- Nance, Jim 1996. The Why and How of Logic. In Repairing the Ruins: The Classical and Christian Challenge to Modern Education, ed. Douglas J. Wilson, 117-132. Moscow, ID: Canon Press. (All rights reserved. Reproduced by permission from Canon Press.

What First

BEFORE DISCUSSING THE WHY AND HOW OF TEACHING LOGIC IN THE secondary school, we must first discuss the what. Read the introduction to ten different logic textbooks, and you will find ten different definitions of logic, all of them presenting a different perspective on this particular discipline.

Logic may be defined first as the science of the formal principles of reasoning. A science is a body of knowledge systematized from observation and analysis. Most sciences (namely, the hard sciences) observe the more tangible aspects of creation: chemistry observes the properties of substances and analyzes the changes they undergo as they interact with each other; biology is the study of living organisms; astronomy is the study of the heavens. From their observations, scientists seek to discover the natural laws by which God governs His creation. This is true whether or not they recognize the Governor.

Logic as a science observes the human mind as it reasons. In his book The Laws of Discursive Thought,1 James McCosh says, “The discursive operations proceed in a regular manner, that is, according to laws. By carefully observing the acts of the mind in thinking, we may discover what these laws are, and express them in language or formulae. In doing so, we are constructing a science.” For example, we can observe Paul’s proof that there is a resurrection of the dead: “If the dead do not rise, then Christ is not risen. And if Christ is not risen, your faith is futile, you are still in your sins! . . . But now Christ is risen from the dead.” (1 Cor. 15:16-20). We observe the Apostle applying a certain law: If Christ is not risen, your faith is futile; you will not admit that your faith is futile, so you must admit that Christ is risen. We can express this law as a formula, If P then Q , not Q , therefore not P. We call this law modus tollens, and recognize it as the same reasoning as: If division by zero is allowed, then you can prove that one equals two; one does not equal two, so division by zero is not allowed.

Logic also observes the mind as it recognizes fallacious reasoning. Consider the argument,2 “All students have two legs. All gorillas have two legs. Therefore all students are gorillas.” Now, you recognize that this is funny because you know that it doesn’t follow. God has created us with the ability to distinguish between correct and incorrect reasoning. Logic, among other things, studies the laws by which we do this. Specifically, it seeks to discover the laws which may be used to distinguish good reasoning from poor reasoning. Logicians try to answer questions like, “Does the conclusion follow from the premises? Is the argument sound?” They try to discover the rules which can help us answer those questions.

But if logic is viewed only as a science, it may be interesting to people with an analytical twist to their psyche, but it is not very practical. We must teach logic not only as a science but also as an art. We must teach the skills of logic.

Mortimer Adler, in his essay What is Basic About English? Defines logic as the art of ordering what is to be expressed in language or of judging what has been expressed. When you try to communicate anything in the form of an argument, logic as an art will describe the order you must follow to guarantee that your argument is valid. Logic as a science discovers rules. Logic as an art teaches us to apply those rules in different situations. The art of logic provides us with practical skills to use as we reason, discuss, debate, or communicate in all ways. It also provides us with rules by which we may judge the arguments of others, to determine whether their reasoning is good or poor.

God has made us such that we reason by means of language. Logic represents the reasoning inherent in language, breaking the language of arguments down into symbolic form (much like story problems in mathematics3), simplifying it such that the reasoning inherent in language becomes visible. Thus the structure of arguments is clarified by removing every possible extraneous part, like a biology student dissecting a frog to see the structure inside. By studying the structure of arguments in this way, a separate, symbolic language is created, a language which has its own particular syntax and rules. As a symbolic language, logic is useful only insofar as these rules accurately reflect the reasoning of the language it represents. Logic must deal with the vagueness and ambiguity of languages such as English. Different forms of logic deal with them in different ways.

Branches on the Tree of Logic

Given these definitions of logic, we should briefly consider the divisions within it. First, I would divide logic into two main branches: informal and formal.

