to be a necessary condition for graduation. To say that a condition is necessary for a result means that the result will not occur if the condition is not fulfilled. When "p" constitutes a necessary condition for "q" we may symbolize the situation by "∼p ⊃ ∼q," which, as we know, is equivalent to "q ⊃ p." In section 6 we noted that "only if" is the converse of "if": thus "q ⊃ p" can be translated as "q only if p." This is another way of saying that "p" is a necessary condition for "q" -- e.g., you will graduate only if you pass English.
To say that "p" constitutes a sufficient condition for "q" means merely that "q" will obtain if "p" does -- in other words, it means simply "p ⊃ q."
i] If you are decapitated, then you will die.
This is tantamount to asserting that decapitation is a sufficient condition for death. It is not, of course, a necessary condition. If decapitation were a necessary condition of death we could assert that you will die only if you are decapitated, which obviously is not the case, for there are many other causes of death. Moreover, to return to our example of a necessary condition, passing English is certainly not a sufficient condition of graduation, for you must pass many other courses as well.
At the beginning of this section we pointed out that the biconditional is equivalent to a conjunction of two conditionals, i.e.,
j] "p ≡ q" is equivalent to "(p ⊃ q)·(q ⊃ p)"
where the first conditional says "if p then q" and the second says "only if p then q." Together they say "If and only if p then g." or more idiomatically, "p if and only if q." Using the terminology just explained, the first conditional says that "p" is a sufficient condition for "q," whereas the second conditional says that "p" is a necessary condition for "q." The biconditional therefore states that "p" is a necessary and sufficient condition for "q." Being evenly divisible by two is, for example, a necessary and sufficient condition for a number being even.
To assert "p or q" in the exclusive sense of "or" means that one or the other but not both of these statements is true; this can be symbolized as
k] (p ∨ q)· ∼(p·q),
which is to say, "p or q" in the inclusive sense of "or" with the added qualification that "p" and "q" are not both true. This same thing can be expressed by the formula
l] ∼(p ≡ q),
that is, by the denial of a material biconditional. It would be good practice for you to construct a truth table to demonstrate the equivalence of k and l.
Our emphasis, so far, has been mainly upon arguments. It is important to recognize, however, that there are certain statement forms that represent truths of logic -- forms that become true statements whenever substitutions are made that transform them into statements. One basic type of logical truth can be established by means of truth tables; the forms and the statements that result from substitutions into them are known as tautologies. As a classic example of a tautology, consider the famous law of excluded middle
a] p ∨ ~p.
We construct a two-line truth table:
|p||∼p||p ∨ ∼p|
The truth value pattern of this formula has nothing but T's in it; consequently, there is no way of assigning a truth value to the single constituent "p" that will make the formula false. Whatever statement we substitute for "p," the form "p ∨ ∼p" will be transformed into a true statement. Thus, "Either Newton was a physicist or Newton was not a physicist" is necessarily true, as is any other statement that has the form of the law of excluded middle.
Let us construct another truth table in order to look at some additional tautologies:
|p||q||p ⊃ q||q ⊃ (p ⊃ q)||∼q ⊃ ∼p||(p ⊃ q) ≡ (∼q ⊃ ∼p)||p · (p ⊃ q)||[p · (p ⊃ q)] ⊃ q|
In column 4, we find another example of a tautology; it represents one of the so-called "paradoxes of material implication" (see p. 41). It says, in effect, that if "q" is a true statement, then any statement "p" materially implies it.
In section 11, we established the equivalence between a conditional statement and its contrapositive. In columns 3 and 5 of Table VII we have repeated the identical truth value patterns for these two forms; in column 6 we have worked out the truth value pattern for the biconditional joining those two forms, and we see that it is a tautology. This illustrates a general principle: Two truth-functional statement forms are logically equivalent if and only if the biconditional between them is a, tautology.
Column 8 contains a tautology that bears a close and special relationship to the argument form affirming the antecedent. This tautology is a conditional whose antecedent is the conjunction of the two premises "p" and "p ⊃ q" of affirming the antecedent, and its consequent is the conclusion "q" of that argument form. This tautology again illustrates a fundamental general principle: A truth-functional argument form is valid if and only if a certain conditional statement form is a tautology: namely, the conditional whose antecedent is the conjunction of the premises of that argument and whose consequent is its conclusion.
Tautologies constitute a basic, though severely restricted, class of statements (and statement forms) whose truth can be established by logical considerations alone. This is an evident consequence of the fact that the truth pattern of a tautology consists solely of Ts. In section 33 we will consider the nature of logical truth in more general terms.
Having discussed truth-functional arguments, whose validity does not depend upon the internal structure of their simple constituent statements (i.e., statements represented by single letters) but only upon the ways in which they are combined, we now want to examine a fundamental group of arguments whose validity is determined by the internal structure of simple statements. These arguments are known as categorical syllogisms. In order to prepare the way for their discussion in the next section, we must first explain what is meant by a categorical statement. There are four forms of categorical statements; any statement that has one of these forms is a categorical statement. Traditionally, each of these forms has been denoted by one of the first four vowels; here is an example of each form.
|a]||A:||All diamonds are gems.||E:||No diamonds are gems.|
|I:||Some diamonds are gems.||O:||Some diamonds are not gems.|
The statements in the left column (A and I) are affirmative; those in the right column (E and O) are negative. The statements in the top row (A and E) are universal; those in the second row (I and O) are particular. The forms are as follows:
|b]||A:||All F are G.
|E:|| No F are G.
|I:||Some F are G.|
|O:||Some F are not G.|
Each categorical statement contains two terms, a subject term and a predicate term. In the examples a, "diamonds" is the subject term, and "gems" is the predicate term. In the forms b, "F" stands for the subject term, and "G" stands for the predicate term. Each term stands for a class of things, for example, the class of diamonds and the class of gems. Definite categorical statements result from the forms b if words or phrases standing for classes of things are substituted for "F" and "G." The content of a categorical statement depends upon the terms that occur in the statement; the form of a categorical statement exhibits a definite relationship between the two classes, regardless of what classes they happen to be.
Since English is somewhat ambiguous, we must be precise about the meanings of categorical statements. The form A is most troublesome. The statement "All diamonds are gems" certainly implies that there are no diamonds that are not gems; that is, if anything is a diamond, then it is a gem. This statement might be taken to imply, in addition, that there are such things as diamonds. Statements of form A, however, do not always carry this latter implication, even in ordinary usage. The statement "All deserters will be shot" does not imply that there will be any deserters; indeed, it may be asserted in order to prevent desertion from occurring. Its full meaning is given by the statement "If anyone deserts, then that person will be shot." This statement may be called a universal conditional statement. It is a conditional statement that is asserted to be true of anything whatsoever.
In treating categorical statements, especially as they occur in syllogisms, we shall adopt this interpretation for all A statements. "All F are G" will be taken as meaning, "If anything is an F, then it is a G." Thus, the statement "All diamonds are gems" will be taken to say, "If anything is a diamond then it is a gem." We shall, in addition, construe the conditional statement as a material conditional (see section 10), in this way building with concepts that have already been analyzed. It is, however, essential to note an important consequence of the use of the material implication in constructing the A statement: The A statement does not imply that its subject term refers to any existent objects. "All F are C" does not assert that there is anything of the type F; "All diamonds are gems" does not imply that any diamonds exist, even though we all know that they do. An A statement does not assert the existence of anything answering to the subject term; it is possible for A statements with empty subject terms to be true -- for example, "All nineteenth-century astronauts were male." Indeed, if we reflect for a moment upon the meaning of the material conditional statement, we see that any A statement with an empty subject term must be true. A material conditional statement is true if it has a false antecedent; since there were no astronauts in the nineteenth century, the statement "If anyone was a nineteenth-century astronaut then he was male" must be true of everyone, for it has an antecedent that is false for everyone to whom it might be applied. As we have said, the terms of categorical statements are terms that refer to classes. A given class may or may not have members; it is perfectly meaningful to refer to classes that have none The class of deserters may have no members, the class of thousand dollar bills in your pocket probably does not have members, and the class of nineteenth-century astronauts certainly has no members. The subject term of an A statement need not have any members and still that statement may be true. The A statement does not imply that its subject term refers to a class having members.
There are no particular difficulties in interpreting the E statement. One point needs to be made concerning both the I and O statements. The word "some" is taken to mean "at least one." The statement "Some diamonds are gems" is interpreted to mean "At least one diamond is a gem." In spite of the plural form of the I and O statements, they are not construed as implying more than one.
