[I'd like to insert my thanks here to John Robbins of the Trinity Foundation for graciously giving me permission to put up a valuable (and exemplary) chapter of Dr. Clark's book and also his own excellent introduction to the book...

You can check out the Trinity Foundation web site by clicking HERE.... -Tony Castrataro 

Also, their mailing address and phone # is:

Trinity Foundation 

Post Office 68

Unicoi, Tennessee 37692

(Phone) 1-423-743-0199 ]

     Contents

2. Informal Fallacies 

3. Definition

4. The Beginning of Formal Logic

5. Immediate Inference

6. The Syllogism—Diagrams

7. The Syllogism—Deduction and Rules

8. Historical Remarks

9. Other Forms of Argument 

    1. Sorites

    2. Hypothetical Syllogism

    3. Dilemma 

10. Truth Tables

11. The Deduction of the Syllogism

Glossary

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By the same author:

Readings in Ethics (1931)
Selections from Hellenistic Philosophy (1940)
A History of Philosophy (1941)
A Christian Philosophy of Education (1946)
A Christian View of Men and Things (1952)
What Presbyterians Believe (1956)
Thales to Dewey (1957)
Dewey (1960)
Religion, Reason, and Revelation (1961)
William James (1963)
Karl Barth's Theological Method (1963)
The Philosophy of Science and Belief in God (1964)
What Do Presbyterians Believe? (1965)
Peter Speaks Today (1967)
The Philosophy of Gordon H. Clark (1968)
Biblical Predestinati6n (1969)
Historiography: Secular and Religious (1971)
II Peter (1972)
The Johannine Logos (1972)
Three Types of Religious Philosophy (1973)
First Corinthians (1975)
Colossians (1979)
Predestination in the Old Testament (1979)
I and II Peter (1980)
Language and Theology (1980)
First John (1980)
God's Hammer (1982)
Behaviorism and Christianity (1982)
Faith and Saving Faith (1983)
In Defense of Theology (1984)
The Pastoral Epistles (1984)
The Biblical Doctrine of Man (1984)
The Trinity (1985)

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                             Why Study Logic?

These are questions that deserve an answer, but the answer may not be exactly what you might expect. Because many people disdain logic, it will be necessary to understand the relationship between logic and morality, for example. After all, many people think one should not study logic. "Life is deeper than logic," we're told. "Life is green, but logic is gray and lifeless." The poets tell us that "we murder to dissect." Many believe that one's time would be better spent in prayer, Protesting, or preaching. Or if they are naturalistically-minded, they might suggest contemplating one's navel, or the sunset, or performing experiments in laboratories. So why study logic? Perhaps if we understood what logic is, we could better answer the question.

What is Logic?

But could there be something more basic than the three basics? Something so obvious that most people do not see it, let alone study it? What is there in common between calculating, reading, and writing? The answer, of course, is thought. One must think in order to read and write. Thinking, just as everything else, is supposed to follow certain rules, if we are to think correctly. Sometimes we make mistakes in thinking. We jump to conclusions; we make unwarranted assumptions; we generalize. There is a subject that catalogues these mistakes, points them out so that we can recognize them in the future, and then explains the rules for avoiding mistakes. That subject is logic.

Logic is not psychology. It does not describe what people think about or how they usually reach conclusions; it describes how they ought to think if they wish to reason correctly. It is more like arithmetic than history, for it explains the rules one must follow in order to reach correct conclusions, just as arithmetic explains the rules one must follow to arrive at correct answers.

Logic concerns all thought; it is fundamental to all disciplines, from agriculture to astronautics. There are not several kinds of logic, one for philosophy and one for religion; but the same rules of thought that apply in politics, for example, apply also in chemistry. Some people have tried to deny that logic applies to all subjects, for they wish to reserve some special field—theology and economics, to name two historical examples—as a sanctuary for illogical arguments. What results is called polylogism—many logics—which is really a denial of logic.

But in order to say that there are many different sorts of logic, one must use the rules of the logic there is. Let those who say there is another kind of logic express their views using that other logic. It's as though one were to claim that there are two (or more) sorts of arithmetic—the arithmetic in which two plus two equals four, and a second arithmetic in which two plus two equals twenty-two.¹ Anyone who disparages or belittles logic must use logic in his attack, thus undercutting his own argument. This can perhaps be better seen by specifically discussing one of the laws of logic.