Informal logic may include a wide range of subjects, such as common sense reasoning, defining terms, informal fallacies, analogies, and puzzle solving. A course in formal logic should have some informal logic scattered throughout to add interest and applicability. Formal logic, on the other hand, analyzes structured arguments in a more rigorous, limited way.

Formal logic itself may be divided into inductive and deductive branches. Inductive logic deals with arguments of likelihood and probability. It makes conclusions from specific facts or experience. The conclusions of inductive arguments go beyond the premises, but these conclusions can (at least in principle) be tested by further observation. Inductive reasoning is the logic of the experimental sciences: “On ten different occasions I have heated water at different altitudes and found that, the higher the altitude, the lower the boiling point. Thus, all other things being equal, water will always boil at lower temperatures in higher altitudes.” Inductive arguments are either strong or weak, depending on how well the evidence supports the conclusions.

Deductive arguments, however, are either valid or invalid. If valid, the conclusions necessarily follow from the premises. In valid, deductive arguments, if the premises are true, the conclusions must be true. Deductive logic itself breaks down into many branches, such as categorical and propositional.

Categorical logic (also called syllogistic logic) deals with syllogisms, which are arguments having two premises and one conclusion. The classic example is, “All men are mortal. Socrates is a man. Therefore Socrates is mortal.” Such arguments are analyzed and proven valid or invalid by various rules, Venn diagrams, and counter-examples. In categorical logic, individual words are represented by symbols, usually capital letters. The argument “No Hindus are Christians, but some men are Hindus; Therefore some men are not Christians” would be symbolized “No H are C, Some M are H, Therefore some M are not C.” Many arguments can be put into categorical form and analyzed for their validity.

In propositional logic (also called symbolic logic), entire sentences or propositions are represented by symbols, along with logical operators such as AND, OR, NOT, and IF/THEN. Paul’s argument from First Corinthians 15 was a propositional argument. Such arguments are analyzed for their validity using truth tables, among other means. Much of digital electronics includes the study of propositional logic.

The Value of Logic

As we consider the value of studying (and teaching) logic, I must make one observation. Logic, as we shall see, is as much a part of our thinking as grammar. Yet I have noticed, usually in the introduction to logic textbooks, an almost universally recognized need to defend the teaching of logic in schools. Rarely do I hear a similar defense of teaching English grammar, or read such in introductions to English grammar textbooks. Given the parallels between logic and grammar, I do not understand why this is. Everyone who thinks and speaks uses both. The study of both is an aid to improved speaking and thinking.

Some may assert that if we use logic naturally, without being taught it formally, we have no need to study it. But is this not true of grammar? All children use grammar before they are taught it in English classes. Why study grammar? The answer, of course, is that we study grammar to recognize proper grammar and to correct improper grammar. Even so, we should study logic to the praise of proper logic and the rectification of improper logic.

Perhaps one reason formal logic has fallen into some disrepute is that historically some philosophers and logic teachers have become enamored with “pure” logic, emphasizing logic as a science and neglecting it as an art. As a result, administrators, teachers, parents and students no longer saw it as practical or useful.

Others may put logic into the same basket with sophistry, believing it to be used as a tool to “prove” error. I agree that this has indeed been attempted. But in such cases logic is being abused, not used, and the abuse of logic is no more a reason to reject its proper study than the abuse of drugs is a reason to neglect the study of medicine. Deductive logic, used correctly, does not say anything beyond the premises. Rather, it simply unfolds in a more explicit way what the premises already contain.

Still, “Why study logic?” is an important question to ask. It will certainly be asked by our students and their parents. Implementing logic into a curriculum is not a trivial task. It requires the hiring (or training) of a logic teacher. It takes the place of some other class the teaching of which could be defended. But even more importantly, this question must be asked and answered because as teachers or administrators we will be held accountable to God for what we teach in our schools.