In view of our interpretations of the four forms of categorical statements, an obvious and important relation exists among them. An A statement contradicts the O statement having the same subject and predicate terms; an E statement contradicts the I statement having the same subject and predicate terms. "All diamonds are gems" contradicts "Some diamonds are not gems"; "No diamonds are gems" contradicts "Some diamonds are gems."5
We have presented four rigid forms of categorical statements. As we would naturally expect, there are many other forms that are equivalent to the four we specified. The situation is similar to the one we encountered in discussing conditional statements (section 6); indeed, because the A statement is a universal conditional statement, many of the variations of conditional statements have counterparts as variations of A statements. To indicate the range of variations of A statements, we may list the following equivalents to the statement "All whales are mammals":
|c]||Every whale is a mammal. |
Any whale is a mammal.
Whales are mammals.
If anything is a whale, it is a mammal.
Anything that is a whale is a mammal.
If anything is not a mammal, then it is not a whale.
All non-mammals are non-whales.
A thing is a whale only if it is a mammal.
Only mammals are whales.
Nothing is a whale unless it is a mammal.
No whale is not a mammal.
E statements also have a large number of equivalents. Many of these appear when we observe that an E statement can be translated into an A statement. For example, "No spiders are insects" is equivalent to "All spiders are non-insects." This A statement then admits all of the various translations of A statements just given. "No spiders are insects" has the following equivalents, among others:
|d]|| All spiders are non-insects. |
All insects are non-spiders.
No insects are spiders.
Nothing that is an insect is a spider.
Nothing is a spider unless it is not an insect.
Only non-spiders are insects.
Only non-insects are spiders.
If anything is a spider it is not an insect.
If anything is an insect it is not a spider.
I and O statements do not have so many variations. We shall give a few examples. The I statement "Some plants are edible" has the following equivalents:
|e]|| Some edible things are plants. |
There are plants that are edible.
There are edible plants.
Something that is a plant is edible.
At least one plant is edible.
The O statement "Some philosophers are not logicians" has the following equivalents:
|f]|| There is a philosopher who is not a logician.
Not all philosophers are logicians.
The foregoing lists of statements equivalent to categorical statements are by no means complete, but they should give you a feeling for the sort of variety you may encounter. You should go through these lists carefully and satisfy yourself that the statements are equivalent, as claimed.
There is one further variation in the forms of categorical statements, as they are found in ordinary contexts, to which we should call attention. In all of the examples given so far, we have used a standard verb, "is" or "are," to relate the subject term to the predicate term. Sometimes a different verb form occurs, as in the statements
|g]||Some dogs bark.
All students have textbooks.
No horses have wings.
All storms create problems.
Nevertheless, they can be construed as categorical statements, for they can be paraphrased in standard form:
|h]|| Some dogs are animals that bark.
All students are people who have textbooks.
No horses are animals that have wings.
All storms are events that create problems..
When such statements occur in categorical syllogisms (sections 14 and 15), it is best to translate them into the standard forms. This enables us to see clearly and unambiguously the relationship between the two classes that each categorical statement asserts. Such relationships among classes are the basis upon which the validity or invalidity of syllogisms -- to which we now turn -- rests.
Additional contrast between the traditional and modern interpretations of the syllogism occurs in footnote 6. p. 57.
The concept of self-contradictory statement is explained in section 33; the relation of contradiction between two statements is discussed in section 34, where it is carefully contrasted with the relation of contrariety between statements.
Categorical syllogisms (which, for convenience, we shall simply call syllogisms) are arguments composed entirely of categorical statements. Every syllogism has two premises and one conclusion. Although each categorical statement contains two terms, a subject term and a predicate term, the whole syllogism has only three different terms. One of these terms occurs once in each premise; it is called the middle term. Each of the other two terms occurs once in the conclusion and once in a premise; they are called end terms. The following argument, consisting of three categorical statements, is a syllogism:
|a]||All dogs are mammals.
All mammals are animals.
|∴||All dogs are animals.|
"Mammals" occurs once in each premise; it is, therefore, the middle term. "Dogs" occurs once in a premise and once in the conclusion, establishing itself as an end term. "Animals" occurs once in the conclusion and once in a premise. It, too, is an end term.
There are many forms of syllogisms -- some valid, others invalid. Like any deductive argument, the validity of a syllogism depends only upon its form. The form of a syllogism depends upon two things: first, which of the four types of categorical statement each statement is; second, the positions of the middle term and the end terms. In a, both of the premises are A statements, and so is the conclusion. One end term, "dogs," is the subject term of the first premise and the subject term of the conclusion. The other end term, "animals," is the predicate term of the second premise and the predicate term of the conclusion. The middle term, "mammals," is the predicate term of the first premise and the subject term of the second premise. Letting "S" stand for the end term that is the subject term of the conclusion, "P" for the end term that is the predicate term of the conclusion, and "M" for the middle term, we may represent the form of a unambiguously as follows:
|b]||S A M
M A P
|∴||S A P|
The fact that each statement in the argument is a universal affirmative is indicated by the "A" in each line.
There are three simple rules that may be used to test the validity of any syllogism. In order to present these rules we must however, introduce the concept of distribution. A given term -- for example, "mammals" -- may occur in different categorical statements, and it may occur as the subject term or as the predicate term. When a given term occurs, it may be distributed or it may be undistributed in that occurrence. Whether a term is distributed or not in a given occurrence depends upon the type of statement it occurs in and whether it is the subject term or the predicate term in that statement. A term is distributed in a categorical statement if that statement says something about each and every member of the class that term designates.
The A statement "All whales are mammals" says something about every whale -- namely, that it is a mammal -- but it does not say anything about every mammal. Therefore, in an A statement, the subject term is distributed and the predicate term is undistributed.
Notice that the A statement does say something about the class referred to by its predicate term. "All whales are mammals" says that the class of mammals includes the class of whales. But it is one thing to make a statement about a class, as such, and it is quite a different thing to make a statement about each and every member of that class. A class is a collection of entities. When we speak of the class, as such, we are speaking collectively. When we speak of the members of a collection as individuals, we are speaking distributively. Some things that are true of a class as a collection are not true of its members as individuals, and some things that are true of the members of a class as individuals are not true of the class as a collection. For instance, the class of mammals is numerous; that is, it has many members. It would be nonsense, however, to say that the whale Moby Dick, a member of the class of mammals, is numerous. Likewise, it would be nonsense to say that each and every member of the class of mammals is numerous.
There are two fallacies that involve the confusion of collective and distributive statements. The fallacy of division consists in concluding (distributively) that every member of a class has a certain property from the premise that the class (collectively) has that property. For example,
|c]||The Congress of the United States is a distinguished organization.|
|∴||Every member of Congress is a distinguished person.|
The converse fallacy is the fallacy of composition. It is the fallacy of concluding (collectively) that a class has a property because (distributively) every member of the class has that property. For example,
|d]||Each person on the football team is an excellent player.|
|∴||The football team is an excellent one.|
If there is an absence of teamwork, the premise of d might well be true even when its conclusion is false.
Every categorical statement says something about each of the classes its terms refer to, but these statements are collective. Furthermore, a categorical statement may speak distributively about some, but not necessarily all, members of a class. Sometimes, but not always, a categorical statement speaks distributively about each of the members of some class; in such cases, the term referring to that class is distributed. Consider our A statement again. It says collectively that the class of mammals includes the class of whales and that the class of whales is included in the class of mammals. Distributively, it says that every member of the class of whales is a mammal. It makes no distributive statement about every member of the class of mammals.
The E statement "No spiders are insects" has both terms distributed. It says that every spider is a non-insect and that every insect is a non-spider. Collectively, it says that the class of spiders is entirely excluded from the class of insects.
The I statement "Some plants are edible" has both terms undistributed. It makes no statement about every plant, and it makes no statement about every edible thing. Collectively, it says that the class of plants and the class of edible things overlap each other.
The O statement "Some philosophers are not logicians" says nothing about every philosopher, so its subject term is undistributed. Surprisingly, perhaps, it does say something about every logician. To see this, consider the equivalent statement: "There is at least one philosopher who is not a logician" The statement does not tell us exactly who that philosopher is, but it does assure us that there is one, at least, so let us agree for the moment to call him or her "Phil." Phil is a philosopher who is not a logician. It is easy to see that our O statement says "Every logician is distinct from Phil." Another way to put the point is this: Our O statement says that every logician is different from those philosophers referred to in that statement. Thus, the predicate term of an O statement is distributed. Collectively, the O statement says that the class of philosophers is not entirely included within the class of logicians.