The first law of logic is called the law of contradiction, but recently some people have begun to call it the law of non- contradiction—the two phrases refer to the same law. Aristotle expressed the law in these words: "The same attribute cannot at the same time belong and not belong to the same subject and in the same respect." The law is expressed symbolically as: "Not both A and not-A." A maple leaf may be both green and not-green (yellow), but it cannot be both green and yellow at the same time and in the same respect—it is green in the summer, yellow in the fall. If it is green and yellow at the same time, it cannot be green and yellow in the same respect; one part, however small, will be green, another yellow. Greenness and not-greenness cannot at the same time and in the same way belong to a maple leaf.

To suggest another example: A line many be both curved and straight, but not in the same respect. One portion of it may be curved, another portion straight, but the same portion cannot be both curved and straight.

The law of contradiction means something more. It means that every word in the sentence "The line is straight" has a specific meaning. The does not mean any, all, or no. Line does not mean dog, dandelion, or doughnut. Is does not mean is not.

Straight does not mean white, or anything else. Each word has a definite meaning. In order to have a definite meaning, a word must not only mean something, it must also not mean some thing. Line means line, but it also does not mean not-line—dog, sunrise, or Jerusalem.

If line were to mean everything, it would mean nothing; and no one, including you, would have the foggiest idea what you mean when you say the word line. The law of contradiction means that each word, to have a meaning, must also not mean something.

What do this law and the rest of logic have to do with morality? Simply this: When the Bible says, You shall not covet, each word has a specific meaning. Attacking logic means attacking morality. If logic is disdained, then the distinctions between right and wrong, good and evil, just and unjust, merciful and ruthless also disappear. Without logic, God's words, "You shall do no murder," really mean: "You shall murder daily" or "Stalin was Prince of Wales." The rejection of logic means the end of morality, for morality and ethics depend on understanding. Without understanding, there can be no morality. One must understand the Ten Commandments before one can obey them. If logic is irrelevant or irreligious, moral behavior is impossible, and the "practical" religion of those who belittle logic cannot be practiced at all.

Something even worse, if anything could be worse, follows from rejecting logic. If logic does not govern all thought and expression, then one cannot tell true from false. If one rejects logic, then when the Bible says that Jesus suffered under Pontius Pilate, was crucified, dead, and buried, and rose again the third day, these words actually mean that Jesus did not suffer, was not crucified, did not die, was not buried, and did not rise again, as well as that Attila the Hun loved chocolate cake and played golf. The distinctions between true and false, right and wrong, all disappear, for there can be no distinctions made apart from using the law of contradiction.

The rejection of logic has become very popular in the 20th century. In matters of morality, one frequently hears that "There are no blacks and whites, only shades of gray." What this means is that there is no good or evil; all actions and alternatives are mixtures of good and evil. If one abandons logic, as many people in this century have, then one cannot distinguish good from evil—and everything is permitted. The results of this rejection of logic—mass murder, war, government-caused famine, abortion, child abuse, destruction of families, crime of all sorts—are all around us. The rejection of logic has led—and must lead—to the abandonment of morality.

In matters of knowledge, we're told that truth is relative, what's "true" for you might not be "true" for me. So 2 plus 2 might be 4 for you and 6.7 for me. If logic is abandoned, then that also follows. Christianity is "true" for some—Buddhism is "true" for others. One result has been a growing antipathy toward Christianity, which claims that all men, not some, are sinners; and that there is only one way to God, through belief in Christ. Absolute truth—which is really a redundant phrase— has been replaced by relative truth, which is really a contradiction in terms, like square circle. But once logic is gone, truth is also.

The use of logic is not optional. Logic is so fundamental, so basic, that those who attack it must use logic in order to attack logic. They intend the words they write, "Logic is invalid," to have specific meanings. The opponents of logic must use the law of contradiction in order to denounce it. They must assume its legitimacy in order to declare it illegitimate. They must assume its truth, in order to declare it false. They must present arguments if they wish to persuade us that argumentation is invalid. Wherever they turn, they are boxed in. They cannot assault the object of their hatred without using it in the assault. They are in the position of the Roman soldier who arrested Christ, but they do not realize, as the soldier did, that their position and action are dependent upon rules that they reject. They must use the rules of logic in order to belittle logic; he had to be healed by Christ before he could proceed with the arrest.