In her essay The Lost Tools of Learning,4 Dorothy Sayers begins to answer the question in this way: “Neglect of formal logic in the curriculum is the root cause of nearly all those disquieting symptoms we have noted in the modern intellectual constitution.” The symptoms she lists include the inability of students to do the following: 1) resist propaganda, 2) argue well, 3) follow a sustained argument given by someone else, 4) distinguish between scholarly and slipshod writing, and 5) learn on their own. Sayers argues that the re-introduction of formal logic into the secondary curriculum will help solve problems such as these. With that, let us look more closely at the reasons for studying formal logic in the secondary classroom.

Logic is Foundational to All Learning

No learning, and in fact no reasoning of any kind, takes place independently of logic. Logic is an inescapable concept. Suppose you are talking to someone who denies that he needs to use logic. “Logic is a human invention and would be best left forgotten in the archives of philosophy,” he asserts. If asked to defend this assertion without using logic, that is, without giving any reasons, he of course could not. The denial of logic is self-defeating. We cannot decide “not to use logic.” We can only decide whether we will use it well or use it poorly. God has made us in His image, as creatures who reason. If someone self-consciously refused to use logic, he would limit himself to either silence or nonsense. And even silence and nonsense are attempts to be consistent— which acknowledges the authority of logic.

Logic is foundational to all learning; it is the art by which all communication is ordered. Specifically, logic is one of the three liberal arts, along with grammar and rhetoric. These three arts are present in all communication, each depending upon and interconnected with the other two. Grammar without logic is meaningless words; logic without grammar is empty order. And both need rhetoric in order to be expressed.

Logic is inescapable. You can’t hide from it and you can’t kill it. You’d best make friends and get to know it well.

Logic is an Aid to Improved Reasoning

We must first note that the study of logic does not impart the ability to think any more than the study of grammar imparts the power of speech. You do not have to be trained in formal logic to be able to reason correctly any more than you need to constantly take medicine to be healthy—unless of course you are ill. And I believe that much modern thought is indeed ill and in need of a dose of formal logic. But given two men of otherwise equal capacity to reason, the one who has studied logic is more likely to reason correctly and skillfully, catching his own mistakes and the mistakes of others. This is true for many reasons.

One reason is simply that logic is intellectually rigorous. It expands the mind in the areas of argument, proof, and comparison. Logic makes students think in ways they have not thought before.

When taught as an art, logic provides students with specific skills in arguing properly and analyzing arguments critically. Students learn the importance of defining terms in debate and specific techniques for doing so. They improve their ability to express ideas more clearly and concisely. They practice forming valid arguments and discovering and refuting invalid ones. They learn to distinguish between premises and conclusions, both expressed and assumed. They learn to think before they speak.

I should point out that in teaching students logic, we are handing them loaded guns and training them in their use. This is good, except that initially all the guns are pointed in our direction. The teacher finds himself standing before a class of students who not only like to argue, but who now know how to argue well. Thus it quickly becomes apparent that all the teachers should also be trained in logic, and all the students should be trained to argue graciously. They should be taught to argue in a Christ-like way.

The study of logic also helps the students to distinguish between valid and invalid reasoning. Dorothy Sayers says, “Indeed, the practical utility of formal logic today lies not so much in the establishment of positive conclusions as in the prompt detection and exposure of invalid inference.”5 This ability to detect and expose invalidity becomes very evident once the students learn the names of the informal fallacies. Students delight in discovering equivocation in the letter to the editor, circular reasoning in the science magazine, and the false cause fallacy in history textbooks. The power to recognize and name popular fallacies protects the students from illegitimate, authoritative and emotional appeals so prevalent in mass media today. They can be confidently objective as they are taught to look at the facts, not at the speaker. They learn to think, “What he says sounds good, but is it valid? And if valid, are the premises true?”

Logic also prepares students to learn on their own. They improve their ability to read what others have written and listen to what others have said, and understand what they mean. They are taught to look for an author’s main points and come to terms with him in order to understand and evaluate his arguments.

Logic is an Aid to Understanding God and His Revelation

In his essay On Christian Doctrine, Augustine said, “The validity of logical sequences is not a thing devised by men, but is observed and noted by them that they may be able to learn and teach it; for it exists eternally in the reason of things, and has its origins with God.”6 Logic originates with God. It is an expression of His unchanging, orderly, truthful character. God Himself is logical in His thoughts, and our logic is valid insofar as it is a reflection of His.