Actually, it is not necessary to understand the concept of distribution to test syllogisms for validity; it is necessary to remember which terms are distributed and which are not. This information is given in the following summary:
These results may be further abbreviated. The subject term of a universal statement is distributed; the predicate term of a negative statement is distributed,. All other terms are undistributed. The four letters "USNP" (standing for universal-subject, negative-predicate) may be memorized as a formula to determine the distribution of terms; a mnemonic device such as "Uncle Sam Never Panics" may be used to fix them in mind.
The three rules for testing the validity of syllogisms can now be stated. In a valid syllogism
These rules should be memorized. Any syllogism that satisfies all three rules is valid. Any syllogism that violates one or more of these rules is invalid. Rule I states that the middle term must be distributed in one of its occurrences and that it must be undistributed in its other occurrence. A syllogism in which the middle term is distributed twice is invalid, and so is a syllogism in which the middle term is not distributed at all. According to rule II, a syllogism cannot be valid if it contains an end term that is distributed in the premises and not in the conclusion, or if it contains an end term that is distributed in the conclusion and not in the premises. For a syllogism to be valid, it cannot have an end term that is distributed in one occurrence and not in the other. Rule III covers three cases. A syllogism may have no negative premises, one negative premise, or two negative premises. In order to be valid, a syllogism that has no negative premises must not have a negative conclusion; that is, a syllogism with two affirmative premises must have an affirmative conclusion. If a syllogism has one negative premise and one affirmative premise, it cannot be valid unless the conclusion is negative. If a syllogism has two negative premises it cannot be valid, for by definition a syllogism has only one conclusion, so the number of negative premises cannot be equal to the number of negative conclusions.6
Let us apply the three rules to argument a. To do so, we rewrite the form b as follows, using subscripts "d" or "u" to designate a distributed or undistributed term.
|f]||Sd A Mu
Md A Pu
|∴||Sd A Pu|
Each statement is an A statement, so each subject term is distributed, and each predicate term is undistributed, in accordance with e.
Rule I: satisfied; the middle term is distributed in the second premise but not in the
Rule II: satisfied; the end term S is distributed in both of its occurrences; the end term P is not distributed in either of its occurrences.
Rule III: satisfied; there are no negative premises and there are no negative conclusions.
We have shown that a is a valid syllogism. Here are some further syllogisms with their forms.
|g]||All logicians are mathematicians.
Some philosophers are not mathematicians.
| Pd A Mu|
Su O Md
|∴||Some philosophers are not logicians.||∴||Su O Pd|
Since the A statement is universal, its subject is distributed; since it is affirmative, its predicate is undistributed. Since the O statement is particular, it subject is undistributed; since it is negative, its predicate is distributed.
Rule I: satisfied; the middle term is distributed in the second premise but not in the
Rule II: satisfied; P is distributed in both of its occurrences, and S is undistributed in both of its occurrences.
Rule III: satisfied; g has one negative premise and one negative conclusion.
Since g satisfies all three rules, it is valid.
|h]||All Quakers are pacifists.
No generals are Quakers.
|Md A Pu
Sd E Md
|∴||No generals are pacifists.||∴||Sd E Pd|
First, you should check to see that the distribution of terms has been specified correctly. Then the rules may be applied.
Rule I: violated; the middle term is distributed twice.
This proves that h is invalid. It is not necessary to go on to apply the other rules. To illustrate further, however, we shall apply the rules anyway.
Rule II: violated; P is distributed in the conclusion but not in the premises.
Rule III: satisfied; there is one negative premise and one negative conclusion.
|i]||All green plants (are things that) contain chlorophyll.
Some things that contain chlorophyll are edible.
|Sd A Mu |
Mu I Pu
|∴||Some green plants are edible.||∴||Su I Pu|
Rule I: violated; the middle term is not distributed in either occurrence.
This proves that i is invalid. Again, however, we apply the other rules:
Rule II: violated; S is distributed in the premises but not in the conclusion.
Rule III: satisfied; there are no negative premises and no negative conclusions.
|j]||Some neurotics are not well-adjusted.
Some well-adjusted people are not ambitious.
|Su O Md
Mu O Pd
|∴||Some neurotics are not ambitious.||∴||Su O Pd|
Rule I: satisfied; the middle term is distributed in the first premise but not in the
Rule II: satisfied; S is undistributed in both occurrences and P is distributed in both occurrences.
Rule III: violated; there are two negative premises but only one negative conclusion.
Hence, j is invalid. With practice, you will see at a glance when rule III is violated; in such cases you do not have to consider the other two rules. In fact, it is a good strategy to check first for a violation of rule III. If such a violation occurs you do not have to bother with distribution at all.
The examples we have used so far to illustrate the application of our rules have been given in standard logical form. Needless to say, arguments found in ordinary contexts seldom have such form. Premises may be missing, and the order of the statements may be scrambled, as we have seen in dealing with other kinds of arguments. In addition, as we would expect, variations of the categorical statements, such as we discussed in the previous section, will be encountered. Usually, then, the first step in dealing with syllogistic arguments is to translate them into complete syllogisms of standard form. This transformation involves three steps.
After these steps have been completed, the rules may be applied to test the syllogism for validity.
Here is the sort of thing we might find:
k] Not all diamonds are gems -- industrial diamonds are unsuitable for ornamental purposes.
The context or tone of voice would make it clear that the first statement is the conclusion; what comes after the dash is offered to support it. The conclusion can be translated into the O statement "Some diamonds are not gems." The premise can also be rendered as an O statement: "Some diamonds (namely, industrial diamonds) are not suitable for ornamental purposes." To complete the syllogism we need the premise, "All gems are suitable for ornamental purposes."
|l]||All gems are suitable for ornamental purposes.
Some diamonds are not suitable for ornamental purposes.
|Pd A Mu|
Su O Md
|∴||Some diamonds are not gems.||∴||Su O Pd|
Example l has the same form as g, which we have already found to be valid.
m] Spiders can't be insects, because they are eight-legged.
The conclusion is the E statement "No spiders are insects." The given premise is the A statement "All spiders are eight-legged." When we supply the missing premise we have
|n]||All spiders are eight-legged.
No insects are eight-legged.
|Sd A Mu||Pd E Md|
|∴||No spiders are insects.||∴||Sd E Pd|
Rule I: satisfied; the middle term is distributed in the second premise but not in the
Rule II: satisfied; both 5 and P are distributed in each of their occurrences.
Rule III: satisfied; there is one negative premise and one negative conclusion.
Hence, n is valid.
|o]||Many people think abstract painting is worthless because it does not present a likeness of any familiar object. They believe that aesthetic merit is proportional to degree of realism. This principle is completely false. We must realize that artistic merit does not depend upon realistic representation. Rather, it is a matter of structures and forms. Only pure studies in form have true artistic worth -- this is the fundamental principle. It follows that a painting does not have true artistic worth unless it is abstract, for nothing that is not a pure study in form is abstract.|
From this passage we glean the following argument:
|p]|| Only pure studies in form have true artistic worth. |
Nothing that is not a pure study in form is abstract.
|∴||A painting does not have true artistic worth unless it is abstract.|
Translating into categorical statements, we get the following syllogism:
|q]|| All paintings that have true artistic worth are pure studies in form.
All abstract paintings are pure studies in form.
| Sd A Mu |
Pd A Mu
|∴||All paintings that have true artistic worth are abstract.||∴||Sd A Pu|
We see at a glance that this syllogism is invalid because it violates rules I and II.
|r]||In his famous essay "On Liberty," John Stuart Mill (1806-1873) deals with "the nature and limits of the power which can legitimately be exercised by society over the individual." He tries to establish the general principle ". . . that the sole end for which mankind are warranted, individually or collectively, in interfering with the liberty of action of any of their number, is self-protection. . . . The only part of the conduct of anyone, for which he is amenable to society, is that which concerns others. In the part which merely concerns himself, his independence is, of right, absolute." Mill then explains the method by which he intends to establish this principle. "It is proper to state that I forego any advantage which could be derived to my argument from the idea of abstract right, as a thing independent of utility. I regard utility as the ultimate appeal on all ethical questions; but it must he utility in the largest sense, grounded on the permanent interests of man as a progressive being. Those interests, I contend, authorize the subjection of individual spontaneity to external control, only in respect to those actions of each which concern the interest of other people." The principle of utility is explained and defended in another essay. "Utilitarianism." According to the principle of utility, acts are right insofar as they contribute to the general interest; that is, all and only those acts that promote the general interest are right.|
If we understand self-regarding conduct to be conduct that does not affect the interests of anyone other than the agent, the following syllogism emerges:
|s]|| All right acts are acts that promote the general interest.