In the first chapter of the Gospel of John, John wrote, "In the beginning was the Logos, and the Logos was with God, and the Logos was God." The Greek word Logos is usually translated Word, but it is better translated Wisdom or Logic. Our English word logic comes from this Greek word logos. John was calling Christ the Wisdom or Logic of God. In verse nine, referring again to Christ, he says that Christ is "The true Light" who lights every man that comes into the world. Christ, the Logic of God, lights every man. Strictly speaking, there is no "mere human logic" as contrasted with a divine logic, as some would have us believe. The Logic of God lights every man; human logic is the image of God. God and man think the same way—not exactly the same thoughts, since man is sinful and God is holy, but both God and man think that two plus two is four and that A cannot be not-A. Both God and Christians think that only the substitutionary death of Christ can merit a sinner's entrance into heaven. The laws of logic are the way God thinks. He makes no mistakes, draws no unwarranted conclusions, constructs no invalid arguments. We do, and that is one of the reasons why we are commanded by the Paul to bring all our thoughts into captivity to Christ. We ought to think as Christ does—logically.

Why Study Logic?

To return to our first question, Why study logic? Our first answer must be that we are commanded to by Scripture. Without learning how to think properly, we shall misunderstand Scripture. Peter warns against those who twist the Scriptures to their own destruction. A study of logic will help us avoid twisting the Scripture and trying to make it imply something it does not imply. The Westminster Confession, written in England in 1648, says that all things necessary for our faith and life are either expressly set down in Scripture or may be deduced by good and necessary consequence from Scripture. It is only through a study of logic that we can distinguish a valid deduction from an invalid deduction.

But logic is indispensable not only in reading the Bible, but also in reading history, botany, or computer programs. It is applicable to all thought, and mistaken arguments may be found in every subject. The study of logic will help us understand all other subjects better, not just theology. Therefore, as God said through the prophet Isaiah, Come, let us reason together.

                                                                                       December 26, 1984

¹Do not be confused by different bases in arithmetic. I am speaking of ideas, not words.

                               Chapter 1
                              The Definition of Logic

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                                              Chapter 2
                                    Informal Fallacies

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                               Chapter 3
                                           Definition

                  Click HERE to buy the book

==============================================================

                                 Chapter 4
                        The Beginning of Formal Logic

The arguments in the chapter on Informal Fallacies were all somewhat complicated. If we are now to formulate a method for testing the validity of all arguments, in respect of their logic if not of their English, we must begin with the very simplest form of argument. The word form indicates that we shall pay no attention to the infinitely different subject matters of arguments, but rather consider their forms alone. Instead of saying, All men are mortal, we shall say, All a is b. The letter a stands for any subject; and the letter b stands for any predicate. All a is b is the first form in formal logic.

The reason it is possible to construct rules of validity for all inferences is that the forms of assertion are very limited in number. Take all the declarative sentences in the language, and you will find that there are only four types. The first form is, All a is b. It stands for All dogs are canines, All storks have two legs, and All revolutionaries risk hanging.

The second form—they are called categorical forms for no sufficient reason—is, No a is b. It stands for No dogs are cats, No Christian is a secularist, and No cooks are perfect. The third form is, Some a is b. Some dogs are pets, and so on. The fourth form is, Some a is not b. Some dogs are not pets. Every simple declarative sentence can be put into one of these four forms.

Incidentally, most logic books do not talk about declarative sentences. They talk about propositions. There is a difference between propositions and declarative sentences. In English one may say, The kick-off was caught by the fullback; or he may say, The fullback caught the kick-off. These two are two different sentences. The subject and predicate are interchanged, and the voices of the verbs are different. But they mean the same thing. A proposition therefore is defined as the meaning of a declarative sentence. Some sentences are not declarative, such as commands in the imperative mood, or exhortations in the well-nigh extinct subjunctive mood. Questions, or interrogative sentences, also are neither true nor false. Only declarative sentences are true or false; and it is this common character that is important for propositions. Of course in English rhetoric there are questions that are intended as propositions. They are called rhetorical questions. They are an embellishment of style. They spruce up a speech. But logically they are propositions. A question that is intended as a question is neither true nor false. It can play no part in an argument.

Let us now return—an exhortation, neither true nor false, but one which it is hoped that the student will follow—to the simplest propositions and the simplest form of arguments. Some additional modifications are necessary to reduce propositions to manageable logical form.