Paul writes that “God is not a God of disorder but of peace” (1 Cor. 14:33). God is orderly; He has reasons for what He does. Order implies rationality; where there is no rationality, there is only randomness and chaos.

God’s word is truth (John 17:17), as opposed to falsehood. His word is noncontradictory. And God Himself is noncontradictory: He cannot lie (Heb. 6:18), and He does not deny Himself (2 Tim. 2:13). He does not act contrary to His promises. God is holy; there is nothing in Him which contradicts His perfections.

John Frame, in his book The Doctrine of the Knowledge of God, identifies these things for us, and then adds, “Does God, then, observe the law of noncontradiction? Not in the sense that the law is somehow higher than God Himself. Rather, God is Himself noncontradictory and is therefore Himself the criterion of logical consistency and implication. Logic is an attribute of God, as are justice, mercy, wisdom, and knowledge. As such, God is a model for us. We, as His image, are to imitate His truth, His promise keeping. Thus we, too, are to be noncontradictory.”7 Thus God is logic,8 just as God is love and God is light.

Logic is given by God for the purpose of ordering His revelation to us. The ability to reason is necessarily presupposed in every revelation. God has given us minds that use logic for the same reason He has given us eyes that see, in order that we might grasp His word. Charles Hodge writes in his Systematic Theology, “Revelation is the communication of truth to the mind. But the communication of truth supposes the capacity to receive it. Revelations cannot be made to brutes or to idiots. Truths, to be received as objects of faith, must be intellectually apprehended.” 9 Without logic we could not obey God’s commands, since they are given in the form of universal propositions: “All men must repent. I am a man, therefore I must repent.” A denial of logic is therefore disobedience and sin.

This is why the writers of the Westminster Confession wrote, “The whole counsel of God concerning all things necessary for His own glory, man’s salvation, faith and life, is either expressly set down in scripture or by good and necessary consequence may be deduced from scripture” (emphasis mine). In order to comprehend any doctrine, we must use logic, since God has communicated doctrine to man by means of language. But also, we must note that the Bible was not written like a confession of faith, and some nuggets of truth take some digging to get at. The truth that there is one God eternally existent in three Persons, though clearly contained in Scripture, is not contained in one place alone. The Trinity is a truth which requires a godly, submissive use of logic to see.

Logic Provides the Foundation for Other Disciplines

Many areas of study relate directly to logic, and thus the study of logic is the proper foundation for studying them.

First, logic is foundational to rhetoric, especially as logic is taught as an art. Many aspects of logic are carried over into rhetoric: the practice of analyzing arguments quickly and refuting them effectively; the ability to organize arguments in a mental outline; the use of different argument types such as reductio ad absurdum and a fortiori.

Logic is obviously at the core of philosophy, along with aesthetics, ethics, metaphysics, and epistemology. It is foundational to these disciplines, because it is used in studying them.

Other disciplines in which a grasp of deductive logic is essential include theology, mathematics, law, computer science, electronics, and education. Students of formal logic would have a good start in the study of these disciplines.

Logic Hits the Students Where They Are

In our school we teach logic as a full year, five days per week course in the eighth grade. This finds the students in the middle of what Dorothy Sayers calls the Pert stage, which is recognizable, she says,

so soon as the pupil shows himself disposed to Pertness and nterminable argument (or, as a school-master correspondent of
mine more elegantly puts it: “When the capacity for abstract hought begins to manifest itself). . . It is characterized by contradicting, answering back, liking to ‘catch people out’ (especially one’s elders) and the propounding of conundrums (especially the kind with a nasty verbal catch in them). Its nuisance value is extremely high.”10

Junior high students like to argue. Too often as teachers we want to stifle this impulse. “Don’t argue with me!” Instead, we ought to take their natural argumentativeness and mold it to godly use. If done correctly, this does not teach them that to disagree is automatically good. As Douglas Wilson wrote in Recovering the Lost Tools of Learning, “If you encourage disagreement for disagreement’s sake, then you will get disagreeable children. But if you teach that it is good to question (provided the questioning is intellectually rigorous and honest), then you are educating.” We will teach with the grain if we take advantage of this trait.

