by Steven B. Cowan-
“Isn’t logic the invention of human beings?”
“Does reality really have to be rational or logical?”
“Why can’t there be square circles?”
These kinds of questions are frequently asked in our postmodern culture. There is a suspicion, given our relativistic and “tolerant” climate, that logic and rationality are optional; that we are free to think however we like; that the so-called laws of logic are simply the invention of white, European males. The rigid constraints of “Western logic” must give way to the greater freedom of “feminist logic,” or “Buddhist logic.”
Even Christians are often led to think this way about logic. Long before the advent of postmodernism, the ancient theologian Tertullian (who uttered the famous question, “What has Athens to do with Jerusalem?”) and the more recent philosopher Rene Descartes, viewed logic as something created by God that he did not have to create. In other words, if asked whether God could create a square circle, or whether he could make a stone too heavy for him to lift , these Christians would have said, “Yes!” They would have been uncomfortable with the idea that God was somehow bound or limited by the principles of logic.
Elsewhere in this issue of Areopagus Journal, Jay Wood has rightly stressed the importance of pursuing the intellectual virtues-good intellectual habits that facilitate the seeking and finding of truth. 1 His discussion, however, presupposes that the rationality that the intellectually virtuous person achieves is necessary for the good life. This presupposition is correct, but is often challenged today as I have indicated . Some people think that being rational or logical is unimportant; they deny that there is any correct way of thinking or reasoning (usually because there is no single, absolute truth that can be found).
Christians ought to reject this kind of muddle-headed thinking. In this article, therefore, I will first provide a case for the absolute necessity of logic, showing that logic is not a human invention , but reflects universal and eternal truths. Then I will outline some of the basic principles of logic and introduce the reader to the discipline of valid reasoning.
WHY LOGIC IS NOT OPTIONAL
So, why can’t there be square circles? The fact of the matter is that we simply cannot do without logic. The laws of logic are not merely human inventions; nor are they something that God arbitrarily created that he could have made different. The laws of logic apply to reality and are necessarily binding on how we think. They cannot be other than they are. To see this, let me introduce you to the most basic logical principle, the Law of Non-Contradiction (LNC). The LNC states that:
A proposition cannot be true and false at the same time and in the same way.
LNC does not assert that a proposition (i.e., a statement) cannot be true and false at different times. Obviously, the following statements can both be true so long as they refer to different points in time:
“Bill Clinton is the current President of the United States.”
“Bill Clinton is not the current President of the United States.”
If the first proposition were uttered in 1997, then it would be true. And if the second one were uttered in 2006, it would be true. All that LNC rules out is both statements being true at the same time. Likewise, LNC can allow that statements can be both true and false even at the same time if the statements have different meanings. For example, both
“Steve Cowan is cold”
“Steve Cowan is not cold”
can both be true even at the same time as long as the word “cold” means something different in each case. Suppose that in the first statement, “cold” means emotionally cold (i.e., distant, aloof), while in the second statement, “cold” means physically cold (i.e, cold in temperature). In this case, obviously, both statements can be true simultaneously. All that LNC rules out is both statements being simultaneously true in the same sense .
Given this understanding of LNC, we can see at least three reasons why everyone, even Christians, should embrace the necessity of logic.
1.Logic is rationally inescapable. You cannot deny LNC without presupposing it. Just try it! Say to yourself, “LNC does not apply to reality.” For your statement to be meaningful, for it to say anything at all, the Law of Non-Contradiction must be true. Consider the fact that every thought and every use of language presupposes that terms have distinct meanings. For example, if I point to my neighbor’s pet dog Fido and say, “Fido is a dog,” I believe that that proposition has a distinct meaning that both my neighbor and me can understand. We both, that is, know what the term “dog” means.
But if LNC does not apply to reality, then contradictory statements can be true at the same time and in the same way, which means that “Fido is a dog” and “fido is not a dog” can both be true. But, this further means that when i say “Fido is a dog,” and thereby include him in the class of dogs, I am not excluding him from the class of non-dogs (all the things that are not dogs)-if I were, then “fido is not a dog could not be simultaneously true. But, if Fido is not excluded from the class of non-dogs, when what have I said when I say that “Fido is a dog”? Absolutely nothing! If contradictory statements can be true in reality, and thus true of Fido, then saying that “Fido is a dog” is no more meaningful than saying, “Blikkety blik blak.”
So, if LNC does not apply to reality, then no term has a distinct meaning; any term could mean one thing and its opposite at the same time. And if a term can mean one thing and its opposite and the same, time, then no term will communicate any meaning. Yet, the person who says, “LNC does not apply to reality” is attempting (we assume) to say something meaningful. But, as we have seen, no statement can be meaningful unless LNC is true. So, denying LNC is self-defeating because you cannot deny it without presupposing it.