In order that logic be as simple as possible, it does not use the verbs of ordinary conversation. Instead of saying, All the track men run well, logic says, All trackmen are good-runners. Instead of saying, No dogs eat hay, logic says, No dogs are vegetarians. The only verb in logic is the verb to be, the copula, is or are. Premises and conclusions therefore consist of subject-copula-predicate, plus whatever relationship is needed, an All, No, Some, or Some...not.

Now, for practice sake, the student should try to cast some ordinary English sentences into categorical form. He may have been surprised that declarative sentences have only four forms. Only four forms in all the books in the library! He also needs practice if he wishes to analyze ordinary arguments. It is easy enough to change English verbs into predicate adjectives, or into somewhat awkward phrases. For example, "children run to school" becomes "All children are runners-to-school." Of course some children may not run. Without the All the English sentence is ambiguous. Does it mean all children run, or some children run? If the latter, the categorical form will be "Some children are runners-to-school." The awkwardness of the English is no difficulty in analyzing arguments. It sounds stupid, but the sense remains clear. When the English sentence contains dependent clauses, relative pronouns, and prepositional phrases, the categorical translation will be extremely awkward; but with hyphens, or by putting phrases between parentheses to make them look like one word, the sense is clear. For example, All (those who have been born in the United States and are at least thirty-five years old) are (legally qualified to run for the office of president). Perhaps now the value of a simple All a is b has become evident.

But there are other difficulties in producing categorical forms out of English sentences besides these awkward expressions. Can the student put this in categorical form? Only good students get A's. Does it mean that all good students get A's? No, for some able students goof off and flunk, or at least get only C. What it means is, All students who get A's are good students. Now, this statement may be false, for some bad students may get an A by mistake or by cheating; but the translation given here is the correct translation. Similarly, "None but the brave deserve the fair" can first be translated into, "Only the brave deserve the fair." Then this becomes, "All those who deserve the fair are brave."

Even logic textbooks make mistakes. One author used the sentence, "An elephant has escaped." The zoo or the circus wanted to give the alarm in good categorical form. The author then translated it, "Some elephants are creatures-that- have-escaped." But this is not really what the zoo keeper meant. He meant "All Jumbo is an escaped-animal." To be sure, the author was not completely wrong, for it is true, under this condition, that Some (an undetermined number) elephants are escapees. In logic some can mean one as well as many or few.

However, when the main idea is certainly one, such as Socrates, the logical form requires All. Socrates was in a class by himself, and so we talk about all that class. We surely do not mean "Some Socrateses."

Other English expressions are harder to manage. For example, business firms for advertising purposes may run a contest in which their employees may not participate. The language is: All except employees may enter. Now, this sentence is not hard to understand, but its use in an argument presents some traps. The trouble is that this apparently simple English sentence is two propositions. It means (1) No employees are eligible, and (2) all non-employees are eligible. Now neither of these two propositions implies the other, for not only may employees be ineligible, but others, non-employees, living in the states where contests are illegal or taxed, may also be ineligible. And also, but not usually, the proposition "All non- employees are eligible" does not of itself imply that employees are barred. Now, what may happen is this. The sentence with the two meanings is taken as a premise, and with some other proposition introduces a conclusion. Then someone who takes the major premise in only one sense may decide that the syllogism is invalid, although it is valid by reason of the premise's other meaning. Or, in the opposite direction, the sentence with two meanings may be taken as a conclusion; in which case someone might consider the argument valid because one meaning (he has missed the other meaning) validly follows. If the premises are both simple categorical propositions, no double meaning conclusion can be validly drawn.

There are other English sentences that cause difficulties. Suppose someone exclaims, either in admiration or disgust, "You always squirm out of an argument." The grammatical subject is You, but it is not the logical subject. Clearly "You" is not squirming out of an argument twenty-four hours a day. And of course the categorical form cannot be "Some You." The word always is the logical subject, though not in its literal sense. To get the logical form one must take the always to mean "every time you get into an argument." Hence the categorical statement will be. "All times you get into an argument are times when you squirm out of it." A proposition is the meaning of a declarative sentence, and colloquial English must be studied to ascertain what that meaning is. Then we can put it into categorical form.