No act of interfering with self-regarding conduct is an act that promotes the general interest.
|Pd A Mu|
Sd E Md
|∴||No act of interfering with self-regarding conduct is right.||∴||Sd E Pd|
You should verify that this syllogism satisfies the three rules and is, therefore, valid. Notice that an invalid syllogism would have resulted from taking "All acts that promote the general interest are right" as the first premise. The remainder of Mill's essay "On Liberty" is devoted to proving the truth of the second premise.
Before concluding this section on syllogisms, there is one additional type of argument to be considered. We shall call it the quasi-syllogism. Although the quasi-syllogism is not, strictly speaking, a syllogism, it is very similar to the syllogism and is often treated as such. We shall adopt a different approach that seems more straightforward. Consider the classic example
|t]||All men are mortal.||Socrates is a man.|
|∴||Socrates is mortal.|
This argument is certainly valid, but as it stands it is not a syllogism because neither the second premise nor the conclusion is a categorical statement. The term "Socrates" is not a class term; it is the name of a man, not the name of any class of entities. It would not make any sense to say "All Socrates are mortal" or "Some Socrates are mortal." The second premise and the conclusion can be translated into categorical statements by barbaric circumlocution, but we shall not do so. Instead, we note that the first premise is an A statement, and as such it is equivalent to the universal conditional "If anything is a man, then it is mortal." Whatever is true of all things is true of Socrates. We conclude from the first premise "If Socrates is a man, then Socrates is mortal." This statement, along with the second premise of t, gives us the following argument:
|w]||If Socrates is a man, then Socrates is mortal.
Socrates is a man.
|∴||Socrates is mortal.|
This argument is an instance of affirming the antecedent (section 7). Consider another example:
|v]||John Doe must be homosexual, because he favors laws that prohibit discrimination in employment on the basis of sexual preferences.|
This argument has a missing premise. It might be made into a quasi-syllogism as follows:
|w]||All who favor laws that prohibit discrimination in employment on the basis of sexual preferences are homosexual.
John Doe favors laws that prohibit discrimination in employment on the basis of sexual preferences.
|∴||John Doe is homosexual.|
This argument can be shown to be valid by the method we used for t. The trouble is, the premise we supplied is surely false. However, if we supplied a somewhat more plausible premise, the argument would be made invalid.
|x]||All homosexuals favor laws that prohibit discrimination in employment on the basis of sexual preferences.
John Doe favors laws that prohibit discrimination in employment on the basis of sexual preferences.
|∴||John Doe is homosexual.|
Treating this in the manner we have adopted, we get an instance of the fallacy of affirming the consequent (section 7).
|y]||If John Doe is homosexual, then John Doe favors laws that prohibit discrimination in employment on the basis of sexual preferences.
John Doe favors laws that prohibit discrimination in employment on the basis of sexual preferences.
|∴||John Doe is homosexual.|
Example v is exactly like example z of section 7, and we could have treated it in the same manner by supplying a conditional premise directly instead of first introducing a categorical premise. There is, however, some merit in treating arguments of this sort as quasi-syllogisms. Generally speaking, the only grounds we would have for asserting a conditional statement like "If John Doe favors laws that prohibit discrimination in employment on the basis of sexual preferences, then John Doe is homosexual" is to derive it from the corresponding universal conditional. For this reason, it is realistic to translate v into a quasi-syllogism.
6 These rules reflect the modern interpretation of the categorical syllogism. This interpretation is a consequence of the decision to construe A statements as universal conditional statements (section 13). The rules are easily revised to fit the traditional (Aristotelian) interpretation as follows:
Although the rules given in the last section for testing the validity of syllogisms are quite easy to remember and apply, they may have had an air of hocus-pocus about them. They work, but it is hard to see why. In this section we shall present the Venn diagram technique.7 It has great intuitive clarity (for which reason you may prefer it to the rules for checking the validity of syllogisms), and it can be applied to other types of arguments as well as categorical syllogisms.
As noted in the last section, the subject and predicate terms of categorical statements may be taken to refer to classes. The categorical statement itself can then be construed as a statement about the relationship between the two classes. Thus, the A statement "All whales are mammals" says that the class of whales is contained within the class of mammals. Similarly, the E statement "No spiders are insects" says that the class of spiders is entirely excluded from the class of insects. The I statement "Some diamonds are expensive" asserts that the class of diamonds overlaps the class of expensive things -- that is, that the two classes have at least one member in common. Finally, the O statement "Some animals are not carnivorous" states that the class of animals is not entirely included within the class of carnivores -- that is, that there is at least one animal that lies outside of the latter class.
These relations can all be shown diagrammatically, starting with a basic standard diagram of two overlapping circles -- one for each class -- enclosed in a box (Fig. 1).
The diagram in Fig. 1 does not make any assertion at all about the existence or nonexistence of members of the classes for which the circles stand -- it makes no commitment whatever about the existence of females, students, female students, female non-students, non-female students, or non-female non-students. The basic diagram is simply a standard starting point to which we can add information about the inclusion, exclusion, or overlap of classes. If we want to assert that a class has no members, we may do so by crossing out the region in the diagram that represents that class. Hence, if we wish to diagram the statement "All whales are mammals," we may cross out the area that stands for things that are whales but not mammals (Fig. 2): This is the standard diagram of an A statement.
To represent the assertion "No spiders are insects," we simply cross out the region that stands for things that are both spiders and insects:
This is the standard diagram of an E statement. Notice that each diagram starts from the same basic diagram and then proceeds to assert that some area or other stands for a class that is empty.8 This does not imply that the regions not crossed out do have members; the diagram makes no commitment one way or the other. Consequently, this way of diagramming corresponds to our interpretation of the A statement as a universal (material) conditional. There is nothing in the diagram that makes an assertion that the subject class has members; the diagram leaves that matter completely open.
To assert that a region represents a class that does have members, we place an "x" within it. Thus, to diagram the statement "Some diamonds are expensive," we place an "x" in the area standing for things that are both diamonds and expensive (Fig. 4), thereby asserting that there is at least one such thing:
This is the standard diagram of an I statement.
To diagram the statement "Some animals are not carnivorous," we put our "x" within the circle that stands for animals but in the portion of it that lies outside the circle standing for carnivores:
This is the standard diagram of an O statement. We now have standard diagrams for each type of categorical statement.
The categorical syllogism is an argument that involves three categorical statements and three distinct terms -- the middle term "M" and two end terms "S" and "P." In order to deal with the syllogism, we construct a basic diagram that contains three overlapping circles (Fig. 6). The way they are drawn enables us to represent every possible relation of inclusion, exclusion, or overlap among the three classes for which they stand. The basic diagram looks like this (again, with numerals temporarily added to refer to the separate regions):
The three circles have been drawn to create eight distinct regions; it is essential that they always be drawn in that way. It is an easy and disastrous error to draw the circles so that region 5 is missing.
The basic three-circle diagram (Fig. 6), like its two-circle counterpart (Fig. 1), is completely noncommittal concerning the emptiness or non-emptiness of any of the classes represented therein. It is the basic framework into which we can put assertions about the relationships among the three classes mentioned in the syllogism. And this is precisely what we do to test the validity of a syllogism. The syllogism is an argument containing two premises. We begin by diagramming the statements made by these premises. Then we look to see whether the content of the conclusion has been placed in the diagram in the process. The rationale for this general procedure is the second fundamental characteristic of valid deduction (section 4):
All of the information or factual content in the conclusion was already contained, at least implicitly, in the premises.
To see how that works, let us reconsider example n of section 14:
|a]|| All spiders are eight-legged.
No insects are eight-legged.
|S A M
P E M
|∴||No spiders are insects.||∴||S E P|
In the Venn diagram technique we can forget about the distribution of terms, for the diagram takes care of that matter automatically. We may begin by diagramming the first premise (Fig. 7). Because that premise asserts a relationship between the classes S and M, we may pay attention only to the two circles representing those classes, and we may ignore the circle representing P. Since that premise states that all S are M, we cross out that portion of the S-circle that lies outside the M-circle (regions 2 and 3 in the basic diagram [Fig. 6]).
Next, we diagram the content of the second premise (Fig. 8). Since that premise says that no P is M, we cross out that portion of the P-circle and the M-circle that overlap one another (regions 4 and 5 in the basic diagram [Fig. 6]). In this step we are focusing on the P-circle and the M-circle, ignoring the S-circle for the moment because the class S is not mentioned in the second premise.
We have now inserted into the diagram all of the information supplied by the two premises of the argument, and we can simply inspect the result to see whether the conclusion has been diagrammed in the process. The conclusion says that no S is P -- that is, it says that the area of overlap between the S-circle and the P-circle (regions 2 and 5 of the basic diagram) is empty. We see that this area is, indeed, crossed out in Fig. 8. In diagramming the two premises we have diagrammed the conclusion; consequently, the syllogism is valid.