2.Logic is practically inescapable. There are people in the world who deny that logic applies to reality. Zen Buddhists are a case in point. The Zen Buddhist believes that reality is non-rational, that truth can be contradictory. But the problem is that no one can actually live as if logic does not apply to reality. For example, the Zen Buddhist still looks both ways before he crosses the street! But why? If he is correct in his dismissal of logic, then there can both be a truck coming and not be a truck coming at the same time! And the postmodern deconstructionist-who says that any text means whatever the reader wants it to mean regardless of the author’s intention-changes his theory of interpretation when he reads the label on a medicine bottle!
It may sound pious or “tolerant” to deny the laws of logic in theory, but in actual practice no one really does it. Everyone behaves every day as if LNC is true. A worldview that cannot be lived out in practice is probably a false worldview.
3.Logic is indispensable if we are going to know anything at all about God. If logic is not something that applies to reality, then it does not apply to God either. And, as we pointed out above, some Christians have been willing to say that LNC is a law that God could have made different-that God can make square circles if he wants to!
The problem is that Christians believe lots of things about God. Indeed, we think we know some things about God. For example, we know that God is good. And we believe that when we say, “God is good,” we are saying something true and intelligible . But, just as we saw above with the dog Fido, if LNC does not apply to God, then contradictory statements about God can both be true. This means that the statement, “God is good,” is consistent with the statement “God is not good.” In other words, when we say that God is good and thereby include him in the class of good things, we are not excluding him from the class of non good (i.e., evil) things. And this means that when we say, “God is good,” we are not saying anything informative. We are saying nothing more than if we were to say, “God is blikkety blik blak.”
Consider also some other absurdities that follow if logic is merely a contingent, created thing. God could make it the case that he both exists and doesn’t exist at the same time! He could make it the case that Jesus is both God and not God at the same time, and that he is both the only way to salvation and not the only way to salvation at the same time. In fact, if logic does not apply necessarily to reality, then Christians can never have any objection to other religions. For, if contradictions can be true, then both Christianity and Islam can both be true (and so can Hinduism , Buddhism, and Satanism!).
The point of all this is that we simply must affirm the absolute necessity and truthfulness of LNC and other laws of logic, at least if we want to claim that we know anything about God and about reality. But, does this mean that logic is somehow above and beyond God; that God is dependent on something outside himself? Not necessarily. Following Christian philosopher Gordon Clark, we can say that logic is simply God’s way of thinking. i Logic is an essential aspect of God’s very nature. It does not exist independently of him, but neither is it something that God arbitrarily created that he could have made different. Logic is like God’s holiness-an expression of his eternal and un-changeable being.
Given this understanding of logic, then, it is important that Christians care about logic and learn to think logically. In what follows, we will sketch out some important logical principles.
A PRIMER ON LOGIC
Logic is primarily about the construction and evaluation of arguments. An argument is a set of propositions or statements which purports to prove something. ii As such, an argument has two parts: (1) the conclusion of the argument is that proposition which the arguer is trying to prove, and (2) the premises of the argument are those propositions that provide reasons for accepting the conclusion.
Arguments come in two main types : deductive and inductive. In a good deductive argument, the premises (if true) provide conclusive grounds for the conclusion. That is, the conclusion of a deductive argument follows from the premises with absolute certainty. Consider the following deductive argument:
All men are mortal.
Socrates is a man.
Therefore, Socrates is mortal.
A moment’s reflection on this argument will reveal that the conclusion follows from the premises of this argument with certainty. If the premises are true, then the conclusion cannot fail to be true.
Inductive arguments, on the other hand, establish their conclusions with only a degree of probability. The premises, that is, only imply that the conclusion is probable, not certain. Here’s an example of an inductive argument:
Out of 1,000 people surveyed in Birmingham , Alabama, 70% agree that capital punishment is morally permissible.
Therefore, it is likely that 70% of the citizens of Birmingham , Alabama, agree that capital punishment is morally permissible .
Again, only a little reflection will make it clear that even if the premise of this argument is true, the conclusion is not certain. The premise makes the conclusion probable, but does not guarantee its truth. Inductive arguments form a significant part of our daily lives. However, because of its brevity, the remainder of this article will confine itself to a discussion of deductive arguments.
Validity and Soundness
In evaluating deductive arguments, the first thing to consider is whether or not the argument is valid. An argument is valid when its conclusion follows logically from its premises. More precisely, a valid argument is one in which, if the premises are true, the conclusion must also be true. It is very important that the student of logic understand what validity claims and does not claim. To say that an argument is valid is not to say that its conclusion is true. Nor is it to claim that any of its premises are true. In fact, an argument can be valid even if every proposition in the argument is false. Consider the following argument:
All people have brown hair.