Students often take great pleasure in figuring out puzzles when they occur as parlor games or as challenges in self-improvement tests in popular periodicals. But when it comes to class work, they are not usually willing to look up additional trouble. Hence here are a few examples to which the student ought to add others; but of course he probably will not. How ever, can the student state the meaning of the following sentences?

Only freshmen need use the back door.

The poor always ye have with you.

Except the Lord build the house, they labor in vain who build it.

When you understand what these sentences mean, you can easily put them in correct though awkward categorical form. If you cannot put them in categorical form, you do not know what they mean.

Accordingly, all propositions are of the form of All, No, Some, or Some is not. The simplest inference, then, must be an inference that has one proposition for a premise and one proposition for a conclusion. To be sure, very few inferences in ordinary conversation are so simple; but the student must learn about them because more complicated inferences are built on this simple foundation. The problem therefore is to discover how many such inferences there are, and which of them are valid. Such a set of inferences is called immediate inference because there is no middle term. "Middle term" will be defined a little later. At any rate immediate inferences are composed of two propositions with two terms. They cannot have three terms, for clearly it is nonsense to infer that some snakes are poisonous on the ground that some poisons are minerals.

Now, if we wish to test the validity of an inference, we must first know what the term validity means. We want no equivocation in logic. Strange as it may seem, it is best, in order to prepare for the accurate definition, to give an inaccurate definition first. So we say, An inference is valid, if the conclusion is true every time the premises are.

This inaccurate definition points in the right direction; but as stated it is pretty bad. For example, it would declare valid the following obviously invalid nonsense: George Washington was our first president; therefore, roses and apple trees belong to the same botanical family. Surely the conclusion is true as often as the premise, for they are both always true. But the one does not imply the other.

There is a second and more surprising reason why the definition is bad. The example just given consists of a true premise, a true conclusion, but a wild inference. But what about false premises? Can they validly imply a true conclusion? A false conclusion? Any conclusion at all? Try this example: All Presidents of the United States have been Roman Catholics. Obviously false. Cardinal Cushing was a true Catholic. Obviously true. But not merely obviously true. This true conclusion also follows validly from two false premises. How can that be? Can falsity imply truth? It surely can. But although the bad definition can defend itself by saying that the conclusion, being true, is true as many times as the premises—for it need be true only once to be true more times than the false premises—this does not explain how we count the "times" a premise or conclusion is true. A true statement is always true. It is not true three times and false five. Hence the bad definition must be amended. It served only to show that there is some relation between the truth of the conclusion and the truth of the premises.

When, however, we consider the forms of inference, the absurdities vanish. The correct definition will be, An inference is valid whenever the form of the conclusion is true every time the forms of the premises are.

It is now required to show how many times these forms are true. To do so, we must consider in how many ways two terms may be related, whether they be dogs, liberal theologians, or rose bushes. There are five such possibilities. First, all a can be b and at the same time all b can be a: that is, a and b are coextensive, even if different English words are used. Second, all a can be b, but not all b will be a. For example, all candy is sweet, but not all things sweet are candy. Third, some of the a's but not all of them, can be b's, and at the same time some of the b's but not all of them can be a's. Some books are interesting, but not all are; for example, logic textbooks. And furthermore not all interesting things are books. Fourth, all of the b's may be a's, but not all of the a's are b's.. Then, fifth, none of the a's are b's. No cat is a dog, and conversely.

These five relationships among any two possible terms do not correspond to the four categorical forms in a one to one relationship. But there is a very definite relationship. The following diagrams, invented by the mathematician Euler, will show how many times a form can be true.

The student will now make an accurate copy of this great work of art, hang it over his bed, and gaze on it in rapture every night.

He must note that in two of these five All a is b is true. In only one is No a is b true. Some a is b is true four times; and Some a is not b is true three times. An inference will be valid if the form of the conclusion is true every time the form of premise is.

Underneath the circles four lines have been drawn and labeled A, E, I, O. These letters stand respectively for All, No, Some, and Some is not. A(ab), All a is b, is true twice. O(ab), Some a is not b, is true three times.

Now, these four capital letters are not just arbitrary letters of the alphabet. They come from the Latin verbs Affirmo and Nego. The forms A and I are affirmative forms; the forms E and O are negative forms.

We could now start to count the number of possible

immediate inferences and test their validity. For example, I(ab), Some a is b, is true every time A(ab) is true. I(ab) is true in the first four diagrams and hence it must be true in the first two. Therefore A(ab) and I(ab) make a valid inference. But though we could now proceed to test all immediate inferences, it may be well to defend the legitimacy of this definition of validity against some opposing views. If this defense becomes too complex, the student may skip the next ten paragraphs.