Do not be confused by the fact that some other regions besides areas 2 and 5 are crossed out. The conclusion "No S is P" asserts that the area of overlap between the two classes S and P is empty, and this is shown by the fact that regions 2 and 5 are crossed out. The conclusion says absolutely nothing about the emptiness or nonemptiness of the other regions, in particular regions 3 and 4, which were crossed out in the course of diagramming the premises. It does not matter what happens to them. Look back at the standard two-circle diagram of the E statement (Fig. 3); the area of overlap is crossed out, but the remaining areas are unmarked. Because the E statement says nothing about them, the fact that they are partly crossed out in our three-circle diagram has no bearing on the validity of the argument.
The foregoing example had two universal premises; let us now try an example with a particular premise (example g of section 14):
|b]|| All logicians are mathematicians.
Some philosophers are not mathematicians.
|P A M |
S O M
|∴||Some philosophers are not logicians.||∴||S O P|
As always, we begin with die basic diagram (Fig. 6) and proceed by diagramming the premises. Since the first premise says that all P are M, we cross out all of the P-circle that falls outside the M-circle, ignoring the S-circle while doing so. Next, we diagram the second premise, which says that some S is not M. This requires us to place an "x" inside the S-circle but outside the M-circle. This "x" must go in region 3 of the basic diagram. Obviously, we cannot put an "x" in a crossed-out region; crossing out signifies an empty region, whereas putting an "x" into a region asserts that it is not empty. To assert that the same region is both empty and not empty is a flat contradiction. Hence, the "x" cannot be placed in region 2. Moreover, the "x" should not be placed in regions 5 or 6, for we want to say that some S is not M, and those two regions are within the M-circle.
Having completed the crossing-out for the first premise, and having placed the "x" for the second premise, we now inspect the diagram to see whether it yields the desired conclusion. Focusing attention on the S- and P-circles, we see that there is an "x" in a portion of the S-circle that lies outside the P-circle. The diagram does assert that some S is not P, thereby showing that the conclusion follows logically from the premises.
In,writing down a syllogism, it is obviously irrelevant, from the standpoint of validity, which premise is stated first. It does sometimes matter, from the standpoint of convenience, which premise is diagrammed first. If a syllogism has one particular premise and one universal premise, it is always advisable to diagram the universal premise first. Go back to example b and see what would have happened had we tried to diagram the particular premise first. Since none of the regions would have been crossed out, we would not have known whether to put the "x" in region 2 or region 3.
We could have handled that problem by using a "floating x" -- by placing an "x" in each region and connecting the two with a dotted line to which a question mark is attached. In diagramming the universal premise, we would then have seen that the "x" must go into region 3, because region 2 must be crossed out. But it is much easier, and less apt to lead to confusion, to diagram the universal premise first.
You might wonder what to do if both premises are particular. You can diagram both premises by using the "floating x" device, but you will always find that the syllogism is invalid. Without a universal premise to cross out some regions and thereby force the "x" into a definite compartment, there can be no valid conclusion. As a matter of fact, no syllogism with two particular premises can be valid.9 The easiest way to deal with such syllogisms is just to remember that fact, but if you do go ahead and diagram them you will get the right answer.
Both of the preceding examples have been valid syllogisms; let us now look at some invalid ones.
|c]|| All professional rock musicians (are persons who) have green hair.
No presidents of banks are professional rock musicians.
|∴||No presidents of banks have green hair.|
Starting with the basic diagram, we cross out all of the M-circle that lies outside the P-circle; this takes care of the first premise. Then, by virtue of the second premise, we cross out all of the area common to the S-circle and
the M-circle. It does not matter that region 6 is crossed out twice, once by each premise, for that is only to repeat that the area is empty. When, however, we examine the relation between the S- and P-circles, we see that the region they share in common is not entirely crossed out, so the conclusion "No S is P" does not follow validly. Here is another example:
|d]|| Some people with high earnings are smart investors.
All professional athletes have high earnings.
|∴||Some smart investors are professional athletes.|
Even though the universal premise comes second, we follow our earlier hint and diagram it first, crossing out the part of the P-circle that lies outside the M-circle. We then proceed to try to place an "x" in the area of overlap between the S- and M-circles, but we find that there are two regions in which it might be placed (regions 5 and 6 of the basic diagram). In this case we use the "floating x" to register our doubt. When we try to read the conclusion, we find that our "x" might be in the region that is common to
the S- and P-circles, but it might be in that part of the S-circle that lies outside the P-circle. It may be true that some S are P, but it may not; the conclusion is not necessitated by the premises, so by characteristic I of valid deductive arguments (section 4), the argument cannot be valid.
We have now provided two distinct ways of testing syllogisms for validity. Is either method more adequate? Not as a test of validity of syllogisms. The choice between them is a matter of personal preference. The Venn diagrams have a great deal of intuitive appeal, once you get the hang of them, and the rules seem arbitrary by contrast. My own preference, nevertheless, is for the rules, because it is easier to carry three simple rules in my head than it is to carry a pencil and notebook (to draw diagrams) in my pocket.
Before abandoning the diagrams, however, we should take account of the fact, noted at the beginning of this section, that Venn diagrams can be used to handle nonsyllogistic arguments. Consider the following example:
|e]|| Everyone convicted of conspiracy will be imprisoned or fined.
Some who are convicted will be fined.
No one will be imprisoned and fined.
|∴||Not all who are convicted of conspiracy will be imprisoned.|
This argument is obviously not a categorical syllogism, for it has one premise too many, and the first premise is not a categorical statement. The Venn diagrams will, nevertheless, provide an easy check on its validity. Three classes are involved -- those convicted of conspiracy, those who are imprisoned, and those who are fined -- so we begin with the standard three-circle diagram. To diagram the first premise, we cross out that portion of the C-circle that falls outside both the I- and F-circles. All three circles are involved, for the first premise mentions all three classes. Recalling an earlier hint, we skip the second premise (which is particular) and go on to the
third premise (which is universal). In accordance with the third premise, we cross out the region in which the I-circle and the F-circle overlap. Then, by virtue of the second premise, we place an "x" in the only available area shared by the F- and C-circles. Finally, inspecting the diagram, we see that it does sustain the conclusion that some who are convicted of conspiracy will not be imprisoned, which is equivalent to the statement offered as the conclusion of e.
This example hints at the powers and limitations of the Venn diagram technique. The Venn diagrams are far more versatile than the syllogistic rules. They can be used throughout the logic of classes, a logic that is far more general and important than the logic of categorical syllogisms, chief limitation on the Venn diagrams concerns the number of basic classes involved. As we have seen, they work admirably for arguments involving only two or three classes. They are much trickier for four classes (circles have to be replaced by ellipses), and for five or more they are positively unwieldy. That does not deprive them of their status as an outstandingly useful item in the logician's tool kit.
8 In constructing the diagram for the A statement we do not begin by drawing the circle for the subject term inside the circle for the predicate term. Likewise, in constructing the diagram for the E statement, we do not begin by drawing the two circles completely separated from one another with no area of overlap. We always begin with the same basic diagram, indicating inclusion and exclusion by appropriate crossings-out. The reason for insisting upon the standard basic diagram is that it makes the checking of syllogisms easier and more accurate.
9 This rule can be derived from rules I through III of section 14.
In dealing with the categorical syllogism and the simple logic of classes, we were treating statements about things and their properties -- for example, "Anything that has the property F also has property G" or "At least one thing has both of the properties F and G." The fact that we often talked about classes F and G instead of properties F and G is inconsequential, for each such property determines a class, namely, the class of things having that property -- for example, the property intelligent delineates the class of intelligent things. It is only in much more esoteric reaches of logic that we need worry seriously about whether each property does, in fact, determine a class.
Things not only have properties; they also bear relations to one another. Our language is full of relational expressions: "John loves Jane," "This car is more expensive than that one," "World War I was earlier than World War II," and so forth. Relations are not simple combinations of properties. Jack may be shorter than Jim, even though neither of them is short -- they might both be basketball players who top seven feet. John and Jane may each be married, but whether they are married to each other is the crucial question if they want to file a joint income tax return. Because so many important considerations involve relations, it will be advisable to look briefly at their logic.
Although the foregoing examples all involve relations between two things -- so-called binary relations -- other relations obtain among larger numbers of objects. The word "between," for example, as it occurs in "Arizona is between California and New Mexico," denotes a relationship among three things -- a so-called ternary relation -- as does the word "gave" in "Bill gave Thelma an engagement ring." The statement "Bob sold his car to Hal for three hundred dollars" involves a four-place or quaternary relation. In principle, it would be possible to have five-place, six-place, etc., relations. Having acknowledged the existence of these more complex relations, however, we shall now confine our attention to binary relations.