All brown-haired things have four arms.
Therefore, all people have four arms.
Every statement in this argument is obviously false. Yet this argument is valid! You can see this if you pretend for a moment that the premises of this argument are true. If the premises were true, then the conclusion would have to be true as well. To say that an argument is valid is to make no claim about the truth or falsity of any statement in the argument. It is only to make a claim about the structure or form of the argument. A valid argument has a structure which is such that, if the premises are true, then the conclusion has to be true. Another way of putting this is that a valid argument preserves truth. True premises preserve truth through to the conclusion.
Claiming that an argument is sound makes a stronger claim than validity. Soundness is a property of arguments that are both valid and have true premises . In other words
Validity + true premises=soundness
If an argument is sound, then its conclusion is true and (assuming that you know the argument is sound) you must believe the conclusion. Knowing when an argument is sound is much more difficult than knowing if it is valid. To test an argument for validity, one has to simply inspect the form of the argument. To test an argument for soundness, one has to evaluate whether or not the premises are true or false. Sometimes that is not too difficult, but often it is very difficult. As a discipline, logic is mostly concerned with validity and that is what we will focus on in the following section.
Some Valid Argument Forms
Let us distinguish between an argument and an argument form. An argument has some definite content as in the examples we used in the previous section. An argument form, however, is kind of like a skeleton or blueprint -it is a pattern that an argument can have. There are some common argument forms which are known to be valid. If one can memorize these forms, then one can often easily discern whether or not an argument is valid. Most of the arguments that we are going to look at are called syllogisms. A syllogism is an argument (or argument form) that contains exactly three propositions (two premises and a conclusion). There are three types of syllogisms that we will discuss.
Categorical Syllogisms. This type of syllogism is so called because it contains only categorical propositions-statements that assert or deny inclusion in a given class or category. There are many possible categorical syllogisms (256 to be exact), but only a handful are valid . Here are three such argument forms:
- All M are P.
- All S are M.
- Therefore, All S are P.
- No M are P.
- All S are M
- Therefore, No S are P.
- All M are P.
- Some S are M.
- Therefore, Some S are P.
Any arguments that correspond to these forms are valid . Here are some concrete examples:
- All dogs are animals.
- All collies are dogs.
- Therefore, all collies are animals.
- No dogs are cats.
- All collies are dogs.
- Therefore, no collies are cats.
- All Christians are heaven-bound.
- Some Baptists are Christians.
- Therefore, Some Baptists are heaven-bound.
If you suspect that a categorical syllogism is invalid (i.e., you suspect that its conclusion does not follow logically from its premises), there is an easy way to prove it. You can construct what is called a “counterexample” -an argument that has exactly the same form as the argument you are evaluating, but which has obviously true premises and an obviously false conclusion. Such a counterexample shows that any argument with that structure is invalid. Every invalid argument commits some kind of formal fallacy –a mistake in reasoning involving the form or structure of the argument. Consider this argument:
- All political conservatives are self serving and greedy people.
- I All capitalists are self-serving and greedy people.
- Therefore, all capitalists are political conservatives .
Regardless of what you might think about the truth or falsity of some of the statements in the above argument, it is definitely invalid. Note that the argument has this form:
- All P is M.
- All S is M.
- Therefore, all S is P.
Any argument with this form is invalid as can be shown by the following counterexample :
- All dogs are animals.
- All cats are animals.
- Therefore, all dogs are cats.
This argument has exactly the same form as the original argument , but its premises are clearly true and its conclusion is false. Since valid arguments preserve truth from the premises to the conclusion, any argument that has this pattern must be invalid.
Hypothetical Syllogisms. These are syllogisms that have at least one hypothetical or conditional premise. A conditional is an “If. . .then ” statement which has the form, “If P then Q.” The first part of a conditional statement (the part corresponding to P) is called the antecedent , and the second part (corresponding to Q) is called the consequent. There are three valid hypothetical syllogisms :
- Pure Hypothetical Syllogism
- If P then Q
- If Q then R
- Therefore, If P then R.
If the Steelers won Superbowl XL, then they are the 2006 NFL champs.
If the Steelers are the 2006 NFL champs, then they are the best NFL team.
Therefore, If the Steelers won Superbowl XL, then they are the best NFL team.
- Modus Ponens
- If P then Q
- Therefore, Q.
- If the Steelers won Superbowl XL, then they are the 2006 NFL champs.
- The Steelers won Superbowl XL.
- Therefore, the Steelers are the 2006 NFL champs.
3. Modus Tollens
- If P then Q
- Therefore, not-P
- If the Seahawks won Superbowl XL , then they are the 2006 NFL champs.