Many contemporary logicians say that there are four distinct types of validity or implications. They hold that the words if...then are ambiguous. For example, one logician gives this list:

1. If all men are mortal and Socrates is a man, then Socrates is mortal.

2. If Mr. Black is a bachelor, then Mr. Black is unmarried.

3. If blue litmus paper is placed in acid, then the litmus paper will turn red.

4. If State loses the Homecoming Game, then I'll eat my hat.

Now, it may be that "if...then" has several English uses, and it may be that example four is not an implication at all. But it could be, even so. The argument would be, The loss of a homecoming game is something that disturbs me very much; I am such a peculiar person that when I am greatly disturbed, I eat my hat; therefore, etc. Or it could be understood as a rash prediction, similar in form to one about Hitler and a monkey's uncle, explained later on. The third example can be an enthymeme and when its chemical omissions are made explicit, it will also be a valid implication. In example two, the fact that bachelor and unmarried man are synonyms does not make the example any less a valid inference. Trivial no doubt, but still valid. Of course example one is standard.

Nevertheless the author of these examples immediately asserts, "Even a casual inspection of these four conditional statements reveals that they are of quite different types." That is the trouble: The inspection has been too casual. This leads him to invent more than one type of implication. He does not say merely that the words "if...then" sometimes do not indicate any implication at all; but rather he says "The four conditional statements...are different in that each asserts a different type of implication."

Not only so: The author goes on to introduce a fifth type. He says, "Not all conditional statements in English need assert one of the four types of implication previously considered. Material implication constitutes a fifth type," and he proceeds with Hitler and the monkey's uncle. He even gives the correct interpretation of the example. But what he fails to see is that implication and validity are each identical in every argument.

This fifth type most contemporary logicians call "material implication." But then he adds, "In translating conditional statements into our symbolism we treat them all as merely material implications." This is an admission that logic does not need, in fact cannot use, five types. One is enough. Another author, after saying much the same things, concludes, "Briefly, material implication works and works well."

Another logician makes it a little clearer. Examples one and five will suffice. One: If the weather remains warm, we shall have a picnic next Saturday. Five: If anything is a horse, then it is an animal. The first of these, he says, "has an antecedent which implies as a matter of fact the particular consequent connected with it. There is nothing in the antecedent itself which necessarily entails the consequent." In proposition five "the consequent is warranted as directly implied in the meaning of the given antecedent... - The term 'horse' includes the meaning of 'animal.' "

No doubt this is so; but it furnishes no reason for listing two or four different types of valid implication. The meaning of the English word inference is very broad. It includes the wildest guesses. Many people have inferred that next winter will be unusually cold because the caterpillars this autumn are more fuzzy than normal. Though this is a psychological inference, it is not a valid implication. Even so, it could be a valid implication if to it were added the premise "Every time the caterpillars are fuzzy in September, the following winter is rough." So also the picnic on Saturday. One might say, We have decided to go on a picnic next Saturday if the weather permits. Therefore, if Saturday is clear and warm, we shall have our picnic. All this is common English. It is not a great logical discovery of a new type of validity.

We insist therefore that an inference is valid if the form of the conclusion is true every time the forms of the premises are; and that this definition is sufficient for all logical purposes.

Perhaps something should be added relative to the supposedly absurd implication: If Hitler is a military genius, I am a monkey's uncle. This is not really a queer and different type of implication. As an argument it is an enthymeme and is perfectly valid. Expand the English and it becomes: If Hitler is a military genius, I am a monkey's uncle; but I am not a monkey's uncle, therefore Hitler is not a military genius. This is a form of the hypothetical syllogism, called modus tollens, later to be explained, and is perfectly valid. Of course a person using this type of argument may make false statements and historical blunders, but logic is not history. The validity of an argument does not guarantee the truth of any of its propositions. It is valid if the form of the conclusion is true every time the forms of the premises are.

Very likely the confusion in these authors is due to the fact that they are thinking in terms of the incorrect definition of validity, given above. They are thinking in terms of true and false propositions. All this confusion disappears when we say, An inference is valid if the form of the conclusion is true every time the forms of the premises are.