It is easy to describe some important characteristics of binary relations, but a little symbolism will be helpful. When we want to say that a relation R holds between two objects a and b, we simply write "aRb." If, for instance, "a" stands for Harry, "b" for Peter, and "R" for the relation father of, then "aRb" translates as "Harry is the father of Peter."
The converse of a relation R is a relation Rc that holds between the objects b and a (in that order) whenever the relation R holds between a and b (in that order) -- that is, "bRa" is true whenever "aRcb" is true. Thus, for example, less than is the converse of greater than, parent of is the converse of child of, and is loved by is the converse of loves. However, father of is not the converse of son of; can you see why?
Some relations are symmetrical -- that is, whenever the relation holds between two objects a and b it also holds between b and a. The relation sibling of is symmetrical; if Bill is a sibling of Jean, then Jean must be a sibling of Bill. Other familiar examples of symmetrical relations are at the same time (Susan finished her exam at the same time Dick finished his), equals (the number of men in the logic class equals the number of women), and married to (John is married to Jane). A symmetrical relation is identical to its converse. The relation father of is asymmetrical; if Harry is the father of Peter, then Peter cannot be the father of Harry. The relation brother of has neither of these characteristics; if Scott is a brother of Kim, Kim may or may not be a brother of Scott (depending upon whether Kim is a boy or a girl). Such relations are called nonsymmetrical. Loves, alas, is also a nonsymmetrical relation.
Using the letters "x" "y," "z" as variables that can stand for any objects, and importing the truth-functional connectives from earlier sections, we can offer the following definitions:
|a]|| R is a symmetrical relation if xRy ⊃ yRx.
R is an asymmetrical relation if xRy ⊃ ∼(yRx).
R is a nonsymmetrical relation if it is neither symmetrical nor asymmetrical.
A relation is said to be reflexive if objects always bear that relation to themselves; for example, any number is equal to itself, any triangle is congruent to itself, and one is as intelligent as oneself. A relation that an object never bears to itself is irreflexive; for example, greater than is an irreflexive relation because nothing can be greater than itself. Relations that are neither reflexive nor irreflexive are said to be nonreflexive. Loves is a nonreflexive relation, since some people love themselves and others do not. These concepts, also, can be formally defined:
|b]|| R is a reflexive relation if xRx.
R is an irreflexive relation if ∼(xRx).
R is a nonreflexive relation if it is neither reflexive nor irreflexive.
It is easily shown that an asymmetrical relation cannot be reflexive, for suppose the relation R is asymmetrical, i.e.,
Then, since the foregoing is true for any y (including whatever "x" stands for),
But this is a reductio ad absurdum (section 8), for the assumption that R is reflexive leads to the conclusion that it is not.
Another important characteristic of relations is transitivity. Consider, for example, the relation older than. If Aunt Agatha is older than Aunt Matilda, and if Aunt Matilda is older than Cousin Nellie, then Aunt Agatha must be older than Cousin Nellie. A relation is transitive provided that, like the older than relation, whenever it holds between one thing and another and also between the other and some third thing, it always holds between the first and the third. (I have deliberately stated this definition without using the variables "x," "y," "z," in order to show how awkward and contrived the formulations become; from here on we will use the more perspicuous formulas containing variables and other symbols. Furthermore -- a point that was not clear in the verbal formulation -- the variables "x," "y," "z," etc., need not refer to different entities. In the next paragraph we have an example in which "x" and "z" refer to the same individual.)
The relation mother of is intransitive; if x is mother of y and y is mother of z, then x cannot be mother of z. The relation friend of is nontransitive; if x is a friend of y and y is a friend of z, then x may or may not be a friend of z. The relation brother of is also nontransitive; if Jack is a brother of Jim and Jim is a brother of Joe, then Jack is a brother of Joe; nevertheless, though Jack is a brother of Jim and Jim is a brother of Jack, it does not follow that Jack is a brother to himself.
We can define these concepts formally:
|c]|| R is a transitive relation if (xRy · yRz) ⊃ xRz.
R is an intransitive relation if (xRy · yRz) ⊃ ∼(xRz).
R is a nontransitive relation if it is neither transitive nor intransitive.
Transitivity is used to define two extremely important types of relations. Ordering relations such as greater than, earlier than, and more intelligent than are transitive and asymmetrical: as we noted above, they must also be irreflexive. If, moreover, every two distinct entities x and y are related xRy or yRx, then the transitive and asymmetrical relation R establishes a simple order among the entities it relates, similar to the order established among numbers by the relation greater than.
Relations of the second type -- known as equivalence relations -- are transitive and symmetrical (and, we must add to avoid trivial exceptions, reflexive). The relation of congruence among triangles, learned in elementary geometry, is an equivalence relation. Any triangle abc is congruent to itself; if triangle abc is congruent to triangle def, then def is congruent to abc; and, if triangle abc is congruent to triangle def and triangle def is congruent to triangle ghi, then abc is congruent to ghi. An equivalence relation breaks up a domain into a set of nonoverlapping equivalence classes. All triangles congruent to one another belong to one equivalence class and are equivalent to one another with respect to the relation of congruence, but triangles that are not congruent to one another belong to different equivalence classes. All the triangles that belong to one of these equivalence classes, defined by the relation of congruence, are alike with respect to size and shape.
Another familiar example of an equivalence relation -- this one holds among classes -- is the relation having the same number of members. Under this relation all classes having two members are equivalent to one another -- a pair of shoes, a team of horses, a married couple, a set of twins -- and these are all nonequivalent to classes having three, four, etc., members. Don't be distressed by the fact that these equivalence classes are classes of classes Such classes are perfectly respectable, and it is sometimes very useful to talk about them.
If we perform a slight redefinition of the term "sibling" to allow one person to be a sibling of oneself, then the relation sibling of becomes an equivalency relation -- namely, the relation of having the same parents. A given equivalence class, under this relation, is simply a group of children of common parents.
Ordering relations and equivalence relations can be fruitfully combined to provide a basis for quantitative measurement. To establish a method of measuring weight, for example, we might choose an equal arm balance as the basic instrument of measurement. The equivalence relation having the same weight then holds between objects that balance each other precisely; this relation defines equivalence classes of objects that are equal in weight. Next, we introduce an ordering relation heavier than, which applies to a pair of objects when one goes down and the other goes up as they are placed in the opposite pans of the balance. Although this ordering relation does not provide a complete ordering of all the objects we are dealing with -- it does not order objects that are equal in weight -- it does establish a simple ordering of the equivalence classes defined by the first relation. Once these two relations have been brought to bear, we need only choose some object to be the standard unit of weight, and a complete scale of numerical measurement emerges. (The definitions just given of "equal in weight" and "heavier than" are operational definitions -- see section 32.)
The foregoing properties of relations -- especially symmetry, asymmetry, and transitivity -- are often employed in arguments, usually in unstated premises. Most of the time this causes no problem. For example,
|d]||Alice Green earns more money than her husband William.|
|∴||William Green does not earn more money than his wife Alice.|
When we recognize that the relation earns more money than is asymmetric, we see that the argument, which can be spelled out as follows, is deductively valid.
|e]|| Alice Green earns more money than her husband William.
If x earns more money than y, then y does not earn more money than x (by definition a).
|∴||William Green does not earn more money than his wife Alice.|
This argument is so obvious that it hardly needs stating. Suppose, however, that because of his lower earnings, William feels inferior to Alice. One might conclude that Alice does not feel inferior to William; indeed, one might conclude that she feels superior to her husband. Two mistakes are involved here. First, the argument
|f]|| William feels inferior to Alice.
If x feels inferior to y, then y does not feel inferior to x (i.e., feels inferior to is an asymmetric relation).
|∴||Alice does not feel inferior to William.|
is deductively valid, but it contains a false premise. The relation feels inferior to is not, asymmetric. It is entirely possible -- in fact, it probably happens frequently between two people -- that each feels inferior to the other. But that is a matter of psychological fact, not a matter of logic.
The second mistake involves a misconception regarding the converse of a relation. It probably goes something like this:
|g]|| William feels inferior to Alice.
The relation superior is the converse of inferior.
|∴||Alice feels superior to William.|
The problem is that the first premise involves not the relation is inferior to but rather feels inferior to. The converse of feels inferior to is the relation is considered superior by; thus, the only conclusion that can be drawn from the first premise is that Alice is considered superior by William. Since is considered superior by is by no means the same as feels superior to, g cannot plausibly be made into a valid argument.