Junior high students also like to catch mistakes, as any teacher of this age group will testify. We should teach them to identify different kinds of mistakes: mistakes in facts or deduction, ethics or understanding.

Finally, junior high students like to solve puzzles, not only puzzles in games, but puzzles in daily life. They ask the same questions as philosophers: What is truth? Where do we get our standards of right and wrong? What is the value of learning? They like to question and search for answers; we should give them the proper techniques for doing so. We should give them time for questions and in-class debates, teaching them the methods and ethics of good argumentation.

We should teach students logic when they are ready for logic, and they will not only learn it, they will love it.

How Logic Should be Taught

Should logic be taught as a subject separate from other subjects (which is how it is usually taught, when it is taught at all), or should it be included as an integral part of all subjects, as some people advocate? For our school, and I believe for most other schools, the answer is both.

First, logic should be taught as a separate subject. This is the most practical method for teaching the details of logic. For example, at our school we want the students to learn the names of more than twenty of the informal fallacies. We want to teach them a number of special terms, such as syllogism, validity, genus, species, self-contradiction, and tautology. We want to give them many specialized techniques for determining validity, such as counter-examples, Venn diagrams, truth tables, formal proofs, and truth trees. We want the students to use these names, terms, and techniques in all of their classes, but it would be impractical and inefficient to have the students learn them in all of their classes. If the details of logic were taught in every class, some would be skipped over, most would be taught redundantly, and quite likely different teachers would want to teach the different names, terms and techniques differently, to the confusion of the students. We also do not want to require that every teacher be necessarily trained in formal logic, any more than every teacher is trained in mathematics or history. If you have separate classes at all, you must have a separate logic class.

Having said this, let me now hasten to promote the teaching of logic as an integral part of all subjects in the junior high and early high school years. The teacher of each course should emphasize the ordered relationships of particulars in their subject. The logic of each subject will naturally take a slightly different form according to the material covered, though the teacher of that subject should attempt to use the terms and techniques in the same way the students learned them in the logic class. Briefly, it could look something like the following:

In English, when students read books of information, paragraphs should be examined for syllogisms or arguments and sentences in those paragraphs as premises or conclusions. Students should be taught to come to terms with the author and to follow his line of reasoning.11 Students should learn how to write their own persuasive essays using proper argumentation. They should reduce the essays of others by fifty percent or more to teach them to recognize the author’s primary line of reasoning. In speaking, there should be many debates in any and all subjects, becoming more formal up through the rhetoric years.

History will provide ample material for discussion and debate. The linearity of history should be emphasized, along with the reasons for this or that decision and the causes of various historical events.

In math, this is naturally the time to teach the more abstract techniques of algebra and geometry. Geometric proofs should be seen as logical proofs, starting with universal postulates and making specific conclusions. The teacher should emphasize the elegance of the logic of math, especially in problems which start horribly complicated and end up simple and sublime.

In Bible the students should have all the basic facts of biblical history and doctrine at their fingertips from the previous grammar years, and should now be taught the techniques of proper biblical interpretation and the development of more detailed Christian doctrine. The teacher should point out that only in the word of God do the students have an infallible source of truth, and so the distinction between truth and validity becomes especially important. They can obtain in the Bible infallibly true premises, and use proper reasoning to deduce biblical truth by good and necessary consequence.

Formal logic should be viewed as the deductive logic course, and the remainder of the science courses as the inductive logic curriculum. Students should be self-consciously making inductive conclusions from observations and experiments, recognizing the validity of inductive reasoning from a Christian worldview.

The Teacher of Logic

Beyond the qualifications for all teachers, such as a love of their subject and a love for children, the logic teacher should meet some special qualifications.