We now return to further study of the forms themselves. It may surprise the student how much is to be learned about such a seemingly simple statement as All a is b. The next thing therefore is to ascertain what we mean by saying that All a is b is an affirmative proposition.

Someone is sure to say that an affirmative proposition is one that affirms, and a negative proposition is one that negates. Do you see why that would be useless? Its defect is that it defines a term by itself. In a definition the term to be defined must not occur. How then can affirmation be defined without using the word affirm or a synonym of it?

It can be done by distinguishing between a distributed and an undistributed term. A distributed term is one modified by the adjective all or no. An undistributed term is one that is not so modified. The all and the no are often explicitly written out in the proposition; but sometimes they are hidden or implicit. One sees right off that the subject of the first form, the little a after the capital A, is a distributed term. By looking at the second diagram one will see that no statement is made about b as a whole. It is therefore undistributed. In the first diagram it is possible to make a statement about all b: All b is a. But for b to be distributed, it must be modified by all in every applicable diagram. Hence the subject of A is distributed and its predicate is not. One will also note that the predicate of I is undistributed. It is impossible to make a statement about all b that holds in the first four diagrams. Obviously, if the b in the second diagram is undistributed, the b cannot be distributed in every one of the first four. Now then, we have our definition: An affirmative form is one that does not distribute its predicate.

The student can now easily guess that a negative form is one that distributes its predicate. But to see this in the diagrams is not so easy. Of course, since No a is b means precisely what No b is a means—no cats are dogs and no dogs are cats mean the same thing—it is clear that both terms in E are distributed. From No a is b it is possible to make a statement about all b: All b's are non-dogs.

But the case of 0 is more difficult. The problem is to make a statement about all b that will be true each of the three times that O(ab), Some a is not b, is true. If some books are not interesting, can you make a true statement about all interesting things? First look at these two diagrams. The third or last diagram is not needed here, since the previous paragraph took care of it.

These two diagrams are numbers three and four of Euler's set of five. Here they have been shaded so that the shaded portion is the part of a that is not b. In the two diagrams there is some a that is not b. Call this "some a." Then it can be seen that All b is non-some a. Or more clearly, All b is unshaded. Hence it is possible to make a statement about all b, from which it follows that 0 is a negative form. English examples are not frequently encountered because there are few English words to correspond to non-some a; but if we put ourselves imaginatively back into the year 1860 we may use this one: Some United States citizens are not northerners means that all Yankees are non-southerners. The diagram would be:

Even if we use the present extent of the United States, and refuse to call Hawaii and Alaska either Yankee or southern, the same result is seen in this slightly more complicated diagram.

Accordingly, the predicate of O is always distributed, and O is a negative form.

In addition to classifying the categorical forms as affirmative and negative, they must also be given the cross-classification of universal and particular. A and E are universal, because we define a universal form as one that distributes its subject. I and O are particular because they do not distribute their subjects.

The explanation of these terms may seem tedious. But logic begins with what is simple and easy, and builds up to many complexities.

This is mainly a chapter of definitions—definitions by which we become familiar with the characteristics of the simple categorical forms. It is therefore the appropriate place, before we count the number of immediate inferences and determine which of them are valid, to explain three other features of A, E, I, and O. Illustrations can be had from these other things; but eventually it is the forms that interest us most.

Some relationships are reflexive. A reflexive relationship is one that holds between one of its objects and that object itself. For example, equality in arithmetic is a reflexive relationship because two equals two. In logic, implication is a reflexive relationship because any proposition implies itself. The relationship "is less than" is not reflexive because two is not less than two.

A symmetrical relationship is one which, if it holds between two of its objects, a and b, also holds between b and a. In family affairs cousin is a symmetrical relationship, for if John is the cousin of Mary, Mary is the cousin of John. Sonship is not symmetrical because if John is the father of Frederick, Fred is not the father of John. Consider, now: Is "the brother of" symmetrical? If a is the brother of b, is b necessarily the brother of a? Good, you figured it out! B may be a's sister. In geometry "is parallel to" and "is perpendicular to" are symmetrical, but in time "is subsequent to" is not symmetrical.

A relationship is transitive, if, when it holds between two of its objects, a and b, and also holds between b and c, it holds as well between a and c. If line a is parallel to b, and b is parallel to c, a is parallel to c. If moment x is subsequent to moment y, and y is subsequent to z, then x is subsequent to z. Now, if John is the first cousin of Mary, and Mary is the first cousin of James, is John the first cousin of James? Or, again, if a is the brother of b, and b is the brother of c, is a necessarily the brother of c? Be careful, now.