Let us consider another rather transparent argument.
|h]|| Arnold is stronger than Bruce.
Bruce is stronger than Charles.
|∴||Arnold is stronger than Charles.|
This time we are relying upon the fact that stronger than is a transitive relation. More explicitly,
|i]||Arnold is stronger than Bruce.
Bruce is stronger than Charles.
If x is stronger than y and y is stronger than z, then x is stronger than z (by definition c).
|∴||Arnold is stronger than Charles. ,|
It is safe to assume that is stronger than is a transitive relation, but there can be somewhat trickier cases. Here is one from the 1982 matches at Wimbledon:
|j]||Billie Jean King beat Tracy Austin at tennis.
Chris Evert Lloyd beat Billie Jean King at tennis.
|∴||When next they play tennis with one another, Chris Evert Lloyd will beat Tracy Austin.|
Those who have much familiarity with individual or team sports would not be inclined to regard j as a strong argument, for it is fairly obvious that wins against is not a transitive relation. Better players sometimes lose to players who are not as good, and stronger teams sometimes lose to weaker teams. Thus,
|k]||King beats-at-tennis Austin.
Lloyd beats-at-tennis King.
If x beats-at-tennis y and y beats-at-tennis z, then x beats-at-tennis z.
|∴||Lloyd beats-at-tennis Austin.|
is a valid deductive argument, but its third premise is not really plausible. Whether feels inferior to (example f) is an asymmetric relation, and whether beats-at-tennis (example k) is a transitive relation, are questions of fact, not of logic. Logic does, however, have a bearing upon the foregoing sorts of arguments, for it provides precise definitions of such concepts as symmetry and transitivity, and it helps us see clearly what premises are required to make such arguments deductively valid.
Not all questions about transitivity are as simple to answer as the one about winning at sports; indeed, it is easy to make false assumptions about the transitivity of relations. Let us conclude this discussion by looking at a couple of examples that have some significant consequences.
|l]||Suppose you have a batch of paint, and you mix another batch that is indistinguishable from it in color. Suppose further that you mix a third batch that is indistinguishable from the second in color. It might be tempting to suppose that the third batch will be indistinguishable in color from the first batch. Experience shows, unfortunately, that the relation of perceptual color matching is not a transitive relation; sample one may match sample two, and sample two may match sample three, but sample one may fail to match sample three. If you mix three batches of paint you must make sure that all three samples match one another by pairs -- that is, you must not assume the transitivity of perceptual color matching -- if you want to avoid noticeable color differences.|
There are many such examples of perceptual indistinguishability relations that turn out to be nontransitive, and hence do not constitute genuine equivalence relations. Indeed, in our example of equality of weight, we had to assume an ideal balance that would discriminate perfectly between objects of unequal weight. With any real balance, we would face the same problem we encountered with color matching.
Another significant area in which transitivity sometimes breaks down is the realm of individual choice and preference.
|m]||You might easily suppose that the relation of preference is a transitive ordering relation, that a person who prefers x to y and y to z will surely prefer x to z. But this supposition is not always correct. Given a choice between owning a particular book and attending an opera, John might choose to attend the opera. Given, moreover, the choice between attending an opera and having dinner in a fine restaurant, he might choose the delicious meal. The same individual might, however, when presented with a choice between possessing the book and enjoying the dinner, choose to own the book. This is not to say that it is reasonable to have preferences nontransitively ordered, but it can happen.|
Notice, in fact, that John can get into trouble if he meets an unscrupulous operator who is aware of the nontransitivity of his preferences. Suppose John has a book and I have an opera pass that John would prefer to have; in fact, he is willing to give me the book plus one dollar for the pass. (If he really prefers the pass to the book, he will pay something to exchange the one for the other.) Suppose, moreover, that I have a meal ticket that will entitle the holder to an exquisite dinner, and that John would prefer the meal ticket to the opera pass. In fact, he is willing to give me the opera pass plus one dollar in exchange for the meal ticket. Now I present him with the choice between the book (which he traded to me earlier) and the meal ticket, and true to the nontransitivity of his preferences, he gives me the meal ticket plus one dollar for the book. The net result is that John has his book back, I have possession of the opera pass and meal ticket that I had in the beginning, but I am three dollars richer than before. It is not likely that anyone would allow oneself to be victimized in such a transparent way, but people do have nontransitive preference rankings, and our imaginary barter shows that there is something quite irrational about them.
When we analyzed A statements in section 13, we said that the statement "All whales are mammals" could be rendered "If anything is a whale, then it is a mammal." Had we felt inclined, we could have gone on to analyze the latter statement, "For any object x, if x is a whale, then x is a mammal," using the letter "x" as a variable in expressing our universal material conditional statement. Using standard logical notation, we could then have abbreviated the phrase "for any object x" as "(x)," calling it a universal quantifier. Abbreviating "x is a whale" as "Wx" and "x is a mammal" as "Mx," and using the horseshoe symbol as the sign of material implication, we could have symbolized the A statement
a] (x)[Wx ⊃ Mx]
The E statement "No spiders are insects" could then have been symbolized
b] (x)[Sx ⊃ ∼Ix]
We did not introduce all of this notation in treating the categorical syllogism, for nothing much would have been gained at that point. Ordinary English, with a few abbreviations, worked quite well.
In the last section, we found that the introduction of variables "x," "y" "z" was virtually indispensable, especially for the purpose of keeping track of several different entities that bear relations to one another. Thus, we used our truth-functional connectives and some variables to define such concepts as reflexivity, symmetry, and transitivity. Had we added universal quantifiers, we could have defined transitivity as follows:
c] Relation R is transitive = df (x)(y)(z)[(xRy · yRz) ⊃ xRz]
It is useful to introduce another quantifier, the existential quantifier, which can be used immediately to symbolize I statements. Using obvious abbreviations, the statement "Some diamonds are expensive" can be written
d] (∃x)[Dx · Ex]
where the existential quantifier "(∃x)" may be read "There exists at least one object x such that" or "For some object x." The O statement "Some animals are not carnivorous" can likewise be symbolized
e] (∃x)[Ax · ∼Cx]
We now have simple symbolic rendition of each type of categorical statement.
We noted in section 13 that A and O statements are contradictories of one another. Consequently, the O statement is equivalent to the negation of the corresponding A statement; i.e., "Some animals are not carnivorous" is equivalent to "Not all animals are carnivorous." Hence,
f] ∼(x)[Ax ⊃ Cx]
must be equivalent to e. You can make a truth table to establish the equivalence of "Ax · ∼Cx" to "∼(Ax ⊃ Cx)," thereby showing that e and f are both equivalent to
g] (∃x) ∼[Ax ⊃ Cx]
The symbol "∼(x)" is thus seen to have the same import as "(∃x)∼"; a similar argument, starting from the equivalence of an A statement to the negation of the corresponding O statement, shows that "∼(∃x)" is tantamount to "(x)∼."
The formalism of quantifiers is rather pointless in dealing with categorical syllogisms and the simple logic of classes, because one variable is all we need to formulate any argument. It is only when we get to the logic of relations that the quantifiers become really useful, for in this context we need to keep track of several variables. This can be illustrated by some very simple relational statements.
h] (x)(y) xLy Everybody loves everyone.
i] (∃x)(∃y) xLy Somebody loves someone.
These two statements are quite straightforward, but things become somewhat trickier if we have a mixture of universal and existential quantifiers in the same statement. Consider the statements
j] (∃x)(y) xLy Someone loves everybody.
h] (y)(∃x) xLy Everyone is loved by somebody.
These differ from one another in the order of the two quantifiers, and differ radically in meaning. According to j, there is some one individual who (like Albert Schweitzer perhaps) loves all human beings (including, incidentally, oneself). Statement k, in contrast, says that every person is loved by someone or other (one's mother perhaps), but different people may be loved by different individuals. In no way does k imply that any single person loves all people. Analogous remarks apply to statements
l] (∃y)(x) xLy Someone is loved by everybody.
m] (x)(∃y) xLy Everyone loves somebody.
Statement l asserts that there is one person who is the object of love for all people, whereas m says merely that each person loves somebody or other, without implying that any one individual is the recipient of all that affection.
Each of the statements, h to m, differs in meaning from every other member of that group. Statements j to m illustrate a very important point: In statements that involve a mixture of existential and universal quantifiers, the order of the quantifiers is essential to the meaning. Failure to keep this principle in mind can lead to a rather common fallacy, which may aptly be called the fallacy of "every'" and "all."