He need not have a degree in philosophy, though that would of course be helpful. But before he attempts to teach such a course, he must have a good education in formal logic. He should not try to teach the material by staying two weeks ahead in the textbook. He will not know the material deeply enough and the students will pick up on this very quickly. He will need to have thought through many basic questions: What makes an argument valid? Can a valid argument be made up of false statements and an invalid argument be made up of true statements? Does All unicorns have horns imply that Some unicorns have horns? Why is All S is P not equivalent to All P is S?

The logic teacher needs to have God-given skills in abstract reasoning and organization. He should have a knack for solving puzzles. Logic is a symbolic course. Training in higher math is useful here, along with some artistic talent for drawing diagrams.

The teacher must be able to discuss and debate, especially with younger students. He must be level-headed yet outspoken, able to argue without becoming heated. He must be able to face a roomful of young people with strong opinions and take advantage of that in class discussion. He must be able to see logic everywhere so that he can teach his students to see it everywhere too.

Available Materials for Teaching Logic

Books are first, of course. There are a number of excellent logic textbooks in print; a bibliography of them is included at the end of this section. Different texts are more appropriate for different age levels. Some are profitable for student use and others only for the logic teacher’s self-study.

To see examples of great reasoning, students should be given the opportunity to read great books. The best books in each subject should be read. If studying physics they should read in Newton and Einstein, if theology they should get a taste of Martin Luther and John Owen.

For examples of bad reasoning there are all the other books, especially modern ones, along with any other form of modern communication, written or otherwise. The opinion page of your local newspaper is probably an inferno of formal and informal fallacies. The same goes for verbal information. Listen especially for fallacies made by students themselves (which is only fair, since they will be listen for fallacies made by you).

There are also a number of games the students can play to exercise their minds toward logical thinking. For the class as a whole, games such as Mastermind (also called Pico Centro), in which the students try to figure out the order of a three- or fourdigit number by means of certain clues, are good practice for precise, orderly thinking. On the less mathematical side are situation games (like twenty questions, but don’t limit the number), in which the students are given a general statement from which they must determine the details by asking yes or no questions. Many other similar games are available.

For the individual students, consider allowing them to play chess, solve matrix logic puzzles, or perhaps even play games like Clue. All these are good practice at making conclusions based on premises, and will form logical tracks in the students’ minds.

The best activity, though, is keeping up a constant dialogue with the class, questioning them, challenging their assumptions, regularly pointing out their own reasoning, both good and bad. This is what will make logic real to them.

1 A great little logic textbook published in 1873 by Robert Carter & Brothers, New York.

2 In logic, the term argument does not refer to a heated disagreement, but to a set of statements, one of which (the conclusion) appears to be implied by the others (the premises).

3 Though I mention mathematics here, I will avoid the perennial debate as to whether logic is a species of mathematics, or vice versa. But you should know that such a debate exists.

4 This essay can be found in Douglas Wilson’s Recovering the Lost Tools of Learning (Wheaton, IL: Crossway Books, 1991), pp. 145-164.

5 Wilson, Recovering the Lost Tools of Learning, p. 158.

6 Augustine, On Christian Doctrine, Book II, chapter 32.

7 John Frame, The Doctrine of the Knowledge of God (Phillipsburg, NJ:Presbyterian and Reformed Publishing Co., 1987), p. 253.

8 Compare John 1:1, “In the beginning was the logos, and the logos was with God, and the logos was God.” Gordon Clark, in his Logic textbook (The Trinity Foundation, 1988), translates logos directly across as LOGIC. I do not feel as comfortable with Clark’s translation as I do with his understanding of logic.

9 Charles Hodge, Systematic Theology (Grand Rapids, MI: Eerdmans, 3 vol., 1986), Vol. 1, p. 49.

10 Wilson, Recovering the Lost Tools of Learning, p. 157, 154.

11 Techniques for reading in this way are described very effectively in Mortimer Adler’s How To Read a Book (New York, NY: Simon & Schuster, 1972), chapters 8 and 9.