Some relationships are none of the three types. Some have two or even all three characteristics.

If, now, the student has completed his genealogical studies, and knows how many grandfathers he and his cousin have, he may now return to formal logic.

The relationship of implication is not symmetrical because if x implies yy, that is, if y is true every time x is true, this does not guarantee that x is true every time y is. If, as we shall see, All a is b implies that some a is b, it does not follow that if some books are interesting, all must be. So, implication is not symmetrical. We saw just above that it is reflexive. It is also transitive, for if x implies y, and y implies z, then x implies z. Transitivity becomes very important in the construction of the syllogism.

The relationship "All is," is reflexive because all a is a. It has to be. Obviously. But be careful: The phrase business is business does not always mean what it says. It usually means that shady practices are excusable in business. The term business in this phrase is equivocal. Lewis Carroll, the author of Alice in Wonderland, met a gourmand in one of his poems, who defended his overeating by the assertion that Dinner is Dinner, and Tea is Tea. Lewis Carroll deflated (well that is perhaps not the most literal term to use) the gourmand by replying

Wherefore cease;

Let thy scant knowlege find increase:

Say men are men and geese are geese.

The three relationships now explained are not the only ones that are important for the categorical forms. There are four others, and we cannot do without them. The four are contradiction, contrariety, subalternation, and subcontrariety.

As was said a few paragraphs ago, the student may well be amazed at how complicated a simple form like All a is b, is.

Here are the definitions, and they have been pictured by the lines drawn under Euler's diagrams. It was indeed a great work of art.

Two forms, or two propositions, are contradictory if they cannot both be true and cannot both be false. Since the lines under A and O meet without overlapping, that is, they exhaust all five possibilities, they are contradictories.

Two forms are contrary if they cannot both be true but may both be false. The lines under E and A do not overlap, which means that they cannot both be true in any instance; and since they do not exhaust all five possibilities, they may both be false in a given instance. In the instance that some books are interesting, pictured in the third diagram, both A and E are false. They are contraries.

Subalterns are two forms that may both be true and may both be false. A and I are subalterns. E and O are subalterns. You can put your finger on a diagram where both are true; and you can put your finger on a diagram where both are false.

Subcontraries are forms that cannot both be false, but can both be true. O and I are subcontraries, for the two lines both overlap each other and exhaust the five diagrams.

This diagram is called the square of opposition.

The student should familiarize himself with these relationships by constructing numerous examples. After so doing, he can try to answer the following question. But, watch out, it is tricky.

Suppose the debate team wants to destroy its opponents' argument. To do so, it must prove its own argument. But what should this argument be? Suppose also that so far as the subject matter goes (though this is not always the case) it is just as easy to prove the contradictory of the opponents' position as it is to prove the contrary of the opponents' position. No more research is required in the one case than in the other. Now, the question is, Which of the two, the contradictory or the contrary, most effectively administers defeat?

This chapter has now fairly well exhausted what must be said of each of the four forms singly. At the start, the student could hardly have guessed that so much could be said. People who have never studied logic never guess it. Because they are not aware of and alert to all these possibilities, they make the craziest mistakes in argument. Even when they know concretely that All Yankees are Americans does not prove that All Americans are Yankees, they still make this very blunder in less well known subject matters. You may not believe it, but it is true: In fifty years of college teaching at least once a year, and often once a semester, some college student has committed this blooper. Even a recent theological book contained the assertion: "If a proposition be true, its converse must also be true." Pardon me if I do not give the author's name. He is a friend of mine; but unfortunately he was never a student of mine. I hope that none of the students who look through this book will ever be so irrational.

It is now time to consider categorical forms in combination.

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                               Chapter 5
                                 Immediate Inference

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                               Chapter 6
                                The Syllogism--Diagrams

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                               Chapter 7
                  The Syllogism--Deduction and Rules

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                               Chapter 8
                                 Historical Remarks

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                               Chapter 9
                              Other Forms of Argument

1. Sorites

2. Hypothetical Syllogism

3. Dilemma

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                               Chapter 10
                                        Truth Tables

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                               Chapter 11
                         The Deduction of the Syllogism

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                               Glossary

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End of Book

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