- The Seahawks are not the 2006 NFL champs.
- Therefore, the Seahawks did not win Superbowl XL.
There are a couple of hypothetical syllogisms that closely resemble Modus Ponens and Modus Tollens, but are invalid . The first of these is called the fallacy of affirming the consequent. It has this form:
- If P then Q
- Therefore, P
Whereas the second premise of Modus Ponens affirms the antecedent of the first premise (which is valid) , this fallacy affirms the consequent in the second premise. To see that this is invalid , consider the following example :
- If George Washington was assassinated, then he is dead.
- George Washington is dead.
- Therefore, George Washington was assassinated .
Both premises are true, but the conclusion is false. Clearly this argument form is invalid. The second invalid hypothetical syllogism is called the fallacy of denying the antecedent. Unlike Modus Tollens, in which the second premise denies the consequent of the first premise, this fallacy denies the antecedent of the first premise. Thus it has this form:
- If P then Q
- Therefore, not-Q
- If George Washington was assassinated, then he is dead.
- George Washington was not assassinated.
- Therefore, George Washington is not dead.
Again, both premises are true, but the conclusion is false-a sure mark of an invalid argument.
Disjunctive Syllogism. The disjunctive syllogism has a disjunction as its major premise. A disjunction is an “either-or” statement which has the form “Either P or Q.” The two parts (P and Q) of the disjunction are called “disjuncts.” Thus, the disjunctive syllogism has this form:
- Either P or Q
- Therefore, Q.
A disjunction claims that at least one of its disjuncts is true. Since the second premise tells us that P is false, we know that Q has to be true. Consider this argument:
- Either Miami or Birmingham is in Alabama.
- Miami is not in Alabama .
- Therefore, Birmingham is in Alabama .
Caution is in order, however. The following argument form is invalid:
- Either P or Q
- Therefore, not-Q
The reason it is invalid is because the disjunctive premise only claims that at least one of the disjuncts is true but both could be true. So knowing (via the second premise) that one of them is true, does not allow us to say that the other disjunct is false. Consider this argument:
- Either Montgomery or Birmingham is in Alabama .
- Montgomery is in Alabama .
- Therefore, Birmingham is not in Alabama.
This argument is clearly invalid and it commits the fallacy of affirming a disjunct.
Constructive Dilemma. One last valid argument form that we will discuss is the constructive dilemma. This form of argument is often used when an arguer wants to force his opponent to choose between two unpleasant options (thus the title “dilemma’.’). The constructive dilemma has this form:
- (If P then Q) and (If R then S)
- P or R
- Therefore, Q or S
Notice that the first premise is a conjunction of two conditional statements. The second premise is a disjunctive statement in which the disjuncts are the antecedents of the conditionals in the first premise. The conclusion drawn is a disjunction of the consequents of the conditionals in the first premise. Here is an example:
- If Hillary Clinton wins the next Presidential election, then we will lose the war on terrorism ; and if Condoleeza Rice wins the next Presidential election, then our economy will falter under the burden of huge budget deficits.
- Either Hillary Clinton wins the next Presidential election or Condoleeza Rice does.
- Therefore, either we lose the war on terrorism or our economy will falter under the burden of huge budget deficits.
Though the premises of this argument are question able, it is relatively easy to see that the conclusion would be true if the premises are true.
There are many more types and forms of deductive arguments, some much more complex than those we have outlined here. If the student can master these simple argument forms, thought, it will go a long way in equipping him to evaluate the arguments he encounters in everyday life.
This short primer has only scratched the surface on the subject of logic. There is much more to learn. It is my hope that when the reader has mastered the material in this article he will go on to read and study more in-depth works on logic. A list of resources for further study is given elsewhere in this journal.
Steven B. Cowan (Ph.D.) is Associate Director of the Apologetics Resource Center and editor of Areopagus Journal. He has contributed several articles on logic to the forthcoming Holman Dictionary of Philosophy.
- See W. Jay Wood, “Virtue and Knowledge” (p.8).
- Gordon H. Clark, An Introduction to Christian Philosophy (Jefferson, MD: The Trinity Foundation, 1968), 67-68. Clark is well-known in this context for his somewhat controversial (but plausible) translation of John 1:1;”In the beginning was the Logic, and the Logic was with God, and the Logic was God.”
- Sometimes the word “proof” is taken to mean an absolutely conclusive demonstration (i.e., an argument that established its conclusion with certainty). Consequently, to “prove” something means to show beyond any doubt that it is true. So taken, there are very few if any real proofs in the world. Most philosophers, however, use the terms “proof” and “prove” in a less stringent way today. An argument constitutes a proof if it is valid and the premises are known to be more probable than not. In other word, one can “prove” something without having to achieve absolute certainty.