Statement l might be translated from symbols into fairly idiomatic English as "Someone is loved by all people," and m might be rendered as "Every person loves someone." Obviously, l does not follow logically from m, for m does not say that it is one and the same individual that is loved by everyone, whereas this is precisely what l asserts. In general, it is a fallacy to reason from "every" to "all." If we are reasoning about concrete things, this fallacy is not apt to occur. Given that every member of the country club drives a sports car, we are not inclined to conclude that all members drive the same one. Let us state the fallacious argument explicitly.
|n]||Every member of the country club has a sports car to drive.|
|∴||There is a sports car that all members of the country club drive.|
The premise asserts that a certain relation, namely, the relation of driving, holds between members of two classes, namely, the class of people who belong to the country club and the class of sports cars. It says that every member of the former class has the relation to some member or other of the latter class. The conclusion states that the same relation holds between members of the same two classes. It says that all members of the former class have that relation to some single member of the latter class. The form of the argument may be given as follows:
|o]||For every F there is some G to which it has the relation R.|
|∴||There is some G to which all F have the relation R.10|
This form is shown diagrammatically in Fig. 14.
The x's are the individual members of the class F and the y's are the individual members of the class G. A line connecting an "x" and a "y" represents the relation R holding between that member of F and that member of G. Notice that the premise does not require that every member of G stand in relation to a member of F, nor does it exclude the possibility that a given member of G may stand in relation to more than one member of F. It is evident that form o is invalid; the premise diagram can hold even if the conclusion diagram does not.
Consider the third proof for the existence of God given by St. Thomas Aquinas.
|p]||The third way is taken from possibility and necessity, and runs thus. We find in nature things that are possible to be and not to be, since they are found to be generated, and to corrupt, and consequently, they are possible to be and not to be. But it is impossible for these always to exist, for that which is possible not to be at some time is not. Therefore, if everything is possible not to be, then at one time there could have been nothing in existence. Now, if this were true, even now there would be nothing in existence, because that which does not exist only begins to exist by something already existing. Therefore, if at one time nothing was in existence, it would have been impossible for anything to have begun to exist; and thus even now nothing would be in existence -- which is absurd. Therefore, not all beings are merely possible, but there must exist something the existence of which is necessary.11|
This is an excellent example of reductio ad absurdum, but we are not particularly concerned with the whole argument. We are interested in the italicized part of the sub-deduction. It may be written
|q]||For every thing, there is a time at which it does not exist.|
|∴||There is a time at which all things do not exist.|
Clearly, q has the form o. The two classes involved are the class of things and the class of times. The relation is the relation of a thing not existing at a time.
The same fallacy sometimes occurs in philosophical arguments designed to prove the existence of an underlying substance. Substance is sometimes taken to be something that remains identical throughout change. The argument for the existence of substance, thus conceived, follows this line.
|r]||Change is a relative notion. It requires that, throughout each change, there must be something that remains constant; otherwise we would not be justified in speaking of one thing changing, for there would simply be two completely distinct things. For example, one changes in many ways as one grows from infancy to maturity, but there must be something that is constant and unchanging, for otherwise there would be no ground for regarding the infant and the mature person as the same individual. The world is full of change at all times. Since each change requires something constant and unchanging, there must be something that is constant throughout all change. This thing is substance.|
The beginning of this passage is the justification for the premise of the italicized argument. This argument may be rendered:
|s]||For every change there is something remaining constant through that change.|
|∴||There is something (substance) which remains constant through all changes.|
Again the argument has form o. In this case the two classes are changes and things; the relation is remaining constant during.
10 In the examples dealing with the love relation, both F and G would be the class of humans.
11 The "Summa Theologica" of St. Thomas Aquinas, Part I, trans. Fathers of the English Dominican Province, 2nd and rev. ed. (London: Burns Oates & Washbourne Ltd.; New York: Benziger Brothers, 1920), pp. 25f. Reprinted by permission. Italics added.
Modern deductive logic is often called symbolic logic or mathematical logic because it employs a symbolic apparatus much like that found in mathematics. In our treatment of basic argument types, we have found it helpful to introduce some special symbols. The notation we have used is quite standard, although there is some variation in usage. Some authors use an ampersand (&) instead of a dot as the symbol of conjunction; sometimes an arrow is used instead of a horseshoe as the sign of material implication, and a double arrow (↔) instead of a triple bar for material equivalence. Such variation is trivial and easily accommodated.12
Anyone who has taken even elementary algebra in high school can appreciate the value of the special symbolic apparatus. Given the use of letters as variables and a few operation symbols, relatively intractable "story problems" can be translated into easily manageable equations that can be solved by applying certain clearly specified rules of operation. Problems in deductive logic can be handled analogously- -- and just as in elementary algebra, we often find that the hardest part is the translation of the problem from everyday English into the language of symbolic logic. For this reason we have taken considerable trouble to discuss different ways of formulating a particular kind of statement (e.g., the conditional statement) in ordinary language.
One advantage of special symbols is brevity; "(∃x)" is much briefer than "There exists at least one object x such that. . . ."A further advantage is precision. It is of crucial importance to distinguish, for example, between the inclusive and exclusive senses of the word "or." It is essential to understand the logical affinity of "and" and "but" as well as their psychological difference. Likewise, we must distinguish the truth-functional meaning of "and" from its temporal "and then" sense. The words of English are ambiguous and equivocal; for logical purposes we need a precise language. A still more significant advantage of the symbolic language is its perspicuity. We are not merely developing a shorthand notation, nor are we merely trying to make ordinary language more precise. We are trying to display clearly the logical structure of statements and arguments. It is important, for instance, to see that the A statement can be construed as a universal material conditional, even if such a construction is neither briefer nor more precise than alternative formulations. When we defined the truth-functional material implication, we deliberately departed from the everyday "if . . . then . . ."in order to have an operation that embodies the logical core of conditional statements.
By far the greatest advantage of the symbolic system lies in the fact that it employs a relatively small number of exactly defined symbols that can be manipulated according to a few simple and precisely formulated rules. Affirming the antecedent is an excellent example; it says that you may assert a statement of a given form (namely, the consequent of a conditional) if you already have available as premises statements of other forms (namely, the conditional itself as well as its antecedent). There are other rules such as substitution and generalization, but we shall not discuss them in detail. If you are interested in seeing the development and elaboration of a full-blown logical system, you would do well to consult the references at the end of the book.
The symbolic apparatus we have introduced comes surprisingly close to being adequate for a complete basic logical system. We might want to have proper names as well as variables for individual objects. We probably would want to have the logical identity in order to assert, by denying their identity, that two individuals are distinct. And we would certainly need relation symbols for ternary, quaternary, etc. relations. But such supplementation is modest in comparison with what we already have available. With logical systems of this basic kind -- so-called first-order logic13 -- it is possible to analyze and evaluate arguments of amazing complexity and subtlety, far beyond any of the examples we have dealt with in the preceding sections.
12 A much more radical departure is found in the so-called Polish notation. In this system, "∼p" is written "Np," "p·q" is written "Kpq," "p ∨ q" is written "Apq," "p ⊃ q" is written "Cpq," and "p ≡ q" is written "Epq." For example, the formula "[(p·(p ⊃ q)] ⊃ q" would be written "CKpCpqq"; beginning with the innermost implication we write "Cpq," then we conjoin this to "p," writing "KpCpq," and finally we make this the antecedent of the main implication by writing the final formula "CKpCpqq." This notation has the advantage that it can be typed on any standard typewriter, without the need for special keys, and it dispenses entirely with parentheses and brackets for the purpose of grouping parts of formulas. The notation we have adopted is, however, the most widely used, and (to me) it exhibits the logical structure of formulas more clearly (though this may be no more than a result of my early training with the non-Polish notation).
Some electronic calculators use a similar idea to simplify the execution of a series of operations. It is called "RPN" for "reverse Polish notation." Suppose you want to add or multiply two numbers. Instead of entering one number, then the operation, and then the other number, you enter the two numbers first and then specify the operation. This is just the reverse of the Polish logical notation explained in the preceding paragraph; there we entered the operation first and followed it with the statements to which it is applied. Let us use "A" for "add" and "M" for "multiply." Suppose we want to add x to y and multiply the result by z. We would enter "xyAzM." The calculator would give us (x + y) x z.
13 Logical systems of this kind are considered first-order because they contain symbols that refer to individual objects and their properties, and quantifiers that apply to individual objects. They contain neither symbols that refer to properties of properties of individuals nor quantifiers that apply to properties of individuals. In first-order logic we can readily handle such statements as "All Boy Scouts are trustworthy," "All Boy Scouts are loyal," "All Boy Scouts are helpful," etc., but we cannot symbolize "No Boy Scouts have any faults" because this statement refers to all properties of Boy Scouts. It states that no property of any Boy Scout is a fault (a bad property).